Measures and Probabilities (Universitext)

Con"""""""

The r;r.nl ra] 1.l m lt Tb,....,..m

III

· · · ·

'"

31 2 31 3 316 3 18

• .'\20

CwLz •• 17

Onler S'"')S'ia 17 I Dell";,;"" o f .be

344

C'trrJer Srar ;"

17.2 Con.u",,""" 01 lbe Empirkal

IV 19

"

344

~IMI""

Operations on Radon Measures

3T7

... Adequa" Faro' ly or M"""URS 19 I Indu..ffl Bodoo Mea'urr 1?2 u,DmIe Familln! 01 C<>ml!OlCt SetS

3-"0

193

"

jcs

SliD 01 Bodoo M....... ltftl . .

IIJI.II&S of Radon t.l euW'ft and P rodtIC' Measures 21, ) 21 ,1 21 3

Im.og .. olIU.donM.... u~ . .. o...:om!>O!ition of a M~ In Slic8 PreMln"« Bodoo

21.3 21-4

Produtl 01 Rfsuw 1.1"""""", Cilium: of Variable Fonnula.

Me·.,,_

H u r Me. ." ....

23 I

IDu na"! M ..asures

'"

..... ." 4]]

. 421

. .n

425 426

xlii

24

C-nnvnli"iop of !!.lea. " .....

24

i

0"",01 ...'* M-w,,,.,.

24.1 248

466

." ,

,

De6njl \op aCGdfA,.., I>I jr

.., '" .e!

AOA Symbol Index

..,

Copyrighted material

Part I

Integration Relative to Daniell Measures

,.

Copyrighted material

1 Riesz SpaDes

I..e< X be

I,

fur>
11."._ ol_i~ ......

x , 1_ 1(1) _ f. "')
6 _. M..... 'a. if I . _><' S tulib."q ... , on [-.bI ...... r.1.(,)o'I d, ... 6,.. h,... to doll"" tho! _ of pooit, ..

1.

_w_"

..,.1" .........

:nt o i n a _ .. , .. ,-~

1. 1 TIoio fi ... ooc<"", ; ~tl'Od .... tho "",ion of <><der-od &J"'UJII and 1at.1o!oo. It "'-"", to. -.., to ~ br. millod ....... ..II ~n ~ . ....:I> .. Z 0< R or tho """"" Clio, III of .......1""" .-~ ... fur>«"- O D [0.61 with .heir ........... O<dor!nca1.2 It io do/W'" to. obort .. Oldy of Rlooo. opoceo, thai .. ot ... p111;_ fu " " - "". ' . d bownJ 101. equll'JM>'l -.itb ita "",,,,,01 <>rdo:ri~ • POll"'''' ~ Iorm on E Io_'n"",,". U W. &nalb dof· D=- 'el' a ,.._ A Dan;,!, _ u .. io • ~_ Io<m [, on • _ _ cl "'... _ ff. _ hu • "SDk • ...u..ion' (DeIinltion I.U) aad oudo

pOOi.,...

.... IM_. to.

M

,10K L{t.) - 0 .hot •• , ... I. debt .., 0 0U>CIerw>0d .... ."...i..w,y oondi' ..... A _

In

11* . no. ~ ,be eumploo

""""';0;"" . . . . ld

,ho, ,hot " .

~.

boo ma,r

...... IOIooop '" mind.,., tho !tit ann ~ ........ , od . . . ",*, ;.e U_ Jona "" tho"'" 01 ..... tin ...... fIt"",1ono "" (0.111 aDd aloo II,. p(IOiIi~ ~_ Iormf t . : X ) I _ l {e).boA. E

".111.

l.l

Ordered Groups

Defin it ion 1.1.1 Lot G be. oommut6,i~ IfOUP wri't.enlddld""ly. An ordot-r ",ruc\U", on G, ~ h,- :!O, Is.-id to boo cornpo.'ible with tho group $I."",,"" ' ...... ol G .. ~ It ..,,,,,,,," lhe 1oIIowi", axiom: Irz:S:~, lhen z+: < ~+I lor &II : E G.

n.. group G. ~ ...;tb • O>
Taltfnl • -

- z, • _ -II • .00 : _ - (z ~ ~l !\><:( ' . 'i ~ly In Definit ion 1.1.1, _ _ 11Ide 1Iw, in G, lho i""'luallliN % II, 0 < II - %, Z - " :!O 0, and

:s:

....

- II SO - z "'" equi.. l.nt FOIl" uamplt, R. w\t;h III \IIIl.l&lloddillon and lUI usual om." , "' 6Il oodt
P ropolition 1. 1. 1 S"¥P<>« IMI (z;)'!o'S. I z, :s: /I, /".. all I :s: i :s: ... z, + ". +:I:. :S: II, + .. . +11.0 . FIo""""",-, %j + ... +z. < ", + ... + 11.0 ....-..ewr molt." 1 :!O i S n I'" ""'Idt. ". < ".

..u.

n.-

u..n.

""""ions

The z, + Z) :!O %, + lOt and "" +" S "+,, [mply %, +:0:, :s: ", +lI':t. On II.. 01",,", hand. if z, +z~ - VI +11:1 . • hen z, +"'2 - z, +11:1 -Ill +/1>. hencoo. :r, _" and z, -Ill. n...gro...-.! P Jl()Ot";

Pint. •• .. ox ."" .. _

t6IIe k>[1owo

2.

t.,. 1lld\lC\Im en ...

0

For CY"'Y' EG,'ho t ~lon" - z+ ..... ::, ,'tlI thoOlrderofG. ,""", if . ,.....,.mpty fa.mlly {z,ku 01 dc"'''''tI of C has. oupromum V .... , z,. tben tho r.m.uy (. + z,)"" aloo It.u •• upntDum. and V,.;/(. + z , J ~ I + VrE' r" A similar stal<:"kot boOdis fOO" the Infimum N.. !be mappl", z _ - z reo .. _ the (Itder of G 010 ,hbt • nootmpty family {~)'
11..,:0,.

(- z;}.., It.u t.u lDfi.=, and thell A...,( - :r,) _ - V,..,z,. Dell nitlon 1.1.2 In 6lI oodw\d Jf"O'Jl) G, ,, e C ill -' 0 ("",p«ti"'lJ',:t < 0). If Gil.." ood",ed """". tho eel. 01 1'- pOOIiti>1l eIocIllC!",- ,..;\1 ..u&!1y "" dmoted by So n (- G'"J .. CO) and C+ :J C+ +

cr.

cr

cr_

1.1 O"I""d G"",p" Pro~ltioD

5

1. 1.2 UI. comml>1Gt.." group G lie gtven, "lUIld P lie • fUbuj

"JO ...:10 tMl p n(- p ) .. (OJ alUl P+ P C P_ The ..."'1;"" %::;

~

if 01Ul mlr

if, - '" E P "/ina on <mJ~ .1"",",", compal,bIe ""th the grnp .lnIdM... "I Moretl~" G ;, t<>UoIl, M"rhrtd if ,,>III <>nI, iJ G _ P U (- P).

G.

o

PROOF: Thill is obriouo.

An ","",ed oe\ E io oaid 10 be din!CU!d up"'ard (..... pecti""'y. down ..vd) .1>0.......... to. all % E G. V E G. t~ ""OIlS z E G .w.\sfyin& :t ., z and ~ ::; ~ (..... pectively. % ~ ~ and ~ ~ ~). EvelY onieT
P .... ~ltlon 1.1.3 An M"IkrN gro~p G " d'r"&':led ",,>III .".J, if it ;, gen. a 'ed br G~ , cr, if """"'- ./mI... j oj G .. 1M: diJJ'~ oj I_ pomi ... ,*",...t..

.q,,,""/.e,,u,.

"""""let

PROOP: If G iI di"",Ied, lor ~""ry ~ E G. tl>o.re """"' .. % E G· lhan Z; _ : t .. ~-(~- :t). Coo' .. ody, if % .. ,, -v and~ .. ... - 1 ......... M. v, .... I _ all j)OIili"", then % S" + to and, S" +.... 0

P .... poo.ltlon 1. 1.4 UI (~,) ... , lie" ......empt, fimte Jamil, ift a di....,u./ gmup O. TIIo. ....... v; _

""".1."". tab! . ..

~"

0

An ""I",ed oe1; E is called .. lallia! whenevIII 1:. , E G lIe~. A. ,,_.Mry "M ~ OI>fId,t",,, u...t z " , .,...1 .. Il101 % V , tzUl. :nw:" z+~ " % " ,+z V, _

PROOF: Su,...,.... that 1: " , ""iota. lbm

:r +, - :r '" _ z + , + (- z) V (-,1 _ z V,. Tbe proof IX 11>0.

<:Onvtnle

group. :nw:" G " 0 /aUt« if "lUI oj tile loilftliftg c.c>ndIlU'''' IIfIUU:

Pr-<>pWltlon 1.1 .6 S..",.".., 0 .. ""

..... if 0 .. O· -

cr- ,,>III <me

onlemj

(.) E....,. pa;~ oJ.1ernctU ,,10+ IIu 0 (6)

o

io j im ... ....,..

E....,.,..... oj ~ ,,10·

...,....". ..... .

IIu on irVim ..... '"

0·.

I'

ale

PIIOOF : n- cond.iliooo aM otMowdy !>Hpl",,),. Con,~y, undo.-r hypoth. ",is (Il) (" "I_li""ly. (b)), eo.cb pili, of ~Iommts % E G"'" , V E C + h.M a IUrw-um ( rwpectl""ly, an Infi mum) in G equeJ to il& 'UpEeit/urn a (_p"o" Ii"",>", it.l infimurn ~) in G"'" . Th" .. abvio:>o..os lOr 0; 1Or~, lei. ~ E G boe alcN-."" bound for % and ,. Thoen l!>ere c:xiwl .. E G"'" such lbat : +" E C + (boa. ..... G .. G"'" - C· ). Nooo 11>f.,..(z+u, V+ uj" greater than ~+" and ea." be writtal ~+ c+ II (c 2: 0). Si .... b +c < % and b +c < V, c _ O. lIenc:e b+ "" Info. (z + "., + u) 2: Z +'" &Dd z :5:~. Nat, lei. " E G. V E G boe arbitrary and U E G+ boe ow:lt IMI z + U 2: 0 &Dd W+" 2: 0; UOOtt hypot'-"_ {al (...... ~i""ly, (b)), z +" and , + u have a ",,,,_mu.m (l"
a

T~m 1.1.1 ~1DP<>"1~1o'.' Le.mnlll) Ld (". hs<s., (II;),;YS. . . 111'(1 fi7li.t~ ~ IP\ (;+ HJUfIIi"f L,s.s. "; " L '!i1S.II, rn G , Wn, IMre ......11 • ~ ~ (z,.; ),,,,;,,,• . '!.IS. #Uth I1Iat z, .. LI!i1 "'....; ~.II j mwI'I" L' S'S'''J lor.,J1 j. PROOf':

I. Ftrsl, consider 1Il00 ~ P '"' q ., 2. Put ',., ., .... p(O . %, - 1')) and .,., .. "1.1 - (Zl - n)· Sinc-e %, - I/"> .. 1/1 - ,.., io .maller than '" &Dd 1/.. '1.2 .. '" - ".' &Dd >7.1 - 1/1 - " .' ftre s-jli ..... M",,_. + z., = ... , + :',ll, and '" .. + :•." II> - "'> + "1.2-

",., Z,., ",.3.

".1

2. Next, _ .. ,~ that 1Il00 theorem holds lor p < m o.Dd q .. n (_ilb '" :> 2, 0 2: 2) and prove lbal It balds for p .. m and q .. n. By

( L ,,) ••• - L .,' 1'S"-'

19S~

Since.he......:lt boIds lOr p .. 2 and q .. n, _"..." find t ..... """'Iue....... (~j,:r;J"'~, (:f) , ,,,, s o In (;+ ... tlsfyillg

r;

&Dd II, ,. oj + lor all 1 :s; j :s; n, Now. since the t~m 10 ' rue fur p _ m _ 1 &Dd q .. ". d ..re "' • dou~ -....... "" (1<; ... j,~.~ .. _ I .I $j~~ ;" c+ ....:II tbat %, _ LI$ih "<'; for oJl ! !: ; :s; '" - 1 ond <j ..

L,<j< __, IIfJ fur all 1 :s; j S n. Pu~ "'; - ....; for oJl 1 :s; ; :s; '" - 1 -.rod';"; .. r; (I :s; j S oj. Then the double -..""n"" (.,.;),""" .... '",IS _ hu the desired propertlell.

, 'I'

ate

3. Int....dw!&l"lllhe roIeo of the %. and the

~, . "'e

find similarly thal if lbe cheoiem holds lor p .. m and q < .. (m ::!: 2... > 2). lben it holds for p .. '" and q ..... The tt.eowu now 101101111 by double lnductioo.

c Corollary ~

Lei

, .%,.... , %~

........ find~;

/Ie .kment.r olC- "",..","9 ~ 'S E,s;s.%'. EO C+ ( I S ; S .. ) Iud> Ihdt ~ 'S %, lor all; ~nd

, .. E, s.s~~ · P ROOF: Apply """""m 1.1.1 to the ~""""" (%, ),S'S" and the ~ ra ..>ad from lhe I...:. elemems /I and • '" ~ ,~. %,) - ,. 0

(E,

~lI nlt lon

1.1.3 For % E C. tM e~t r v O (_Pft'\ively (- r ) VO,"'" "pea;vely (- :r) V:r) ill called tho pooiti"" part of % (h P"'1i>-dy the nqalive part, ,etpec:tively tM abololute value. or variation, of %) and ill .. rittel! :r+

(,...-th"ely %- , )%Il. NOle that . accordi"ll to t t - definit ....... tM nepl.ive part of:r ilia poIitive element. CJearty, :r- .. (- %). and ) - %1 -14 NOle abo tM Iormulu :r V ~ " :t + U - %). and ~ ~ (/I - %) • . "The liM fol1oon ;mnwlia' rly from 1M Cat:\. that lhe ooduins of G .. preoerved under tranol.t;"n ; the ecoond if a ooneeq_ofPIt)jI,,";tlon I.I,S.

%" .. -

Tbeoo e m 1.1.2 (. ) ., .. " . -.,-

"j

~II
"+ ,, r - .. 0, lor e...ery rEG.

(/I.)

01 % E G '" A diiJ.moct. 01 boo ,....;titte elemenI.r, " .. ". + '" ,,11<1 V " ,, - + IU. ~ ... . . .. " .. . ., 'S) if ~ '" ..,- S , . "n./.,- ::!: ~-". 1" 1 .. ,,++ ,,-.

M

I" + , 1 :S 1" 1 + Iwl

(c)

For

An)

~

" .. .. -

U

for all "./1 EO G; ""'.. 9<'''.....11, I E' s's ~ ", 1 S E, ~ . s . I:r, l /or An, jiniU ItqWnOIi ("' )'s. s ~ G.

(J) )%I - Iwl

:s I" -

r l for all:t.

;"

/I

E G.

PROOF ' We ..ill prove (.1 and (b) lirnu)' ......... "'iy. If z .. .. - W. .. lot", " , ~ are politi.", then .. ::!: %. , anti ...... - %+ ill positive. FurtIw:[I,ooot. z + - :r .. %v o - " _ (%- z )V(-:r) _ r-, bmce:r .. ". - :r- and ... .. w- %-. """ ; e G be amaJltr than ,, - .. ~. - " . Then,, :S "ooot>O.tr, if z S " ... then:r· - % ill poIitive. II) " . :S " . - '. and z ill nqalive. n""eb'(, "'" fmd that:r+ " %- .. 0 and . by ,r...wation. t~"'" " .. .. ",.

,,+ -;;

•

{( j ' Let z , WEG beP""'" If,, :!> ~ . tbm snd :z - ~ , - . ConV
",. s: ....

,,+ S w" Md :. - ,.. , - .

r :5~· and

,,+-:r s:

-t :5(-%)+ _ :r- .IIO

~+

- , - .. , ... bene>~r

s:

(d ) : z" v ;r > 1z1, btca. - :r :5 ,,- ""d - :E :z- . NIt. If .. ill an upper bound lor % and - :r , Ibm .. ~ 2: z".
,,+

Slnot ,, - iI poeilive and linee Int(,, · .r )., O. the La6/. '100 inequ.l!itioet imply .1;0.< ,,- .. 0 and .. .. ,,+ ; tbe /i.a t"", ioequa!it;'" .hCD live a 2: z+ V " - .

Honee z + V,, - .. Izj. Now

+"

,,+V ,,- .. :z+ + ",.

by Propoeilion I. U ' .

s: 161 + " I/)(a._ s: 1"1and " s: IPI· Similarly, - z - , s: Irl + 11'1 · 1I_ 1z s: Izi + l.t, (f): ~II& z and ,In (e) IJ1 ~ and ~ ve Izl - 1, 1 s: 1%- l'I; (e), :z -+ ,

llimilariy,

...

1, 1- 1"1 S Iv -

:t

% -

r l ..

Iz -

,.

....

III.

C

Noce lhal:r .. z · - r"" .. 0 if 1z1- z ·

+,,- .. 0; t hUl I:r] > 0 lor all" E G,

""""rnpe,

Pl"Opoooit ion 1.1. T Ld {", I'E1 /It. Q j"",il'J .n C .../11 "" ;,vtmM'" " .,." lei • E C lot ...-ti1N". n.. .. 1M ,amllv (: V :t;j'EI 1141 "" mfi'""'" "M ..... '{, v ¥.) .. : V(/\"ol " ,j.

p Rfl()r: We ...-y _ _ that , .. O. Noor 1/+ s: z; £or . U i . If a ~ ,. Iooftr boo .." k>r (r j).( /. 111m a :s :r, + :r, for al l i Ii' 1. Now z, fo"',. ( I, i E J ; tb"" a s: :to + ~- lor I E I • • "d a s: II T ,,- .. ~ • . " "' .... lIo
.. -os I<>

s: ,-

e_,. be l booom.

0

I~,.: "

and v. _ obta.in • limilar poOp04h lo)n. In p&rbeular. if {1<.).c, it. nonemply family with • 1lUpr=wn " in G. tben ,,- .. I\;,[ /:r;- . be<:auM (-%). _ (""" - :J,) . _ "",/ ( - ,, ;) . , Dcllni llon 1.1 .4 T-.o elemenu :toII in G are oald 10 be disjoilll. or mutually $ "1'\1 ..... .. hene .e. ~I "t.! _ O. In Ibill ...... _ shall.loo>M)' lhal: % iII diajoim from , . If A ill . "'~ ck;;>ent
_tmplr !in;/.t, /-u.·u

PIIO()J> By induction on 111 and IJ I, it suffi<:el to Iihow Iht.t :J "

(~ +.)

s:

",,'+%,,; for ~I:r . r . : E G"'" _"OW put I " :r " U. + . )_By n-em I Ll. s:

I <:M be.......n~ I , +~, .. be..e O I , and ~ S %" :, and l be ""uk &:>I~

s: II and 0 s:

I, S '. Then I,

s: % ,, ~ 0

'I'

ate

Corollary I If", E G U diIjoin.! I>
Jrr!m

I"" All/I + 1"'1AI' I .. O.

0

Corollary ~ If (;r, l IS'S'" (l/;lISlS" ft~ II." fi.w.e lel z; ;, diIjoim Jrr!m ..w. 1/,. th .... z, + ... + "'.. Gild 1/, + ... + lIa a~ ~t.

PROOf: T his fol~ from Corollary L by indu.etion tIn m and n .

Corollary 3 ··· + 1"'.1·

lIz" .. .. "'. ""' "'hl....!1y diljc>in.t.

PROOf: If "', , are disjoint. tben 10 are ... (:r + ,l+ and ", - +,- .. ('" + 1/)-.

0

th .... lz, + ... + "'.1 ..

I"', I+

+1/" and ",- + 1/ -. Thus "'~ +,~ .. 0

Propoo.ltlon 1.1.9 U I (z ;l'EI bo ft ...,_plr fGmjJ~ in G /unoi"fl G nII'ft" m...., z. 1Jom allllll E G IMI;, diljt!illl fram..w. z; ;, oW dVjotnl fram

•• P ROOf: 1, 1" z; '"' 0 for eoocb i E 1, 10 (P,oc-ition 1.1.7). MOl ..."",

1, 1A z·

.. v .u (lwl A :tn .. 0

1111Az- .. 1111 A(/41:0",) .. A••A IIII A",,) .. 0, .. 1/" disjoint from both

1.2

",+

and "' -. and

""DOl! from 1z1..

", +

+ "'-.

C

Riesz Spaces

DefInition 1.2.1 Su..."...., ,hal. E: II a .-...I _ :S oa E .. Mid to be compatible wilh lho: _

IJW'l' An order relation IJW'l' H""'un: of E i f lhe

IOllowil\l two conditions hold: (al '" :5, impl;'" '"

+, :S 1/ +,

for all

,E E.

(b) '" 2: 0 l",pI;'" AI" >0 lor all mil numbo:n J. > O.

_.par:e.

A real_'J)ke ";Ih .. O)mpatible order II""Ued "" OIo:\t. M By definItion. a ru.... IJW'l' II." OIo:\t ,ed _ rpooce which II. lattice.

,

10

I. RiMa ......

Let a ru.u spw-e E ba

P""1l.

For IUIy f'Ola! nwnbcr A > O. ti>c Ofder of E mnailll invt.rit.DI under the bomothdy ~ - .\.:; thUfl ....p(.u,.\.') _ .1.1tIp(:r,,) for .U :r, , € E _!n particuW, (,I.... ). _.\.·:c and (b)- .. .\. . :. b CYt't)':r € E and br ~ .\. I1 «i""" intt&'"" .. 2: ~I ,:r and n1' ar(l dil>jc>int and Inlll a-I A~. It I'oIIowI lhat, If A It a I>(W"oen'p"y .... boet 01 E. lhe diljoint complement of A 011 . _ IUhIo,*",

of E.

Oeflnitlon 1.2.2 E It MId to be Oedekind oom~ ... heo~ _h DOlltmpQ- WM 01 E 11>0.1 .. bouode<:l al>oo'e bM a suj:ftmwn. &tuMoI~nUy. E .. Dtdttind oom~ w~ t..:h -.empty ..,beet of E that 101 bo:>w>dood

t t'JOoo I>ao an infimJm Pl'<>po<ion 1.2.1

E;, DtdU:iM """"'I~ if 4M 0fIIJ 1/""" 0/ u.e I~

""" .......i"".., /otIiJ,: (. J E.....,

~pl,

01:.( ......

(6)

.

Mid A 0/ £+ Uuol if dirtcktI _,... and 100_""""

~ .. m.

ew..,. ........1,01' .......'

A 0/ ~

thGl if d,,~UtI.w--..d Iwu on inji .

~

PItOOr. Clewly. dIll conditionll uel\b ''Y. Con_I!, _ _ t hai Qi')Ddillcill (a) holds. and lee. 8 he • l>Oiiotwpempt7 Mite P6N 01 8 o:.ooKitute a ed C ~ 101 dir««d upwW_ Let .. he <)Of! olitll tlo!:roetltlI, and deIU>e C. _ (:r EC: z > 01_ Tbta C. - G. \.bat It direcud 0JpQrd and bounded abo;oo.e, hao .. ""~ •. Now " + 6 It lhe 1OlPf'!""lIn of C., and ao 01 B .. ...,Il. In floC!, condition (b) imp/a condition (_) indeed. If 8 101 .. noaempty .ut.et.ol E· , di~ upward and bounded abo.M!. and If c ..... uppoer bow>d. 10< S, then c - B .. dlrec:led downw.....!_ Dtt>o\In« b)' m .he inRnwm of c- B, we _ thalC-- m It l be IIlIpremwD of B. 0 E""'Y _ _ w bsl*'" of

otdooorfld _ If*"
,..1 ..

lUI

n

:s 1rIOI'e(Ml'.

Dcflnl'lon 1.2.3 Let E he an ",d",«1

,"«:to, ~ &nd V, IV

he two .uppIo",...t,a:y _ ",t*P"'dI 01. e . We...,. that E .Ihe oode. «! direct ...... of V and W ~ thecaoonlcal map (z , ,)- z + r f..... the o<de,ed _ _

,

1.2 RIMa " -

II

dc,w_to<

lpooOo V " IV O
Deflnltlon 1.:.1 .4 Let E tIt; . RioI> 0fJI"0e. A _ ~ Fol E ill c.lll: F for ..u :r E F .00 , E F . F ill <»lied an ideal wbene...,.,. E F. , E E , and ItII S 1:r1 Imply 11 E F . F • call<:'tID"m In E of. n<)nO':mp(y ""beo:t
"",""",pie. gI-. B top<.>ioci<:aIOp&<:1': n. let F (n . R ) be lhe _ _ . ..... of..u ~.alu<:"alued functiom on n. Then 8{O.R ) .00 C{fI, R ) _ RIea ~ of T(O. R ). B{n. R ) io an Ideal of F (O. R ). but C{O. R ) 10 lOX. f inally. B(n. R ) il lIOl a band in .1'(0. R ). e...... t"",,«b it ill ~ eomplete. Fl:Io- ttlt; "....·1........ <J! thie e<'!<."ti
--

N OIflIMl ~ bat>d B in E 10 Do:dcld. N(IOI' let A be a ... beo:t
Propooltlon 1.:.1.2 Lot D "'" 4

...,....mpl~

• ..u.1 01 E'" .1OCh /JI4I

(. ) D+D c DGnd ~)

:r E D."" 0 So , So:r impj, 11 E D .

Lot M "'" 1M od 01 "'preIIIG ollllo~ ........ mpl~ ~" in D IMt G.. "'-dtd E E· om "'" --;U.... ~+ •. 00\(.£, E Mu u.., "'preIII"'" ...... E E"'" .. diljoi1lt frcrn M .

. . . . . ~ <WF)I:r

.1(. E D: v So "I

P IlOOr. By P1opoo1llon 1.1.9. it ....1Iico.I toloo,.,. that ... l: - ~ ie~ from DOt. tqUi...Jmtly. that Ii .. ; 1.1 .. 0 for all t E D. Now. for all ~ E D Ntil(yinJ .. S " , _ have v So ,. aD
as "

11.

Theorem 1.:.1. 1 (F . RIea Dec:om""';tlon Theorem) Ld A k G """,,I ./ E. n.c... tlw: iio;coi1lt ......pkme.u A' 0/.4 ;, • Ioond. TIo~ ~I """'J'Iemml.4~ 0/ .4' it tmcllr 1M knd ~toI .. A . • ...t E ;, 1M ~ &fId ",m.JA'.A~.

f llO()r. By P "'P'JI'it loo. 1.1.9. A' and A" &llI l)onda. E""'J :;t: E ~ ..... be written , + " wb.re r E .4' I.Dd • E A" _ ~llive (P"'P'JI'ilion 1.2.2). Th. 11.,'" that E _ A' + A~. F'Iut)"'''IlOI"e, A' n .4" .. (OJ , II) tMI A' , A " are

I.

12

ru-s-

oupplem«ll.ary """"'" ~ti,,.

of a

A'.

(ku",ul

of E. F ,ually, fir>ee the OOlnponente in A', A~ positi~ . .....,..., .h •• 15: ioI.he 0«1"'00 direct sum of

sut.~ ~

A~ .

I~

remaiDl '0 be shown that A~ ;" ex&d its diQoinl oompl ,,,,,,,,t 8'. Now it C B , 10 8' C A'. On the ocher Iw>d. B C A~ lU>d E 1'1 the din!<:t JUm of A' . A"; thlJ6 8 .. it" and 8' .. A'. 0

E.

Note, ,ncldto>Ut.Lly. that -S E di!ljoint from it if.nd only if It is disjoint from B (bol
P l"OpI)eT bo1IJod oj tile J_ !41 1-s,1Jtn" ofoo;k JamiPJ ('1;),0 ;n A. Ltl M, be ~ clGM ol 8lq>l"t:fnO oJu..os.. """"""'PIt ."bod, oj M , !hal ""' 6oundtd . . . . in E . 1'1II:n At. _ B;' . PROOf': Indeed, M. C B . Now. ~ti,,« l.l"' be ,be d illjoint comp&.,rnent of A (... , "'Iui>-alettlb". of B ). and taki,,« D .. At, in Pro~tion 1.2.2. we lee that ......,- (1",,_ of I!:"" ill tho. IIUIII of an cle""'nt of M. and an ~ of l.l"'. Beta..... E ill ilic dlrttt Ill'" of B and l.l"', tbe proposition MUon.. 0

knd gcnenLl<'lli" &r .."".. JiKd a E E emf leI bo! 1M .1;0;'" COI I+leill~' oj B •. 1'111:", lor eK>'}t r E e'" , ~ "",",pone'" oJ", ill B• .. ..,....: 10 oup"" z' inf{ .. lol, z).

Th«>renI 1.2.2 Ul B. /10

e:.

~

PftC><)P: "-un\
,.netUed

If nEE, ~ E E .... diojoinl: . ,Ilea ,"" b6Ild.! A . B by ". b, ",,»«:livoely, "'" dOUoinL Inc\eo!d, • ~ to the d.joinl oompk" ..", it' of

(n} ... B C A'.

1.3 Order Dual of a Riesz Space DclInl.J.on 1.3. 1 A 1i!>eV Iorm Lon.., oro.oed _ ~ti"" If L(r ) 2: 0 (in R ) for all % E E+ .

If L ill JXI'Iili~, the n:btlon P~itlon

% -:: ~

implieo! L(-s)

"pace E Is Mid 10 be

':S L(').

1.3.1 Ld E ... ""...-dtml _,tn" 'J'I'<'I' and L .... ~jlmc­ tionjrom E in.to R ~ng L(;,:) 2: 0 Jor
1.3 Order 1>-1 of.

ru... Spoa

13

P I'IOOF: Additivity of L implies thai L( - : ) ,. - L(: ) for II!'m')' : E E, and ., ... .......:I ooly treat thoe cuo in ... bich ), 2: O. N""" L(n.) .. n · Lj. ) for.U "ENand,.]I.EE, ~

LG ') " ~'L(')' Theedoot. L(r:) .. r · L(. ) for"]l ralioccal numben r :<: 0 and ,,]1 • E E. Now , If r , r' ~ tatioccal num~ satWyin,g r ~ ), ~ r', ~ ha,,,

r ' L(z ) ~ L(.u) ~ r' · L(. ). But r · L(z) and r' . L(%) "...y be IUen .. clooie to), L(:r) .. dtIIi"'f E #tIdI tJwu C + C c C .red JC c C I.". ~/), > 0) """ ""'icb E .. C - C. Lei M '"'" A ""'""" frr>m C into R MlU/)cfiIJ M (", + ~) .. AI(", ) + AI(p) "'" all"" ~ E C AM AI (.u) .. WI"')

f.".""" % E C ami Ctoer)"), > O. Then

t/c.erc: aU,", A ",,_lin...- ~

LoIMI<>E. P ROOf': EIocb • E E can be .. ritt..., % - ~, .. h: If % _ ~ - "I II aoot.ber decompOlitkla of. (~, " e C ). then % + " .. >:' + 1/ • ., M (", ) + M ("I )" AI (>:' ) + M (,I). The mappi"ll E:t. _ L( . ) io obriowoIY. linear form on E . 0 1.3.3 Ld E '"'" A lli..",W "",/.or 'P<JCO, "mlltl M .. a....."..." fr=o E· into R + ,1IGh tJwu M (%+~) .. M (. ) + M {II) /.". 011 • . II E £+. Then M ""'" ....iqw PQ4iI;'" I;""'r~ I<> !hI,,..\f E . P.~ltlo"

P lU)(lf: The acne &fIUIl>I'UI OIl lhat of P 'OJIO"ilkla 1.3.2 pro_ tW AI b.u .. unlquoe addili"" nullIIion L 10 E. Ptopoaitinn 1.3.1 then Jboon that L io llnear. 0 For thoe mnIlDdcr oflbio..ction. Ie!. .. ru-",*", E be p,....,. The _ Q 01 pOIill.-e U....... formo on E io ... ut.et 0I1.he ~ dual £' 01 E ("'*'" 01 all tiMar fonno on E). ClearlY. Q + Q c Q and ), . Q c Q for oJl cui Qben ), > 0; mocWIU, Q n (-Q) .. (Ol beeaWlt, if L and - L both beIooI to Q. then Lj",) 2: 0 and L(.) 0 for oJl • 2: 0, ., L .. 0 (Propooilkla I. t .3). Thio " n . lhal Q ddil>ee &II ordtt ...u.tkla OIl £' : L S AI if and only if /If _ L .. .. pOIIili.-e Untar fonu, or L S!If if and only If L(. ) < !oft"' ) for "U % E EO. £' io tb"" &CI ",deeed """""" "I*"', and lUI ~Itl"" t lc""",UI an. the pOliti"" "near fuo rna: thlf fact toCO)UUUI for thoe ~ooIoey p~y IDtnxloced.

s

DefInition 1.3.~ A liDt&t form L On E is6&id I<) benrder·bounded whene-teI" L It. bowxIed on bE E : Iwl $ z) for every ,. E E-. The :IpAQe of .oJ1 orderbounded l i _ f("". 00 E Is c&J1ed the nrOcr dual nf E. Tl>eore m 1.3. 1

fa) A. Imt4T form L "" E

01 ~ poniI.... (l)

u <>nI"u_Ioo ..n.Ud if ..nd <>nI, if il;, ~

mota>" I~·

diJJ~"'fUY;

n.. C>f"der i....t n QI E u" DtIIiebn4 ocmpkk~ .. 'I"'«.

P!tOOP: If L .. U - \I , w~ U. \I ..,.., !;orO po&itj~ linear 10,,,,,, on E , tbeD, lor all ,. E E+. the ~ion - ,. S II S ,. lrnplio.'e -U(,.J S U(r) U(..) and -V( .. ) $ \I (W) \1(.. ). lIeno: ILII,)I S Ulol ) + \I (ol), and"" L is oro. ....

....

""'Conveu.el.r, ....

s.

s

that L Ie ordeo--bounded. Wt need onI)' find .. pOIIiti~ ti~a:r bm N tud1 thu Nlz) ~ L(,.) 10, eed> ,. E e+ b ill lbe diKe .."". of t..-o pOIIiti~ Ii.,.,.... bmo. Put M(.,) .. ""Po;s.rS-' L.(u) fur all ,. E £ +. Then M (,. ) + M (:r') .. IUPoS~ • .o:s:"S" L{r + Y') $ AI(.. +:r') for &l1" , :r! E r . Nert, for ..U < ~ 0 S : S z + r , ..., find 1/, Y' E e+ ,ueh tlu.! 0 S. , S z . o S V S :r! , and : .. (mn;illary I<) Tbo,o~ 1.1.1 ); then 1,«) .. L(u) + L{y) S M (,.) + M (r). wlotoce M (,. +:r') S M (,.) + M (:r'). Since MI"+:r') .. M (%)+M(:r')1or 0.I1:t.:r' E f:+ .• be m&P~"'z - ""I'oso$. 1-(1/) frnm £+ into R hAIl .. pOIIitl~ linta:r ~ M \.(> £ . n .. p«M'!I tbe fir3t pArt of tlot Ibtotem and ~ O>Or.~r. that M is the ouP"'mum 01"0 and L In n. [t «'oWIIS to be ))«M"Il that n 1$ OedekiOO <:omplele Qt, eQuivalently, that ~ry """"mN oet of pOIIi.;~ linear forms.hat is dl"",1Cd upward and bounded abo<><: (in E' ) h.u .. su"",mum. Ail thil ill. in flirt, .. = ?'luetlce of tbe 1oUowi."l! lemma. 0 \Ul'lOl

,+ "

"*"

LemmA 1.3. 1 Ld H .... '''''~ply ..,.-rd.iireold ,..bn! f>1 £". II ""I>..oN " lol) .. fi>ta" Ir>r 0./1" e E+. Ill"" JI IWo .....,..,..,.. .. '" u in £'" . on.o:l .-(:a-) .. ,ul>..oN .. (,,) lor all " E

r.

P1tOOl' : For ~ry" E .F;+ , put v(,,),. SU""(HU("'), Tlotn vI""' )" .l.u(z:) lor 0.11 " e £+ and all .I. 2: O. Font... """,,, vI,. + ~) = u(,.) + "(W) lor o.Il "', ~ to e+, tM:<:t._ H 10 di-=lCd u~, The lemma n...... foILoon from ~too \.3.2. 0 fUo:tJI that L +(:1') .. '''PosrSo L(w) fo< all LEn and toll ., E ~

"""<:too,, II••,

(L V M )(,, ) _ (L II M ){s) ,.,

...,_.

IUp .~o..~

.~

.. ....-. inf

~~

r. From this

L(II) + M (:) L(,)

+ At(: )

(n

,

1.30rdt
for all L, M E 11 and for ~ r E £'". In particular, tbe IiroI. of tt... bmuIM, we _ ILI(r) ..

~1lI

15

M by - L In

that

'"'....

. ~,. ~ ,

L(II-

~).

~.

Obeen'e that I~ - % 1S r in thil IMt formul&: &100, tbe ",w.ion 1.. 1S :r impl;" L(.. ) _ L(... ) - L(M - ) S ILI(" · ) + ILI(" - ) .. ILI( I.. I) < ILI(:r), wbellC! ILI (r ) _ 'UP!-I!Oo L( ..). In particular , IL (:r)1 S I LI(~I) for all % E E . ProJ><-ltion 1.3.4 T_ ".,litillt: lillftlT /omu L, !oJ "" E an: dj;jo;'il if Irwi ~ i/, for """l':r E E· .nd ~! > O. tMr< """1 II , z E £+ #...:100 /Nil :r _ w+ z .nd L(W) + M (z ) S ~.

Paooy: In view 01 tbe """""" formula in Eq. ( I ), tbill conditioo L II M _ O.

m

F7 no that [J

P~ltlon

1.3.5 Let L . . . ".,IEtlut: Ii..... r lorm ... E. A "...;m.. linmr form !oJ .... E Iodoftg. t.> the !Iond ~ 6,1 L .n 11 i/ and MElr i/, lor '"""'Yr E £'" """ ewry t > 0, tMr< .noll a raU nzunkr~ > 0...d>1ItGl the ...",..., O:s , S r arwl L(, ) :s ~ M (r) :S !.

,,,,pl,

PRoof': First , we obow that tbe condition ill ,.. "y: if At 2: 0 beloap 10 tbe '-d snzeraud by L, tben M _ V8~ «(nL) II M) by Tbeonm 1.2.2. Now U• .. At - (ilL ) h M ill IX"iti~ !or all n E Z·, and since 118 U. .. 0, Lemma \.3. 1 poo_tbat U.(:r) co.,," .... to 0 .. n _ +00; thlll tr.:.; Il oudI that U. (r) :s t/2. n havilll Men c:n.:..n , ...., _ that the f1'latioo :s , :s " [",pl;"

""m-

o

M (W) S t/2 + «(nL) II M )(W) S t /2

+ IIL(,).

L(II) S t /( Zn ), .. ..-as w be . hoom. W~ """" pro,," that tbe conditioo II I Ul!icimt. For """f)' 1X"i11~ lioear form M on E...., can wri~ M _ U + V , where U I:>eloocI to the band ~ted by L , V II diljoint from L, and U , V "'" poo;iti~ (TI>eon'm 1.2. 1). NOlI', If M Mtiolieo tbe condi.ioo of .be I>f"OP<*tioo, then 10 0, 'boo_ 0 < 6 :s , Iudi. that \.be f1'iatlonl 0 S .. :s :r and L( .. ) S ~ imply V( .. ) S t; for ....... ~ , z e £+, ...., ha.., :r _ ~ +. and L(,) + V(.) :s 6 (J>ropo:.itioo I.H). Thus V (r ) .. V eil) + V(. ) S t + 6 S 2t-. Since t ill ... bill"lry, V(:r) .. 0 for all r E £+ . Hence V .. o. 0

Thus M (, ) S !

w~r

For example, let T{O, R ) be tbe ordenld """""" 3paoo of all ....:I·valued

function, on .. tel. fl. "' ... t . : / _ 1(0)

Gi~

distinct " . ~ in fl, ...., _ t hat the pos;tl~ U_ I _ 1(6) on T (fl, R ) "'" diljoiill.

"""!. :

••

1.4

Daniell Measures

()efu~

.1"(0 . C) (""'pecl:iV'Oly. F (O. Rl ) to be the spo.co: of complex· (respoc. ti""ly. reel· ) valued fu...:\kms on 'giw.n nonempty 0Ie1 O. UId. For I e F {O. C ). denote by ~ I UId 1m I the re.ol and in"'tinalY part.5 of I. ILl '/itO, C) be a _tor .ubopo.ce of F (O.C). euch that Tk/. 1m I . and II I belong 10 11:(0 . R ) _ '/i(O.C) n .F(O. R ) for any I e 11:(0. C ). Since

···J "">'V' ,. I , ) _I, + h+2 I" - h l

and

. ' (I I )

""

"

-=

I ,+ h -ilo - h i l

I" I,

€ '/i(0, R ), .., 900 t llal 11:(0. R ) is a lI.ies> subopo.ce of .F(o, R). MOii,OOt r. "'" _urne t41, for all I" I, in 11: + _ 'Ii(O, R )+ and tll 9 in H (o.C) IIUCh thal 191 < I , + h tb
for all

19.1 S 1,. I9'>IS h . Bod"~ + 9'> - g. n..-n. fix"" & Cli".,.,!Qnn" on 1I:(0 , C) &mo\ml, to fixirq! &1'1. R,..llMar mappirq! r from 'Ii(O. R ) Into C . " and ... """tOO by ,,(I) . ?(~ fJ+iT(I",f) lor tll I e 'Ii(O, C ). For IhlI lEI "". "",..-l not disti"l!uish C li""", on 1I:{O. C) from R,.. Unear "",p~,,&, of 1-1(0 , R ) inlO C . J:le6ne a CI"-r for m " OIl H(O .C) ... reel if ,,( f) ill reoJ ror eva)' I in H (O, R), and pojilive if ,,(I) ill pooiti ... ror f:6Cb I in '11 +. 'The"", of R.li"""", forllll on 'Ii(O. R ) will be 0""'..00 br tbe rel.lIon 1', S 1" ... ~ " , (I) 5 1'.(1) ioo' &II I € H+.

f"",.

Ooflnition 1••• 1 A Cli,...,. form I' On 'Ii{O. C ) .. 6It.id to be of finitfl vario;.. tion wi......:"" L(fl • ""Pri'''''III.C).'''I~ 1 11'(9)1 is finlte roo- every I E '11 +. TMorem 1.4.1 ~II' ''''' C ·/......... '''"'' on 'Ii(rl, C) . 01 finik wNlioft. :n..:n u..: ""'pp11Ig I _ L{/) o~ H+ h4I a ~niqw C ·Ii........ ulennon 11'1 ((If' V,, ) to 1M """"" 0/'l1{0 . C l · M"..,.,... •. 11'1 ~ 1M .... 4Ilut olllll ""';Ii. II ....., lornu " on '/i{0 , C) ... t~fri", 11'(g)1 5 "II, )) lor 411 , e '11 (0 . C). 11'1 it m/kd u..: obol..u oo.Nt, or u..: -"olion, 01 1'. PROOF'. Lt< T be tlot unit~i,.de in the (:<)ntpltx pl&ne. ILl 10, h e 1(+ . If g" ,. E '/i{0, C l an:M>Ch that 1911 S I, and 19'>! < h . tben Itt + ( ,.1 5 1, + h for all (E T ; but ...., ~ Midc • • function 9 e '/i (O. C ) , lTd> thal Igi S h· By hypothesi', l boa exlol ",,. e 'I1 (rl.Cl 08tisfyi"l! 1901 5 ', . 19>1S h . and, - " ... ,.; I hUll 11'(1)1 S 1I'(g')1+ 1,,(9,)1 S Ll / ,) + L(h)· SiT"'" 11'1g)! is .... bitrarily do6e 10 h ), we _ thAt L(!, + h) 5 L(!,j + L{/. ), and

,,+

L{!,'"

.. L(f, + It) - LIft) + LIIl). By Propo6ition 1.3.3, L Ms " H (O.C) . oo 11<1 ill

pOIiU~.

Iu-r ,",,10M Ii')'! 11'1 to the .,boIe of ~....sl of the proof _ follow$ easily. 0 UD~

1.4 o.m.!l Mo.umI

17

n-"em 1.4.2 ut " lit ~ C ·b'"""" form .,." ?f(fl . C ), and _me tAal,,;, ....w. ut ... "" the H·Ii........ fr>rm .,." ?f(fl . H ) c:om!Ipondi"f t<> " . i71rm ; 1nMftII"' . ~(f) .. lU~rE "(n.A).t.Is/ ll'(J)1 (10 IkIIt 11'1 ~ t<> II'!). P IIOO'" The 6m lUlemenl iI obvious. ,,"ow . for fixed

a ..

. up

• f ..m....I.I.I:Sf

f"

11. +. pUl

1,,(, )1 .

I'or e¥ery 9 e 1( fl.CJ salillfyi", 1, 1 S J. IM~ v:iol.l a cnmplex nurnt.o.r ( witb moduli.. I .""" that ~ (,)I .. ,,«(g). Then 11'(, )1 .. ,, (Re(C,» a. So !l<1(/) Now the .... , .... i""'lualily is ~k!arly lrue. and lhe reoull

s

s ...

rouo.-.. c

The.~

QM (fl. C ). of I""'" eli ....... forms 00 1((fl. C ) dial ha"" fini'" .......Iion • • a _ _ IU"~ 01 1M al3<:braic dual 11.(fl. C )" and """Will lhe lei. of Il<*ti"" U-.- for"", on 1( fl. C ). MOito ~'.lpl_ p!or any Il<*ti"" li-.- form (by Tbeortm 1.4.2). II' , + 1'.1 s 11', 1+ 11'.1. and lol'l .. In l'II'1for alII',. 1',. I' In QM (fl. C) and all " E C . I'o r any linear form I' on 11.{Il. C l. lhe linea. form Jl : J _ ,,0) • called 1M canjuple of 1'; Re I' .. {II + Jl)/2 and 1m I' .. (p - p)/('li ) an. calk!
DefI nition 1.• •2 A I' in Q.\.i (fl. C ) is said to be a Daniell - FZ".m! if 1'(/. ) <:OII·. u ... to 0 for ~ dec , 56", sequence (/. ).~ , in 1(+ admitll", 0 as 11.1 Jow.er en""'ope. Equivalently. p e QM (fl . C ) is a lAnieli metllNl'II if 1'(/.) taIdo to I'U) for evo-ry iDC.--i", ""'I""""" in 11.+ o.dmilti", I E n + as il.l upper ~o,-.Iop< . Call M (fl . C ) the -=tor rubol*'" of QM{fl. C ) <:<)08isti", of the Daniell " ., ,.,..,. and QM (fl. R ) (, !l pectivdy. M (fl, R » the H-woctor ~ 01 1""'" I' in QM{fl. C ) (""pediveiy. in M (fl . C » IILaI an.......t .

P ropo.lt ion 1.4 . 1 M {fl. H)

u ..

baM

In

QM {fl. R ).

P IIOOF: Let p E M {fl, R ), and IeC. (I. ). Z' be an iDCreuilll!: ""'I"""'" La 1(+ upper ~o,~ I heloo># to 1(+. Tbe ""'I""nce (p+U. »o?: ' inc> F ,., and II ...... ,Nkd. 10 .. .. ruP.Z' I'+U. ) • finite. For ~ > 0, lhere in "H.", , :5 I , ouch tlw. p+(f) :5 pIg) H / 2. Now the ""'I""""" (tofU., , l) . z , iOCiCT E and hzoz LafU, jI) .. fJ as it, upper en,-.Iope. Thill ,,( infU~ , fJl) ~ to I'(gl, and ,hel'll ill an n for _hid> ,,(g) :5 1'(lnfU • •,l) +t/2 1'. (I..) +~/2 + ~/2. FlntJly. " . (I) +~, ",Ilich ,," 0_ t lw. ,,+(1) .. a ;

_"'*

ex.",

:s ..

s

:s ..

,

.. be""", we _ lhat 1" II. Daniril mea8U~ &nd .M(O. R) is. 1Ueoz .... ~ 01 OM eO, R). tiat. Id A be. nor",,"p1y ou'-t or M + ., M (O. R )+. di~oo upward , bootInde O.lbere uillU ~ E A -n that (p - vHt.) < Then (I' - ..)(f. ) ~ ttl ro. all " > ! N<>-. for " larK" eDOU«h. V(J.I ~ ,/2. 110 ,,(f.) ~ t. lIe"c~" Os" Daniel! ,.-ure. 0

Ln·

Ob.enoe thai 1/11 belouc> to .M{O, R ) for ~i)'" E M {O, C ), b«&1.IIIe !PI < I&1'1+ IIt''l'l· In ]n>pO we IIhalI ex"",i"", !alt<. If I' € OM (O.C) and ,, € QM+ ....,...m thal!P{~lI < v(h) lor all h E 1i'*' , t'- I,.{gJl S v(lIi) lot 0.11 g E )t(O , C). For" E OM (O, C ), lOOt 11'1" SUPGET Rt(QI' J.

I

Leo E .... " JAddr '-.j -.~ R-. oJ*)t. A 610."" ...... :F .,.. E .. oaId '" bo _ .......... bo,.e'd thtn ttiouo X E T o.tw 00 bourriod.".,... uoI below. Tlw:ro _may _ .... ho _ IiIDiI (n ' S _Im.ty........ Iimil) <Jl F . wri._Iim_F C. LLro<' Mly. li",kIf F) • ..,. , "" "",,,,,II In/",, (... p X ) I. S"' .iwlr. _" (Ia{ X)) 01 E, ,.,hen X " " . - ....... ' M ~ B 01 000· .......... _ X E :1': ."""

w....., .ho,

;;""1aC1' S "",,,,,,,F. F "- ........ ~ ...;. ill O. "" II (",Jo •. ) . ( _fOat. If lim kif F • 11m "'p:l'. "Ib.o " " " _ ...: ... 01 ~ .~~.Io:mtntI! .. thorn d : ' Old ..,. ~m 1'. and .... _ .ha< F """'.€Fi <0
I,

",,""'IF""

...

T 1I~'It.et_F_£ . _u..,

101 (oup X - Inl X ) _ U" ....p F - lim iIIl F ,

,.

Let / be ... "' .... "" ~ I...... E ""'" on i1MI of. D...lddDd

...... pIook 1UtM""""" F . If" fi""," balOo F "" E """',,,, ..., _ o.tw tho ~" ... _ /IT) "" F .... "cox.. _ .ha< liID/(F) - / (ti1llF), 3. I.e< A be. 'II "P'} ",<1.. «1 ott, d!rocted "po<&«j. _ S.be filter bMif (p E .... : 8 i! ,,) (a .. E .... ). Tho 610 .. "" S .. <we. 4. I.e< CE; )", . be.·n , •• ply family of DfIdeklDd """pIook Riotm ud lot E be tho " ' _ """"'" opooo E" s_ .1Ia< " I:n«ndtd fiJter ....... F .... E .. - ........ ax •• ot 11_ only If. 10, -.,. i .• be AIW ...... JI"", (.Fl II ",
.... . ....... A...... ...,01""'_ 0<1..,. *"

A,

boo·...w

n,••

'''''.0

"*"',

r

2

"*"

Lo< 0 be. D<dotiod ...."P'o<e Rioa ...:I lot (6' ..... bo. 1&miI:J 1m E. Ordot< 1>1 lod,...,.. ."" <:t.oo F{t) of 6nit ....- . 011 . _ put 0" - E ,,~. ball H E F (I) (.. .. 0 ). Tho IomilJ {6 ,) .. , 00 oald.., be ( _ ) ... ",.. ·h. if

E",,"c·

Jor Ch'P""" I

IV

I ON , H E' T{I)) io boIoa.cled aDd il ,be """Wi"3 H _. " bu • limit ...,... ,ho ~pwanl-
wri,"'"

I.

"""'" .1Lot a family (r, ),~, if .ummablo if _

ODIy if ,ho

IoIIo-ins toad ..

lioot Idd: (a ) Fa-......,. H E' T(I ). ,be .upremum r" in E . (b) Iul"" . ...m ... _ 0. (U .. part 1 of Exercioe I .)

~

11..:1 : K

E T I l), K ..... H ..

I,

hat a

1.

"""'" IILot IWIJ' oublamily of a oummabIe family io .....,_Ne .

s.

Lot (r, ),., be •• ummabIe family iD E _ (I.) ... be a partition of I . Fa-=!> ). E L , put .... r, . Show ,bat the family {" )' H if ."mm,b!o

E,.,. aDd ,bat Eu-., ... E ... , z •.

4. Lot (z,),., be a family ia E _ (I . ), ... be & &1Oi", partit .... of I . S u _ ,hal.be family (r .).. , • • O\Lm ..... bIo Jor an ). E L. Sb<>o-.hal {r, ),.., 110 . "m_Ne I . Lot (z,J.. , _ {.. I .. , be , _ ...... mabIe famitx. in E ';Ib lho _ Iade.~, 1. p..,... 1....1 (r,+1' )"" lIoownmableaDd tIW ,{r, +.. ) ..

E.•

E.. ,,,, + L,tI"· 6.

Show u.a. a family (r ,).. , of ~Iiw ok ,..,.......",m'b!o if aDd oalr if" tho ~ I.., : H E T (/)) • boIondod _ _ _ .......... lb. __ . E,., z , _ ... p"....m .... Lot (.. I... , be. r.... iJJo ... E .ucl> IILot 0 S .. :5 '"

Jor oJl ; E I . Show that .be io iD E ............ , (Iz,I)" , 110 .......

_ aDd ..... , iD.hiII .... IE .. , z,1 S E,. , lr,l· .. Lot (z. )... be. family ia E a/ mn• ....u,. diojoil1!. .Iot .... _ "'wet.b.ot liz,) : i E' I ) • bounded . - . Show ,bat (Iz,I) ........ m ..... bIr, prow, in fact, ,bat (L,u"')* .. E ... , z: aDd (E .. , r,)~ -

L.., ,,;,

... 111M S

IE,.. %,1_ E ... I",).

Lot E be • o..dNiDd OOIIDpiote I.

rur.. "-",.

Sbow that , ............. lamll,o (u. j ..., in E a/ .. rictly ';,b , ... 1ollo.i"3 "" \ ,;..,: (. j (b l

~ . ;.."

.Ioroe,,"

.." ... ..-e dio;om. .. be ...." .. J;' t . Fbo-......,. ,, :> O. t bore ...... i E , ...... Ilia< lul(" .... ):> O.

(V" Z f E I , aDd",,_ tllM ' Iz. ): i E I I .. .....' k.! _ _ ia E. Show , ...., lot: ......,. ; E I, r • • ,ho
E..,

••

"

loo:< A b
,haI." "

I . Pro>" (PI V ,.) _ (z " " ) V{z " " ) Io<..n z, " . " E A if and oa.ly If z V{ .. " " ) '"' (ez V .. )" (z V 1'» for ..n cz, " , .. E ..t (in .h...... A io Mid to 110 " ",nbuti ..).

2.

Stow- 'hoolollowi,..; O>O>diliono {., (b ) (e)

or<: O&Iioliod:

A io " iitribooU ... A 1&00 ... "' ~ _m. D. U>d .... pr=um, I. A-"",,, E A 11M • «01 z' ito A, ' 11M io, """"Y z f' A , _ "";"to ... r € A _ ,bat %/l z ' _ 0 &ad z v r _ I . I

, , , ....

(Tbt io, A io . &c It " • ....1..... ) Provt ,bat ~ z € A ..... ...,;qu., .1 -r and ,ha< (:00: V _ :00:'" r/, _ fez /I ~ )' '"' z· v r/ ro. oil z. ,E A.

,r

«wor!o5

I•• " . ' +00;1001 _ •• c~ _ io ........bat io boob <: ~ . I U>d opeD. AH."..!__ II' _ 5 io <&Ilod ...-odi.... . _oaI if ,1>0 .""",,, tot. Iotm • _ lot ,1>0 "'1>'1 C '" So 5 io II.... toOdorl!" io ...-odi ......... "

"'*'"

I.

Let 5 110 ..... ·Ily d _ _ ..... . """"_ H· .....c,If .............. 6{S) be 'ho c!.M of .." ED I to of 5. o.nd -10(5) to, Incl...,.,. p",.... ,t.u. b(S) • • _ _ .1. b....

2.

.... 5, i be Mo 10(5) """" "(T) . P ....... bo, ,be«: ""'.... I..... " .... pIo .... ., '" 5..., T _ ,bat ! W ) _ (o>(cz) ," f' UJ for ..n U E 1>(5) (if. E 5. po.I\ 8 _ (U E 10(5) , • e UI. &ad obow 1M< (""\"u ! (U) io • .. "II ....... ).

3. Lot A be. 11.0 b , . . , _ A- -..II, d jeooq_. T io ' .

_

S"- _

'" A"

....,.. .... s..-, I 0 of< I. " ""boo:< J '" A 10

6{T) &r
....... , .....'" A I\u •

"''''' ~''m. by ~

to .1.

if A _

2).


sw.. _

w• .....,. _ .....

Ia'u.:.. w..

(uniquo II!> ",,<WI_ to tho ""'"

~he«I

{.,

calk
IEJ.

(b) 011'1, and (e)

lor

..n", ,E A,>: "

,E J if &ad

""'J' ifzE J and, E J .

.... 5 bo ,110 elMo '" """"imal ~11erS and, 100- - , . z f' A. put S. _ (J f' 5 : z E JI. p,...., ,bat (5.)' - 5 . , &ad .ha< s. ~. So " So U>d 5.v. '"' s. u So lor 011 :00:. J In A. S __ lha< tho _ So korm 'ho _ '" • H '. h lf~""

-

s.

4.

.... Ai C Abo - " ,Ioat 5 ;. U"Ai 5, lor all ~nl"' • . . - . A. of Ai . DoaooI. by 1 tllo of 1 _ • E A b- .,hldl .bent .,,;0,0 • fig ... tot A. C Ai ...... lhal • aI. I • • all ... and _hldo that 5 .. """'pOd by _ " , , .... 5 f- U ' EA, 5. (~Zcr~·. Ienuna).

""*

Ex,,", ': I 10< CMplft I

21

5. $""",.""".110 map ~ _ S. io alai.;.", ioo<'.....iom frooo, A 0ftl0~S) (_ ,.., ""","poet_ of $) . IS

A Boo!

A io
~ ~

~~ .

I.

A H'''wIQrfl" _
X 10 oaid to'" """d'''!) dio<:nn""""'" if ,hoi ,h• .,... hold:

, _) E""l" open"", """ _ (h) I~)

~

(lo..

clo>oi>o",.

E~ """" .... U wdI ''''''' U _ i"'{O) io clos>tn. If U. V "'" diojooint _!OJ' to of X •• _ ti n P ....

."".a..

Pt-ow. .""" ;r X 10 1y d~ • •Iw:n X 10 ..,.&11,. dio<>on"",,,<.l. 2. Lott X ..... t.ota.ll,. d _ _ ""'. ~ H........... . ~ »~ ' 10M /( X ) 10 Dede):!nd -..pIece if ond Gilly if X io ""~ dioooo~. f

IAt X be .. ~., _ _ An LSCfwo<."ti<>oo. on X ("'pood i.-.!y. .. USC fu",," tion 01> Xl io an .t.t. .. iati<>oo. f<>r. _ ( _.~. a wn) .. jrnp;i......,. "'....100 on X. If A io .. _ ... _

A" jrot ""Y " E X .

01 X o<MI I • fiu>cti<>oo. _ ~i1i .....

n.... M' r _

X o.IW,i"!l ...;~) 2:

/ (' ) ·

......... A

..

'0"

inl ,

":or)"

A into 1t .b.a put ..p I I, ) , ... ...,""

01._

U.........1eot USC fu...:t.io ... I" .... I I,) f<>r 0.11 ~ E A. ...:I , io o:aIled l ho USC ~oriuIloo

" J. II< I.

""',I<. -...... ''''''' X 10 """'"poet...:l

0l
kt I ... an lSC fu""'"" on X ond , ill USC recu~i<>oo.· For ~ a E R ... poi ... " E X ..... h ~""" A") :> a doeo '"'" I>olon« to ~M_'" of A _ (r E X : .,(p) < " j : to, 10 l.5C 11<..... , .. <1>n\i....,... For """'"l' pair of ..... i<>nal N,rnhen <. ' < < •. """"" .""" the <~ I .... I" E X : I I,,) :S <. 91") 2: . / .. ..... bdte tJo.a'I _ , OIl toido .. ". . ,,, , ""'.

Mdt."""

2.

~. Ie< ~ bto a ful>Cl.ion"';'h "01_ ia

Ii. dofined and """,i. """" "" the 'X _ Mof ._he<. _ ... IoI. Lot/ ....... lSCf....... m..n X ;nIb R .bat _ "';'h ~ on X - M and ito equal to - 00 "'" "'. Show.hat.ho USC ,.....~ioc, of, _ with ~ "" X - .ft. u.d to "" X _ lot , 'M' ... ohow ..... A h.oo .. tutiq ... ""~ to Denote to,. c-' (X, of . _ <:O<&tla ....... IUII«_ I I""" X ;nlJ> It fo&- ..-~idt r '(+«o) and r '(-00) "'" _booo: .... .,...,."';~ .hlo.,.., ... put A .. I + , Slmll&rb-.... deli.,. ~I lot all ~ E R ond all I E c-'(X . C""{ X,ltj . "*"",,ittally .... &.0..:1. io. R.;e,;o _ ~ .""". if H ..........."pI,. .._ of C"'( X, 1£). .....,nd-! 000 ... and if I io 1.. _ _ ."' zI ..... (I(z ) ..... ~ .. ~z ) 100 oil z E X ). ,ben ,be USC 'OS ,)...I _ito&> <~.I

,

m........

n-

""".in...,...

x.

m.

g c( / io Ib.I '''P'''O'Urn of 11 I~ C"" (X ,Ii)_ Cot>tludo iU ideal C(X , R ) ..., Oo.IokiDd """pieI.t.

•.

,h.o., C-( X ,II:) Md

~ .4

boo ... """" ... t.d c( X , and let H e<moiot of tho Indicator funo:tiono Iv c( ,bo urn ofHlmC(X, R ).

8

~

1.

E be •

[lor'

If " E E ;,; . lhal ( ~ , " E Z +) ;,; bow>d''' ,bo. " ..

""PI i " 2..

liD<.

_

,

lbol 1..t""N

((I/ n)..) .. II rot...,.,.

~ E

£',

I~ ... E: 10 Ac
d.....

9

~

E be.IJood.kjDd alID~ R_ Opooo oDd B" fh 1_ boondo in E: Mliafyin& B, C ih . lei z be ... . k"..,,, c( E: ond :t,,:t> III< oom_1I< in B,. ih_ S ' that . " 10 lho ...... 1· ht.l~ c( z, 1m l bo _ B, «I B, .

10 Let" > lI_li_ln.OredrI
,

;,; • norm

:s • :s

1. A < E C ... ho comp'"''.''' of Min . booDd of C. if and only illl u and i..t( .. " _ oj _ 0. S"",," ,Iw. lbo _ A ot ,''''''' eIetoet1.. c 10 • l><dokiDd oom~ Px J · · ~ ~ and io o ~ i in ,be nonD
a.

C. .

For ~

~

let .. bo lbe oomponect <>fC. _ _ by (~" -z)-. SIoow .hat .. be,"",", ~ IIw. C>. .. II ro, A -U" H, ....:I ,ha< .. .. w lor A > n% l. For.11 ~ E R, P"""' t bbt t ho _ 8. In C•• by •• ,;,; _ lJ- .he band in C. ",,,,,,,,ted by ( A.. - %)-. Fix

%

im

OUI>.~ i..t(~~" _

_.ted

in R,

%) +, ..) ot .. In ,bo _

:s "" ..

:s ".

:s

• . Wi,b ~ _ -"1\00";0 ~ J" " , - .haI ,!>t .., "'~ of z in B. 10 k . - (.I, .. - z )· . ....:I 00 ;. .rn.ller ,han ~ . !I c bolonp to A. ;f < S <., and If B 10 lhe t.nd in C. _ ~ by .. pr
r..

Let a , B E R ",ilb a < 0, and I.. Po , '"~ be l ho projcoctlono of C. <...to . be boondo _ ..... by <:0 and '" mlpectivdy. p.",.. Ibbt (p$- Po )(" ) ~ (P. - Po ) · (".(,,)) ;,; ..-...JI..- IbM tI(.. -e.) Now _ by a;. lho """'PI<ment c( ( "" ) in c., and p... B .. 0:. "" 8 . 5i""" the "",,_nl (a .. - .,)- of ., - "" im 0:. 10 pooilive, t be """,_I
""p'"

&.

Let , > 0 ...... let >... ~ ,._ , .•.I.., be t-..l nurobo:n.ud> .hat .... _ 0, c .... ~,.. and 0 < A. - A._, :S. Sot.tJI I::: j::: n . For.,.....y II :S j::: n. d _ by '"

F .. i"b~1

Jt,.. - ,,_. )(z ) -

~ (CA.

-

CA • ••

11 'f 1'(":0..

-

c>., •• )

boll l S i S n . and 10 O~ - r: ,~,S' ~(eA. - <••• ,)1 'f f. Oed""" that , .... _ 00 1"1_ V of C •• ~. i
Ld S "" II.. Stoooo 0p000 of A. 6< I"" indiuto>r fu...::tion of II.. r c-.wy c E A (D O. Fu' ....

r.-.: ....

.... nu..v..r. .I.o . l, . . . . . ....... """ that .1.0 'f I~f, .. f{fl . ..... > ,up,U ,t.). ond 0 < ~ - ..... 1 'f ~ b~ I 'f j 'f n, PIlt A, '" I' E S:f{t) <~) b 011 1 'f is'' ond B, _ A" B.... A,n CA.., b 0112 SIS n . " that If - r: ,~.~. ;1.,1 • .1 'f •. no... W io _ In C(S. R ).

&.

p",.... thai. &

1I_ QI)OnI,j"", ....

E", l ,c, of .IE"..,... . ,

Ifond
.. _ wile, : i E Hl v (.. -

L

c,_

ZH

-

e, : i

I " ",

E A io

pOOili~

..... d •• by

I HI) ;

L

I ••

HVUJ ... H

H
lor j E r}. Thuo tbo:rt io • uniquo ioom<>rp/Uom ' from 1M .. due
Ld (z. ).. ~ , be. Co.ucby ..... _ la C... &nd let I" •• I. ~, ........boteq" n.....::ll t ..... I~ • • , -z •• 1 'f 2"""' boll. 2: I. Put \II'" z ... , _z •• boll . ;?: I. Pro... . hat the fami ly ("' )'l' i ......... mm ..... la I> ODd _ thot. i f , ... (z " )' 2' __ z . ~ ;-z .. , la ..... 2, C. (10 C. io . 8Mod1 0p000). COlldudcl that 9 _ to an _ .. try &om C. i...., C(S. It).

,,,,,,o>o
E.

\1>. '_

""pi '"

LeI z E C•. Pf...-..haI.,. ... 9 io pOOil ;'" if and
II

Lee £, ,,> O. C. be .. in Exeftilo 10. aI>d M!I: B. be lbe band ito E

~

. , o.

I.

B:.

Foo- 0Wfl' z E pu. ~ . ... laf(....,.,) b all I~"" " ~ O. aI>d .... i~ .... 1.... USC fOC\Ilarisat .... of lboo uppor ... . <1",. of I.... h .. "" IbM iz ill """,I~ ___ It, boIt:oIco ... aI>d " h ._ lbat 9, Iml (z, ODd tbtrfti;n lbao, S z . O""h"", .hal ijz ~ ... C-(S.R:\ &nd • • boo in C-(5. Ii) of lboo h • . 1.1", ,", . . .. h ... ,up.,.c•. O$.o'S' h' .

c:

"'pr
:s

:s .(

,».

1.

~ IhM Ih.er. io • uniq ... 1;""'" ""'I> j from C. into C""(S. Ii) .1lCb Ih&t I (z ) _ fr Iot.a % E B: • ..-d _ 1h4.t ~ ;,.., ..... .,.phism from B. """" all <>r.!enId _ oubipta of C""' (S. 'ft) (oboenoe IMt if z. ~ E B: _1Ji s: Iz. ~ z).

1_ s:

J.

4.

s: s:

Let rEB: _ g E c'"'(s. R) be V-O. oud>.hat 0 9 9>:. """ ....y i
p,..,.. lbe

....

btatino t"""""'n: th..-.: ",,>ott • Rita ""'" tt....! ,
ioomorpbiom ~ from B.. on!<> &D idoooJ 0( C""(5. 'ft). lot all • in A _ I mapo C. on\o C(S. R ).

l>.

Ib C. (eurnln<.N:""" ....... do ""... .,n 60l Ex.ercioo 10). s;..,. z '"' oup'ZO ibf( n~.z) 100- ~ 1: E 8 :. j .. uniq ... *"-1>""" from lbo...-de"'" _tor .pooo B . OI>to .... ideilI '" C'" (S, I£) S_.IIM V io ...de< _

oudt. .....

i2

t'"

1(.1- I .. fOr oil. E.A..

""""maI._

Let E be .. Dodddnd <»mplo<e Riees'!*'eo C.... )a. A II In E oJ Olrittly pooiU .... diojoiDt -' ", aDd, fOr - . " E A • ........ e by bMd _ated by ..... E io _ I hlc 10 I'" ide&! of n... AIl.. """ ' 10\\"11: of 11_ (2.)..; A oud>.haI (lz. ) : 0 E A) io bou_ olw:rvo ( _ bettioo J) . fO< ~ o E A. let K . be tbe St<-"-'" of (:0: E 8 : 0 % o . 1:o1(z .0 - 1:) - OJ ond I. ,boi" .phiom~ted ~ from B.. ""'" .... i
a.. ....

s: s:

E>orn:iooe

llJ.

I"'"

"*'" '"

Let X be .... !<>pOI onm .,rl'" K .. C""(X. 'ft) lbe II.,.. ......... ,..,... fu .... _ I from X imo 'ft: b ,"'h>ch r ' (-hx» and r ' (-",,) do_ in X . and "H( R ) .... """""
x.

....,_bore

"'u....

i.

If B;' .. ""-,,poty .u.... of E, boun<\cd above. and if I io , ... upper ........ lope of ..... (:tl. b z E B.' - tha, * (oup B ) io .be USC """larizati<>c Of

2.

by F (A } .hoI .... of finite IlL....... of II . If r e ". _ 1: _ o z. is.hoI of:t t.loo.!! tbe 11,...... a. fiDr all" E A, ..., boo.,., Za _ O"l'.~ inl(" ..... %a ) . .. _ Inl(" .... ,z.) bolonjp to ~ - . ("If·). _ % _ ouP••lA :o:. io tbe OUl'f1'lllu,", of tho u~.di_ted .... B ,, - ' iotlAt! oI t"" '~P""J 'nf(" ..... r a ) (J e F (A )• .. po>oi. i"" i " " -). 0.,. d"...bat ."" &1\.0< (; of -,10.. ol B...-do:< coa ... , to r ond.,.......;... ~

E..

_.p<JOit.i<m

• -'("H' ). 13

t.et E bu RiooI>. 'POO!!. and (U,).." • """""'P'Y family '" po>oili .... Ii ....... _ " " E. Aoo""", .......... 'opoloc T . dofi""" bf "'" ......i _ U,()zl), io a H· ,...,.....,

-.

i . S"",", ,hat "'" """"""" r _ )z) is un.ilormly <XKLtin ...... I""" 8 ...10 E ond ~ in "'" ~ T .

,ha, r

... ..

'I'

ate

E

LC:<'

.".

Ch.apt
l . S u _ that E io DodtkiZ>d oompi
g

3.

s_ 4.

"'*'"

H• ...,.loct b, we .._ Ihr.. the ~ _ _ E ;. (Dmplc< .. N"", Itt A be. & _ 1><1 upward-how 1hr.1 "'P..,A I (z ) _/( .... pA ). o-I...... hat E _ DodtIdZ>d oomplot.e.

or

<»<>I"""'" .....

".p

Le\ F be. "" order ........ Sit ...... f: wi.b Iimi, % . Let A be u.. ... 01 u.. .... p X - iDf X. ~"""' X • . . - _ 'M ~_ of l:>owodod .... ;;, F . Not.e Wot A io dir«t ....... , _ E.

'I'

ate

2 Measures on Semirings

",. l boor)r

lot

~

or ~,~ _u_

whd> .. 1... r<>(t
in , ......... p'.. . 1o_i.a1,

......,

m ,1>0 Iheoty or ptOboohili,y_

n.

U C;-, .. _ _ P'J"'" 0, c.ho _ oeI ol """ipptd .. ith .ymmoOf'eIy .. -urinl! (Orinit.,., 1.\ .2). The " .ri.. of HUm.:;. _ ODd ,be .. "rboa '" ~l .... (W.iU.,.. 2. 1.6). ,,, a ~ .paD!! n , ..... on-. .be _ impor<&D< eompIM of ,...~ _ . _ and ..ill be widely . - I

,_'ion ... ..

'~" tl.. book. 2.2 In . hio ...,.;0.. .... finot dJi_ quMl-_ ...... 'bon _ .... .,.. _~ It. lullC\io>n " on .. -Uri.,. S .. .. _~ if and only if, lor &"f . ' '1'' .. "r
r:

UA.

s. •

L"eA,) """........

bot_ ~nct ' c ' 5 m.... deli""" on a oomiri", S ar>d D&nidJ - r. ...... <106_ 011. S4bnplo fun«1ono ( ~fI""".",.,. ).

W. ;0..""1\",,, J ct " " .. _ .... _ ' ,.. dellJIfId ..." . . oj I.. ~ "'.~ iI>d..dod iD _ interYal/. 2,3

2.1

Semirings, Rings, and

0<1

the

....u.UIf; of a.II

o- Ri n~

Let 1'(0) be .he peA" oct of .. D(II>tmpty tel. n. For all ;t, B E 1'(0 ), tbe dilfe' , ' '''''' of A and 8 , A - B , is defined ... A f) 8"; lhe 5ymmdrie ditfert"llCl!

A t:. 8 _ (A n a<) u ( ACn 8 ) tonaists of the elemoontl that lie in A Of B but not In both. for eKh A E "P(O), denote by 'PA the mappilll from 0 into the rilll Z/ ZZ C ~ua1 tAl I on A , tAl 0 on A • 'Ibm 'P"-,,s _ 'PA + '1'8 and 'PNl8 - 'PA' 'P8 lor aU A, B E "P(O). So the operations t:. and n !lYe "P(O) the st"""ure 01 . uniUty rilll· Dellnlt lon :U.l A elMs R. c "P(O) ill called .. rilll ohubeets 0111 (or eirnply • "'" In fl) if it •• rnbrilll 01. "P(fl) in the .... ua1 _ . Equivalenlly, "R •• ""I .. hen it. IIOOmlpty and when A t:. 8 . A n B lie in R. for aU A, B E "R.

PcopOIIIltlo" 2 .1.1 A n
~nd


if

PIIOOf": The conditioo. D<' " ')" boa""" A U B .. ( A t:. 8) 6 (A n 8 ) and A n a< _ A t:. (A n 8 ); It iIIrnllicient btet.""" A 6 8 _ {A n a<)U (ACn 8 ) andA n 8 _(A u B)6(A 6 B). a DefInition 2.1.2 A no)I"Iempty cw.. S C "P(I1) • caIlcd .. aecnirin& If, lor aU A. 8 E S. then: • • finite colleaioo ol mutually d.joim S _ wJK.e union t. A n 8. &lid if the...". pr
I.

Now the in\<;nooctloa 01 • coIleaion 01 rillp ... ri",. So _ m.y ddine the rinr; ~ by C to he the ...... 11..• "'" mnUlnin« .. Ii"'" C 01 P (fl).

..,1><:_

Propoaltion 2.1.2 LeI S W .. nmiriJog in fl. 'nIcn E E P (O) IOu in 1M ";!If "R ~ ~ 5 if and <>nlr if;1 ..... finiU ""...... 1>1 S,~t.. oJIic.\ .... ~ W ~

dUJoi>tl.

PROOr: lei. R.' be tbe claM oll"'- IUbeets 01 0 ... hicb can he finitely par. tIIO:w:d by s.-u. We fusI. Jbow It...! e¥ery IInloa 01. linl\<; I-.mlly 01 s-ts (no!. '. ' rily diojoim) Iieo in "Ri. The proof ill by induction on tbe number 01 elecnenll 01 the f.a.nlily; IUppoet it illme if lhil number iI n - I (n > 2), and let ( X, ),,.;.,.; _ be .. l...mly in 5 with n el<menll. We an write

z_

U I ";;";.

X, -U

U

X,)n x:Ju x •.

1";''';. - '

N.".., by the Inductloa hypotbtolls, U,,.;.:>.._, X. _ U, ~<. Yj, ... hen: YJ E 5 and YJ n Y. - , if i ;. .t. MOIW"",, I"J n X: .. U' '';G'';., Z.... where Zj ... E S &Dd ZJ ... n Zi'/! _ , if (t " fl. Theldote, Z _ lU,~~U':5"";., Zj.... ) u X. , &Dd _ _ 11w ZJ .... n Z• .Il " 8 if i" .I: or Q;' fl, and that Z; ... n X • .. . b oJl j, Q. n.;" _ lbat Z 1... In R. i , &Dd t be proof is comp\Hc.

,

N.... A U B and A. n B I~"" ~' for aU .4., B E 'R'. It rern.ai ... ro be ~ tbAt A n IJ< beIonct to 'R ' _By the ... tu<:!illll N'gUII'Itnt , i t IUfIlceoI to conoM" .. tile CMe in .. tuo:t> A !MiIt 10 S. 'The> let A be an $-eM and B,., U's.'S~ B; be .. ~nJ«o union of S·~ A n lJ' _ n, Sis . (A n 8:'), wheno An n: lw in 1\.' ; bcoot A fl lJ' ill an 'R'-... a

_"'pi,

P ropn 2.1.3 Ld Site m......iring in n . F"., ~nr {int S·.a. B. (k E K , K /iniU) ,lid! tIw .... .4., " •• ~ 0/ ........ o/Ih. ,.11 B. _

en.,.

C n.,

PIlDOP: E~ _ A. ) n Al'l lim in 'R for any DOI'""pty IlUbtet J of I , ... -=II £1 can be linMl y pUtitlono:(O) ad "'<>It &p ... lilt . /00 .. 01 .11 nobtll .In dtid .." .. mUm (\u A., ~ f it fin;~ and".,.,...,..pfJ .... eI n~ A. ... A~ Iiu in C '"' .... i e I . ..mil al /trill, _ A. me. i'htll ... it a _',;"g, ."'" 1M "'If ~k4 !of ... it eucll, ~ ring ~uJ '"

C.

PRO(IPC For.u £ • niSI"., A,.,- aad F - ('''!.IS" d ,,"nJ' lie In 4r . On tbe;,u.,;... hand,

U

U

A~ .. _

IAt,n(

A.,.,- In 'i' , the let E n F

n

A, .. )I.

where tbe A!j .. n tn, soSl - ' .4., .. ) ~ diojoim ; ...

U

E n J'C .

Ie

n

A,,,)nA; ... n(

ISiS., ISiS_,

.. the

fini~

n

A. ,. lI

'S. :5j- I

a

union of disjoint ..-au..

Let C and • be .. In P r<>po)Oition 1. 1-4. Denot
V the cl.a88 nl finite

of C - ~ (A,)",I be a li nite _ p t y ,..."ily oIlUboIeU of 11, IPUCh u..t at Ieo.ot _ A; lod"'gII to C, and "",h tlw "'tbH A, « .4~ liM I"C for all IE I. PlOt J ..,(1 E 1:.4, ECI . n..n ia«!~

- (il iA.). Il .0

L

K C/_J

(1- 1,0: )

",' _ J

(-I )I KI . l i n.

, )",,, ..

\I ~. ~

A:)

io a line.r oomhil\t.tion of tbe I" (B E V ). If F Ie An R-_lO< $pace and I a rnappi", from Q UIt(I F 01 tho. !arm I:;EI c, , 1,0. (I finite, ... E F. A; € >t). thea, by tbe ... rnmark. J is . . . 01 tho! form L jH II;· 1,,_ (J finite.

-u'"

"'E F. B, E V ),

Dellnltlon 2.1.3 A Mlbclaloo R of P IO) iI oald 10 be. coun, ·hIe unio<1 of ~ iI an R.-ed.

• ri""..oo.

a·ri"l!"~

it iI

H"""". R C 1>(0 ) iI • a -ri", if it .. , ......."'pey. if A n B" 11M in R lOr aU A, 8 E 'R, ..00 if , count&ble union of ~ Is an 'R-1It't. In this eue. , counu.bIe ime, _tio<1 of ~ iI aIoo IJI 'R-_, "'""" 51'

.. here A _ U. ~ A" 'or IJIJ' - . _ (..t.)' N of 'R-eeU. M befor.!, then! Is. omrJio:st a·ri", containi",. civm MlbclM:! C of P IO). c· 11ed the " _ri,,, ~5", .. ted by C..00 oftw written ,,(e) 0< C. P ........ ltlon 2.1.5 LdC 600 a ... /PCI", "f P(O). If E iI an, let in ,,(C), r.V:n ~ uti" a ,»..."dle .ft'600'"11 V ofC Ada u...t E E ,,(V ). PItOQV: Tloe union of tl:oo:o a .. u~ of a(C) tlw. _

by tome a ,untable ruM •• of C iI. a-ri", a>ntr.ini", C ..00 contained in a(C); il iI 11.. i
,em

a

With not&l.looro .. in Pr........,;tion 2. I.S, the elM:! of tl:oo:o 55eU in n «>o(r«l by • CUU/lUlbie union of C_ iI . a-ri",. So every elePPlent in a(C) is contained in • munt&ble union of C-u. DefInition 2 .1.4 A lP'-ayatcm in n ilulul P of oubseuoCO sucb thaI An 8 iI.finite or muntablo union of dilj<>int P_. for all p·oeu A, 8 . Deftn ltlon 2.1.5 A

,.,.~

in 0 iI • clut C of subseu of 0 with t be

roilaorillll poopullM:

(. ) F _ E 11M in C, for all C-eeu E, F sucb that E c F. (b) A countable union of disj<>inl C-eeu itlJl

Obeerve IIw. if . it ...

"·ri,,,.

~Iem

c...t.

Ie ck H :l1rith rESptCI to finite imcr_ tionl. then

Tbeo..,m :l.l.1 (lP' _ A Theorem) LeI P 600 a lP'-.,.1eno """ C !he A·.,.teno ~ 6r P (~ Df all A•.,slmu
.,..riI!f ~U
aU 8 E P . N"", CA coutairro P , 'or eeeb A E C; thus CA _ C, ..hlcb that C Ie clo cd ";th _pect to finite Imcr_tiano.

,

ptOVES

0

Glvm,. !IOm;,I", S In O. l he ~lemenu of ...(5 ) ..·m htncd"orth be o:alIo!d the 5-BoreIlieU, or ~mply tb.. Borel ~ when the", is DO poIifIibility o f amfllllioa. A oemirin« (l'5pfId.ivoe\y, " "11& " ... _ring) R in 0 is Sllid to be. !IOmlt.~lxa (h!IIIpoocti...ty. fln ~ .... _. btaj ..!.me''e' 0 I>oeIon&! to R . The OrDB!Ieot ~t". (~i~y, .... ~) ODfltainiDg" p ven .ubcw.s C ,,£ 'P(O) iii called the ' IVi... (mlpectively, ... -~bta) ~ ...ted by C; its eIemenu "'" the X £ Rand lbe X ' . for X E R, wIleR; 'R is lbe lin« (..... pooctively . ... ·rin«) generated

", C Defl nillon 2.1.6 In alOp<:llop;ial opeoe 0 , the ...·ring 8 (""l*1ively. BII ), gmenuod by ,be eta. '" ali open _ (. ppoleI.ively. <:If all CQmI*I aeu), 10 called lbe Borel ",~" of 0 (no6pCCt;vely. ,be HILl""" ". ri", of 0 ). and iu elorno:nt& fln!l the Bon:I oeu (..... ped;vel,y, t be HaI..- lieU) in 0 . Prop
",,·.bn.and LJ...,

P?tOOP: t:. - I A c o : A n x E 8 K for all X E B", I"' 00QI.n. the cx-lsut-u of fI, PICI t:. :l B. If A belonp to B lUId it OOQIained in lho! countable union of o::>mp8
bece._""""

2.2

tn..e

Measures on Semirings

o.,Onitlon 2.2. 1 A maWing I' from 5 into C is Mid to 1M: ...Jdit;..., .. homeve. II ( A,) II (Ao) for """'Y fini te bmily (~).." of mutually d isjoint 5.-. _bo8rl union I... In 5. We define 11 (') to 1M: O.

14E,

L4'

Tho!o~m

2.2.1 Gi_ 1M. &dditMjI...dll'" II'S _ C . lAo... auto. 1I.. iqw. "dditi... ,.rocli<m II, from 1i """ C ...AD.. ",otrictWft UI 5 it II.

Eaeb Z E R it of the kmn Z ,., U'S 's~ X , . ..Ith X, E S and X, n X. _ I ... ho .... u i '" k. Hmoe _ may put II , (Z ) - E,s's' I'(X, j. To pr<MI that II, io _1kIeII!M!d, _ .......1 sbow that, if Z"'?toloo of lhot Jonn Z _ U,Sjs, Vj , with Y; E 5 and Y,ny, _ . wbtDO)"" j '" I. thot> E ,< ,s~ II( X,)l;:,.s;s, j« Vj ) . N<M. k>r """IY 1 :5 i :5 .. and ."..,.,. I :5 j :5 P. of, n YJ It l he d~1>1 un"" of 5-. Z;J.~ ( "';th I :5 " :5 m(i,j» ). PICI PIIOOI'".

'I'

ate

whl'".,.

we _

,hal LL S ':S:~ " eX, ) .. L;.:[,:s:. plY;).

We prove IlOO< that II, is additive OQ 'R: Ld (X ,),,,;;,,; , be .. linite f.omily of mut....:Jly dilljoint 7i.-8eu, .00 pul Z .. U'S ';'> " X i. We have Z .. U' Si S, Yj. with Y; E S and )oj n y, .. 8 .. hene>~ i 'I- I. &.cb Xi n lj iii the disjoint union of S-1ItU Z'J..,( I :5 (> ~ ,,1{;, j)), 110 Yj .. U, s .,.;. U' :S~ :S o.('J) Z,,;,.. lor

.u1 :S i S p. and

Finally.

.. L L p,(x ,n YJ) ' s.s _ ':Si S. .. L p ,( X; ), 1:5' :5 _

-P~IUon

2.2.1

S~PPO"

th<J.I S;, ~ ring ~nd Itt ,, : S - C W 4" aJditive

J.,....or finite "''''''' (....l,u 0/ S · ..,.,. """"" F ( / ) ;, lilt: po"" ....<»lkd m.:Julon·erd"';on

'orm").

It!

0/ I

(IN.

PROOF: T he..,.wl ill obvioull ror 1/ 1_ I. Let n 2: 1, .00 ""me tbe formula io...:lid ror IJI:S '" N........, ....:.. , hat 1/1_ n+ I. and d _ &rI ele",.,," ~ 01

,. """ _(UA')-_'Aol+_ ( U ,,)-_( U'A.nA,'). ""

,El- I_'

iEI - I _)

t«a'1H I'(A U 8 ) .. II(A) + ,,(B ) - p( A n 8 J for iii A. B In S. Nel
Oeflnit;"n 2.2 .2 A n the ~n,;tl ng S is IUl addit;"" function from S II'Il<> C $UCh that II< I(A) .. oup L""II« A
V" if tMdilive / .... ~nr qt
A E S MId ( A. )oE' be a finile po.nition of II into S ..... Clearly. LIE, VI' (A,) :S VI'{A). Con........ty, ~ (Bj)jEJ be an arbitrary ftnllol pal' !ilion of A ;nlo S....u. For cw.'l' (i.i) E I "J. II, n Bj III the ...,;On .. finite l&mily Z'J" (where t E K'J) of d;.uoint S __ . Henoe III{Bj)( < LIE' L.E .... ~ (1'( ZiJ.• )( ro. all i E J , and PROO~ Let

or

E (II( Bs)1 JEJ ....benQO _ _

s: L ... ,

(:L L

that V;<{A) 5

IEJ

Ip(Z,J .•

. (lr,.,

11) :S L VI'( A.), ""

4, V p(A,)

o

VI' 1I1hf: sm&IlfIII. Uitole Idditi~ functions" from 5 Into R + "'Ii$(yi~ I,.(A)I S..(A) fo,- all A e 5 . Jr 11 10 al VI'( II;) :S Vp( A ) Ior e....-ry - . _ (A')."2; ' of diojolnt S _ whnoe unloo>" contal...,d in an S....,. A. l~ , ioo" ......:b Inl<#f n ;00 I, A n (U'!j'~~ 11;)< "M be partitioned into finitely m4llY S _ 11.(,1, E K ), foe 4IIY lirul~ po.nllion of A. (I :S ; :S n) inlo 5~ A;" (j E J,). thol .... J (lSi S n . j E J.) and t be Ai. Ct E K) form. pBrtlt Lon of A, ~

." DeAnltion 2.2.3 A functioo I' from 5 illto C ill .. ,,-"lire ...h....·...".

C.l

II II. ".....;._""

(b) II is ,,·odd!t'''' (i.e., ;<{A} .. L.2:L I'(A.) 10< """'l' d;,,;o;.lIt S«u ",hoet union A ill 411 5_).

""'I"'''''''' (A.I.2:' of

from 5 mto Cis .. m........., if (..lid onl, if). 0/ dirjotml 5 · ..." _u.imd m Gil A E: 5, 1M. ,~

ThM .... m 2.2.3 A jtoMNm I'

f<w

Oil, "",WAa! (II;).~I

E '2:1I'(A, ) """"".... . .."d if ,,( A) ~ E. ~ I lil A.) .....,~ A - u. ~, A•. PROO", For M Y eeq.......,. ( A. );>I 0( disjoint S·$tU (OnIaint<\ in M A E 5, tbe _leo L;2:' p(A.) ill comolui"... ;""ly c:onverg<mC . "" E. z I II<{A,1I1!l\1!11 be Snlte.

'I'

ate

S«t A, put VIl l A) _ IUp E;E,I,,(Adl, .. here tbe l upremum io I'm over tbe d_ 0I.6nite pwtitioos ( .... ).H of A by S-. We b"" to . . . t~t VIl l A) io 6nite. r im, 01 >e thaI., If thll JIFOIl«1y is II'\Ie for all .IooG""" Ao 01. ODe finite partition 01._ A by S«u, then it boklo For A,.nd V,,( A) s E.." V,,(.... ) (br tbe ~ 01 'lbeormI 2.2.2). Now .......,.. that tho prapmy io w.. lOr ODe A E S. Then there aillto .. finite »II"ition ..tJ (I El,) 01 A InlO S«u sud! lhat E"" . r,,(~ )1 ~ 2 + I,,(..t)), r.nd, by tbe poe.io\Is obeetvatloo, tho ",ope.ty is f.aIae for at!eM1 one AI, (i. E Id. M~.

ror

eoft')'

L ,U,-/'"

"'(~) I

-, •••L

1"(~ lI-I,, ( A:,)1

2 + I,,(A)I-I,,(A)-

,2

L

,,( ~)I

L

/I.(.t!)I,

"',-;'') + 1,,(Ali-I,,(A)I-

""-I" ) Ai. (:foil be

~/' _I", I,,(AJ)I ~ I. Fo< tbe _ _ I"UIOII., S«u A~ (i E I .) 10 tW tho pi operty io faloe lOr _

10

pwtltlooeod by ..t:, Mtiofyina; ,(P(A?>I ;>: I. ~l", thi:8 poor • lOA) lind d~t S«u A!' (t ;>: I, i E I. - {i.}) .uch thallho .. tieI E. z • E"". _ I;. IIIi1A~)1 di""l"J'III . .. bleb II .. coiltrr.dktion. 0

L:...,._I..

T t-..em 2.2.4 V" .. " ...........e lur P IIOOI': Let (A. ).Z'

be ..

et>efJ'...........,,, "" S.

of disjoint S«u ... bote unloo A 11M ID S. We ..-I only """"" IIw VIll A) < r:. ~. V,,(A;). BUI. if (BlhEJ io ... ",bilrfory finlte pr.rtlti.orl of A IDto S«u, Mod> A, n BI (I;>: I, j E J ) 10 tho diQoint unlool of S-«u Z'.i" (J: E K,.j. K'J 6nite) . 10 ~....e-ntt

~ I,,( BI)I

-

~ ~ . ] ;.. "{Z.J..)t

s LL L

1,, (Z'J .. lI

o Propooltlon 2.2.2 UI" k" ...................., (_,;«If, " ..........e) "" S "",, '" tJ\.e ""~ Gdditi .... ~ 01 " to"R.. 17Kn '" ... qIOUi·,..e . " (-rupt>etiwl), " ..........e) "",, akft4I

V",

V".

,.

.

:s

P ROoF' Denote b,- '" u.. addll j,,, exlmsioo of V I' to R. Tt.en I",(AII .... ( A) for all A E ~ . !It) 1' , ;s • qU8OJi._ure U>d VI" S ..... Mut..,nr, ViiI (A ) 2: VII(A) .. l'l{A) for all A E S, 80 VI' ,(A ) .. " ,{ A) for &II A E S. llen<:o: V" , and "I. which agree on S. are id.ntical. Nat, _ _ume that" is. m FE w re &nd ~ that PI ill Also. 9'Jre. LeI. (A;)QI boo. fIe loogI to R. and IK (8i),~J boo..., ",bltrary fini"" p8dil io<> of Z into S-toe1.& ~ A. n ,B i ,(i ~ L; E J) is the union of . fa,nily ( Z'j" )'~ K.~ (K'J finite) of d l!lJoiot Thm B, .. U,2:1 U'E K'~ Z;J'" ..hence ~8j) ... L; ~,L.EIO:.~ I'(Z,J.. ) .. E,q, ,,,(A, n 8 S), 1'0. """ry j E J. ~fore. ",(Z) - E .i€J :L;i!!' p,(A, n BS)' $i...,. -=It family u.,(A;n 8j));<: . ill bummable, "" is (1<. (.4, n B;») '
Ii'"

s-.

"""1'

ba.~reon S and (A.)~al • OOq~of S..- <1«. ' i"" to A in S. Tben ,,(A.) - I'(A) .. n _ +00; Iud d , p, (,4 I _ A") .. E, <,<._, p,( A; ..t... , ) to I' ,{A, - A) .. E.<:I I',(Al - A,.,) "" .. - +00. Similarly, If (A,.),,<:I ill..., 1...... ""'I' ........ of S..-. wb.oee union A be'ongo to S, lbea ,, (A~) _ ,,(A) .. n _ +«>. Corm!r.eQ-,,,,,,,,,,, t hat S .... and Io!$ I' be .. q ua/;i-measure 011 S . If ,,(A~) _ 0 for e>Uy 8equenoo\ (A. )02;' of S-u which doc, unto e. tbet! "

Q)rl'"''''

5

""It

"n&".

"a m " . "<e.

s..,,.,oc

PI"Op<>IIltloo <1.<1.3 I/>d! 5 it .. .,.· n"!!. ..... t..o.ndtd rh.. ruPAE.I V ,,( A) < +«>/_

TII~n

all

m.....",... " 011 5

P!tOOf'; SUppeoIIure" on

S which is not bounded_ T Mn. for """" 1nt""'" n ~ 1. there is .. B.. In 5 ouch t hat V,,(B~) ~ n; __ • putting 8 .. U,2;' B •• _""" V,,{ B) ~ n for e>Uy n. 1!.Dd"" V ,,{ B ) .. +«>. 0

S io .. "" , .. ,," if ..nd only if it io.,._ additive. For any 8eq""""" (A')'2;' of S--. (00\ ,w 'rily disjoint) """ for A positi"" function" on .. ..,...;n .,,;

s.-

u,
:s

B Indudcd in A.. In Ihla cue, ,,(B ) E.2;I I'(A,). Iv', : j , any Io!c " , be the unique _reo e>::tet>ding " to R.. and. for every intqer n ~ I. pul B... .. B n Ao n (U,<,< ~ _ , A.t TIw':n . &l!>Ce B ill l be di!Jjoint union of the 8.., _ haw ,,( B ) .. E~, ", (8~ ) ::: E o 2;' " , (A. ).

Definition <1.<1.4 LH F be .. real """"" "1*'0. A mapping / from {l inlO F ill S-.impie If it is or the form / .. E.JO~i' I"" ",here J io finite, Ai Jieo in S lor ~'(1 1 J e J, and c, bdoop t o F. The 0Iet of oJl S .... mpie """Wines from n in'" F will be
wriItt"" '" LO€ K (l;.l, ·1-,. lar .....14IoIe memlle.. £I, .. 01 F .

••

LJ_O.

P ROoF: E...:h / , Qlb be .,ritltn"" C;.j ' 1... ~ . 8 y i'TOposilion 2.\.3, there uisl d>.joint 5«u B~ (* E K, K 6nltt) tuch thaI e.d> A;J (i;n /, i in J I) is .. union of fOe of lhe B" : A;J _ UItf:I<,~ 8._ Then

f. -

L
,0.

Itf:K. ~

- «.K L ( UEI•.LItf:K' J I <,), •. a Now Iel.

5 _ C bt; addilh~, "",d Ic\ F be a BanadI "IW"" (real 01" c:omp6ex, .. It tabs OIl mol o. com plex vaI_). II f ., ColA, " ..., Saimplo rnapp;ng from n into F. lW define the wintognJ oil 10 be d le _ JI(f) .. f f dp .. E ,vC;jt(A, ), This definition m°w." ..""" beo:a_ the _ r:.." e;I'lA,) III il>depen
E"",

L.uC; lA, " LIEJSl,,;, !here exiM disjoillC ~p'r 5..-. B. (t E K . K finite) rud! llult ...:II Ai and each is .. unino of oome of the 8 •. T betl If

J ..

Ai

L ., 4 1'(A, )

- L., < L

1'(8. )

B. C A,

- ~ L~. e;) - &C~. s) - L~ ' L -

",,(B. )

"I'(B. )

I« B. )

jO

B, C A;

:L S· ,.CAj).

~,

The functioon ji: I ..... [/Ct~ from 5 inU) C ...... 0Ul identlfy E and thl: ~ dual SI(S, C)" of SI(S, C ). !ltacd.. t b .... 0>n!Iid0:r only SI (S,C) [...Iead ol tho: moor.. ~0.1 )f(ll, C ) of Smion 1.4.

"'*'"

lomdiDn " ""'" S inU! C .. Q "",",,"""""lin: m...,.,..j if """ onitI if ji h~ fin;~ ",,"';clUm (,...p"di ... I~, ..

Theorem 2.2.5 An IOdJili«e (~,

Q

c Daniell m _). MO' II!:>LC' , (foj _

ii;.

PItQQ,; 1'\nI.. IlUI'P'*l that ji hall finik variIIt;c.,. Let (A,)~, be a fini te po,rtltioon of A € S inU) $ ....u.. For e"o'ef)' i € I , t~ .... a oomplex number Co .. ill> modul ... 1 .. tisl"yinc I,,{A,)I ., '" ."(,,, ). Then I _ I ... ta1w;". III $ I.. , "" E.."I1<{"'ll" ii{/) " IM!)I < !PI(I .. ). Thus" is II quas>rneasurt and V,,( A):S (foj( l .. ) br 0.11 A E S. Ox..., oely.llUpp\* that" if B quas>-_"', CIe&rly.l [ 1'4<1 $ [ !tIdV" lor all. E SI(S. C). and [ IrlV" lilion 2.'2.4). The .. !Pu!) ., !lUP~$f(S.G"l.lflS / I [ 1'1,,1 $ [lrlV" br ~ry IE SI+ (S). So ji halllinlte variation, and Ijil(I ..) - v,,{A) for 0.11 A € S. This jJ<'MI that liil - ii;. N~, ~_ that ;:. It a Daniell """",ure, and let (A;)02:' be.~_ ol diojoint 5-oeto .. ~ utUoa A lioelln S . For ~ry integer " (I. ). !, doc, .. 5 • to 0, ii(l.. ) - I'( A) - E,:s's....( A. ) taIdfI U) 0 .. " _ +00. and bffla! " Is II _u~. F'lnaUy, _un", lhat I' III • 1I>eIIOiill O. Then

i"''

E"" '"

s.. .. (~ € £I : I.(~) > e} u..ln"llloo: alln ~ I . and the ao:querIClI! (8' )" 2;. dec_ to lhe empcy """ ... VI'. ( B.) _ 0 ..... _ +<:.>. N<M

J

l ..dV,,:s M · V".(B.,) + e· VI' ,(A) .

.. t.c ... M

- SUIl,... It ("') and A., (~ E fI : I, (z )

.-.-lim

> OJ , thUil

¥P(I.. ):5 e . V" ,(A )•

tim

ii;(f. ) -

o.

o

"'*'"

Call QM (S. C ) ( . I"" th-ely. M (S.C)) I .... o:>mpIex vector of qu&lll-" , ",.-ell (reoptcUvdy. _ ) 011 S. By Theorem 1.3.1 and P"-'I'O-

tition 1.4. 1. th$ cruer..I ¥fICIQr 'poo<:O:' QM (S . R ) ol.....! quui.meuureI 011 S ;.,. Do!do!kind cornpIote It>-"p6Ce, and M {S, R ) _ QM(S. R ) n M {S , C ) ;., a b&nd In QM{S. R). MO' Ilc"cr. for e¥eT)" I' E QM(S. R ) and for o:YeT)'

A

E 5. I' +(A) ..

H I'{ A)+ VI'( A))

~IU.P

..

L

PI" ) 8 ~PC A)

(1'18)+11'(8)1)

L

fUp ,,(H)+, PC") "£p(,,)

..

.. hen: PIA ) at.endII.,...... lhe clUi 01 aU finite panillorio 01 A In\O 5«ta; li m.. 1.dJUon 01 s..:tlon 1.4.... ""til!!i.od ... hen ..., ~ 51 (5. C ) (l"'lwd oI1i{fl. C )). 1looo.,.,..., ol ~ti>O!" ' '''... 011 5 io "";tlen M +($}. For the rf:IDaInder ol this I<:t " boo: • pOSilive meMu,"" 011 S. Denote by 07+ lhe lei. 01 upper et\~1opeI 01 inu : , dl\l ..... ue, ..... In 1i+ .. SI+ ($}, and put 1" (f) '"' ... p,o; ... . •,S.I ,, (g) for all I e 07 +.

"""'ion.

P~itioll 2.2.:; l - fJ ""''''' le:r 'er Ali I eo7+ and 9 E H· ... (~ fJ S I· 111f{f, gl, g), I +~, ond M Io>t.lot.og to 07+ , /M" aliI , 9, II e

:r'

"'1'
w'"" .

oiWI/cr <011 " > 0; ...... I"{f +g) .. ,, ' (Il +"' (~l ond ,,' (M ) _ ,,·,,' (11). Finall). ."..., i...".."..;,.g ."""en.. {f.)~/: , ino7· lou ill """"r "" ..... 1 in 07" , ond ,,' {Il- "'P.~. ,, ' {f.l·

POWOF-. E •..,.,. I e 07+ is the upper ~Iope of an in<;. : 1;"11 ~_ (/. ). /:1 in"K"'". For ev.ry g E ){" 8/Iot;,{ying 9 ~ I. the ..... _ {f. - inf{f•. ')). /:1 loci 5 5" Ie I - g, 10 f - , bdotlglr \0 ;r; !leX1 t he 1Oq""""" (Inf{f•. g)),,/:, i'..... ~\O9.80

.--

I' (g) .. 11m ,,( inf{f.,g») S

.up ,,(gj ......·.ri.f

.--

.. lim

11 m ,,(f.);

I'{f~ ) .

.--

:1+ and I ;!jj UP!""" e", elope. For ~ n ~ I, ~ (g. ... ).. /:, be. ao::oqueoce;n H+ ;nc""";,,« \0 I. , and put Noot,. Itt (/.).", be an lont ' "1111 !equen« i n

h., .. 8Up{g, .... 92 ... . .... g..... ) for till in~ m ~ l. Then g; .... ~ II .. ~ / .. for aU l S i S m, 10 Ii - .uP"!' iii... S "" P.. /:, h .. k>r .-.y i ~ I , and , S

10 ... T ills ~ lhatlbe!lKi\leDCe (h.. ).. ~ , i""~ 10 J. Mou:.".... , IlUP_2:1 ,,"(j... )
o

2.3

Lebesgue Measure on an Interval

Let 1 be ... ...,....."ply l.D~rvaI of R. PUt 0 _ inf(/ ) It.Ild ~ _ $up(/), 110 lhat 1 .. (4.~). Consider the """,iri"l 5 in 1 a)"'; ...i"l of &II inlervalo 1 0,/31 ouch that 0 S in J , &z>d, ... boo .. bel.oop 10 I , of ai! inlervlt.la (0, jJJ such that fJ ~ to I. 5 is oalled the lIIIluraillemiri,,« in I . The ,,·ring ~ by S ill _IT "..., 10 be lhol Borel ".o.!&ebr'" In I.

fJ

Defin!tlon 2.3.1 LH p. be ... complex m" . 8' .... on S. A function F from I inlO C is oalled It.Il indeliwle in!"I!ra1 of I' ... F {Pl - F (o ) - 1'(Ia-, M) &II 0 , {J E I oatWring
bene..,.

roo-

f!~: :_:;:~~..": ~'fJ: ~!

J:

dp+

[~+

I:

~

1'(1<>,0])

01"

dp_O

f:'

br all deIinlt.e i~ of 1'. Now F (z ) _ F (:ro) + til' &II" E 1 &z>d ro.- &:\)' other indefinlt.e !otewa! of 1'; 110 we Bee that ind:tnile intqrab of " .,.., equoJ. up 10 ... <:OnI\Nl1.

r

roo-

Pro~it lon

2 .3. 1 Ltl F : 1 -. R iIo! m."..,...;ng, eM rig/lt-..,."tin ....... on (.., 10{. TIl ... 110......... "".................. I' on 5 cdmilting F ... gn ind~fini~ m/qrnJ eM ",tUfyi,., p.({o» .. 0 ..... ~ g /!dong. to ,. T he uwqumees of Bud> ... _ure" otJl'io ...; 10 prove lu exio8~, ~ defir.e ... fuD
L 1'1 (<>.. ,fJ,,)) < p.(o,PI) • ,

.

~

1>0.:.,..., I' is a

LeI , > 0 be «I""". A. F is rlgh\.-a)ntinoous at + 6.tJI· Additiooally, roo-~. n 0 ouch that "P. l S p.«o.,Jl..I) + , /2- ', If .... put Jft - (Oft ,Po + 6,,1n /. N.,... tbe quao,;..~.

exi",

interior J: of J~ rtlMi...:: \(I f COOlainol (n •. A.). Md, since the cburt of J io! indinu..~, J:.lhen:~:cisl H, ~ I , .. "n, that J C J., U· . ·uJ., . From p(J) 5 p(J.,) + ... + p(J~. ) 1oI1m.'l! IIQW p({Q,81) :s:: ~, I'«n • . P-.ll +t . .... . io! arbitralY. tJl) .. L~.:!; , lI{(n • •.o..)). and $I> jOlt. mU9'",. 0

1'«0.

°

Dellnl t lon 2.3.2 The unique meMure I' 00 5 ~uch thn p()o,Pl) _ fJ _ JOoall .. , fJ E I aato.t:yiD« 0 :s:: (J, . nd ouch that p({a)) .. 0 ..""... ·,..,r 4 ""~ to f , is called Le~ ,IIOU"", 011 I . To IlOO that I' ""isla, tal.. F{z) _ z in Proposition 2.3.1.

Eu,clse.f for Chapter f I

I .. R ' , ",O(U0p, n '~' ~''''''''J (G,,,,, lor I :!i ; S t ) io ""lied" 1Iq...... Go io iodtj>£'ode,iO of •. Let S "" .be c_ " ~ ioIinc of """ ompIy Od ot'Id 01.l000000 1 I q _ .. htJooo ...... _ h&ve .... '.,..·1 Show ,Iw. S 10 " oo:n>lri .... """'" 'Ioouch A n 8 """Y """ Ile in S lot ubitr..,. A E S. 8 E S.

"""""'i_

.. be ..... , .. _

l

c.Jl S """"" _iril\ll if it """I ...... ,hot""'Ptr.tOt ....:I io' ~ j ui>
S·..u At. . ... A. ouch u..t A .. At. c A, C ... C A... Band A. -A, _, E S lor 0.11 1 :s:: t s: ... Let ~ : S _ C I>< " fwooet.,.. ...... that jO(A U 8 ) .. ,.fA) + ~(B) .. be ...... A. BE S, A U 8 E S, and A n B .. t, A ~ ni'" ~ioco ( Eo ) ... , of E E S IntO S·..u io called • ",-,i''''' if ,,(;In 6) .. )~. ,,(A n 5o) .,. &II A E S. If {Eo) .. , and { F,),~ , .... bil. ~_ 01 E E-S-';'IO 5 _ lben (E.),~, io called • ""ht-tit.,.. '" {F,)!" ............. _ oeI; E. 10 .,... .... t>ed no """ '" tl>< """ F, .

~""t

\, S - lbal. if" oubpwtiliOli of" - 'ilion {E, ),., 10 a

", ~it .... >

u....

( 50 ).. ... "",p&rtic.ioo>, II (Eo)... """ (Fi),.., _ ,_ ",p&rti' ior>o of E. . - ,10&0 {Eo n F,)(•.r')<'.' 10 • S. If E. Co C C, C ... C F. whtR C. E S lor &II 0 S. i S. n, """ if C. -C, _• • D. Iiooo u. S lor all I :!i i:!i n. _ lhat (E .Do .. . " D.) 10 • ... ~it .... ", F . 7.

".putit""'. c.. •

4,

s- ,ba,......,. hita p&rtitiooa '" aDd «*:Iudoo lhat " io addil ;'"

<)
Wi

S-oet E

I". S ..... 10 •

",~i';o".

S.

:J Let R bo ..... fine _ ' i . , . m'bo fiQi •• ...,J ooIini~""" in ... uno:owrtablo n. Doli"" II' "" R- b:r Wine II'{A) 10 bo d." n~onbef '" po;w In A if A 10 finite, aDd - " ....... " io , be Q ~ _ cl pOinU in A' ;r A ill ooIini\e. S...... ,bal II' io ,.·oddith" "" R, "'" 10 '"'" a_un.

t.;

'V'

ate

3 Integrable and Measurable Functions

rod_.""

fund. !" ,aaJ _io:)nool nocIlcit.lt, ''''<pabIo,.oo-.. ' .....-' to " Ji-- ___ nr.. <1_ of Hi", D o g Intqrablo _ _ ........ koeIy ........ (d. Seot ..... 7.6) .. wu d' -...... In U.. nl _ _ h _ury th......p 1M otudy 01 Fourier _ _ fur in'" ... In ......,MI. the <'- ol fu...h :........ let "~ io qu~ Iarp. !ndety_ J. could be O&id thAt lIIit .. 1M ' - ' oJ. .1>0 _""" ud I. it ,,1Ia! to nndof'OW>
ThIo ch~ i... .......bIo fuDCtioao

""~

_,,,0l>I0

"Cl"''''

am. ...

,--

1 1 Cr.... " 0... .... " r F'",I'. \he Uppof ;~ 01. pOOl. ;.., r...n<:lIoo. with roopot', I<> V" ill defh.ed A p...l1 ... fur>« ...... Mid 10 be ,...nqliciblt If itl uppot in..,01 io ....n. n.io 0I1owo .. to oX6... 5PI' ltIIbIo_ aDd tho _10<> o f _ 'y ,,,..--..• .,..' y",!wn" . We Ibm I"'I"N • low imp>nu.' .-.1.. 0U(:h . . Bt:ppo !.-i', , hz .. _ It' ",W 3.1 .1). F'Mou'. he (p..,,,... "ion 3.1.2), .00 tho ru--Fiocbor h iD (Theot . 3. 1.3) "" ~_ of £'11<). 3.2 One of 1M ....... limn.at_ of rue.-"".I~teVtJ In ,he """ \h&I i. _ yiold ....ibnt nsulto ... _ it ....,_ to the ;""",01 01_"", .. Of ...... cl f"""'-' F.,..,.....,pM. if " .....Iormly _nd
t'"

m

,!

o

3

....,.. .."bout _ _

cn.-

... "." mI.Oclt • ••ono1a1.o '1i.1t.ooo'" .....,

l ... '-';"";'1 ond

m.

1_

di l!_i&.bi~ 1.Y

of an

in~&l

~.

n., _ • •_

wilb 1 : ...... to.

pu-..-

(P, - ".... U.3 and Ttoooo ... , 3.2.3). ue of """"'......... in ... olyoio..

•

3.3 In ,hlo obon""",ion ... Iocuo our ."... , ion to .... "',.,.,. , ....... fw>cI.ioonI; &Dd dolno 1'.... ter;rabIo _ and ~ u_ leU. U Tltioo -.ct;'" ioo to ,he deIi.l..,., of "·....,..ur-,,",, ioo. ..c. <. ~ "cd Ifritb ••,..i., &Dd 'ho _ion of _"",t;,k ~ botWl!Ctl ''0'(> oudo _ I, it iIuportan. to _ioo ,bat. ,1UoI _ion 01 _....toIllly _ _ depend

de..-,'"

.•'e

"** ,ha,

OD._~

en.

.1>.

3.$ ~ maiu ","ultof -'ion io .1oo ioIlowi .. iDrm of ~'• •l.oo..= ......" 3_5.1) , . OOQ_ 04 ,.._u,-"",, mowO .... f....... n iO>to" . hid> """,u ... IotaIIJ _ ewry ... here to I """-r" uniforml, to I on .100 """,. ~) on' of ..... bitnoiilJ ....all ~obIo .... Fin.all, .... Pl'O"O .10.& •• I'· _urablo ",_Ion from n iDto a Bu d _ ioo U>tq:rabie If and only if ito _ imqr&1 .ith , _a to VI' . finite (Thoo<.m a . ~_ l). n. I .1"1 it>tqr&1 flO",,"io ... ioo~, ......11 .. boundod m< "ra 3..7 FirS..... deIi .... ho _ and ..,.,.. i"'qr&1 of . poOll"" f...... _ I ioo in.. P"bIo If aDd 01017 if ito _ and Iooow i .............. IWte and ... iUiI. Notioo ,hal il l !o IUlqroblo"" ioo VI: m .b. ''-Y. •here it DO "imp ......... i ...... aI". We tbon _ J........ '. 1-..!11y (Tbooo . ", 3.7.3) .,hid> Ia.., impOrt ..... _ ~h in.1oo of ptObobility and in &aa.l"';O.

a.'

,....,;...bIoo "*'"

or

t-,.

po_

""'_04_ . . . .'-"").

" .... I' Ia . I'. in~ ... whk:tl """ .....m&II... (omaJlo.-and _ _ botb in A _"no wI. _ _ ioo Mid to too dIlr••<; ~ _ " ' " ioo &I:t _1>10 of_ & " · 'UI'O. 0.. .ho otbor _ . & '1'",. Mid to too _Ie: if -=to IiOI1nrs1~bIo a....:r&blo

38

intu it i..oiy. 0lI alOIn lew. _ . ,, ~

n.. ......., 1l1li a

•• 000.ho oemiria,s of MU ouboot.o '" N 10 ... ex&DIJlIe (t! s..ctloao 6.3). If I' 10 and I Ia . l\lllCtion from n i _ • .-rizabIo _ , I ioo ....... ...-.bk If &Dd <>DIy if It ioo .,.,n' .n, . ... 010 _k atom ... <Mit&i ... an

&tom.

.">mir

( 110"", .... 3.111). A Daniooll • ' ''''' I' ~ ...... _ ..... it on ,ho ,1l1li ~ of inlqr-""""'" alkd 'bo maia pr<>IonpIioo of 1'. Siaoll.&rlJ. I' drIi............... ,.. caJl&d .ho_i&I ",·ok I ...... of 1'. "" II>< rlati ~ '" _i&IIr mtqr&blo -.to. W•• hea $ttod;t .....,... ..I& ......... ic>o bot ......... boooo 3..9

3.1

_u_

Upper Integral of a Positive Function

Let 0 be • 0>0DefI\p(y ~. F (O. C ) I"" ......,. of complu. vaIl>Od funct __ on O. and C ) a ....::tor wbo~ of F {O, C ) web thal rk/, 1m I, and !I! bdoq to H (O,C) for &11 1 E H (n. C). A.wnune ,It&t, for.U I ., I. E H + and Eo< all, E H (fl, C ) OlUi"fyi", 1, 1 :s " + hI"" .... etist , •• 9'l E H (fl. C ) ~ud! tlt&t lg,l S I ., 111>1 S I. , and, _,. + 11>. MQo "" .~r. "'!'P'* IIt&t iufU . I) liM in 1f+ br aliI e 11+ (Stone', Q)Ddition). Now If:< ,:r- be. lei. of functlQ"'I from fl into (0, + ooJ ...·ilh t be foIlowl",

nco,

pt OJ>ttt~:

(a ) 07+ :J 11+ """ 1 - 9 boelongtl to 07+ f(>t &11 I E 07+ and 9 E H + IIU:I:h

.hat, :5 f(b) inf{/.,), ..,P{/, , ). and I + 9

bdor!«

to.1+

br &111, 9 E

r .

(e) 1<1

€.r for """y ~ > 0 and e""'Y l e:r.

(d ) E""'l' 111<1" 'ing ~""noe in ,;,.. haoI i'" upper tnw:lQpe in

(e) For ~ Daniell" . , ''''''

<)D.

1(0, C ), if we put

V,,"U ) .. lor all

I E

r.

IIIp V,,(,) ,.,>(·.#$.I

,r,tbm

(i) V,, ' U + I l

.. V,,· (/) + V,,' (f)

(ii) V,,"(IUP.i!: ,I.) .. in :7+.

for all I . I e

IUP..1:1 V,,·U.)

lor

,r;

e'oel) l""~og ~ueo<>':

We encountered aucI: a cl&8l:7+ in Section 2.2. No. Ie\" be a DanIell "....U'" <)D. 1(0. C). Detlnltion 3.1.1 Fot """y fW>C\ion I from 0 into [0, +wI, t),. number inf",p ... U V ,,' (1:0 )( +<><> llIhen: i8 DO 1:0 E :7+. 1:0 ~ !l io called 1M upper integral 011 with .... poet to V" &Del .io ...
r

r

.,.., lwo IUDMion:: frotU {} IDto (0, +wI aucI: lbat I S P. Ibm V,,' U ) s V,,· (fl · If I ,. h. I .,.., II,,~ functioo:: from {} into [0. +<><>Jand ~ > 0 is a real 1PU1Tlb«. then V,,' tIP + J.) S V ,,·U, )+ V,,"(h ) and V ,,' (1
T heorem 3.1.1 (Beppo Lev!'. Theorem) F{)r U~ )~:t ,

~

me • . :;'5g

~""

QI fo-ticru frr>m {} inUo (0, +001. V,,' (IUp I . ) .. sup V,,' U~ ).

P P\OOf: It eul!ia:o; to Jhow

thaI

ily clurf)< bold:: if sup V,,· (f.)

n..

V,," (.up/~ ) S IUp V,, "U. ). l.t>oqual. .. +00. n.en.r.::ft, _ ""'y ' '!I5!''nIOl tbat

sup V,, · (f. ) < +00. We tI>ao- 11:.t.1, lor _ h t > 0, tl:.oerec eris'" an In<:~ iog 55Ilq_ (g,,)~> 5 In :7 + ruo:b that I. S ", &Del V" "(,,, ) S V,,"(/.) + t. If I is tile up. per t melop<: of 1M ~_ (g"),. ~ , , _ I:.t.,.., V,, "(g) .. tupV" ' (,,, ), flO V,,' (g) S Wp V,,·U.) + t. S;""" 1Up I. :5 II and t Is arbilnory. I),. thwl(m wlllloUow. lbr IhlI, oo.e..... that lhere exi$U It.. E :7+ 8Ud! that I. S It.. and V,,· (h..) S V,,·U. I '" r/'l". Tho fun.;.-tiono ", ..... p(Io ,. I., •. , . •h~llie in J+ . f
I_ I.

and

A...
1UpU._I,h.) _

,.-" .. inf(g,._I. h~) ~. _ . + II. , it k>I~ .hat ~

~ I~_ I ; lina:

infu.. _I,h. ) +

V,,"(g,, _,) + V" "(h.) - V,," ( ioflg,. _1.11.,.)

V,, ' lg,. ) _

:s

V,,· (g. _I) + V,," (h.) - V,."(f. _,l

< V,,·U. )

+;. +€(I - 2"'-, )

V,,·(I.)+t( I - ~)_

:S

11.00. V,,· c,. l :S V,, · (f. ) + { for IlIl n

P ropooltloo :1.1.1 For ' '''"'11

~

l. ILl ~red.

0

~

(1,). 2:1 ollMI>Ctlono 10, ....1. _ ........ V,."O:::'l l !oJ:S E' ~ I V,,"(f. ).

PROOf', If woe puL '" •

E, S":5"- I.

/rom n

ill/
for ~ "
V,."{g.):S

L

V,,"U. ),

IS· S· V,,'

(L!o ) - . up V,.· (g. ) :S L '~ I

·ll

V,.· (b )·

.~,

o

"'ftC-

P ropo.Jtlon :U ,2 (fiotou'. Lemma) LeI (I. I.l I Ito a _ 0/ tionI /rom n into (0. - 1. ~ .. V,,"(limlof. _ __ I. ) :S IiminlV,,· U.)·

.:j,"''''''''

PIIOO" FQlIlII intet;~n n ~ I . Put 90 - inl.1::'>I•••. The (t.,).l l in<" " . and euP.~ 1 '" _ lim in! I • . For each " oo,.. :S I .... for , .2: G, '"'" ha"" V ,,'lg..) :S Inf.lC' v,,·(1. ... )_Th ...

V,.· (limin! I. ) _ fliP v,.· (g,.) oS liminf V,."(I. ).

o n..tI.nltloo ':1.1.2 A flmctlot> I from n into (0.+001 III oald 10 be ,.-Df'dlclble .."",... e, V,,' U) _ G. A..,bitt. E 01 n is MId 10 be ""lIf3litible, or 01 V ..... rr F7"'.1/"" 0, if '" to ,,-nqilclblo:.

E. l,l •. and .. ..,p/. , III ,...""&Ilclble for ~ """l""""" (I.)oll of,... DO&Iiaiblo! fulldiont from n into {G. _I. Evef)' s,,1Mt, of a .....phle !Itt is ,..."..tlpble , and • union of

"""o..lIIy many ,.·nqlillbk acto III ,..",,11&11>1&

o..6.n ltlon 3. 1.3 II pn>peI'ty .. oald 10 be true alll108t ""'"')'w l=e fOf " (abl:rrev\ated 1.. 4 ". or ~.) if &rid only if It illrut outaide .. ".negligible 10)1.. P ropoo.lUon 3.1 .3 A. nn 'I h, ~n
'''PI'''''"

P "OOI': Tbe oondition .. ,,,, "y. lrl ,d, l luo. I ill Degligibloo. and let Ii: be the oct of pOima % E 0 such tluo!; 1 (%) ~ 0-. then ! ~ 'f Sup~ ("n . 110, I . . . Degligiblt. The <:OI>P""" IIuol. the IO!I E of point.ii % whe~ I does DI)I. , .... ish .. o>e&Iigl,-*, then I 'f sup.~ I nl " . 10 I is ox>gIigibloo. a

P ropoo.lt lon 3.1.4 l..., I ....., g bo ttoo /t".,,~ I", 'U., 1M" V,,"(f ) .. V$O" (g) .

from

n

mto [0, +ooJ. II

P ROOF: I..et E .. 1%EO , 1(%) ~ g(l:j). S;""" in(U .,) '"' 1lUp(j,g) 00 £<, It ",,/6COII to prove 1M pt"OpOIIillon .. 11m I 'f g. n..o let II he the r"nction cqual to +00 on £ and to 0 on E"; W1l Iuo", I 'f 9 'f I + II , and"" (V,,· )(f)

s: IV 1'. XI) s: (V 1'. )(1 + II ) s: (VI' " )( f) + (V $0. )(11) '"' IVI' · )(I )·

a Proposit ion 3.1.5 [I I : n -

if finiu.

......

iO,+<»] u.udI tA

~

I

s:

PROOF : Put E .. 1-'(+00). r OO" every I"~" 2: 1. _ h...", nls I . .. nV,,"(l.) VI'·(f ) Sinu t his".rut lor ..,.bilrarily Wg
s:

a

lIuoxk>rth, F .. ~ A mapping I from 0 illtO F is Mid he i C&n be wriHen I" L.u/.... , """ .. fl· nite iItI I, , and elements ... of F . Such .. rep"""","" wioo .. ailed .. dec:6mpo8ition f. ~ H (O. R ) 1<1> F be ,he .....:tor 'I*'"' of deeoiilP"'! "bIt maWings from 0 lnt(> F. N .....· IUPpoooe Ihac. F is • real ()T complex l!anecl> 9pf1Oe .. I' .. real '" (OtIlplex, and let F' be itll dual (the ...... of ODtlt ioOOlll li-. lor.... on Fl. If L'EI 1;4, is .. dec:6mp<>6it1on of I E }t(O. R )I F. Ibtu L .. I ,,(1,)<>..: 00... not ~ on the penicul..- deoompooillou of I , b .;'( ,,(f,)<>..:) ., ,,(.' 0 f) for &II .' € F' (W . Rudin. f\onclOo...t Anat,N, C"h&pter 3. Corollary to Tboo«'ln 3.4). HeDOI! W1l ....y

Lei

,

3.1 Upper In' pol of a 1'I:Ieit1¥e Funcl.ioD

dMote tbe _tor

_~. I f

0&5

E... ,,(I<}4; by f 1dJ., clllled tbe jl-intevaJ of I. FUrther--

1<411- SUP.-t"r ,''-IS ' Il'U 1<4111 (ibO:!., Corollary 1.(1 ~ U ),

beDtt

~, II(l' I1''''1- .-tr.''''IS'

0

1)
""pi 1%' IldY" S I"I/ ldV". 0

.'

r

wbHe 1/ 1.tV,, :s ~/ I ...I · Vp(lf;1) ill finite. I n what foIlf:>,.... , "" fOOl> 116 we o::onsIder \.be t!><pre!I8ion I <41 (or lbe similar ",,~ic)n lId" of s..ctic)n 306), It shall be ~"'~ lhal F "" & -.l or alrDpIo:x &n.ch "P""", as" Is -.l Of complex. mappOn&'! I from n 1.111.(1 F """" that I/ ldV" < +00 form.-.,.. ~ T}II'I. and the function I I/ leW" '" & ...,,;norm Of>

f

n.e

r

r

Fj.II').

I",(n •

DefInition 3. 1.4 Do:no::M by C~II') the cklllu", of 1f(n. R ) iii F in the ""m;" .... ,ot ,' !"","Il>"" the ,,-I"~ "'ppi~ lrom 0 ;1110 F. N_ 0>0Slde0- C~v.) as • ....::1.0< ",bel*"" of 'M iJemlnonno:d SpooO! .1JII'). Tboa ,be "imq..J

"I· Obeco .c ~hat I f Id,,1S N,(I) fOl" 0.11 I E C~v.). A &It.. \hat cnn....1"!"1 in F}{I') Is MId to amvcrge in lbe "",&n. P r<>pmltloo 3. 1.6

II U ;, ......lin .......

lilim, map /n>m

F

imo II Bomadi

qo«G 1mIIi/ I E C ~v.), ~"Uol liu m 4(P) and f (Uol}dp - U(f Id,,). Propwitloo 3.1.1 Ld ({~)~> , /It II ~ ... F}(p.)...d< eMt

L N,(I.)

."

1Jw

<:;

+00 .

""rit-. E .2:' 1.lz) ....-,.,..• ...uon.ulW"'e. II "'" JOIil I {z ) -

E.~, 1.lz)

at IIlrno.l
H, frt<" .,// n

~

(I -,~. I.) S .:;', N,({.l

O. TIKro/"",.

~ Krit-. E .2: 0 I~ """''''"'JU

UI I in !h. """,n.

E.2:1 1/.(z li of n into ill,+«>1· Since N,ul S I:..." N,V. ) <:; +00, 9 "' finite • .e., &lid the series I:.. ~, f.{:r) """,ugw at.olulely •. e. Si".,. F 10 """,plete. this...n.. ..,."ugw a.e., and PIt()OF : Conoider ,be function 9 : z -

g(:rI ..

lI("'JI:s e.lk""", N,(I):S N,{g) < +oo.1U>d J!ioei in J'~(P). On thl: ""her h$nd , 1(%) - LL:s.oS~ J,{:t)1 s: L:,~ " ., 1/.(o: Ualmost "''eI')' ... ''''re, for '"""Y intq;cr n 2: O. ThllO N,(t - L ls t s .lo) S L'~""L N,U. ). 0

,..)

E~~,

M

Th~ ~ (1;.(%)). 2; '

N,t!;•• , - / 4) ;, /ir!ik .

"''''''''lJ<'.' 0./"",,/ ~"",,

(c) 1/ "'" ,.., /(.,) _ !1 _ _ _ h .. (:r) acI aImo" .,II pointo % .mc.: 1/>" limi.l aioto, =d t...t. I{:r) arl>iI .... riI~ o~n 0, _ (1;,,) .. ,.,

u.-

I'%Ut.o Z C 0 .UdI tJuu V,,-(Z) ""if<>mll~ to f "" (1 - Z

"""""'JU

:S

!

and tJoe

(e) The ... ..mu a "".." ..... 9 from (1 into 10. +001 oudIlIMI 11',(9) < .nJ If;. (;Eli'S: 9("') /(}r ail '" Iii fl.

+00

""",Iy

PROOF : By indt;cl.ioo on n , dc~ne a ;""..,...1"11: ""'Iu"""," {i,, )"~, of ID~ 2: I, sueh that N,(I<-1... ):S 1- '" for..-chn 2: I and foTeochi 2: i"

By I'roposilioa 3.1,7, IM """"'- .. ilh general term h .• L -I;. ronvesges almost """.,. ... be .... I&I>d in ,he _ . 1(1 ... IlIII.ppi~ h E F H l' j. ThUll the gee etQDeT>Oe (J;),,. . i!$elf WIl'UI'*, Th .. P"J""l$ (al. ( b). (el · rf_. tor """". n in N . pul y~'; I" € n : Ih •• , - /;. 1(" ) 2: 1 /2~ 1. From ( 1/2~)· I y. 1/; •• , - h. l. il IoIIooN Ihat (1/ 2")· V,,· (Y~) N,(/••• , - / •.1. wheooo V,, ' (Y~) 2-~. P~I Z~ _ u...,,~ Y..: By cooostruo;boo.. '~P.en _z. 1/;.. , - /;.1 < IJ2" roo: ..u k 2: n. II:) lhe _ .... t h &""Crallerm f ;•• • - f .. (t 2: I) co"'"'gti6 ~Dlformly Oft Z~ Si nce V ,,' (Z~ ) :5 1 /2~- 1. roo: "'''''- > 0 "'" """ cboooe n !ItICh Ib.u V ,,' (Z~ ) :S c. Then. If N .... """",,IWble >Itt sud> IMt {f;, )>>, o:on'UXCOl lo Ion 0 - N. "'" mo.y ub Z .. Z~ U N I'lId 9 .. If.,1+ L,~ , I/; •., - f .. l. which PO""'" (d). a

:s

:s

n-

T~m

:s

e

3.1.3

( Hie
Theorem )

",mp/d<.

PlWOf: .1J(P) Is (t>mp~ to,- Theorem 3. U . Md .'0) i$ £} (p) ...·hich Is clooooxl in F~()I.) .

0

J.I Vppe< lr>tq>oI 01. _

FIaKtioa

47

Prop / , til"" / € t~CJ.) and (f. ),~, (:On""'!U I<> / in 1M .........

.""piIe'"

''''-1_

P!t()OF: By Tbeorfln 3.1.2, I""~ exist • • (f,. ).~, of. (f;~i!: ' ...hkti ,"",wga a.... to a "..pping 9 € t~CJ.) , ouch that (f.l'i!: ' con ....... to 9 in tho:: mean. Hence / _ 9 '.e_ Q

Prop
(. , (h).~1 ........,u '" 1 in tile "'......

o

PROOF: This folloon from ~'" 3.1.2.

Lemma 3.1 .1 ~I EO.. """"",,...01 _Iqr """'" ol/inite ...-....-..... F.... ('""l" e > 0, tht;,.. eriu lin.... I....,., " " .. . , '" "'" E n.di t1W ( I - oj ' 1% 1::; "'i>ts ls , I~ (~) I .s 1'1 101" ISII % E E . P ROOF: Put B .. {r E E :

1. 1 < 11 and S .. {.

E E :

1.1_ 1}. B ..

<>pw

liN. in B Ior...m • € B and tech ), € R .. Uafyi", I)'IS \. By lbe Hahn- Banach tI...,,",,~ b evoery II E S , there exisu a linear fonn ". on E NCb that 1".(.)1< \ lor ali. E B aDd ~.") .. 1 (W. Rudin, FImcli<m.al Anal..... CbapU!' 3. Theorem 3.4). Then 1 - ( .s t~.(I )I < 1 for all. E S ;n """'" ,uitably d!(sen open ~boe(su........ry pOint of. S liN. in one of the \Ii = 11"" 0 COOYI!X,..oo),.

.s

l'bcre.".;slll • eeq"""",, (f.. l.~ I in 1f(O, III tbat '"""([ilCIi to 1 in lhe mean. Roopiacin« f .. lor I: , if ,+ "y, ""'''''y .,,_ that"""" f. I;"

!'!tOOt':

In 1f ~_n....

o Propooitioa 3.1.10

f lll-dV" .

F.... 1

€ t~CJ.),

1/ 10."'-

'" Cj.CJ.)••n
PROOF, To bogin, lUPI"*' tlutt I Ii... in n {l1. R) 0 F. and iO has tbe form E '5JS_ h ' oJ . ... hen! /; E 1tII1, Rl and aj E F. ld t :.. I), By ~"'''''''' J, 1.1, there ~illt !;nee,: forms M" . .• , u, on the >«C.or t;ubsp.a<:e E of F genmt.1ft! br ab '" , .... , Iud> that ( I ~ t) '1:1-0::: .up,«<. I",(.1I :s; 1' 1for.U , E E. Th"" - -

(I - c) ·1/ 1:s; sup I..,o / l :s; II I

,

s;~ ,

Sut ,be ~,ol ~ to 1t{I1, Rl, iO~. _ eup.<;<, I"."..r_. 111is 1'inu,uable. It J"ftJ>ol'" to trt*! tbe (:8« .. here 1 if not I,," ' mily deo:om_hIe. For Ihls. ore let (f. ). ;!: . be. family in 11:(11. Rl0 F oon,~", to I in tl~ "~",,. For...-y" :!: I . "'" MWl II1I -ll.11 :s; 1/- 1.1. !IO Iil l-I/. II · dl'" :s; 11 - 1.1' .tV,.. Hence the .....""""" 111. 1)..;:, oon"'gf)i! '0 111in tlot """"" , and LlI_ ".in~ tw-a_ eltu.) ill c~ in F:"{jO) FInally, If J$ I :s; N,(jl lor """ry 1 E e~u.l, ""d N,(n., f 1JIdY" ... tboorn In 1.....""'" 3, 1.2. 0

r

r

We aw defi,.. on e ~{jO) the .,..uivalcDO! relation "I _ 9 if 1" /I ahnoet ev SpIW:1:. rill ..Upracti<:al 1"1'P'*"', _ -=d not distl"&Uiosh C~( "l and L ~{jO). A fuDdion 1 frem n IntQ JE II called jO-lntegrable if It is equ.a! a.t. to a ". I"tegrabie func\ioo 9 from n \"to R- Clearly, if 9 and are t""l lueb funetlontl, I~ Jr4< - J ttl", ... it m O' . _ to define Ihe oon>pIe.. number J gdj< .. the In!.e&raI of J. written 1 """'''''''''' 10 1 E e '(j<, ft) in the mean If. for 9 .. . 1>00."', III - , 1·.tV" Ie.... to 0 .. h exl mdll.,...,.. the fil\.e"r F : ,**,1)', t hie dt/inltlQn doeo depend on ,be d>oi
t

J

r

'*

Theo..,m 3.1 .• 1

€:r

r

io Wint.",

,* if a nd onJp ;1 r I · dV" <

+00.

P lIOOr': SUppolee I· dV I' io 6nite. Si~ V,, ' Cl) .. fliP...... . 4~1 V,.(f), for .,.,.,.,. ~:> Olhereexist.lgE 11:+. 9 5 1, sud> l hat v ,,"(!) S V,,{g)+ ~ , Then V,,' (!) '" v,,"Cl - 9) + V ,,(g), .o V,," (f - g) S ~. 0

P ro. -It lo" 3.1.11 lUi'l l ,. h) AM infCl,. h ) Ii
h , 9. , 9:1

..., rml-vaIuod func:ciono on 11 , then

h

E Q. Inde«l, thill inequality evldentb' Idds il thoeroc .... ;.," ; in (1.2) such thaI f«~) _ IUP{/, . h )(z ) and 9;(ZJ _ ,Up(g,. g,)(z). n..ref
for

eYery :.:

..... 1-."" g, (z ) 5 g,(z) 5 1,lz). '" II; - 1;I(z) 5 II< " I(z); 011 the oI.her hand. If 1,(1') < 1,(:.:). toe ha"" 1,(1') 5 1;(:.:) 5 Ijlz ). and II; - Ij l(~) 5 Iii - 9)1(:.:) ..... t.eooe tbe d.sin>
j ; If

1,lz ) 2:

~,Iz) .

;.., two " .int O. thoeroc al8t II In 1((0. R ) and lJ'l In 1/:(0, R ) such tb.at I" - III . .tV" < ~/ 2 ..00

f

f lh - ,..1· dV" 5~/2· n..n

I

Isup(l!. hI - ,upU, .,.)I· dV" $

r

(I" - ,,1 + 110 - 9>1)· dV" 5 (.

3.2 Convergence Theorems Theorem 3.2. 1 (Monotone Convergence Theorem) Ld(/~ )~2:'" ... ir.... . (_p&:tiO(Ir. a ,u"..,."mg) ""VUt""" in .c'CII: It). no.... I - "'P. 2:' I. (_fiO(!~. I - inl.2: ' I. ) u inlt9n -oo). In lIIu _ . (/.,.", """""'l'U to> , in 1M: """,n.

i."

.w"

P ROOF : \V.. "XI"8i(\er the <:a6O'! of fill lr.;rMSi"3 ~_ (/. ). ", In .c'(p.; R ), and SlIpp<.>1e lha • • up.",I l.dII,.. ill Soile. The oequenal U I~' dII,..)~" , 1D<.i :: I . '" It <:OI>Ve~ In It. ~, I II . - 1.I4V,.. - IV. - 1,}dV,.. (OO·.up It> 0 .. p , q _ +0<>. SiD
* """It

.w,.. _

I

,...llMITabie.

0

P ropoeltlon 3.2.1 ( F'lotou '. Tboor<>m' UI (f. ). ", .. a _""'" ill J:'(p. ;R) .Iid! Ih - DIO), , , _ III~ ....... 9 in J: '(p.; R) ........ III..: 1:5 I. A.e. (_,iwJr. 9 2: ,. ",.e.) Ir>r
,.w"

f 1·1fV,, $. f l'dV" $. ~,!:!'J;!.f I. ·dV" (f"eJP'lCti""lJ. 19: dV" 2: I I . .w"

~

Iim,up_ ..... II• . dV,,).

",ill prwe ""Iy the fiN; otaU!fl""'~ For ea.ch ru:..J. II, ,be _ q\lotnC'e (", ... ).~"' when: " .' .. inf. 5.iS'!" deo;,..,astoI. and""e ha~ f 11-.... ciVil > g . .wl' f« all k :!: n. By the rDOn<>tone (;Dn,el~I>O% theorem, I/o .. lnf.~~ "' .. is i"m~ LlkeYioe, the ""'I""""" (9")" ~ L lDtm , aDd ..., ha~ J gn'dV" :!O Inf'2:. f J. -dV I' 'S c:, wben! c .. ~m 10£"_.,,,/,. -dV". ThUll ,uP.;!: . iI... lim Inl,, _ _ In is ".i.o~, a"d f , -dV" ~ / .,w" 'S

PROO':

W~

J

•

a

Domi .... ted Converxen<:e Theorem) 1/ G (J.),.~, ;~ C~{Pl """otI"f<" ~.e. /.Q 4 maPJl'ing / fram 11 inti> F (......'e F to • 8a.JWIdo IJ'G<%), .n.m fI ",I<> 10. +001 ItI, then I ;, " .;Ilt..,Wc ani (/.),.2;1 """,.."u 10 f in 1M ......" .. /:O~UJo. f 1. dJ;.

Theorem 3.2.2

"'I"''''''

( Let..:.~ '.

r

""""".,... II>

J /dj..

f

PlIOOf': It IW!io<:eI to prt>'I'e lhal If, - 1.1. dV" con~p 10 0 .. p .00 q lend 10 +0:>. Fb:: n ~ I. ADd 0>D0ider the 8equen<:tl (g., ..).':!:. where p,. ..... luP"s..d. l/. - 1.1· No. {p.")' 2:" is an illc" "ing 8equmot of ~inln(>toM 1.110..,..,"', {g,. .. )v. converg.. a111X111 ev
<:OIl..,,.,.,.,..,.

°

J ", ..

r

r...-

Q:)." .......

C orolla.y Ld r /Ie • fiIt~r 1<'ith • """,,14b/e ba.>u "" G od <>/ indiou A, (f~ )"'A /Ie • "'milt; "" C).{p), .IUII : 0 _ F /Ie • ",.ppin9. A_",~ th4I, lor I (z) ~ To N_ ,.,.~ III=. U • ,",,,,,I;"" II frr;m 0 mu. [0, +001 •...,. IMt g.dV I' < +00 G"'" 1/.1 :S: 9 ru:gligiWt: od (4eptJld,ng "" oj lor dI Q E A. Tkn I E C).(I') " nd (f,,)~ .........,.,. "" / "" 1M ma ...

.,,,,';:Ie.

r

r

UId, lor eve' )" n :!: I. let o. be an .. bit •.." d.meDt of.4,.. By T """""'" 3.2.2. I ill ,..il>leptble aDd (/• • ).I!' OO.... CZ'"* 10 I la the..-n. By ooptra
P"'"I'WIUon 3.2.2 I" ""'.... tIW I : 0 _ F lot. I'"in.!.,..,bk, il io >IC
r

coodi,ioos ,.,.., oatWied Let (f.)~I! ' "" • !equtllCe in "h:{O. R) €I F . ..e. OOIlVerJiLllIo I, aDd let II that 1/ 1:s II everywhcn: ('I'I.-.", 3. 1.4). P ROOP: SUJ>POI!" tIM!

3.2 Coo,uS""" 11«
NOOI' e.r:b

I. can be ... riUftI I • ..

M

~, ,s. < .. ." •.• . ~ ... , wben! ....... E 1/. and

""" E F. PutU", 10,...... .. lof ("', .• , m(2; - Inf(2g, II .Il)), for each inc.eget m 2: I , tbe "... ... .,.., I"*tive. ".iutesrable. • "d bounded _Iv .. by 1"; ••. The ""'I""""" (h; ...... }.. ~, (u.. indica; and n ~ fi""d) eo.," tl p' to tIM! fund.ion -(J; ... " ...... by

• I ••

I) _ >:

{
it 1/ .{z )! < 2g{z } if 1/.{z H 2: 2g{z }.

By tIM! domi.....ted mnvergmce tl>eor<m, I!>. .... ".integ. able. . . Po> .. ~, ,s.,st. ~, .•. »< •• lies In C}{P} (Proposit ion 3. \.6). Further,

g,. I) z ..

{

I. (>:} if l/ .(>:}1< 2g{>:} if 11.(z )l 2: 2g{z ).

0

NQt, tIM! ""'I""""" (Po>J.~, <:O>I_geo .. ~. to I. and 190 1 !i 2g for ~ .. 2: I . The dominated con~ theorem then.bows ,ha' I • inlqrabk []

3 .2.3 (Contlnui'), wll h Respect 10 a Para mel ... ) wI A ..,,, topoIofior/.,..,.., ...... " potnt i" A, ond tl
t,

(. ) For..w. " E A, 1M ""'wing >: _ 1(>: , 0 ) ;, ~)

willi..,., W•.

Far oabo!.ul ail >: E n . 1M ""'Pl"ng 0 - I {z , 0 ) ;,

O>IIIi" ........ ,

00 .

(e) :n,.,." .not. ~U 0/ .... "nd ./o=tiofI ~,..".. n inIo 10,+0<0) ftdIlMl ~ < +0<0 "ndlh4t, lar ~ 0 E U, 11(>:.0 )!!i g(>:}

r ..w"

""tIUk. "<'!iiigi/>le,.1 (kpending "" 0 ).

~

1M

""'wing " -

J1(>: , 0 )",,(>:) ""'" A

i nto F ;, O>IIIin ....... aI 00.

P ROO', Th. follow. im....,.;l;"I,ly from t he corollary to n.wo.", 3.2.2.

[]

At llUI point ..... in'rod""" """" IIOUtlon and make _ dorfinillona. If E , F .,.., .....t (or cornpIn) nornv:d 'P"""", for e:very i~ n 2: I , .... eaIl L" (E , F ) the ...........:I opaa! of n linear conlinlIOUI mappillP from E" in\.o F , _ .... put LO(E , F ) .. F. On the other hand , .... ioolJ(:l.lvely define ..... ....,.;1 "I*"! L{" )(E ,Fj u IoIJo-n: LIO)(E , F) .. F , and L{. )\E.F) .. L(E , Lt" - ' )( E , F)) for n 2: I . Then tben! iI an iIIOmoetry i. of L - )( E , F) onIO L"(E , F). defined by

(.;.h))(11,.···, .. ) .. n..)· "

_I ..... 11, .. (h(.. )· .. - ,) ...J.

"0 '

N_ let X be .... optn JUboJet of E, I " mappi"l: from X Into F, .... )lOin!. of X , _ n > I .... intefl"r. 1 iI Mid to be n limos dilJrrmtiable.t" _ to have V " )1(" ) u deri""'-ive of on:Ier nat", i f aDd ooJy If

,

(a)

I

II ( n - I ) ,i..- dilfM'mtiabLo! at mcb point of an open V 01 .. (,,1th V e X );

o~hood

(b) lhe tn&Ppillll :.: - D(~- '1/("') f
D" 1(.. ) '" 1. (0(· )I (a)) is alio called the

I is aid to be of claaI C"

",h derlvati~ of I

8t a.

it is n ,imN difJenon&iabLi! al ~aclt point of X and if the mappi,,«! rr I (0 S k S n ), '" _ f? 1 (:.:). are conlinuoua on X . When E _ K ("'ilh K _ R (Ie C j, I is n timel dilferentitJk at a if and ooly if it 11M a dcriva,i...,. p")la), of orokr " a, a (in the usUIII """""j. and then p"l( .. ) _ lY' / (.. )(1, ... , I). W6 _
Tbuoro! an opm AbMt 01 .. ""'"""" 1I'>lI (or compla) E, and let I I>o! c .....pping from (O-N) >< A inlo F, .,""" N .. "I'.~ l\Ib.oel 0/0. For 09"'''' iO\tqlu" 2: I, '~PPO" lMIlM lolk>"';fUJ """"it..,... a~ ...tUfted:

,,,,,,/.or .paa

(oj For """'lI

% "

n - N, 1M m<:Ipping 1(%,,) ; • - 1("". ) ;., 01 clcu C"

~A.

PJ For

md1 z " A, ilIe ""'P"''''' " ... O·(j(", ,'j)(z) fmm n - N ink> l,.· ( E . F)
r.)

For...u.
10, +«>1.-ell tJW. /M'

m.u an

inI."..,Wc ",,,,,Ii,,,, 90 ,...... n - N .nto = .,.jJiei<:ntl~ cl<>n I.... ,

~ D"(J(% ,·))(.)II:::: 9"(%) ~~i"O - N ,

n.o.n ill. ""'Pl"fUJ Il,. _ f I (% .• )dl'(%) /rom A into F .. 0/ duo C". and rrh (z ) _ f rr(j(",.·)) (. jdl'(%) Jor 0110 < k :::: n aM. e A.

P ROO", Let a" A and 90 be ... &bow, .... d let r > 0 be .. real nwnber such that A 0001 ...". the,,~ b&lI B with e<mt<=... and radius T, and such 'Mt, for all : e S, ID "(/(",. ')) (: )1 :::: 90 (:1:) •. e. in 0 - N_ Let 0:.::: k < n - I be find for t~ _ o t , and let g. be t be fw>ctioo 1:'"

L OS~" - · - ,

from

Ilrr" ·(J(",. -ll(a)~ . ; + !Io("' )'

(:::)!

0-

T~Ior'~

N into 10,+«>1. Evideotly, 9. is ilMgrabJe. For any given : e B, IomtuLo. ano... that

,.

for &II

,

~

E {} - N. But

(I - 1)---- ' , . D"(!(%"))(H 8(r- Cl))' ,JIJ • (n - , - I. )

f.

and Ibm!: eDol3" I 'glipble ... btet N. 010 C>(lOU.lni", N oudIlhal, for evuy .. ~ I and for e¥CI)' 1 :!>":!> II, I D-(/(r "))(4 + (~/" )(% gA(.z) for all:r 01 fl _ N.> Thus

anns:

n /.' (H)·-~ ' U

•

0 (n _ J: -I)! , D

(If." ·lj(a + 8(. -
II .:s (n _' k)I90(%)

u

for all :r of (J - N •. In shott, for evuy • E B , ..., ha~ 10> (If>:, ,))(z)1 :S go (.,) Le. In [} - N. Now, ror eodI O:S 1t :S n, dtnou by I)(')/, lbo: m&ppina:' /1)1.")(1(." .J)(.). <4«",) from A lnlG L(O)(E , F ). By ,ho pta;ed,,,« ~t and Proposition 3 .2.3, 0(' )/1 io oonIlnOOUl at ... Next, lei. 1St S n aDd • E B ho: 6n(I , and I '" 0 be ..... I.. IIICh that 1'1:S I. Then

1)1'-1) (If" .,»(4 + It. - "J) - U"-I) (It." .J}(a)

,

-(I.' and tho: _

WO){!(:r"J)(CI + It(. -

&tJUmen1 .. '"'-" .00... tlw

an ·dO) .(. - aI,

i

I~ ot·- I'(J(:r,' ))(4 + I{. -I4») - l)f' - " (!(:r, -Ill")

5"(")'1·- "1

11 - N. The domill&ted """""I'K"BCe tlltoo«!m tben ,,"0_ tlw 0- ') 11 (0 + 1(, -
Le. in

(01

We ~ that otber.......,... ollhis It..... a .. aisc , bo.n "'" shall requi~onl,.

n.....eIU 3.2.3. We conoider lhe foIlawi,. a&mpW, whkb are lntw.dM to lU""ln.te the IdMo "'" Moe Just po: :"L«!. If " .. .. _~on .. "",,,in,. S in n , aDd if,. ill the U~ form I - J Idv on SI(S,C), then "'" put IdV,. _ IrlVII for every I : n - (0, +001, and .. on (.1+ W;m I t in CbapIer 2). G iven .. rwwnpty interval t • ("' ~) 01. R witb IHl wdpoinl. .. and ~ wdpal.nt 6 (... 6La Ji), 1cI.)' he ~ _un! on t , aDd lei. F be .. ruJ

r

r

yr

ate

3.

50(

I~ aDd M~

Fun<. ;"..

.It&.

If 0 and fJ are.wo poinlll of Ii IIUCh a. ~ one of t he in~rvabI ",itb endpoinlll 0, {J I:a lnduded in I and nontmp ty. and if I : f _ F is .uch tlt&t IIH ill ~.inlegrable for one (...0.1 10 e-my) i"~'8111 C J ..lth endpolnt~ o .

p, ~

put

-

".0<8oro>8. -

I ; 1 _ F is..ud to be . ICq> lunctiOll il then! ~xlsu. pwti.ion of J into finlt~ly naQy In~'8Ie J. such lhat I i6 C'OIISIM' on eed> J. ThUll the it~ funaione. I,..,." 1 Into F for m a mal I : 1 _ F .. SIIId to bc ~la.ted .. bent""" On _ h c<>mp.)<:t $ubintcrvtol of I. It i6 tbc uniform limi~ of 8Iql IUn<:Iions. It 18 euily _n t hat I : f _ F is ~lall:d if and only if it ...... righc.-hand limit &I eed> point of (... 6( ""0.1 ,,

_ .pace.

Jef\-hand limit at ~ poin. of 1a, 6}. In po.r1icubr. any MtItinUOOU!l funetiOll f,..,." 1 into F .. regulated. [f I ; 1 _ F .. .-qula\<:< ""'WtiIl.!!; outside 10. fJI. and l>o:t>c.t io k-ln~. L:t I : 1 _ F bc reguJalOi on 1 """ the coroilary to lhe domi .... 'oo c<>n'-ergen'" .~n" " nd il hM. right, lt&nd deri .... tl..., I _ '[""ly," left-hand deriva\i .....) equal to I {r + ) '" lim.,.., . _./IJ) IIe>ll'«tivdy. to 1(<<- ) _ limy ... . .--. / b'll at .-I> point r of (G. ~ (Ulilpoeellvdy, of Ia. 6» . [n po.:1i<:ubr. it is OIXIIinUO\.llt and It hoe a deri...,iw: equal to 1(%) &I. all points % ~&I. do DOt bclon« to """"" munI.&bIo ... ~ of I. n ... ~IOt . , z ..... J:. III)dt is" primitive of I . and_

I!

I:

1{1}dt .. C(P)- G{lated fUn<:Iion I : 1 _ F is integrable if and only il 1/(t)idt .. r il l . 1 1<>"",4~ bas a fioi~ limit ...
a>nclude that

t

I!

l

Id), ..

lim

HI."",

iff

I (t )
0--0 ."_1 "

(br lbe.......tbry "'tbedom.h"'teda>n~nee thec>r-em). Talrl"i I -(0. +<»]. tbe.d:. c, I : z. .... flioz. / z io not ~·intqr&ble ....."" 'bou«.h .In l / ld, """. limit .. z. _ +00. N_ ~ ~-e." eumplc .. ~ lbe dominal.6d OOIl""rgC..... t~ dofto nOl. appIJ. L:t ~ be Le~ _ ....'" on)O. I). Given (> > O. pul I~ _ D O. 1!o," ~1 for teeh n E N . Then I : 10, 11 ~ z _ [1/ z.)O is the I/Ilpremum of the I~ (".~ 11Iz-J desi"",t8 tbe Lnl-ep"al J'IO'Irt of l Iz). If <1 2: I. lben: is tIO ~.in~

I:

3.2eo..... , ....

Tboot._

M

funcUon "'hid> domin"eo a1l l be I •. If (I ., I , fl. dA doeo not con~ to 0 .. n -+oo. NUl . lei A be d .. act of aU fini u. subsets Dr 10. I]. Order A by inclusion. and let :F he the Slter Dr -:tions <>f A. FDf each 6nite IUbeet J of jO, II, deo<>te by I J t he iDdW"M' func!.ioo of J on

1I.Ioa.c:F, but

f

/0, I]. n..n

IJd). .. 0 doeo not tend 10

f

IJ COIl>at>eo pDinlwWe to I IdA ... I. Compare tbill ....ull

with the eoroIla.y 10 lbe (\omin&ted corn~ nce \~

In our IMl example, "'" apply Theotem 3.2.3. Let P, (1wputivdy, ~) b! !be funcUon on R. periodic _il h period I , defined by P,(z ) ... :r; - 1/ 2 (l E!pectl~ly, ~ ( • ) ... "'~ - .. +1 / 6 ) for r E 10, II. Ilt nce p, los I'f&Ulated and ~ ill ClLIa\ioLJOUa. Now lei flo .. (: E C : Re : > 0 ) and fl, '"' ( : E C : Re z > I) . G;...e" • E 0 " .........;.ier the fu nction I ; .. _ (.. + 1)- ' on Writ.e lOr the llItO&faI put 01 z . n..n. for """y n E N,

10. +001.

1..1

,

.. L (t -D·/(.n+ l) - L 1tS .. <·

{

f( rn+f)dt

1tS_ <,, ·

0

- ltS_Lo (~/(m + I ) + ~/(m)) - [ ' I (:r;}d« Q

.. bela

L

I (m ) ..

ItS_ So

~/(O} + I ( I} + ... + I (n -

1"

1) +

I (z }d« + ~[J(O) + I (n») +

"

~/(n) -

1"

I (:r;,,".

[

P,(.. )· f (.. }d«·

0

Leu ;,. n u.Dd 10 +00.

«.).. L (m + W' .. ....!...+ _,1 _ • . r- P,(I)· r ·-' d.I . • - I 1, -~

((.) - ....!...... .!.'1 - •. •- I

1.,

P,(t )· r

·-' d.I .

1,-

P, (I) . 1- ' - 'd.I ill .w6"...t and bolo::>ow"phic in 110. Tbe. dote , theo:e ill • hoIoo:no
....

,

••

3.

I~

ODd

~1eNu.oble

f\".<. ion.

b- all • E i"I(, - ( I) . GIY""- • EjO. II. let , be tbe function t ~ , - . - 1(. from iO, +oo/Into R. Si""" b;t 3· .. - ' bail I E I.., n+ Ii.

Po.

-!oct -

I)

ill pooitiY1l (where 0, .. 1/ 6) and 9'(1)

s

,, /.." . ,/.." "2.

-

(10, - 1\)'(1) ' , (I}dI

-_ .

-

(10, - 1\ )(1) , V(I}dI

1,-

ill . . than ,,-'/ 4 b- ~ n E N . lienee P,(I)g(t)dt" Ieos.han "'/24 . .00 (,(.) • INti thaD - I + ,,"/ 24, ThiI 81,.,.... that , • sttio:lly deoc~n& 01>

10,11·

3.3 Integrable Sets ~,.

he .. Danio!ll _1= ""

DeRnltion 3.3.1 A

a,~

,,,beet E of n

("""'ion I " 10 p-intesr-bioo,

'f1>en

q

1{(n .c ).

,,1IaI
to

be jl-integr&bLt if iUt indicatOr

J I ....." . A . B E i.

put ,,(E) ..

Ca!l i. .be eta. of in~ab'e __ For all

I""" _ oul'< I A . h ).

'" A. n IJ< and A U B b60ng to i. Rcnot i ill .. rilll MoretDOO (A.).~I '" A. be\otlp to X, beo:.tt._

x,n..;:"

.

'n.

A '" . In! ,. ;:" I A,n."""" • ....... ueoce '" -R ,uclt that E.>. V,, (A.) <

(Tbeorcm 3.:2.1 ), If (.4,, ). ;:, . +<00, .be~ u..~. A.. I~ In i.. by ,he

..

tnQDOIont

(;O
, , .,, . jl~.'dV"S L V"(A.)

j1U, • • A. dV" S L

for all "'~ n ;<: 1. Flrany .... uen<:<: (A.). »

. ~\

ofinl~ subett
latqrabloe IO!\. A .. u,,:!:. A. iI " .intqn.bIr;-.........,...... ,,(A) - E.:!:l ,,(A.) If .he A • ...... diejolat.. P ropootido" 3.3. 1 Stt>er 1M dew 0/ Gll;~..w..u oj E . F\,,~, .0( ""'r ~ 1:- , e+ ~ tiIGI e- nE+ '"'. p".! E .. E· U E- .

3.3t~S
51

PIlOOf': u... (F .. ).. ~ , be a fIe(j~ of in\.eXTabie I Ubeeu of E , ,ut:b thai. ,,(F.. ) COIl'U&,* I<> ""PoC II:I'(G) ,.. m - +CO. and pu~ F .. u..~~~ for ead:! fixed illlOSO' m '2: t , the!1C\9 Fi (I C I .. .. IL.l..... m», w

...

"' - Fn(nF.)n ( n ,,) . •u

,U~- l

form, finite: partition P..(F ) of F inl<> int~bIe!IeU. Ot:.w:, ~ t hat Fi' '" (Fi' n f!.. , ) u( Fi' n F.... ,) .. U tf' 80 that each tel In P.. (F) can be partilioned into ~ in P... ,(F ). Dmotc: by £.. the union (Ill",*, Fj' (/ C I .. ) l ut:b that 1'(Fi' ) '2: O. Then ,

r,'" F;;:;i...

for~I S;",m.

Maw..,.,

,,(E... ) '" ,,( £.. U E.... ,) S .. · < ,,(

U E1),

,,-

beca'. -=II £.. U £..~, u ... u E..+> io partitioned by E.. u ... U £"+>_1 and by I"'*' I7' •• of poetdve _ure di:Uoint from E.., U .. . U E.. ... _I. Th .. implioellhal

for

C¥tfy j

::!: I. Put E"'

--- n UE,. _

lim sup Ej

"~J J~_

BlIp ,,(G) .. lim ,,(F, ) S ,,(E"').

CCII:

' - +00

.nd"nee E+ C E \.hiIo 10 ... oq.wily. Finally. InfcCB ,, (G ) '"' infcCB ,,(E - G) .. ,,(E ) - p( £+ ) .. p(E - E"' ) . .nd80 .........ytakeE- .. E - F*" . 0

ObMorve thai. ,,(G) 2: 0 lOr G e E· and peG) s 0 for G c &-. hel!! j , if G c ~ for example, u.e..

Note that Stone'. condition hu oot ye\ I>oeen .-I.

iO.+
I lie

intqrnble fundWn from n inlo R + .. inf(f, I );,~, and r' (]r,+ooj)" an irUtg, hI""1

Propo<lon 3.3.2 Lei

1m

,,,,mkr r :> O.

PIlOOf": ~ ill .. """"'...,. (J').~I in 'H. + cntl""rging \() I in tbe ",..n. Tht.doot, li,,{(t•. I) - inf(J, 1)1 . 0 be .. rerJ number and pmli", A - 1- '0".+001). ,"" func\ions

r

r

II. _ im (I,,,.[/ - r -in! (?,l)j) ~ int.ograble.

domilW«! by fir, aDd """verge point .."", to I" . Thus I" is

a

I~~

Lot / : n -10, +<»1 l>o "

Propooltion 3.3.3

ftin(;.ti(m.

FM' .....,

in's"' n ~ 0, P-' I• .. L,,S,,s.:r-. , Ilk - IllT')' 1.0.,.' ~ A.... .. 1-' {l(k - Iltr. "lrO JQr 1 S k < " .:r and A•.:I'- . , ,_ - I - I ((n. +001). Tht:n tIw:..........,. (I.Jolt. """"""'" I<> ,.

PAllOr: Otw......

a

Theorem :t.3.1 LeI C~v<).

p~

First. let

F""

I : 11

ro»I &.....:h.,..or. Tht:n SI(R..F'l if ~ '"

_ R + be integrable. In ,he notation of p' ......... p*

11M :U.3, lilt f~koo I . 1.. In S r"{R ). 5i,..,. Ibco

""'I"'''''" (1. ). ;;'1 I"", ,· ?, ,

it <XI
L;o.,

N_ let F be .. -.l SanICh iII*"!, / E .c~v. ), and let A be, win~abIe !Iet_ Fo.. ~ t ,. 0, !.hen: ""ISU 9 E Stj'R , F) euc:h lha~ I II - , ldV I' S ~. Tbeo I/ I~ - gl" lclVI' Stand gl" life io StjR. F). W~ mil)" conclu
r

I l ~ ilWI~

Propoooition :u ... "

H ,

I l ..dV I' '" .up.., l1(n.C).I. IS I II gl" dl' l frtr ftI«1J

we_

1(1 A .

r

tn.

PROOf": Glvm ~ ,. 0, Ie!; I E H + be ouch that II .. - I I . dVl' S RepIo.ciD« I by Int{!, I ), if ' 0' - "y, we may IUppOIIe t .... , I S '- Then Ie!; ,E 11:{n ,C) be ouch lil&l.1g1 S I _ II gd..1 2: Il
If", "1

> >

,,+ / ""-' " (J ,clVp-i) -J /

I ln_.. .wI'

~

J

II .. dVl'

-~

·" c Prvposl\ion 3.3.5 oel A .

f

I AdV,. _ . up""s<{R,C).1 f hl"ds
If

P ROOF: Given ~ > 0, let, E 11(O, Cj be ouch thal t91 ::;: 1 and gl .. dpl ~ I".w,. -~/2 (Propo.;tion 3.U ). Since SI(R.C) II den .. In 4{p.). there e:r;iN II E SI(i . C) RICh lb., f (g - hi . II, if '-';> "y, ... hen! P .. doe projection from C (IIlto tbe cLc:.ed. OOD~ IIeI. 1)(0, I) _ {. E C : 1_1:5 1), ..., .....y auppOee Ihl::; I. Then

J

1/ ill"d,.) 2: I

f

gl"dpl-

f

f

(f - 1I)I Ad,, ) ?:

f

1.. . dVp. - f ;

c

.. hence the result.

T'->rem 3.:1.2 Far net) i~ ..1 A . Vp(A) _ SUI>(A.). ~ , I "( A;}I, IOI\erc (A. ).. , t:>tendr _ !lie cJo.u 0/ all fi";~ ".,rtitioru pI A 'IUD mIL" abt. H • •

P ROOP: G;""n t > 0, Iel " E S I(R , C ) be suoh that 1111:5 1 and I JlIt1 .. dpj2: J IAdV p - t . No... h can be .rnt~ I: ,«<~ 18, . ", when: tbe B; _ d;.joint ilJU!Cl'ble octs and lhe" ~ 10 b (o. i). Tben

H..,.,. 5tIP(A.).

obrioos.

E..., ", (A.)I

2:

f

I" . dV,.., and the ""....... lnlquality iI

[]

DeflnlUon 3.3.2 A IUboK A of 11 .. called #-Mod",.te if it iI ....... ;rwlID • muntable union 01 ~inteptble __ A maPlJ'lnl from n Into R Of .. vector .pace II ..-moderate If it .."W- ouwide. ,..moderate "", ,

"

P roposition :1.3.6

II I: n - j(I,+ocI iI,tid! doal

/ iI ",,,,,,,",,,,.

r I· dV" < +00,

do~n

<"

+001'''''''

PROOF': Let" be aD i~abIe functiou from 0 imo (0 . lhat I (Tl>e<:nm !I.l t ). Thm +col) ~ inl.tFd:>ie for~., n ~ I . &nd /

"-'01/1>.

"",,;Shes

""UJi<\e. u..;?;l II-I Ol/n. +001).

0

3.4 ,,·Measurable Spaces The po.i. (RA of . nonemp
" .. .",.-abio "P""" ~Hn ltion

:1.-&. 1 Let (O, F) &nd (O'.F) be 1_ ".~ . pe ...... A m' wi"" / : n _ rr if.ui to be ~Ie F IF ...~r r '(A') E F for -=I> A' E F .

(rr.r). and

I[ (O.F). ~ FIF

, aDd I a

''*''''

(fl". r ) "'" <1· mta/lunble I • mapping """",urabIe F / r , then, ciee..ly, go I ..

ma~

moMUrabk :FIr.

...

P roposition 3.4 .1 l.d O. 0'

«""" ""n~lp "'"

a...t

I: n _ 0'

~

map-

J. UI F Ioc a " . ,..;", in n . 'l"M d4.u F f 0/ nkriA, Lr ,~) i" rr ~1>C1l1iIaI1 ;,

"....,....,We F I P . t . U tPboa<1· nnginfl'. 'J1o,,<:/lluI-l (p ) _(r' (A' ): A'E F'1 LrQ """';"';" n. CIIIItd 1M ",. ,."., go ...",,'" ~ I . I -'(P ) u the 1maI1..1 "' . ...., F in n otId! Ilto.tl;' m64!J'U'fGWcFI P . PJIOOY: ObYlou8.

0

'poI_,

P roposition 3.•. 2 Ld (O,T) , (O'.P ) b( 1_ ",·m_nWlt OM leI C' lit< • 0011«& .. 0/ M"" oJ fI' .1Id! IiIoI F ;. tIw: ",."", goenm>t.:d 6r C' Tlleal it _ruhIo: F I F' u\(.....e1" I -'(A') Ii.., in F l()t 011 A' E C'. PItOO, : In tbe nQtatiou of P r<)p(lIIit;"o 3.4. 1.

Ff

"'>RIal ...

C' ""d , , ..... , F .

c

DeHnit iou 3 •• •2 La n be • """""'pi)' ""' . ( n,. FI»)'~1 be. nonemply funlly 01 <1·-=abk ~~. and, for eadt ; E I , ~ I. be a -wlns from n inu> 0,. The ",.rill(!: F ID 0 ~eratal 1»' {,,'(A,I : ; E I . A, E F;} is ca.l1«I the " . ring g.n,,,.ted by tbe famlly (J. ).~/'

3 .••3 I.. I/o.( ...,/ali"" OIl JHfiniliqn '.-l.t. F ;, 1M orn.dlerl OIl IMM: in fI nidi th4J ...m I, ;, .........,u1e"F;. Lt:I (O' ,.:P) /.Io! .. " ._~rcbl~ ~ h /.Io! .. mapping from 0' """ fl. Then h ;, ~ .:PI F Ii ""J <>nl, if I, o h iI ........nsbI< .:PIF, lor....:/l l . P~ltJ.on

".ring.'

......

PItOOF: By p~tloo> 3.4.2 , h iI meaourable .:P IF if &nd only if U, 01 11)- '( .... ) lioe In F for """'l' i € 1 and eWlr)' A, € F;. 0

If F , ill the .... rin« .....10& generated by

by a cl-. C. of osuboeu of (A, ) : i € 1. A; € C,).

~t.,d

{I,'

0" then T

ill the

'""'''''''P'Y

Deftnltlon 3.•. 3 t..el. (O.}) be a ,,·_ur.bIe lpoo<:e and E a SIt*, of O. T'hi! .... rin« F I E gmer_1ed by lhe canoni
boo a _""upey {&milJ' '" ...· _..ur .... IP'"'* C"""kIH lbe product oet fI ,. n'EI ft, and lbe canook·1 projectlor> Pi hem fl 0010 tbe fador f1; . The ...·rill,(!: F in fI smented by t he family (J>,)I<;' oImawill&l iI called lbe product .... riq of the F, and is writteD ®,uF,. Theorem 3 .• . 1 Ld (n. l,u k a I."'ilr QI 1<». ' sjeW; .e. F.,.. -.It. i, Id 8, .. Ihe &>reI " . •~ 0111.· ALN, 101 0 100 u.~ pr<>d Ie; I bg;,' q<>(II n ;,t I G"d B il6 B«r:1 " .a/gUnl. :nw:.. 0 ' EI S. C B. I" ,..,tic1Jar, if 1 ;, (tJI. ..."II) .......taW. aM ...m I<>po'-'Sim! """'" Iuu .. """"tab/./: 1o.tU, Ih<m B _ 0 "" B,.

n.

n.

P ItC)OI': 5i""" Pi is meuurabIo 8/S. for eo.ri> i. lbe idm!.ily of fI ;, it u.. 11>10 IUI>!oet.s of fI of lbe form p~ ' (V.,) n ... n p;:'(V... ) {i ...... i. € I. V.. e U" for """'Y I :5 .t :5 '" """",iluL.e. beeIs fur the product topoIocy, thlt (countable) built ;, ront "ined in 0 .." 8;. N..,. ewry Open eub&el. of 0 lit _ union ('» I rily oount.&bIe) of di$t1Dct .1.u",u" of thil buII.., il liN in 0 iEi 0

s..

-

"Ibe pal. of a ......"'p'y ~ n and .,,•• is' bra F In n

DefInItion 3. 4.& Let (n ,.F) and I! ite Iloor-.I " .lllgd>ra.. f mo 'J •• bleFI 8'· FT

called._MUtabIe

a " "urable~, 0' a t~!Ipe<:1O , fI - ll' ~ aaid to ~ _urab!e F if It is

~

:

~

Theot'<)m 3.4.2 Ld (0 . F) It.! a mfCIwnb/e

'PIlO<. 0' 4 "",'riusbIe 'PG". an
(t. ).2:1 4 ~ pI ",,"Wing.! frvm n in./(> 0' ttJhid> o?~ "..inl~ /;> a ""'wing J. S~ IA4t ""d I. ;, mmJ1
PROOf': ConsXler <m 0' .. d;"t~ " that, for e"")' open 1>1'-'. V or 0'.

r '(V) C limi .. f I;'(V)

c r'(V).

(1)

",here limiDf 1;'IV) - U~~ , n,~ . f ;'(V). Indeed, if z E I-'(V), Ii""", f Ir) _ lim,,-_/p{:t) and V ill open , I,(:t) beIorl.p 100 V 6:>r" large eoougb , and the firn inclusion obuins_ Similarly, if z E lim inf I;;' (V), there ~ta "" inleger n II>Cb that I,(z ) belongs 10 V for all " ~ n .!K> I(z) " 1; ...... _ _ 1.(0:) lM$ in V. Thio gl_ the second inclwd let B. be 11", union of thooc Open boJlo of ..,.diU$ I / ~ (",~ ~ E N ) _oed 0.1 poiots of F. B. ill open , IWd F _ B• .. n. 2:,J~. For. if z E /1., lhen
rr.

n.<:,

rt.2:'

n r' (B.).. r '( n S.) .. r'( F) .. r'( n b.) .. n r'(lJ.). . ~,

.~ ,

Applying ..,1&lioo ( 1)

1<)

. ~,

. ~,

B. , Ml oIrtain the incl".ioll

r '(B. ) C !!minf I;;'(B. ) c r'(lr.l. So

r'(F)-

n r ' (B. ) en

( Iimlnf I ;'(B. )) c

'2:'

which sayt

3.5

1- '(F) _ rl.2:' ( lim lof I;' (B. »,

n r'(lr.),., r

""d tlris lieI! In F .

' (F ), 0

Measurable Mappings

Definition 3.5. 1 A subloet. A 01 n is said to he p·met.lIunbie if A n B is p.in~ for aU jrin1ql'able """" B-

The d ... M of

,,-II'

? · "'.rable. IICt8

io obviously ..... ·~bra.

Dellnltion 3.5.2 A ..... boe\ Nol.O io Mid 1<) he loo:tt.Uy ".negligible If N n E is oegligible for every Ull
••

DefInition 3.5.3 La F be .. metriut.ble s~. A .... ppb\& I from () Into F 10 aid to be /t" mNI8u ... bIe if (a)

I iI ......u ... bIe M :

(b) W ewry i~ ~ E , then: exists .. rq;ligible atl N e E IUd> tlw the t.opoIop:aI I ( E - H ) • "p&ra~.

"'*'"'

~ IIw. ewry

mapping froo> n into F which '"4Ireee Wilh

I

l.aA. II

/J.meuurable.

A rw . " ry and 1Uf!iciem cooditioo that .. fUDCtion I from fl into R: be " . meuu ... bIe II I l>I.t 1-'{l T, +001) lie in M f.,.. aU rW numben r , 100-1, tbe Bo.-el O'·tJrebra of R: ill'" ,, -rilll! ttne ...ted by lbe c1_ mnoi81illl! of R: and tbe!lets

/1", _

I·

Propoeltlon 3.5.1 Each I

E.r ;, ".~.

PlIOOf': Let r > 0 be a fHl numbe-r and \e\ E E i. T het-e uiou 9 E .r" Iud! tlwg 2: 1.11" and < +00. lfn > r . . . . inl'l"'". inf(f, ng) bdonp to.1 · and iI intqn.bIe (The:Htm 3.1.4). Tb... Iot~. r' OT._1) n E ..

!"g ..rv"

E n lDf(f, ng)- ' (lr. +oo1) iI/,,"in~.

[]

DefInition 3.5." Let E be .. oonempty _ and 5 .. "mial&ebn in E (i.e. , .. Itmlring ronuinilll! E). A .... pping I from E into .. eel rr II Aid to be S" o>p!e .....ne-.-er tbe", II a finite putitioo ( E, ).EI of E lnto S·oet.I Iud! tlw I II oonot..,11 on ~ E•.

Propoeillon 3. 5.2 A .....pping I from n ...... a ....lrUoWc ",..... F ;, p m" ,*,c!lle if an.cl """ jf, /.". mdr. imfJ' die nib .., E 0/0, I ;, 1M limit a.t. ift E 01 a ~ ol 1l.I E.ft",pk mapping>. P ROOf": Let d be .. distance on F (compatible . ill! tbe IOp<:IIov of Fl. If / iii /J'" .. 'J , ..... bleand E .... intqn.bIe""" tben! ....... .. rq;Ii&ible_ N C EIUd> ,,,-, I (E- H ) II oeparable. Then. letti", (~ : i ~ I) be .. <:OUIltabie denoe oub_of / (E- N ). _1>1...... , lot......,. l~n?: I. I (E - H ) c U i!: ' 8 (IIo . I/ n), or E - N c U i!:' (B(~. I/ n»), whftoe B(II;, l I n) d' 'iptee lbe open tJ.l1 with.,..,~ iii and radio. I/ n . Hence there exisu ... in~ "- IUd> thai. tbe V It" IL¥ F'un! of Y• .. E - UI '5' 'S •• I - I (B(~,. I / n») is Ietoo ,Itan I j r. N_ )LUt

r'

Z. _ U"i!:. Y.. , and deli"" an i l E..unple mapping 01,,: oI,,(:t:) ..

I

~ E

.,

F

if :t: E Ao .. Z. .

if:t: E Aj .. E n

",here IiO iii fUed but arbitrary (l...Io:s.SI- ' ..... )· n I - ' {B(", I/ n »)

IorI S i S k" .

"Then tbe eeq_ (o/>.,)~2: ' o:xr.t' g& to I at """" point or E V,IIni!:' Z.) .. o.

n.

lt ,

Z•• and

Con.......ty. JUppCI6e t hat the condition .. satisfied, and let E be an intepbic~ . We <:an find B ""'Iuen<:f: ("';).>1 of k / E-,;mple mll.ppi"3ll from E into F, which ~ to fi E outside B ''''I(ligihle ~ NeE. Fll< , E F and, for...::ll i <: 1, let I"! bo: the mawing from E into F wbich ogreeII witb 'P, on E - Nand ... hK:b 10 , ",,(E - N). Fin.olly, lor ~ -EIoml1 8el. 8 ib F, I-' (B ) n E is integrable fo.. oJl E E k ; I!O I-'( B ) I.... in M . a

"·,,,_urobk

P ropoo,it ioD 3.o!>.3 l.,d I. be ~ "",pping /r<1m n mt
from 1(0)

into" "",triuhle """'" G, tkfl u

0

I ;, " . ..,.." oJUbk

PlIOOf' : Is E be "" inlf«rabie oe\, and let N be. negligible JUMet of E ouch that I.(£:- N )" 8epe.::abie for every- n <: L Fort$Ch .. ~ I. let 8" he tht: Bom
181.>1 (E-

R_

...,...,.,a:

Theorem 3.5.1 (£ SOI"O"V'. Theo.... m) ul (I.l.lI bc G "I ,, _m~ "'appingJ from fl ;nto a "",tri.t.o:~ "nil""" 'JIG<" F, ....... ~ l ~ .e '" G mappinf I . n..... I if " .~ M_ _, lor ~'""lI i~ ..,1 E an4 ....." t > 0, then: aitto an intqrnbi• ..,1 B e E .ucA tJw. V,,(Bl:S t an4 nocIo.lh.ai u.. . t\'U<"Me (I") .~I a>n«ryu "n;j~ to I .... E - B . PROOF: Let N be .. I0<:0.I1)" nqii&ibh: 8et lIueh IhIIl {I. )..~I ">nod&,,! to I ou~ N, a:>d let d be • dis ....... on F. For every PBir of illlq
:s

It mtqrabit, &lid V,, (B) :S I:.>, V,, (8~h') :S ~ . By oonsln.lCt;o", (I.. ).. oomt. gtS "niiormly 1.0 I on E - B-'" D _ +00. Finally, /IE-Ii", ~ble fil E - N. hen<:(: I Is _"",bit; M. ~ow ...., can chooIe Ii 110 d.at eo.ch ME - .... ) "'~' &lid be.><;e ICE - .v) is toepIll1Iobie. a

>.

Propoo.llioll 3.~." U I (1_ ).. bot a I t ~ rrwoppOng I · Ft>r end> m 2: L. It:! (I....~ ).~ I bot 0 ...,....._ 01 " •............obl<: """,pi..,. /mm !l illl<> F, OI>Q"",,.,..., I.o.~ k> 1_. TMn, It>r eodI mOOtt""" (Q(P))'~1 cl ;~ ... 2: I .•1ICh tMl (J,.~(.) ).~,

.....""'JU 10

I

p.t. .... A .

P ROO~:

Fi,..., OUPP<8 .h.t.t A 10 In~. WI :> 0 ~ &i~. F.,.-"""'Y '" ~ I, there «isu &n 1~1e Z_ of A such that V$I(Z_):;: tlr' and ...m thllll"" ""'I.\IeIICI!! {J..... )~~ , co", er"", uniformly 1.0 1_ (Xl A - Z ... : t hen, puttlD« Z • U...;: ' ha~ V /-IfZ) :;: t , and t&d> 1It
,,,beet

z.. , ""

_"('X" unikinnlJ" 1.0 I,. on A - Z_

No.. "" _ 10 the 0_ ... bOe (.4.;).> , of loltpable - . which wt may lake disjoint . By .. ha. woe h.t.~ jlN ";"'n, lhere «k" A do, of inlesrMle fUbetu of A;. Iud>

that

Cil

-

n.2:' Vi ..... oqI~ble:

(b) for - . y m:> \ , , "" ""'I.""""" {I_.~).~I ~ uniformly to eedI A, - Y, ... (with J> ~ 1)_

I ...

(Xl

Put Y• ., U,~ , Y,.... for all J> 2: I. "" that n.~, Y, k nogl\ilbie. By indootioon on p , ...., <»nSI.n.><:t .. Strictly i,.......;"11 &equroce ("(J» )~I of integc,.. tbo.t .1(/,..(.), 1,) :;: II p on U 'S'li'.( A, - Yo ... )- Nftn, ~ Ii he a oqI~bIa ...'-t of A sud> that {I.. )... » comE' .... 10 I (Xl 04 _ N_ 0 ;""" :r; In A U Y.) :r; bf:Lonp A, for A ouil&ble ; 2: I. and then: «itt3 PO ~; """" iliB,:r; d_ not lie in Y... ThEn, for all p 2: po , "" ha", d(J~)(z). 1,(:.:)) < lIt>. lIentt I (z ) is lbe limil oi lhe !,.• (p){:,:). p - +00, and the proof ill complete. 0

2: 1 Iud>

IN (n.>,

J.

w

ul F bot .. rwJ &ruu:A $J>OO<- A mapping / /m",!l into) F .. I'"~ ..n
PItOOt": By PlOpel;"'" 3.5.2. ,boo CODdilion is ~ul6.:ie"t. Com~y, IUI~"" tbo.t I ;" ~ and mo<\err.l.t. and let (o4, );~, bt; A!Isiliooa 3.$.2). FOI ewry n ~ I , ~ g.. to he the ""'Willi from n inl.O F

tlw ogne with I". on eMf! A, for .'hiel> i '5 n , ..00 thal vanisbeo ouWde UL!; '!;~ A,. Then (g.)~~L coo""'!1'6 \Q I a.e. [J

T heorem 3.5.2 ul V Ie a ~ _1M rub..,.,.", til C}v. ), whene Fila rml Ban4dL IPO'OO- A mflppmg I from n iroto> F iI "'rouGI~",bk .. nd " ............k if and on/JI if it illIw: limit ae. 01 a "'I"'enot: on V . P JU)()S": By Propoai\i.on 3.5.5, Thoon::m :U. I . and P ropositioJ\ 3.1.9, e&cl> I in .c}v.) ifll'""~bIe; Ihllll ihe coJ>ditlon ill aufficienl. COh'.'sely, ~ I : n _ F he ""","urahle and TOOde-re.~. B)' P,op:ai lion 3oS.!>, I io the limi~ -.e. or a _""""" (J.. ).. ~L in St(R. F). BUI ...at I.. ill l he !imi~ a.e. 01 . _ueru:::e in V ( P n;>pO)!Iitioon 3.1.11), 010 I is the limit Le. 01 . _""""" in V (Pmpo:wilion 35.4). [J

Theorem 3. 5.3 A ....... "''11 and nfficiert! o>ro0 and I ifl l he limit . .... nf. ~ in 1i{fI , R ) (3) F, So Thaw.", 3oS.3 folJooors hom Theorem 3.5,2. [J

_n<>I fJI - I:.,;>:, r

T I__ e m 3.5.4 wi {f.).>, he a into (O. +<>oj. ~ ra:.~, I~ ) '.w"

" .m~fo<mtWnl

I~ ' dV" ,

/rom n

PJtOOf': F\Lc all p-rrEL"u.abIa fw,c!io, .. I , 9 from fI into !O,+<>ol, ..... Ita... rU+,jdV" - I·0 or 9·.w" - +<>0. and aI80 if I and , ..,.., I'""inlegrablt. Then 3.5.4 Wh.. from Beppo LeY;'" lbeo<em. [J

r

r

r

r n.eo"""

3.6 E",.entially IntegrabJe Mappings By OOCl.Ultlon, ..... "'" 0

~

(+<>0) _ 0 )( ( - <>0) _ O.

DefinI t ion 3.6.1 For all funccioDs

j ' l.w" .

I from fl jnto (0, +00(,

...

oul!.j' 11~.w"

is called ,he upper ' , ."tia! inttgnJ nf I (wil h...,.p«t \l) V,, ). c.:...a.:qt>den.te ~

r

r

r

r

If E iI B I U","", 01 0, 1 o!dV" _ 0 m _ ,hAl E is loo:eJly ~~l.iciblo. If / and , aft two func:\ions from r:l into [0. +<:>o[ ...hd> qree I.... e. , tben

rand/·,\.dV" - r g·dVl'- lf I,g. aOO II a~ t h_ functiona from ,. 0 ;". n:al numbe"

r{f

tben

+ g)· dV" <

r

I ·dV" +

r

r:l into [0,+0::11

g_.w"

j "Mo.dV,, _>.. j · h·.rv,,_ If U.).,~, is an i.uc,.... -ing oeque",," 01 functions from r:l into 10,+00] u.d if 1 - oup"2:' I~, , hen

P ......... lt ion 3.1. 1 Ld I , g. dM h ... u..... fImdio'" fnmo 0 into 10.+00), ..wt ~ dM h wmeo;,v ... hl. Tht" 1(g+h)-dV" .. 1,-dV,,+ Ih -dV".

r

r

!'ROOt': It !lUffictoI.., p ,h(, .hllt

1/.J +h) -dV" ..

t

r

19 dV,,+

t

r

Ih· dV",

51 ..... /19 + hl _ I g + Ih..... ha,..,

t

J{g+h)· dV" s: /

and I. r<:mai", to bo:

~

Ig·dVl' + /

lhal

/ f9 ·dV" +

r

Jh · dV"

s:

r

Ih· dV".

1(g +h)· dV",

We need oaly 0)1: S' v., IhI; ~ in .. hid> J" 1(, +11) -dV" < +00. "Then tbere '""~. E ,,.. th.al " ::: 11.1 + h). P ut ~ _ . '1.1+ 11) .. I...."'.' , + 11 ,. 0 and" < +"." and put ~ _ +oo .. bcre

....:n

r o{g+h) -dVI' r .. -dV",

,+h _0 or .... +00; tho:n It ~ u{g+hl, to> < M.... w 0 and u(..,) < +001 is WmoMUrabIoo and tbe fuDCtion Id, ~) - a/ 6 from [0. +oo[ " ]0, +001 in.., I't "' COIIdn __ . and

benco "IE 10 n..-; 'Urabioo M IE, .. hid> ......... thal " 10 I'-n-..rabIo:. A...., and ,,/,. a:e I'-~. Siooe ~;:: J, ..... ha...,

j "I,· dV" + j "Ih· dV,. s: jO trg . .w" + j "0-11 · dV"

.r

..(,+II)-dV"

OJ,

st . .w". a

•

Theorem 3.6. 1 l.Q.

r-J III;, II

r 1
""I ",cd ' ut~

M Il l ;' ...odtn\te. (c)

into [0, +001

I lot: " fomdion frrmt 11

r l.w" .. J' IdV" .

r l.wl/- < +<:<;!, ....:.t.o ...m u...t / """w.... III..,..,

II

i>I~ leU ,

leI

A. u...t "

G

""""~

L .. ~. ""ojdc A .

.........

1>,

r

PROO' : (. ) ....d (t.) 1I.t.~ been I"",ted preylo tll.t.t /1iVI' - .... P~;: I /I A., dV1'. P ut A .. U~l: ' A~. Then

r

J'

~~r / 1A.,dV" ~

r

/IA
r

I I AdVjJ.

j ' /

r I IA' dV" ..

r I
I e:r. then the IIi!t , -'(10, +<:<;!D " A Ie moderat whence _ _ tIw J' IdV,, !S IdVp. c

Prop05itioll 3.6.2

1'lt(M)l": If , e " .. and f gdVjJ !S

Now Itl F be • Bo n"'h from 11 ioto F cueh that

c

O.

1IpA(Ie. and

1T.U) ..

e""'ll

let ~&o) he tbe

r

IIi!t Ql

r l/I· dV" < +<:<;!. Endow

tbe ""rninorm N; ' I - r il l'
I

~~

T heore m 3 .6.2 / Ioe/ongt I<> ~&o) (ru"",~,. ~(jl») if .."" q..Jy if il ;,

""'", lId" ..

ep4Ilo.e. 10 G f e P,.&o) (",",,,,,,11,,01,, g e .cj.(j.); '" 111;' f pdoo)- I : 11 - F ;, UK>ItWit, ,, ·iPII"9f"Gble il ""d omI~ if il;, ".~~ IfIJV" < + 00. I ;, " . inl .•• =We i/ .."" <mI, if it ;, " •.....km14: UJ'tn,Iiallp jJ.in~_

a"" r

e"" c

I d" .. f I dl'

for all I € Cj.(I" . tbe _ntW ;IItfll'a1 ] I dp. of ...,. I € ~(I') ill \Wall,. .. rillm f Idp.· For e\W)' ~~~.eI: A, fIlA d;< .. then ",ritWO fA I d".

Sinco! ]

•

Obee....e tW. If .. 1tt A iI _nl;.,)lr in~bIe (I.e.. if 1.. iI). TI.eot.", 3.6.1 oboon th.t.t Ail .. union of.., integrable Itt and .. locaJlr ""&li&lble Itt. Noor. put 1111 - sup",>I(n.CI.IfI";' 111(, )1 ,., ~uPIE>I'J";' VI'U ). 10 that 11'1- UVI'I· Dellt, llIo n :S.6.2 ". is saki to be bounded "hEtIC.e. Propoo
11'1 _

u.KJtliAll, "..;"'1';1 : bk.

!

1... dV".

• •up

1".1 is Snite.

r 1 . dVl' . So I' .. """nda/ ;j~,..I ""'~ if fl ..

I!QI .. d".I'5 SUP ! I, IdVI' :S •

~(n.CJ

WI!> '

sup 10" lSI

V".U)-I".I

r

for .....,. intqrable Itt A • ., I . dV". < 1".1 (Propooitioa 3.3.4). ConvttlElr. VIIU) :S 1" 1 . dV" lor ..u I E 1(+ ...tWyi~ 1 :S 1. Hence

1I' I'5 rl .dV"..

[]

Propoo opa(e F.1Mn J lvdl' IJ vdl' hu;" D lor....., ""'JIIrirIf I from fI imo 0,..1....., UIeIIliall, mt~ /ram fI imo (O,+oo! #UdI Iht.t J g 0,..1 19 ;, tue1lt1aIl~ j", ;; .. Wo!. If Id". - I, IMn J I~ Iiu i7I tAe elo«.d CIim_ "'. . . . ofl{fI ). lor ....., tue1lt1aIl, irtlq'uWo! ""'PJ>i1lf 1 /ram fI into. r-...I B
D

I.",,,t>om, 1"

°

PI\OOF: We ~ ooJr J)rO'o'e the fi~ cloirn. Oeow: by F' the dual lpoooe 01 F . Lee ( o. a') '" "'(z) :5 0 (a' E F'. 0 E R ), • relation defini", in F • clMM h.t.lf~ E "hlch """w... D . TMn U{.z),(.z),.') '5 o,(.z) for aJl.z E n, ., (/ 1gd", a') '" /(1" ..') . d" '5 <> J "t"., ..>d J I gdl'1J gp to E. By the Hahn- g·n ..... theorem. D illhe intu_lion 011"," clMM h.t.lf...... "bid! oonw.. it, t.Dd the proof ill mmjWte. []

Propc.-ition 3.B.5 S~PJlOH " II j:IL>Ati .... a,..l kl I : n - (O, +00) .. " . mtIfl.nL",w.,. PI£/. 9(1) '" 1'"(I - '()I, +001») lor ....., raJ nwmlo!r f > 0, and covider~ , I.e ...,... "'........., en)O. +00(. Thcto jd" '" ~f)dA(I).

1"

PIIOOr. FiM,""ppo8fl th.t.t

or {O,I" ... , I.) (with 0 <

I

f,

r

bu. finite, positi"" ......., """",Ir. {I" ... ,f.} < f, < ... < to). Thoen

r

On tbe Oloo hll.ud,

L

g(1)d.\(I) .. I, . ,,' (1 2: I,) +

(I, _ 1, _ , ). ,,' (1 2: t, ).

·';'S·

boca.... g \a ...... he val ... 1"(1 2: I, ) on 10, I, I, the value ,,' (1 2: I.) on (I" 1.1. _. _. If all the; 1"(1 _ I. ) an: fiDi"", t he ",,18 (1 '" I, ) ftre -.:ItiaDy iDI<>grftbIe,

...L

I, ' ,, (1 '" I, ) '"

L

I, (P(J 2: I, ) - ,,(1 2: 1;+1)1 + I • . ,,(1 2: to)

'" 1, -,,(12:1 ,) +

r

L

(1, - 4 _,) 1'(12: 1;);

~ ' S·

r

he"", Idp g(1 ) - ,1.\(1). Evlokndy.• hio "'Iu.olity remains InIO! if 0lIl': of the 1'"(1 '" II) Ie inftnllt_ We ""'" puoo to ,"" ~ra1 cue: ""PI>'*' I n. -=-.bIo:. and D(K>I!~r the ....... function / . . . In P'Q908iIIorl 3.3.3. For ~ t ;> O. the 8eq..en<:e of XU

(U.

;> I)).,!:, ;' ''''

.. n _

+CO.

1 to (1 ;> I), SO ,," «(1. ;> I») converges 10

Thl:n:ron,.

j "l ·d" '"

,,' (U ;> I))

... j ' I.-iJ<

r '" r '"

Urn

.- ~

Hm

,,'(U~ ;> I)) . d.\(I)

P(I ) -
a

3.7

Upper and Lower Integrals

Let Ji(n,C) and:r be '" in Section 3.1 Denote h1 J the sec of t"functionilio from n Into I{ "';th !.he proper\)' thll.t lhen. exists 9 e 11{n , R) oodI thlll h - 9 bo:Ion&» to :r. Let h he .. poeitive ' ''ment of J , and lei , E Ji(n, R) be ouch thaI " - 9 bi' o'ga to.r ; lhen , ' :s 10, and , . - g :s h - g; Ihuo 10 - gO .. (h - g) - (go- _ g) ,"", In J+ , and &l8o" '" (10 _ g+)+ , '. T hitl mealOI that J+ io the IIr:I of pOOIiIive ~lemcnu of J N_ , for~very h E J. if 9 E 11(n , R ) io ouch tbat g:s h. then Io-g ];e. In J , and """"'" I;" In J~ . AIIoo, '" +ho beIonp to J for .on " " ": E J. and M E J b- allh E J and.\ E R ' . Furthe'II"" ' . if h , and 10. belong to J and 9 In Ji(n , R ) 10 dotnilW.M t.,. Inf(h " h.) , then $Up(Io, ,/0: ) -, '" sup(h, - g, It., - p) and ;"r(Io" 10, ) _ 9 lie In Thill J)r'<W<:O that J is ft lattice_ Ileno:
.r.

3.7 Upp« and!.oooer I....... .

71

Definition :1.7.1 For ~ h E.1. p " (h} .. suP"; I'I(Il.R).. s ~pll} iI ailed the upper int~ 01 h.. O~ t hat . .. t..n h I... in .1 • . p" (h.) is iust the Quant ity

r hdp ..hid>

..... _mum in Scdion 3. 1. I! h. E .1 and g E ?t (O. R ) are .udt that /I < h.. then p"(" ) ..

p" (" - g) + $0(/1 ). Hmoo p" (h , + h. ) .. $O " (h,J + $O " (h. } for all h, and h. In .1, and p OeM ) .. ).. p' (h) for e-.·e'l')" ). E R · and e-.'ft)" h E .1. Simlt.rly, p' (ouph,, ) .. IUP.~ , $O' (h A ) for every inc:noui"ll ""'I"""" ("" )A ~ I in .1.

P ropoooltion 3.1.1 A n, ( ..a~ and ~ """",/>on !hal. h E intqnlWl: U !hal. p ' (h ) < +00 . In !IIu .....,. p " (h) .. I h · 4p.

.1 hoe

p-

Suppoee thai; p" (h) < +00. For ~ ~ > 0, t here exisu /I in ?t{fI, R ). 9 S h , such that p(g);:: p" (h) -t. T h<-n p" (h - 9)" p" (h) - p(f) S t, which P IlOOF:

that h is p. i~. Mo.-..er, I lul.p S p(g) + r S p' (h) hdp S p" (h) and. In fact , I hdp .. p" (h).

prO>'OII

I

Put - .1 ... ( _ h , h e .1). Any vaI.- and iI p·lntqrab!oe.

~ti~

+ t , eo

a

(u"",ion 01 - .1 takco only finite

P ropatltlon 3.1.2 A "'MohM1 I Jmm fl ;nl<> It ,. " . inltgrubIe if and <>nIr ii, 10<' everv r > O. then: m.I ,, · inl.,.-obk "'Moho". 9 e - .1 and h e .1 IIOdI tiUII,I S , S hand I (h - 9) ' d" S c . We ""'~ /4U 9 pouiti ... VIm I u pcuitiK. P ROOF: ObriooAly. the modition iII~ . No.- .... ppoee that , ill ,,·inl~ and ot-m: that..., may""'''''' I ill ~ti"". Given r > 0, tllMe uist U E ?t ... . ,..,h t hat II - " I, d" S c/4, and v E .1. such tha\ II - ul S " and dp. S ~/2. 1'0.,.. - v :!> 1 - u :!> ", .. he""" u -":!> I S u + " and (u - v) · :!> I :!> u + v. Th<-n it .ul6c:e. to take 9" (" - tt)· and h .. " + v ( t..cause .. + v - (n - v)· :!> "lv). 0

f' ".

r

Propatltion 3. 7.3 For ettCf}' t~ ",... ti/m I Jmm fl into Ii: (rup«:tiud" inl.. lO, +00)), then: o:rirl .n i~.<9VtIIOI' (g.). >, 01 inlqnJbk ",otdiov 0' - .1 (_tivd~, 0/ ",.iIi..., /10""6",,, 01 - .1} ,,";,J 0 duo , hog ~ (h,, )A~' .., ~e ",... IIOM af.1 ndllhat g. :!> f :!> h" lor oJl n ;:: 1 "nd / .. suP.g" .. lnf. h. """"', ~. PlIOOf' : riM . • u~ thai. / .. p<8it;'.,. For every I n~ n 2' I, """" exiIt an imqnoble v. e .1 and • pooitive .... oJ -.1 ouch thai. .... S / S .... and 1(.... - .... ). 4j. S 1/ ... If ..... put 90. .. IlUp(U' , ... , .... ) and h" .. in( v, .. , . ,VA)' ..., .,., , hat the ""'I .... """'" (9A ). 1; ' and (hA )A~ ' have the dollil'fld pmpert.... l~, iii""" 9 .. IlUPA>I Jl,. :!> / . g is ".lntqrab!oe (by the lhoomm); and ' A)' d" :!> ,,~) . d,. S

""""""'" '""', .. "eo....

1Ii..o. I(J -

1(.... -

I/ n . ...., hIt.... lldo/J - 1gd.jJ - lim,,_+oo l(l- ",,)d'1' _ O. ",hie!> pnPI"eI that ! and 9 are "'IuaI &I...... """'Y"'bere- The argument;' tbe..",., for (II.M ,. In "",."a! . ..., may app/)' tho ... ta>.lilli argument to ""d 1-: B().ben: are two in<:rflOIilli ~n<:oOI (1.).2; " CJ:). ~, ofln",«.abie luno:tioDs of -.1 and tWO decrn5ll1i ltquen<:oOl { ~~ ). ~l' {II:).:>, of In<
r

-

.ox:h that

(a) 0 ~ (., (b)

~

r

~

h:, and t: S - / -

r _SUI','. _ in!

~

peLt g" -

,:,

("") .~, hIt.... the

+ g! r.nd II. S"'ed

~

Jo:;

~O;

alrnool e>'f:rywben::

(d - / - SUI't.; - inf II; Theo>

-

alrnoot e>'f:rywbere. _

II!. + II:. Clearly. tbe IIeCIUI:IlOIOI {g.). 2;'

pL"Oper1\a1.

and []

n_

o..llniUo D 3.7.2 for """ry func tion J : J[, /J'(I) - Inf""J' .... ~J/J· (h) and /J"(f) ,. - /J"( - f) .... caIJcd tbo ul'P'" ioto:gtai and tbe lower ioi"eg.l of

I ~

that /J "(f) Ie the same q\l&l>tlty '"' that in Definition 3.1 .1, _1>w J ill positi""

P ropoo l[

J 1101" far

~"",.,.

"'ndiim I

PROOr'. By ddl.nitio<>. /J ' (f) - Inl (J II 01" : II 2 J. II E .1. II in"'l. able). 10 Inf~J . ..... 'o. lIolp. S ,,' (f ). Thill ..., In&f <:Online out all_ion 10 the c:.w: ",hIwe inf.u..~Jhtlp is not +00. and let. r E R be li&tktly &Jftlcr tlwl this number. 'lbertexisu II 2 I. II iOlqrable.!N<:h that htlp < r . and. by PIO(>Ofitk>t> 3.7.3. there &leo fXIWI II' E,7, II' i~, II' 2 II . NCh lhat h' do" ~ r . Therri:ore. p"(I ) < r . and It.. propooIition bI~ []

J

J

1

Similarly. P.U) - "'I' (J gd/J : 9 S f . 9 integ. ahle

I: """"" /J, (f) S /J'U )·

P ropgo,ltlon 3, 7.5 1/ /, ..... " ""' I.., ",,,,,,-,, frtmo 11 i/oto Ii .....:It t/I<11 " + II if tk/iMt1 <11......1 ~~"' And if /J"(ftl t /J"(II) mllku oenH, ~ ,,"U, + II) ~ /J "(J,) + p"(h)· JI {t. ). ;." if An i""""""IIf "...,,«

0/"''''''''''' from 11 "'-to 'Ii nw:o\ /hal p" U. ) ,. -00 far n ~ tft01OSJO'I , u.~ /J' (OUP~,

f.) -

w~, /J'(I. ).

PROOP: 10 PfD'I"! !be Srst, rtion, it ""16",,, 14> ronsi~ tbe """" ...be .... /J "U,) < +00 and /J "(h ) < +00. If h,. h• ..,.., two in~ fun<:t,;,;,.,.s@ thlt.t h, ~ I" h. ;:: fI , Irt ,;" " it, boe t~ I8l-vaIucd funo:tiono OD 11 w it, + h. 2 " + h .oJ ......t e....ry ... here, ITO /J "(/ ' + h) S /J{h, +h,) - /J{h,) +/J(h, ). and p" (f, T hl :; /J"(/, ) +/J "{/,).

" gUllion, ~ ~ as in e.7 io intqrable &lid that II~ 2: I~ .

For the

I>"

~

T~rn

3. 1.1 . IUI'I""'inc that

a

Theorem 3 .1.1 M\eIIp' {I} " /i'u/e, !htTo: eNtI a .. iftltgrab/e ",,,,,lion /1 ,lid th4l " 2: / and /l{l,) .. /l" (I) ; 1/ h .. ~ " " ",nd .nttgrabk ",,,,,lion nell tho/. h 2: /"ttd /l(h )" /l" (1). !hen / . ~ h olrrw.t ~ A "_'N'''Y ~ttd ~ o:ond>'ticrn rh.o.t/1Io! ;nt es1dle .. tho/. /l , {I ) on4 /l' (I ) N /iniU

""....,.

P ROOF: Suppooc ,,' (I) io finite. ror ev"tf)'" 2: 1. the ... u:iltII lUI inusrable fuDCtIon I>" sud> that II.. 2: / and /l' (f} ~ ,, (I>,, ) ~ /l"{I) + I / n. o-Iy. II _ infn2:' 1>" - infn2:' inf(h , •. ... 1>,,) is intqrable &nd /l' {I) .. ,, (II ). /'0_ Let / " h be"" in tbe II..... • .1tioort. We 1"0,," that /1 _ h .01",,* """")'W1>o. ...: theft iI no ratooion in .... uminc that /, ~ h- for ev"tf)' i E (I , 2). let be • -'-,-..)ued funet~ on.o, equal to /, at poI~!,1w!re /, and h _ IInM, and . up ..... that /, :s J.. We ha"" ,,(h), or /lli. - ft) .. o . ..

i;

i, .. i. u., and /, -

h

,,(i.> ..

u. F\naIly. Let / be. function from n into R: Iud> tbat /l, (I) , /l ' (I} _finite and equal . "rIIo= u:ioI intqrable functions , . " such that 9 ~ / ~ II and " U) .. ,,(h). Then 9 " h .. / al..-t ......-.ywlw!re, 1>0.""" / io inusrable- 0

",to n:

/0""

P~IUon 3. 7.6 f/" ~nd h ~'" ",,,,,tio ... from n .-..ell 11\.01 " + h ;, th/inm .... .. nd i/ p, (It ) + p" (h ) maku -..e, tA.!n /l, (It + h) :S /l,{I,) + ,,' (h ) ~ ,, ' {I, + h )· 1//, ; n "ttd h : n arfttnJ..,.. then 1" (It + h ) - ,,(/ ,) + ,,' (h ) GtId /l, (It + h) .. 141t) + ", (h )·

n: ;,.nltyro.hk

n: ;,

P ROOF: Tto prove that " , {li + h) :S " , {I, )+"" (h ), ...., .....y IUppo:oee ,," (h ) < +0>. Gi""" lUI integrable fu"",1ou II. 2: h. let h. 1>0. • ...aI·valued fuDCtion equal to ~ Le . If 9 io an lntegrloble f""",ion ouch that , < I , + h , _ ha"" , ~ /, + e., or 9 - it, ~ / , u . II foltoor. that /l(' ) - " (,,,) ~ " . (ft). "Jbenof....... p,(f, + h ) - 1'(11.) :5 p, (f,). or " , (f, + h) :s I',{I, ) + 1'(11,). and finally p . (I, + h ) :5 " , (It ) + ,,' Ch )· Tto ",...blilb the lnequalJt)· I' "(/d + ,,' (h ) ~ ,,"Cit + h), ((jllSidel - It and - h . The Iaot .nation now follows easily. a

h. •.

_ R: IIo! neIIlhatl',U) (rupecti~J,. ,,"U») ,:, foniU. 1/, ("""",'i~IW, II) ;, .... i ..';;,£6I. ",,,,,tiM> from n iIUQ ,11(1\ th4l 9 :5 / .nd pU) .. 1',(1) (""f/«filtd). II 2: / and 1'(11 ) _ ,,' U»). theJo 1"(/ - , ) .. 0 atld I"U - 9) .. 1"(/) - 1', U ) ( rupecfuoti•• 1',111 - I) .. 0 "tid 1" (11 - I) .. I"(f) - ,,"(f»).

Prop
ut/ : n

n:

PROOF : By Pt-opo.ilion 3.1.6. 1"(1 - 9) .. I'"U ) - /lU) and /l,(I - , ) 1"(/) - " tI) .. o. Similarly, ,,' (h - f) .. ,, (II) - ,, "(I) and 1', (11 - I) p(h) - ,,- (I) - o. a

,

Proposition 3.1.8 wI / 10: "" ~

Jrmn rt

""",tiq" from fl intQ R. 4nd Id 9

1'[ • ..dI IAGj ,,' (9) .. finit.o:. Then " """".....I")' .. nII' ~ Qm4ilion IMI, "" inl...,....W. .. th4l "U) .. ,,' (g) + ,, ' (J - , ).

/Ie • ",,,.(;tWQ

int(J

PlIOOf': Suppooe that 1'(1) -p'W) + P' U - gl· Si""" / II i~rat>le. ,,' (f - gl ..

-

__ 'c gl .. ,,(II -

But ,.' (f -

,......

ji(fl + ,,' (-g) '" I' U) -

,.' (g) by hypod",sit. 10

1'. (, ).

,,'U) ..

,.. (g), and 9 Is

a

Oeflnit lon 3. 7.3 F(If aU su~ A of fl, ,,"(A) .. ,, ' (I,,) and " . (1,0) are called ,hoe outer m: " u"" and tbe inDO!T ltWUun! of A. Natt that ~ h.o...,

aln;o.d)'

dull ";tb the quantity ,,' (A ) ..

" .CA) ..

r lAd" in

!"'=yiQua 'O'Or'k.

Propooitlon 3. 7.9 F<>r A C n. " . (A ) .. eup(I'(B ): B e A. B ;"1'9' /,/" ..n4 $0' (04 ) .. in! (,,( 8) : B ::> A. 8 ;"4. 101.) P IlOO', Let r < /O. ( A) be. r...J num"" •. T heM t'Ii!l1.S an in~ funetioon f PJdI that 0 S f S I .. and r < f fdJ<- For all in~ra " ~ I. BR .. C" E 0 ./(:0) ~ I f nI ~ In~_ BT d~ domi .... """ con~n"", t'-"tm. f /I". djl (:OD>e'ga11 \.0 J 111." iii n _ +O:::>.!10 tbe"" exiWI n <= I ,uch til&! /1 IJ.djl <= r . For lb. n, B~ .. B- Since f S I,. ..... hay<> ,,(8 ) ..

p""

J

_"ion. that ",' ( A) ill finite &nd let , > ",' (A)

f 18<11' > J /184- :! r, wbicb 1"00<:6 I t.. fir$!

Tb proI'e tho IOIXIOd ......\1"", ",ppooo:; be. -.l numbt<-o ~ exbu I th.at I 2: lA and Id", ~ , The" B _ Iz € f} : I (z ) 2: 1) ;,. !~, <:O
E:r ...m

f

""*

Pro.,.,.ition 3.1. 10 F..-.-ry ,lIMa A % Ih4t II, ( A) <: +00 (rupa:ImeIy, ", -I A) <: +00), til ..... eriot, 0'1 ~w. HI A, C A , re'1'«liIId,. A. :l A) ....,.. tIIaI ,,(A,) _ I'- (A ) (,.,'p«li!1dr, ,,(A. ) _ ", ' (A )). Fqr ~ A, (re.,...:Iiw/f. A.) o/lilu ~ II, (A - A,) .. 0 ond ", ' (A - A,) '"' ",' (A) - II, (A) (req«l~", ,,_(Ao - A ) _ 0 .IId ",' {A. - A) _ ",' ( A) - ",,( A)). P ROOF, Tht! e>:i6tenoe of A" A. follooon from P mpooitioo 3.7.9, &nd the otl,er ~""'(Qi ~ opp .. ",ofP,OjXISition3.7. 7. 0

,...,

P roposition 3.7.11 Lot A, B 6e I"",

d~t .~~

o/f} 4n4C!heir 1Ini<m.

p_(A) + ", , {B)::; ",, (C):5 ",,(A ) + ,,-(B ) S ",' (C):5 ,,'(A) + ,,'(B ).

,

3.1 U_

ODd I-..- I ·'

C..

~

PItQ()F; The mlddIe inequalities folkN from P~tion 3.7.6; the tiM and lui. from Proposition 3. U. 0

S.7.12 LeI A, 8 ". _ diJjoim ,~u oJO ,110\ lllal ,,'CA) < +00, J" ( 8 ) < +00, IIn.d Id C ". Ihti. union. Thtl'l PI"O~ilion

" , (C) - 1', ( 11.) - " , (8 ) :!: ,,' (A) + 1" (8) - J"(C) . PII:OQF: Fin
/1'"' - /1'0,", +/

-

/ J I A,d;<

JI",d" - / JI AtflB,d"

..

/ JIA",,,,dp. S ,,( A.

n 80 ).

". (e) :!: ,,(A. n 80) .. ,,(A, ) + ,,( 8, ) - ,,(A, U 8,, ) :5 ,,' (11.) + ,,' {B ) - ,,' (C).

,,,,,,,raj

We """ ~ to lhe cue, &ad ~ A" 8 , be t .... intqrablo ouboeu 01. A , 8 tf!IIp
" . t(A - A,)U(B - B, II:5 I" { A - 11. ,)+,,' (8 - 8, ) - ,,' 1(11. - A,) U( 8 - B, )).

So, " . (A u B ) .. ". [(11. - A,) U (8 - 8 ,)) +,,(A,) + ,,(8 ,) "" Propooillo.. 3.7.6. and

,,' [{A - A, ) u (8 - 8 ,)) + ,,(A,) + ,,(8 ,) .. ,,' (A U 8 ). Tbudu,~,

.II. (C) - .11 ,(11. ) - 1', (8 ) :!: !.II'CA) - .II,CA))

+ !..II' (8 ) -

.11, (8 )1 + " . ( 04. ) + " . (8 ) - .II' (C).

" , (e ) - 1', ( 04. ) - " , {B) :5 ,,'(A ) + ,,'(8 ) - ". (C).

o

76

3.

In~

aDd M_llfabie Func. io""

Theo..,m 3.1.2 A ~""·,,rv 4nd ~t O>IIdltitm 1h4l4 ,!dI",t A 1>/0 lI< ",.~e to rJ\4I it opIU "," ( i.e. , ,,' C B) _ ,,"CA nB) + ,,' (..1' n B ) for all I1ibKtl B 1>/ 0 ). PROOP, 5uppooe A is I""meatluro.ble. and let B bol an a.tbitrary fUb6eI of {}. We &Irsdy know ,hal ,,-(B) ~ ,,- (An B ) + ",- ( A'n B ). If ,,-(8 ) is finite, .... lei. B, bol an intepable ... oonWninj!: B IIUCh tlul.! ,,-(B) = ,,(8,). Then ,,-( B ) .. ,,(8,) .. ",(A n B, ) + ,,( ..Ie n 8, )

2: ,,- (A n B ) + ,,' (..Ie n 8),

,,-(8 ) "'" ,,-( A n 8 ) + ,,-( A' n B }. eon," ..... ly. IUPII"'" that ","(8) '" ,," (A n 8 ) + ,,"l A' n 8 ) £Or $II integrable 90ts B. Put I = Is and 9 ~ I Ans. 5i""" "U) = ,,-(g) + ,,'U - gl, .... """ that 9 Ie Inte&;rablo! (P ropooilioo 3. 7.8). Henoe A is "·""",,,urM>Io,. 0 go

Dell,,[tion 3.1.4 A fW>C\ion I from {} inu. R is qu"';.integmble Cwith reapect t<> "I if and only If It is ",·"""",,,,,,ble and ,,"U ) .. ",(f). In thiol <:aile, .... put

"U) - ,,'U) .. ,,_(fl ·

This implies lha, is Snite.

I

ioI inte&Jehle if and only if it is ql>11$i-integrable and ,.(1)

ProjXlBltion 3.7. t 3 J.d.1. g lI< t_ 'l"blc '~n(;~ /rom fI.nto R , ~"" '"PP:>''' lloa/ ... (fl ... ,,(g) ma.tu KfU<. Tht:n 1 +9 ,. d.{irwrl <>.I"""" e...... Dkrc, .. quUt.""ttgrUk, lind I'(f + g) -1I(1l + 11(9) ·

P ROOF: Fi$, . ..""'" that "CIl is finite. T ho)n 'upp(IOifc that I t.akroI on tini.., v&I.-. Furth""

I

,,_U+g) '" ,up{ ,,(t l: t s l+g, _ IUp{JI(f + h) : h S g.

is inte«mble , and ...., """1

kinteg' able) h ~,able)

.. I'U) + ",(g),

-

and o.Imllady ,,' (f

+ gl

=

"'U) + " ' ($I), "" I + 9 if quasi-int
in thie

N.,xt, _moe that ,,_ (f) .. +00 _ " . (g). The", e:o:ist inle&n'bIt funel iorl$ /o ,!Jt I""h lhal I, S /, !Jt S g, and lhal ,,(/0) and ,,(g,) ....., atbitmrily I-. We haw. I ~ - 0<) Le. and 9 ~ - 0<) a.e. Henl:k /tmct;(m I , s l Im.!t <>~ "Ilk nllrnbtn ",'(/") 4"" I'-U-) .. {itUt~. "M 1I (f) = ,..- (/" ) - 11-(1-) _l~, kl I lI< G ftuw:ti<m Jrnm {} 'nll> R. ,...ma."....w. aM I"""""'"'u, mch /hat I.ouI """ <>1 the .. umber. ,,-(r) ",..I ,.."(1 -) it {iniu; 11= I ..

P ro»Ollltlon 3.7. 14

0'

qooui.;II~.

F()r"

c",,·

P ROOf: Fif«, SUJ)pOle thai. I ill quui-io~. If pCIl II finite, then I iII~ , p' (r ) ....J "'.(1- ) ..... finlte, ....J ", (I) .. ",(1+) _ p(l - ). [f p, (1l - +00, there erist.s an inuvabk function , < I : iii..... :s , -. ...., uoe ",' (1- ) < +00, whet, p. (r ) 2: 1'.(1) _ +00. If ",' (Il- -00, then ",' (r) Is finite, but 1' , (1-) 2: p,(-Il- - ",' (Il - +00. N_h" I be .. in tbe --.d ' w :,tion. We Pf<'V'I! that I Is quui-in~ First. suppa.: that I Is pooitioe....J bounded. If ( A.. ).~, II an 1.... 0: 11 Inc.tf
r

ru"""""

1',(1). We _ pMI to l be ~ <:Me. We .....y """"""" thaI p'(r) illinlte. 10 that / - Is intqrable. By tbe ab<we arwwnmt, II quul.~, and P".p";lion 7.13 ~ that /_ Is qllMi--in\qrablo and lbat p(ll '"'

r - /-

r

p' (r ) - 1"(1-).

[]

Corollary LeI" C II k p.~ ~nd 1J·",o,1:e Ie. 171m ,..(A. ) _ p' {A ), ~nd p' ( A.) _ ,.' (B) + ,.. (A - 8 1 br 011 ....mil 8 0/ A .

PROOf": ,.. (Al .. ",' ( A ) by PtopoeitKlo> 3.7.14. Then ,. ' (Al .. p' (8)

+ ,. , (A _ 8 J

o U A il .. p-......unbIo lei. ... hlcb .. DOl p-IlIod", .te. II iI p , (A ) -;. ,.' (A) (tee ChapW" 7, Exe"iot: 'j.

-'bIe lhat

Loom .... 3. 7.1 LeI D k a """""'" ...1 in ~ .....J toc:aIlr am"",,, 'po« F. ~nd leI op III! • """""'" II""'~ "" D (.wo. u..t OP('u+ (I - ' ), ) :S "·OP(r)+ (I - ')·op(l ) J.... r , • E D ~mI 0 :5: , :5: I) . LeI L III! 1M. cl.au 0/ r:tmti" .... u afJinr; foptcfio'ru "41 "" F AId! IMt "41/ D :S ",. II D ... d40tJ .mI op t.".,..,. otmiamIinllOu, /.VII op ,,1M. ~ en~ {"4I/ D : ... E L} . 17w: ... me......u Joo/h if D .nd op CIIntin.......

'I,

,,<tpen

P JIOOf : Lot rburbitr&1)' In D, and lei a be. real numbe,..ucb tIw.. < op(rl. Put A _ ((I , I) E D" R : I 2: op(.» if D ill &72 1 and ",!own ~jnuoue, and put A _ H",I) E DJ< R : I > op(l)-op(,)+ a) if D iIIopenandop COOItInuoue. In tM 11m. cue .4 ill .. c...cl ..,.,vex lei. in F J< R , and in tbe teCO<>d cue il Is an open """""'" au'-!. of F J< R. B, the Hahn- Ban.ocb IIdritlu, then! F J< R .c~ byperpIane ~ (r, " ) ",hlcb doeI not mM,t, DameIy H .. {(I,I): ..(1)+.1.1 _ oJ, .. t...-e .. ill. tonIi""""" Ii_form 011 F and.\, 0 ..... t..o real numben. In fan, H .. ((I,!): u(1 - II) +),(t_ Il) _ 0) ,

""".In

beca""" it coot.lns ( ~, a)_ Now .I.

t- 0, becalJlie u.,
~ OO!

he ih H . Therefore, rq>Iacing ~ by .. /(- .1.), .." ""'Y 8Upp IhM H ~ {(.,I) € F x R ; ..(z - ,) ~ 1 - 01_ Si nce u (~) - "'(,) < ..(V) - .... "''I: ha"" .. (~) - \PI%) < ..(V) - 0 foo- e>"I:ry • E D, or " (1 -~) + a < .,:>( z ), ..·bich P"""" the lernml>. 0 T~m

3.7.3 (Jel'l$en', Inequality) A ..""", lMt ,,"(r!) '"' i. Lei F l>< II4dIIJICO', D " COnlinKOU, r.\.~ f Idp I>u it> D, 1''' I is qtoIl.I" ".~, and
""n..,.,

" .....z

f

PIlO(>P: In the Iir$t (till, "'" koo>w that I"" ~ .. ih D (Propooilion 3.6.4) . In the fIOODI)nd C88C, "" P E l7'. The", eo:isU ... <:<>nUnuous linear fonn u Q " r", lOll \I € D. T hen u(/ Idl' ) - n 0 1- n )dl' > 0, $0 /d" t-- p. Thio; p"-""'l6 lhat / I"" I.,.

f

/e..

\0 D.

!.em,,,,,

N<., in On<'! ~ or lhe other, witb l he ..."", oot&I;ion as ill 3.7. I, ~(J IdI'l " f (of>o / }dp S /.1",0 I}. 0

3.8 Atoms LeI. p be ... Daruelll'J>eMIIIe 011 ... 8fMOOI' n (O, C).

Definitio n 3.8.1 A ".ln~ ... bk oet A isa".atomlrV ,,( A) > O."d j f V ,,(B ) is "'lUIII either to V 1'(,1) or to 0 foT """'Y ".in~bIe ,ubeet n of A.

Vp(A ) -11'(,1)1 for

A, by Tboorem 3.3.2. Defj"" all equivalence relalion on lhe class C of atoms as foIlcJws: A , _ A, jf and 001)' if I .. , ~ I .. , a i _ -.ywbere. Oboe,,,,, that, if A" A, lie in C, tben ";I!>er A , n A, is oql.WbIt 0' A, _ .-4.,. LeI. " bt lbe canorucal ,nappillg from C ooto the qllOtient 11*00 C/ _ _ eve<)' uom

Pro""",itioD 3.8.1 11 E ;, an inl~.d and CIi: tho cI4.u 01 1h-oJ. "tor>u u'liGh "'" c:ontoined;" E, th .... ~C-'!' ) it 01 m.»1 c:ourUabk. PROOF; For 1'\"'1')' n 2: 1, let C. be the -""" of tho8e A € C" ouch tbat V,,(A) > (1/( n + I)). VI'!£). Assume that ~c..) hall n+ I ru.tinct ~ ~Al) •. . . , ~A"I) for,."". i~ n 2: I . Thet> A, n A, is nqligihk too diMil>CI i. j , $0 V"IA, U ___ u A• ., ) _ L,<.<e+' V,, (A.;) by the indusior>exclusion formula., w~ V,, (A , U ... U A~ ~-,) > V ,,(£ ), B ront raliictioD.

3.8 'n"","

Itenoe p(c,.) has at _

.. elements, £Or

~

n
o

PT-opOaltlon 3.8.1 S~PPO'" I' i6""'. ul E ~ An inl~ ..,1 _inillf no o>I<>m. n.... {p(F ): F i~ .ndIul oJ E) U A"""'''''''' ilLkrNl in R.

e-

PltOOv: Let f:+, be ... in Propooition 3.3.1. O~ that Vp(F) .. p(F ) ( ..... poUi¥tly, Vp(F ) .. - p(F ») for all intqrable oubooeu F of 6+ ('&J*'" tively. of e-). by ~ 3.3.2. It we can J>Rl'I'" that the Proposition boIdri for £+ and £:- , tben ..... conclude that (p(F ): F imqn.bIe, F C £} .. [p(£:- ), p(6+ )J. be. it sum10 consider tbe <:Me in whlr.b peE ) > 0 and p(F ) O. .... can Sod an integrable I nbed A of E llIdt that 0 < pI A) S r. Let ,. be the suP"'munt of the peA), ... here A e> Inttsrable ,ubeets of E ,...eh that p(A ,) S ,. and p(A. ) S T, tben ,,(A , U A.) < (J, and"" ,,( A, U A,) S T. Coo ,. 4uoently, there Ie an 'iq ill:lquen<:e ( A.. ).;:, of in~ IUt-u aI E ouch that pI A,,) a>D'''rgeo to T ..... _ +00. If _ put A .. U" ,A• • then pIA) ., T. N"", tbere e> T. which a>nI..-.dicu the dmllition 01 T. He""" T > fJ/2. Now. if io an illlqrable 5Ut-t of e ouch that p(C) S (J. then p(e) S T. So pC£ - C) > {J - T. and (by tbe p •.,.;.ed!ng argument) tbHe " an in~able subed 0 aI E - e for whid> «(J - T)f2 S 1'(0) S (J - T. from the fact that _ h&voe p(e U OJ S 1. and ,,(C) + fP - 1)/2 S 1. Hence 1, which proves Ih&t 1 .. (J.

n.e..

,,>(:..

e

ot:!

of

P~ltlon


n. -

;::;'~~:':i;':"'~ 1/2"_'~::r~~, ", ~ . ,"'1""'* ;~ that

construct an

3.8.3 Ld / hi! 0

n.... f ;, ......t4n1

- 1/ 2"+' ), _

mopping frt>m otom A .

p ,~

A.e. in

~

{}

irIlQ A ~ttU-­

80

3.

I~

&r>d M ....... o.ble F\inctioDo

P ItOOF; I .. 1M limll, a.e. in A. of a o;e<jUffiOl! (/. ). ~, of 'iiIA.simple mappinp. Since It is lUI alom, eo.ch , . is OOnMant a.e. in It. and there cxistt .. rqli&ibll: lUb8et N. of A sud! t .... t I . t.e.ke!l the oonsta,u ,ruue~. on It _ N. _ ~ N be a ntFigibit ~bo!et of It OOO1a;niog ali 1M N., sud> lhat (f").~ , con'''rplIlo I on A-N. For all z € A- N, t he 8eqUffiOl! (f.(:t)).l!:' ., (""),,<:' COD>(' &a 10 I (z ), and 10 I ie tOOSUnl on It - N 0

DeHn llio n :1.8.2 p. ;.. Ii&ld 10 boo dilf_ if il has no 1IWmII. ,. atomic if <:ech inl~ fit! .hid! ie not negligible oontail'lR an

.. oaid

10 M

atnm-

T """""m :1.8.1 S~pp<)It. p. ;, "'''''''''" / .. """"r Ih4t Q ",apping from n In.to a IIldri.tMtc qeoo lie ,..-.IOrCIbk. il;, """"......, aM ~ tJI~1 i.I _ _ ~ OCI ..... nl ~ ,,"f. in fad> G~. I ·.w,. - EJ("I" I I"dV,. Ivr "" ",..a>.:>... I /rom n hit" (0.+001 • ..nere P(A) emndo ~ tJ.t H I F 01 dmu ,,/0./......· f Idp. - EI'I"Jf II,,~ lor ~ tlHnti..Jl, p.-'nl<"gnlWe m4""'ng

r

from n in.lo

•

PROO .... Ltt

I he

r

&.....:.11 "....,..

a mappi,,& from fI int(l a metri ... ble a).'llOe . ond ,uPJ>(IIIe IhM I .. _lUll a.e. !n <:ech alllm. Fix an intqrtt.b&e .... C • ."d Id (.t.)l<;/ be a fatnil)' of atoms _iDOd in E, sud> Illal each alOm contained In £ ie equioalcnt to oat ond only (lDIl Ai . n,., $et £ - U o A.. OOlllai ... "" &lOt """'1 j E I, thmo ;"" nestillible ou!.e\ N. of A. sud! Illal I II oo:JSWot oro At - N,. Now Jet; N boo a ,Itglisible suboolt. of E, oonWning E A" No, a Dd U'';)Eh/ .,,.J A, n A,. elcwly, I 19 the

U.E/

lhE,

limit 01> E - N of • ""'1_ of iiI £ -almp\c mappillf\l, which

II

~"-n.bio.

Now 18

I

be a fUDdion from fI inlll

r

l·dV,.S

"",,-eo

lhat. I

iO. +001· Then

L I'I"I~'

r

11,, ·dVp.

e""Y ..hen: lor <:ech illl~ oct £ (F " ie IM,a of ~I "'E" of u.:.. &t.om8 t hBl an: coma;nod in E ). S<:.PP'*' tllal I ill ~_un.ble and ... niobftl outAde a ~....,..urable aDd ~moden.IU,,1; E. Then I ·.w,. - EI'I"IV". II"eN" by Beppo l..,.,.i',

t-:.. . I I. S [;1'1""". I I" a1,OOIiC

f'

r

r I · dV,. - EI'I"w, r I I" . dV,.. _u"'" r I· dV,. < +00. n..:n. kor > O. then: eo:;sU , E:r """" that 9
HI!fII:e

I~

~ f

aDd

~

which P"""'" Illat

r I ' dV,.
".measurable.

r I· dVp +00. Thm! E.1ICIt I . II.

No. "U~ tlLat p.~rau,

[

tel

lP.XiIu a I'.meuurable and loo:aUy dmoet ~here.. So

<:

thac.

L

1·dVI' . j' fl. · dVI'i?

oIA}ET

We ILave thel tfole pio.eo:! that

I : rl_ 10, +001·

r I . dV

j'11"" •. .w"

..r

• L,.,. j'IIA 'dV" . I' • EoIA)€!"

I1A . dV" for e>e.,

Fin.tJly. lot I he .... _tiaUy "'[D~ n1&PP;"II; from rllnu. & ~ ope<-, and let E be a p·meaourable and p·rnodcrate ~ ouch ,hac. I " II. loo:aUy oJmoK e->E'1where. Idp .. ! II.dp .. LoIA)ET.! IIAdp .. EoIA)ET! I IAdp by the domi ...,ed con'dlE"'" 'beorem. C

Then!

3.9

Prolongations of IJ.

on a ouniri", Sinn. and " iI the ~.-c funn I ..... ! IdA Oti SI(S,C). tben we put I . C io. q ....... meaoure. and be"", If A is a

n..

10 a

I'

r

"PiS"'"

pa " " " , , , by

r

r

r

TlW!ooC'D 3.3.2. M"''''' ..... , VjJ.(A ) .. f lAdVp For all A e ii.

DfIflnltlon 3.9.1 jJ. It called tbe maln prOlooptiott of p.

r

r

Theorem 309.1 I · ping '""" rl iiilo .. mc/1"iz4hle Ipoot to p.~ if ani, if il to jJ. .........1It"CIbk.

0""

P I\OOP : [f (§,.).l!1 Ie an i...... . I'i"ll; Ieq........ In sl+{ii) and g . ruP.l!'f".

r g · dVjJ. . "Up..! f" · dVl<" ouP.!,. · dV" • r g. dV/,- ThiI.oo... ,bat. r f· 10,+00]. For tbe """'... inequality. ot.erve that every , e :J. r ,. dV < +00 IIertce is , ,,, uwoer of an I";", In t'-

ltld:llhac.

io

I'-I~~,

and

r

~Iope

ilw;>

r, ·

I'

Ieq''''''''

sl+ (ii) {P'oprow
,,,,-, r r

r r

cru.) of .1'~u.). C~u.) Is tM d06U .... of 51(ft. F) in .1'}(P); thOlS C}u.) -

C,(P). The li_!%1.t.p(linp f - f f "" and f - f f dj. from C~(,, ) inlO F. "'hid> are (OntinUOUil and "'~ OIl St(R . F), are ,n f..,t ilkntical_ a

Now, £or A E 'R:, V,,(A) "" fUP( A,) 4J I,,( A; )I, (A.loe l rMging 0W;r the elM! of finile .... rtitionl o f A inlO ~ [ndfled. ""'ry E e 'if io _ un;"n of an inl~.et and _locally ~;gIbIe.et . n.. ... ro.-.,. p : A - f IA<4< io" m '75 'reOll~. and Vj.(A ) _ flAdV" for "II A e'll Definition 3.D. 2 P lo! c· 11ed lbe -..11&1 ptOionptN>n of ".

r

Tbeorem 3.9.2 I· dV,. ... 1'1' dVl' lor ~ /undW>" I {rom n .nto jO,+CCl). C~(Ii) - 4(1') {k._ is r. ........ ~robok if aowl <'fIIr if i l "I'·-"rsbk.

- lid"

r

be a fwx:&ion from n llllo 10. +00). By tbe definition of I ·d\" il and 1'1 .
I

r

r r

r

l'

l'

r

r

l'

r

r

l·dVr. - J ' / IA·d\"Ii - J ' f'A ·dV,.. .. J ' flA -dV"

~ rl'
r

11Iio """""" that f . dV p .. f . dV". If F io a Banacb "*",, t!>en '~(ii) -l}u.) by the _ 01 Tbeorern J.9.1. n.. ",malnl", . " alono are oIw~

Pro~ltlon 3.9 .1 (. - ,., to - r., !'ROOt",

~

~ment .. that

a

;: - li. an4{o - r..

facti are immedio.le from Tl>tor-emIIl.9.1 &nd 3.9.2.

a

Of particular I n _ is the foliowin,s problem. Oi"", a rneas=e 1'4 on a ... in n , ",!>en ill it true thal,. _ fl. (resp«t;""ly. r. = r.. )1 A panial ill Ii- in P,OpOeitioool 3_9_2 and 3.9_3_

~ &I ........

Pl'OpOOIltlon 3.11.2 u! 41 III! a .oemir'ing of uunl;"U~ ".irlt.,..,blo ..u. and
•

3_9 P.. J. • •pt.... 01 p

ttl

«I A c {} Utou tr.oe A n E On E E . .. llull /oQ:Jl~ p.n~•.

,,~.

E ..

/oQ:Jlr p-~ lor

83

ewr)'

ne..lbCJ'+ ) c lbtJi); ~, f I ,dVjJ• .. f l ·dVjJ ~nd f I~ .. f l · rIp lor..u I E 4:u..). Ftn4l1" p" .. ji. II (and onl. if) mdll«allr p.~ «I

it I«allr JJ+ -~

PROOf": Since SIt •. C ) it dme. In ChljJ ). the

aJ!Umtflt .. I"'-t of P~l ion 3.3,5 and Theo ... ,n 3.3.2 &i- VJJ+ ( E) .. f 1,,·dVjJ lor all E E • . Tbm "' ·dV jJ .. lI ·dV JJ+ ror all upper "' I"eiops It of ."" rine ""'I'''''''''"' ito 51'· (. ), and eo I · dVjJ ~ I · dVjJ. lor all fuoc\iooI I from {} Uno (0, +<><>]. W" COIIdude ,,,,-, 4 (JJ+ ) c li::tJi). and ,1141. f I ·dV /J+ .. fl ' dVjJ and f IrIp . .. Idj.I for all I E 4u..)· If .. _ A io IoeaIly jJ+'''''IIi«i ble, itl If""" A n E an ~ E E . io locally ~necli&ihle, eo A it locally jJ'~i&ible by IIypOlhee.is (b). lienee lhu..) c

r

rr

Arne

r

f

lbtJi).

To linilb l be proof, "'wo- tlw NCb locally jJ' ~i«ible"",- io locally /J+' ""IIiClbIe. If I t ' ....p to 4:v.), it "locally ~a1_ e .... ,.hen the Ulnil of .. Caudty ""'I";""'" (/.)-2;' in 51(. , C ) (which may be "'Pfded eilher .... . ubil*" of 4:CJ'+ ) or of lbv.)). T heo , rir>ee I " IDeally jJ+-aIn>O!it en'1wbm! lhe limil Dr (/. ). 2:" il belc>ngs to !h(/J+ ). 0

Propoeillon 3.11.3 U I • /or ~ oem;,;"g 01 p ·ink., W. Kto "nd k l jJ+ /or tJw. -.we E I,, · dIJ "" • . 5, , _ IIi4l SI (• . C ) it..:e.... in v.). n- £bCJ'+ ) c 4:v.); ~, f IdVJJ+ " fl ' dVjJ "nd f I~ " IdIJ /t)f' 0Il1 I E £hCJJ. )· Firn>JI" [J. .. ,.. if and oml. if eocA J' . ~ K' ..

f

£)"

JJ+-Mgl;gibk. PROOF: A""", III ill

P~tion

o

3.9.2.

The next. rerult . ttIOti ....t«l by Propo;::oait;on 3_9.3, deaIoo ... ith "",,,iI'inp 4o fur .,hicb St(. , C) io de_ in £h(jJ).

u,

_t.

,W.

PropGIIltlon 3.11.4 5 , • W 1000 ,tlJI' D/ jJ · ;nlL!§1 Kto, and lei • W W " . 1"ifIg ~ .,. ... 5_K III
PROOF: lei. V be tbe cloture of 51(. , C) ito £i,CJJ). Given A E • • defuIe T A 10 be tbe cia. at the 8 c A ... t.<.e indicator function I;"" in V . For _ry 8 ETA and for..-y € > 0, then eristI I E St +{. ) ouch 11141. !l" - I I· dVjJ 'S', and tht .. be ouch that 11.. _8 -( I .. - J) )· dV,. :5 t; l hilo ~ tlw 04 - 8

r

r

lies in TA _Similarly. if B" lJ., tore elw>ents o f T... lben: ""isl J.. h e Sf~ (.) $t>dI tlw li D, - /.1. dV" ~ t !'liQr \M!ry 1 :.::: i .,: 1, and then

r

r

11OIP( 18, . I.., ) - OUP(/L ./.11· dV p

:s: t .

wbich ~ 1M! B, U B, licoI in T... Finally. Jet ( B. ). ~I be an ;,..:>cni'l\! 1ItqUC>QII in T A, and. put B .. U.~18"; the...eq..eno:e (I";').~ I ron'~rpll.O Is in 4,{p), "" Is beionglI to V . N"", TA job. ".aIg<:bra in A.. ThiIo ........ thal. t hl! c'" of tboeo:.oo:U E c for which 16nA Ii ... in V for ~ A. E '" iI .. ".fine """'·'nil\! • . Now let £ e S and B E i .. in .be III.&l.eII>mt or t he proposition . Tbo:no io a ~ (04.1. 1: ' of d"'joint ~ "",h thlu 8 c u.~ , A. • . The funttioor>s :c ,~.:s" IA. " B I~ in V and. they Wi""", poinurige to 18 .. n _ +00. 1'1>0 _ _ _ ,g........ I~ then stto..'& IIw I s. &I>d benc:e I • • lies In V. FiDa.lty, t hat SI{S, C) c V and. V - 4 {P). D

n

obw;,..,

EurciIu for Chapter 9 I

<:00.""

If /I.<¥ 1M ........

2

""'I''''''''.

",r<: ""

1Al" boo .. pooi'ti ... 0an0ell .,. ' P 11(0 , C ). I.-:t ".1. ;,> he .. of i~tq:ro.bIo f~I>(t""" from rt In\o 10. +eeJ. ond lot I be .... iDlql'abIo fuooction m;- I'l _10. +j O
f /rI.,. .. n -

+DO, ___ that (/' ).~ I .....~ to

J iIt

~"he

.. pooi.i ... o..r.l.1I _""'.,.. 11(0 , C ). F .. reo! Bana.:h opoce. and .. ""'t_!~ Cj.y.) '--sine .1"...• ......,.,.her-o to / II In .c~ u.). S~_ .... ~ N,un )""'_ ~ N,(f) .. n _ +<:cI. Then _ ...... (In)"'" ....."" .. ~ I ,........ - . tot " ",.hM 11/.1-1/. - I II Sil l). (f.)..~,

3

Lot .. boo .. poei.i", [)o.o.~

I 5Fl' ''' on HC!}. C). Lot ( ... ).~,. (~).~l' (I. ).~, boo.Juoeo _ iD £'IJ.;ft). 5 ......... . ......'-'t "" fwlctioM ... b, _1"'pu:U~. _ """" t hM G. S I. S , aI_« .. o.l>at " . & ..." intogr&blo &ad tIW 4 . , "" to ...w &ad _p
.. n -+oo.

I.

i"

<10<1>"',' f ... J

r

.... ~ f . ldp. '"' 1$ C_ F...... '. Iomm.o). 0.,,11..,. .hM I .. In. ... ~ &ad ..... f 1.4" ...... to J Idp. .. n - +<><>. ~

l . P ....... IIIa.t. it lo. },.z, .... "'.. ,..,. to a \a tbel ~

4

J

,_bon>, s.._ f w".

I

ill . ... _

1_ E- c"'l).

_n o

tho .. (f.). ~ • ..,., ... ,..,.

Lot" ..... poeiU,. ""_jell m ' w .... _ H (n . C). Let (""). ~ I boo .. ,,",,""""'" or MI' I .. of n. Su_ thoro 10 .. _""""" ( B.. ).~, of diojoi .... """" tbot A. C B.. lor ~ n?:}. Prtwoo .hat ". ( U.~ l A. ) '"' L.~, ,,· (A. )

,.·_IlBbIo ....

&ad .1oM ". (Un!,

.t,.) '"' E.~, ".C..... ,·

F

.1("'. lor Ch'p''' 3

M

a,

5 I.-" 1>0; . . . . . fl... · ~ "' ''l oa "H(fl, C) ""'" lot 1; bo, ... lntqr.blo"",," SIoow lMI (P( 8 ) : B ....... &bIo. B e E ) io """'poo:t. ( _ ~"'" U .2).

t.;

'V'

ate

4 Lebesgue Measure on R

A. ,bIo pOi ... iI

--

uampkot.

o.:.a. ....

b,o,,, <1000 in ' "

..tuI to IDol! .. what..., will .., ",,,, ' ,,, ... .-it b ......,.

n. • • • obott,.,lo. ol'

4, 1

_.n!

5 1",,1>" ooioo. of. r<JoJ

tiatr<

' ""pIoo b

in

n....-.n-...

.....t \bf'O\lS ........110 I
of_ d.. ""'"

p'

,"

~ """"

.. III

4_'

bul III ~ .. 0 .,...... . "eo, .bon! (b

~,

. be ..... ' .... of "PMbcioslul" ...... I. 1*rtkW..-. ";"6

4.3 We _

p" fun<.'tioon _ _ \100 ..;.".... 01 • .-.. 8
, "H p'in&.

.. u",).

c.mo.. ......

Base-b Expansions of a Real Number

Ltt 6 ;> 1 bill." '''~~. and:z. . 1'1\&1 Dumber. For --*' In~ n 2: O. ~ p.. be !be pw.UWIllnlqrn" lUot !Ip,.. 1 . 6' · oS %. and "" T~_, r~. for '"'~ n ~ I, tbf: ""I"""" (T.},,~ lnaeas e' l(I z . FOr ...a, inl~ n ~ 1, put " . .. p., - bp". , . Sil>Qe !Ip.. , - 6- ' :!S :t .: 6(p" • • + 1)' 6, n, ..., ha,,, 0 "a 6 - 1. N_ T • .. PI> + '5' S_ ... hence Z .. l1li + L' ;!.I ... /b".

s:

.s

E

'V.

.s

.s

DdInitlon " , 1.1 With III)tatioOD. all -'vM. (po. (10. )'~ 1 1 "' called lbe propo:. ",po."';';" of % (wltb l&\ptd to the bMo 6).

•

( .1 HI ,: bE.""","""" 01 .. Real N"",,*

87

to, I, 2, ... ,~ -

I}. By ..,hat ~" h..,,, just shown, the function ..,: (9o.{ .... )~<:I) .... 90 + E.. , .... /IT. oends Z )( AN onto R. We wiU pM" that, if z e R is not of the ~ "/11' (with k E Z. " :> 0). It lllhe ima&e lII>doer .., of iUl .ole ptOpO:'t uPfol'Sioa. On the other hand. if .z 11M t he form k/ II' , we will tbow lhar. it ill lbe ima&e under.., of 1M) (kme~ ~y, ita plover ""p"noinn.oo aDOtMr exp" .... ion which we call improper. So let (90, ("~)"i!: ,) be such lhal z - 90 + L"i!:' ,-.. /11' .00. lor evuy iolCler Ir\ 2: 0, pul , _ .. till + L I<":5," " .. /1>". CleAtly, ,_ ~ .z ~ ,_ + 1/ ""' , .00 .z + I /~- un_ .... .. 1- I for any" > m. We ooociude lhat r ... '" ' .. 0< r ........ + 1/ 6- . lbe ,""uer equality holding ooly w!oeo " .. .. 6 - I for all i~ " > m. If the oumMr of in~1S n ::!: I .ud> IhaI. tI.. < b - I ill fmite, e.nd if Ir\ io t!oe .mallesl pOsiti ... integer such that ...... b - I for rI > m . t hen :r .. 911 + L ':5 " :5" .... /11' + I /b'" h.oo the form Itl b- (k E Z ). ThUl, if:r ill DOl. 01 1M form k/ ..... (till. ( .... ) .. i!: ,) is .... orily 1M 1"""'" ex.,.nsioo of :r. N_ IUppOot thal z h.oo the form k/II'. TIw: ... an: lwo P<*ibilila: P ut A ""

<._

I. " .. < b - I for infinitely """'lJ' indM::ei n. Then (911. [ .... ).. i!: ,) It tbe proper ""panoion of z .

2. " .. < 6 - I for only finitely many ioojcn rI. Tben ld m be ,he omall..11 ~iti ... intq<:r such WI tI.. .. b - I for rI > m, "" IhaI. " h.oo 1M form Itl b- (with k e Z). When m ::!: I. k io not divisibk by b. ThUll m It 1M omalleot of liIoR inlfFB m' ::!: 0 .ud> that :r has tM form t' / 6... • (t' E Z ). I..:t (p.o. (.... )..i!: ,) he the proper ""p" ...... of :r. If m _ 0, then Po ... 'III + I and ... ... 0 lot rI 2: I. On the <>CMr b&Dd, If m 2: I, tlw:a Po - till . .... _ .... lor l :in <m ..... _ tI.. + I. e.nd .... _ O lor n>m. In abort • ..,hen" hail the form It/ b" , it ill lhe I",. under.., of at most 1M) .",.",,",.00 In fOCI of eunly 1M) and OUt claim ill true. A real number.z h.oo a ~rminating ""panoion" (ruch that .... _ 0 lor rI Jarp CDOU&b) If and ooIy if II has lhe form It/ .... (n 2: O. It E Z ), and in that <:Me iUl t6minating e:xpansioo. ill lu plopo:'l upt.nlioa. Nen . «I", AN the lcrimgl"aphic ordering <:' < (",,) ,,<: , if ..", < "_ !or the omaIleot . m. 01 tiloR in~ .. n.ud> that .... i> tI.. ). T ben tM functioa G ; (.... ).. ~ , _ L~ i!: ' ",,/1>" from AN onto (0, I I iou . . f'urtba ......... for " . W IE ,,"" _Wylng" < " , lhe equalily G(,,) .. G (u) boIds if uod 0IlIy if II. " J«, ".pl>Cti~ly. the ",Opel tot.,.".;oo u>d the iJDptOpO:'t e:x.,.".;on of ........ .z eJO, Ifn B , where B .. {It/ II' ,n::!: I. "E Z}.

.let".",,".

«.... )..

Deflnlt ion 4. 1.2 When ~ ... 2 (,... pecti~y. b _ 3. ~ ... 10). the proper ~ 6 e:xpanSion of" iI called iu proper dyadic (I EE Ji
"'w

88

4.

~

M.aour. _ R

4.2 The Cantor Singular FUnction For aztr nooemp(y .ubintenall of R.

~ shall

denoc.e by .:0 (/ ) its left endpoint

aM by !l(1) i l 5 . endpOi."r,.. De/l1Ie, by inducl.... "" Il, • fam.ily {J...)... of mutually disjoim, ope" Lntel d •. as btlows. The ln~ " u.keo all poeitive ,...tuce; foe o:ach " 2: O. P taJu:. thf: .-at_ 1, 2, 3, ... All Int~ 1 are <:OQWned 10 10, II, and ..., laloe 10.0 _ JI / 3. 2/ 3]. Nat, . t.ppoiIC Ihf: r +' - 1 i~ I...... have been ~ forO S m S n, Ib 8UCh .....y thal. if J~ Ie tbeir uni<m. then 1<1, IlnJ: is thf: ......... ofr'" d J I,,~ K .... (';tb 1 :;; p $ r+ '), mutua.tlyd"'}olnt , all of lengtb 1/3'+'. and suet. lhal a (K •. ,) <: a(K• .• l <: .. . <: a (K. )"",o'). If K . .. _ la,bJ, then "" tab for 1.+, ... tIM: <>pen Interval. J.. + (1) - .. )/3, ~ - (1) -

,r ,

ft ...

.. )/31· OdInitlon "",2. 1 The complemenl K in I _10, IJ of 11M: unioa w tlle 1"-1 io c.lled the Oontor 110\. T hf: 1. ............ 11ed lbe iD«1"Yal!l Ulntilluoos to K .

Ltot ~ be ~ _un, "" I . S;""" ~I I - J.) _ (2/3)''' ' for eve7)" n 2: I , tbe Cantor set K ,. n..~{l l . ) III poneg!.igll*, bec.r:f: tottlty di'loxl~. If n. P ..... 1_ lntq<: ... JUc:b that n 2: 0 and I $ p $ '2"~'. II",,, '2"+' - P caro be "';11"" )',2' + ... + ....... ,2", ... be", A" ... , ....... , ..... uoiquely detennluM elemettU of {O,I). By Induct ion oa n , It is <:tI&ily shown lha,

-

(a) a(K"-I1 '" E ,s;. s .... Z( l - .l.t )/3· , (bl IfUl ~ Pi _ 21+1 - (A," + . . . +.IJ+'2") for ..U 0 S j $ n - I, lhea K .... c K • • ,. "

.1

C ·,· C K~.

Oo:note by • ,be function (z.).~, .... L .
ei..... In'

if 5 n 2: 0 atId 1 $ P $ '2""'. let ("" )'2: ' "" tbe proper lriadic: e>-...... inn ollt(K ...I. N"" let ("")' ~' he a pOint of B atId :r ito image under ~. In o.-der that", IN: In K .... II Io"i« pry and JU/Iiclf:n' lhat """ "r lho: IolIowiu& IIImo DDOd.iIIooo bold:

(al :r. _""

1or~ 1

$ /0 $ ,,+1 .

(b) Q( K~ ... ) '" 0 and ("" )..?I is i;s improper lriadic~.

(e) i1(K ",,) '" I aDd (:r. )o ~, is

il$ proper tri&dio: ""p&It$<m.

In the pArtiot:ula:r calle ,.~ (: to C is strictly it..., " Iill(l aDd ~C) is a>nlaiuM in K. C..,,,"( ,acly, let z E K . For ~ n 2: 0, ",rite.:o A lor t he ori&;in or the K .... (:()Otal~ill(l:l'. TWo .:oA'" - a"

.s

has tbe form t: •• 2/30+ 2 for alUiuble l:~~2 in {O. l }. whma! ... deduce that z liM in ~(C). Therefor., K '" V(C) has t he ~&rdin.oJity of t be cooUnllum. Now dmnte t.,o k \be tel. of \be P( KA . . ). If t: lies in k , thero iU impr~ tn.dic expanelon be' 00p 10 C. On tbe d if there eoiltllan intqer n ?: 0.1lCb that l: . '" 2 for all I: > n + 2, thero ¢( l:. ). ) '" P( K... ) fur a.uitable 1 S p S 2·+1; th..., it:. ila pnint of

K - k , thero

ill proper t n.dic ~~po.osion bekmp \ 0 C. F in.oJly, a nee

' '1

and IUfficlenl. condition that Z E K be tbe i""'«f' ~ ~ of two diotit><:t ~t..m.ntI of B iI that It be an ~Ddpoint of !lOme inl.erval COOti&uouo to K .

I "" 1M ,""",ion /rom K in.l .. [0. 1] nch !hoI I( t>((z. ). ») '" E.~, z, /2' +' I'" t>4dI (z, ), 01 c. ~ I ',LI" f!mt.tion

Propoo
leI

Jr- I

.n.te> I llllaico\ meMo I ond II n 2: 0, I ::; p::;2" ). g

"""'t,,"' ......... mch ,,.If:rwIf... (1IIWt

I

ilour.iec'ive and ;oc, ·i"3. The unlq ... poi .. (z . w) in K x K I ud> that z < 1/ and I (l: ) '" 1(, ) ate t be poin (o(l ..... ),P{I....J), wi. "i(( . . . deduee the e:ristence of g. CI

PlIOOf': ClearlY.

Definition 4.2.2 gil ca!1o:I the Cantor singular function.

om...

NOlI' Id (:r, ). be ... tle"",nl of 8 and :r '" "'((z. ). ). m .. tbe luprallum of u...e integen k 2: 0 such that l: , ..... z, ..", all even (m may be "'IuallO +00). It z lieo in 1 - K . and if ... put q '" E ,s:>s .. (", . / 2). 2"'- > + I , thero z belonp 10 I.... and g(l:) - / (11( /", .. whma! ... _ that

».

ThillMt equality il t tut even ... ben z; liel in K . He""" ... ha,'e obcained , be eoplicit ex pc . on of g(l:) for alll: E 1. Ob8crvo! that 9 iI equal to (2p-I)/2"'" 011 NCb f ..... Pro_Ilion 4.2.2 F", ~ n 2: O. plOl K. '" U,s.s,. *' K . .... .. nd IIIriU I K. "" tJw: indio>,.,.. 0/ K • .... ,. Atoo •• o' io. '" (3/2)'+ ' . I K.. tiM let '" be 1M "'n<:lion l: h,, (t )dt .. h,,1 1O..jds.< on /. ~ 1M "'nepq,.., g• .....P<1l'" ,mifannlr tI> II>~ C~nlM.o"f'l/
/t",,,"""

f:

I

PROOF: If", II; a pOint oIoome I..., (m > 0, 1 ::; p S 2"' ), thero g", ... (%) _ (2p - 1)/2.. .. ' lor all iOIelen I: ?: o.... {g..... - ", )("'l '" 0 for NCb int""", n ?: m. On tbe other haneI. if % iI a pOint 01 ....... K .... (n 2: O. 1 S P S 2"* '), thtrt "'(o(K.... S "'("') S ",(II( K .... whm<e ... dod..... IIw. ", (l:) &z>d, simiLarly. ", .. ,(z ) lie in rIP - 1)/2""". p/2"*'I. So I",.. , - ",1S 1/ 2"+' for all ;DIqe.. n ?: O.

ll

»,

"

.

CleArly, tbe limit of tbe ,.. ill ~ on ead> conootWrl component of

/ - K. N<>w let In ). be an e\o:menl. of C ~nd Z '" \!t( z. ). ). For any ill~ II
o

"'' 'l1lOIit

In slK:rt, , is com.io ....... Inc...... nl':, and has cieri_I", zero elIerywbere, &11<\ ytt ioi tbe. unibrm limit or lbe .b6oIuldy awlnUOUll ful>Cliono g._ By P"""",ition 12. U , 9 h-. deri .... li'" +00 &I. UOOClUlltably """'Y points

on

4.3

Example of a Nonmeasurable Set

For &oy ... boott E ol R and any mol numl:>er ... .." define E + .. as t be IOtt £ EI . Man "'......Ily, lor AI»'''''-I& E, For R , "'" let E + F be tbe....c ( z + __ ; Z £ E, r £ Fl. Fi....uy, we " 'riU, .I. to.: U:bo1Iiguo _ure on

('" + .. ,"

"P rovo-Ilion 4.S.1 lJ E ;., .. '\'·in~ . 0, IMn D(E ) . (z - ~: z ,1'£ E) u .. ~ ¢fO . P RODP"; For AI»' mol numl:>e< 0 <: .. <: 1, lbere iIIllIl ~ a.t U COI1I~ni~ E , suclt thGt .. - .I. ' (V) !: .I.(E} Ieee Theorem 6,1.1) . Then .I.(E n J) 2: .. ' .I.(J ) for &I. ~ _ cuonooo:tcd Q)a.p<>t>
1-

.l.( J)/2 , ),(J )/2(.

[J

PropoIIltlon 4.3 ,2 ~re to • ( """",_"",...) ..-..6Kl M of R ,tfch tII.1 .\,. (M n E) _ 0
./ "P ROOP: LeI. 9 he an i ...tioM! ~r, aOO let. 8 (rtllpo:octl""ly. C) be tbe...c of numbers ollM form II + 1118. wbett m € Z &lid n iIIllIl """" {.... pectiYl!l)i, odd} iIIt.,-, For eod> inl~; "ffI inlqff II; E Z such I,," 0 +;e <: 2; pul z; .. II; + ;e. N<>w, If Jill .. bo:>unded OpeD interval, lei. t € N he such that ),(J) 2: 2/ " : ~ lhe nultll;>en :r. , . .. ,Zt, theN: are &I. \e&et &11<\ :t) . Iud> Ihat "', - z,1is strictly Ieso IhaD 2/ t , and

:s ...

t,...,...,:t,

tben there exisu r E Z for ...·bicb r(I, - I, ) helon&o to J . Thil provts that B ill dtnoe in R. CleM!y, C .. B + I 11 also
-I.n<'gi~

I(t ..Aid. .. ""I a BImIII(/.

PJU)()F: LeI. ; be tbe func:tion lhat agreeo with the Cantor IiD&ular fUDCtion gon t .. (0.1) and that ilequal toOon ]- oo,O) and to I on !I,+oo[. The (Ooott function Il : :r .... (:r + j(:r})/2 ;. III.rictly i...... -inc on R; beDOe it ii_ ""'->"",philm, ... ith in....... Il-'. Sinoo -I(II{/" .. )) .. ~(I" .. )/2 lor all itLI."Fft" > 0 and I :S p:S 2". ~ _ that -1 (11(/- K)) .. 1/ 2. TI.e. dOLt, -I(II(K J) " 1/ 2, e-.~" thou&h the C..,tor eet K i1-1.nql~bIe. WIth notation loS in Propoeition 4.3.2. E .. h -' (AI n h{ K ») ;. -I."""i«\bIe, ,-""" it iI included In K , but ill ima«e M n h(K ) uno:Ier Ills not ~ measurable. Finally, E ;. not. ~I oet, '-"\lie h{E) ill not _ BlIreIIOt. C

Propoe.lt lon 4.3.4 Ld AI Ioe '" in I'topolili<>n , .j.t an
p( (M n Ed U (M< n Eo») ~ ~~"(E,) + ~ ~"(Eo) ~U,

ufinu a " ....u.m.e ,...d ..... p "" T ""'" Ih4I P(E) .. -I "(£) Ivr

tdlEe M . P ROOP : Si""" -I ' (M n E, ) .. -I' (Ed and -I' (.W n E,) .. Eo In M. the pr-,tion iI oIwiouf. Thull

.i. ;. not •

~'

(E,J lor all E" C

maximal extellllioo of ~.

1 Lot II be lhe functioD:r .... :r - ~ + 1/21 .... R, wbo<e (:r + 1/210 fu<>aion % .... ,(..,,) .... . d_ltuuly at -t. (2k + I )/(Zn) (witb k E Z ).

'VI

ale

4.

Lobftl!!"" M._"-", ...

R

SM..bat f : "6 _ E~, g(u)/n' .. ~1aI«l &10<1 that it .. n' .. ~icl>" _ 01 tho lot", (U + II/( ).» · 2. r.... ~ boo • p<>nI. oI.ho form (ll' + I )/(~' I. ADd 21 • in ........ 'e<mo: o ~ (Zk + I )/(~) (";th n
I.

0:..,

:::'!!:::l-.!}';1 ;}E;;;·}/(~..:!:.:~~n' ~ ~·/(8n·)· c...cl..da

l

lhat

1

i""""".

Le\ ~ > I b<."

I. !.ott % b<. """'" lope! t , ' b."pt.nOioo>. Show ,tat " .. . a!\oDaI if _ IH1iy if t lltte uiI. ;·"V" ...
~

bMW

p<,;..die.

3 p,..,.., , bu tho: C· ..... oi .....1aI "'''''''''' 9 iDd"""" • bijoction f""" K n (R - Q ) onI
(Q. l]n(R - Q )

(_~ ...

s_ ~ D(X") .. (,, -

I.

2).

• • ~ E K • • ~ K)

cooot.oJ .. 10. 11, _

' - ,tat

D(K)_ 1_1 . 1!. Oom-",.ho «>11<, ......

2.

5

II> """ 1 ""-th P , opoohloft 4 .3. 1.

01 Propooi.1ooo c.3.2. if G" .•. . G. are poi .... 01 A • • _ .... (U, ~ .""(L +0,) .. 0. 1 _ . An D(U' ~I ,,"(L .,. ... » 0>1>1 ..... 110 ",hoe< t.. ... - 0, (I :S i. j :S 1'). In

~ ~ioII

pol,,,. ....... . I.

Lot { ... ).~ , be an ~""""ion of.ho poiDto in A. I'U - " i ...... p ~ D. I"'t F. " U.~.(L+o. l. Silo-(F')'~ l <10<_ to .humpty_ bu.

,ha.

1I,a! l' (E n F.) .. ...·( E) 10< ""'fY .... _u,abIo _

""u1. in

E _ _'l' P ~ I.

.boo,"""

:I..

C-~

I.

Lot n be • _ _ and " • _ _ .... oemin"" in O. Ale». "" £ be • ".int.vo-blo "'" fuoct""" f...... n iOl<> R ........ hot fl.

,100

put

I wi,h Beppo Levi ',

_!.

10 ,..m.~ Fot '"""Y • > D. """'" tto "',..".,. 01 6 > 0 .""" 'h.aI., 10< o.lI _ohio """itlono (J' )' H of H. iaIt6 s-u oIltn&th ... than 6• I I~ c•. ,,(E n So ' . itIdoep< ..d"". ol It..

.... ...... I{

. ..... ol CO ill

)1

Jo.

w_" LobeoJ.... 01 Ihill .....11. 10

E •••

_u~ ...

r '{ JoJ)1

." in •.....,.J.

Ii ...... ,""",","
5 V Spaces

III .. '........ (J lf r.I!<)"·

A ~ f ....... 1 .. _ 10 be 1m (;'(,0) if tI>o p'" POL " 0( ab60. U hao IIw! .. ruct~ of • _ .poooe....d 1 _ N, (J) _ io . oemi-.., ..... " ~ I . V , whld> io I~ """'..... oJ. £' ..uh , _, 10 tho ...... ioo f _ ~

io. 8 8""", _ U> ~ I ). In , hIo ~ .....m ulef>d ...... of t be .....110 01 Soctio<> '-2 ( ..... ,L . . _ ) to.bceo _ T'be FiodIoe<- Rlon d .... ea I_ r "te_ of ~ I ) mq ""'(""-Io,ed. olio< ,ho .....Inated ........ ," • •Iow< I . . . .boo ..... oj ~Ir>d.""" n . . ' . hoof . ... 01 55: P'II'" ,"""'" In that 1\ oJ.,." _ I<> _ ..n tbr. tooIo of Be" 5 :~ _ _ tlw:<><J. jr&.& ,

,.t... ".

V.,.,

U

!D''''' _10m ~ """"" .....oJ ~,nd·""",tall"""l...JI\"" Fb< "''''mpioo. if,., *'"

I! , _ , / . t ,'",. to (0,+00» . ..... if I . 9 1_ funo\ ..... " " f} _ lhal N.{f ) aDd N,CtI) &aito., .bon N,{fg) :5 N.{I)N.(f) (P, ........... t.Ioo> &. 1.2). 'l"bo<>rom 1.1 ... ......· oli ..... ioo 01. Minlo:Nol<,', I--.litr. if I and ..... e- fuDc:t.iom oo tho duoJMy 01 U aod L' wbeA ,..., 9 .... ~ lhal it, .. ben 1/1' + I/ f _ I. If F io . &naeIo _ . azod" _ , .... <:<>fI,jupte, . ' - . 10< .......,. , EO L:"'v.). 1- {lrJl' io • <»nt;""""" ~noar ,o...""'·! on L~(I') ...nb ""'"" N.!, ) {1looonm ~l..5 . n.. .......... w will t.. doNIt .-itlt ill ' " ' - ' " IO• with 1/ ,

fT _

+-

*'"

ot ."" .....,.....

s..:uo.

U

n.. I>(Ilioa of " " " - " _ iD M "

~

s..

.boot..,..

9 "" io IDll"OdlOChl. Thio Ml
01 UDib"m . _

Tltio Ml,...y t..

_i,,,",, deo.Io ..n it ~DlfonaIy ''''"FabIo_.

,

5.1 Definition of V Spaces Lemma 5.1.1 I.el 0 , {J E)O, I( !(Ita/I. TIooo a~~ =:: 0<1 mol",.,...;.,... a , ~, "'7""Iilr hD/4ill9 if Gild omI~ if"" 6.

+ fib for

all PO""""

PllOOl': The inequality is oI:wiouo for G = 0 or 6 = O. "fherefore ,!RIppo101! thal. tbe funcliGn r _ ~ (rom /0, +«>( inl<) R. Then:ai$ta 0 <: G <: 6, and e E ,,",10( su
""""ide.

{JG"-'.

In ~hat foIlo.nos. lei. fl hoe • ""no:mpty 6d. and f-l _pace "H (fl, C ), which ",m initially be positi ....

~

Oaniell _ un: on B

PropoeJtlon 5. 1.1 I.et - 0 and 6 >- 0 noel> tN.J. af .. bg WoUmo ol _rv=.

'0 r

r

(r

r

r

r

A .. fdp. >- 0 ... d B '" gd" >- O. From the 1""'IUNlly (f I A)" . ul B'I =:: (of)/ A + (h)1 B , It folkrn that P ROOP: We m&y JUJ)p(l!Ie lhat

'f· r l' d" =:: ~ f· ' f·

AO .B'

fdp.+B

yd;J._o+ fJ _l ,

wben<:o: the de!ired Inequality. When, ..odition&lly. / and g an: ,,-me&IIurablc, 1M «iua!It:y boIds if and only If

that is. if and only if

(!),, ·(:f - <>} + ~ oJ_~ry"lm. 01",

if and only if

/ 104 .. 1/ B .IO'!(JSI <mlrywbere

o

functlo ... I from fl into [0, + ooJ, .... put N, (I ) '" (1" I'd,, )" ' . For """'Y function / from fl into R, we ~ br Moo(fj lbe $mal\ollt of thoeol e E" J( such that I (r ) =:: c locally WoJIDOIIl; ~ben!, &ad .... put ",-(f ) .. - Moo (- I). F inaJly. '"" write N...(f ) for the smailf"ll. of tl>oole e E (0. + ooJ such that 1/ (" )1 =:: c loct.lly " ....Jrnoot eve.,...hcn:. 111"" """,(f ) =:: M.. (f ) and N,..(f ) _ M .. (l f l) when"" O.

For &II n:&I numl>en! p >-

(I and oJI

P ropoo!ltlon 5.1.2 If p , q, r ""' ill 10. +coJ, ...m IIuu 11r ~ III' + l/ q, a"d il / , I ""' 1_ ",..me", fr=o fl into jO,+oo( . 1IdI !hat N,( fl end N.(g) e,,"

finite, Ihtll N, (f,l) s: N~(f)N, (j). A n«:tua~ An 0, ~ > 0 -..dI IhiIt .. · f P " 60· t' p-"Imoot , " ~

... .

P ROOP: Fim , 5IIppooe that P &nd q are finitE. T hen

/r '

'11' ill<

s: (/' Pr4<)'" . (/' ,.dl<)'" s:

by Proposition 5. 1.1. Thuo N, (f9) N~(f)N, (9). Clw-ty. tho: latler inequalIty runa;". trut .. hen p" +00 CO" q _ +00. 0

s:

In particular . ;':,(f,l) N,(f)N, (g) ..hen P. q E 11.+001 ~ oonjupte (I.e. , INCh that IJp + IJq _ I ). Thio ill HOlder, i""'lua!ity.

n_

P .... paotJt lon 5. 1.3 1.<:111 ' (0. +001100 I<-~' " n
.....,(1) .

J s: J godl<

Igdv.

< M",, (f)'

f ' 11 _ R

J

grip

m .. (f)g{z ) s: I (z )g{z ) s: M.. (f)9(Z) . t...... e>'eT)'wl>e.., (when

Pl-opoooition ~

5. 1.~

Tho

$d f

1< '" 0), the ""ull is obviouI .

o

I be .. ,",,,d;,,,,, from 11 irlta (0. +oo( ""'io\ ;, ..... p _ 01 tilo,," p € jO,+oo] lor ""'kA N~(f) if /i"iu. u either

1.<:1

_jOt, or".. ~ o/li., anJ. ""'til J .. U'lR ;, ~pf), I I p '''' qN~(f)

;, "".\lOU "'" ( I / p : p € J). N_ """"" IIw I u p. ~; th<-n !he fufLf;ti0"4 p "" N,(I ) u .-Im....... on lAc dwuro 1 011 ....tlllUpOCl tD ]0, +001. aNI it ;, i ...... eo li .. ~ in Ihc inleTior r 01 J ,.Jw. r ;, ..........pf)_

"'u.'" di/!.,

P ROOP, Lee. ~, , be \_ diMi",,! d ement! or J &nd IPt. 0 <: 1 < I be .. ~ nllIllber. Put I / p .. 1/ . + (1 - 1)/" "J"ben '" _ Ip/ r &nd tJ .. (1 - 1),/. wto.I I. By Propo:.ilion 5.1.1, f'~dp /'CI<)" . /' dl< )II, l hat ....

rr .

s: (r

(r

N,(I) < (N,(I ))" (N,(Il) '_', &nd k>g N~(f) S t· k>g N. (f)+ (l-I)-k>g N, (fl· Tbe. eloce. J is eitber empty or an !nr.......J of R. &nd the functioo I/p ....

los: (N.{t))

is con,""x on (I/ p : p € JI . Nut , 'U(>p(lllt that J ", oonempty and fix . € J. From / , _ I'/~ ' ill_ ....... N,.(1l (N.(m"' · N",,(flr..-· I/. for &11 -.I nwnbas p > '. &lid

s:

Ilmsup_ _ ,V, !!) S N""It). Henof!. If +00 1:>1:1<>"9

I.(>

1, J oont.i"" ",bi·

Ifarily ~ nwolle .... which prow!II tl.a.t I is an inte.voJ of 1'(. N(>W .U~ that J is not" pOint , and Let ~ be ill orI&In and , ito endpoint (r <: .:S; +00). At 1/1' - lor N~(J) is ~ on (I/,. , p E J). p .... N~(J) is ronlinUOUll on Ir .•!. Put A _ (% En: l Iz ) ;:: I). Wbm P E J tends I.(> r, f'I A dee" : to r i ll and " I " . iDe.: 5! I to r 1" 0, ""/ l'IAdl' decreuee to /,I"dl' by lbe II>OIICItOnoI! t",""mn.oo / 'l". d" inc ..._ I.(> r IA' dp. Finally, tends \0 f I"dl' + 1... 4" /'dj.. which PfO\'f'II that lbe function ]l "" N,UI is c<>ntinUOUll at r (whenever r > 0). The 54ffiOI ~t...,..\uI for., if . < +<:C . Now _U"'"' +00 and leta be .. r +00, _ ha~ lim inr..... _ N,(J) 2: thal. N.. (I) 2: Um8Up,.-...., N,(I). .." c:onclud/: that 3

J

""" "'g'DOe J /'''''' r·

r r'

r

r

'hill, _

S"

p ....

N,tf);,;

con~nUOUtl.'

•

Finally, oul>P08" lbad' is DOlle"'",),. We I'f'O"" t hat P - f !'dp is inlinlUOly dilf....u iable 10 .r and 111&1 III n'~ den ..... is P - J /". kig" / - dp. P ul A _ (z En: 0 < J{TC) < I) and B _ lz E rI : I(z ) ~ 1) _ For a fixed I> e \c\ r. t be e\e ...""u of J.ucb that r < I> < t, and cIIooo;o. real number 6 > 0 eo t hat r < p _ 6 < p + 6 < •. Thf: fuoctionl! I _ It -Iog~ II from IO. I[ Into R and 1 .... 1- ·· w,g" 1 from [I.+oo[ into R an: bounded by """,,"e>O.Then

'i,"

r,

II .. ·

r ·1og"/1.. 11.. · r-··r .log" I I :5 c · r-··1.. :5 ,,/, 1..

1,, ·

r .log~ 1- I,,· r" -r' -log" 1 :5 c- r+' -1,,:5 " /' la.

eo Ir· k< I I :5 ,,/"1 .. +"I'la for all q satisfying r :5 q - 6 < q + 6 :5 doeIraI a>ndU!lion foI~ bjI tbe ll\andard ...... Ita 01> dilf. ",ntiation i~ sign (Theonlm 3.2.3).

•.

The tbe 0

"00.,

r;. I. ~ 5,,1'_ tMl r ind" ,. I. UII : rI - [0. +<:<>[ II< ,,~ tuu1 "<>I ~_ 11 J _ (I' E[O. +oo[: N~(f) < +-00) .. no .... CIIl>tr, ito infimum .. O. M"""",,,"r, Ihe ", ..dum p _ N,(f ) .. ir>t:muing en J,

P ropooition

Iog(f) .. ~inl S.d'., """ N,(1l ~"'"

IQ

up(

10.•[.

r In8(!)dp) ... " "I

O.

P ROOF: Let • E J aDd p E T1>en .,,, ""d . /(. ~ an: <:OI\juple upot>(llta, "" N,(f'I:5 N. /,(f'I N./C. _,)(1) and N,(!) :5 "'. (f). Thus. if J 10 not.. mpty, Ito infimum 10 0 and tbe function I> _ Np(f) illnc""";ng on J. Put A ,., ('" E n : 0 :5 / (>') < 1) and B,. ('" E rI : I (",)?: I). Foc every u of 10, +00[, (u' - 1)/1> deer : 7' ,," 10 log u as I> goeo to 0 in 10. +00[_11ldeed, fur r , • E +-oo[ oru
10,

u' - I

•

_

,/..•.

-

••

c'
ro ...•

~dll _ ( .. ' - 1)/ • . Now let • be in J . By tbe II>()Il(ltODe comco p'"ce LM<mm, I) / pdp con~rgea to fll Iot!;(f}d" M I' - 0 In 10,.)· On tbeodotr '-d. f AW -I)lpd" con~ to foIA .\otI(f)cf" (Pros»lillcn 3. 7.5). ThIM fW - I )/pd" <:OIl"'",,, 10 f\otl+(f)cf" Loc- (f)d;.1o((/)d;. (PulpO.;t;"D :).7.1 4). SiLlC'e log .... f,"/l / r)(, ncr Fio (1/ , ) .

llf(r -

r

r

IocN,(f) -

~ .\otI(! rd;J ) :!O ~(f rd" -

for all" €jO ••,...hlch Ieods 10 \otI(Lim N,(f»

-

~ \otI( N,(f)) s.

1) -

f /';

!
r

Iot!;(f)
aOO L; m, oN,!/) < ap{f 0log(f):p(p· flo& I· ....... ~ thu ~ ( rlolf/}d,,) _ lim_N,(f).

r

s.

o Lem"", 5.1.2 Ld <> ,b 10: ' - pomn,. rtai n.. mhr:n ,,"dp be <> Itricu, ,...,Ii"., ... ~. Thc-n (a +W S a" +11>" i/O < P :!O 1, and a'+II>":!O (a+W' if

, 2 I. PROOf": Suppt:a! that p ejO, I). Tbe inequaLity (a +W :!O a" +11>" Ie obvlouo for a _ • _ 0; for Q H > 0, tho: a.b<M! ...." be written (0/(0 H))' + (6J{oH)), 2: L .. hich follows from tbe f.-t .hat (../ (.. ""," ~)t 2; o/(a""," b), (6/(0 -+ 6),. 2; b/ (a + b), &Do:! a}(a + Ii) + Ii/Ie. + Ii) ., \. The <:Me whoere p 2: I is t_ted IIlmilarlJ_

0

Propoait lon 5. 1.6 LdO <" < 1 be .. rtai number."" / , illI.... I' ............-.bI.o fwnctl"", from 0 into ((I, +co>l. That N,!!) + N,(9) S N,U + ill.

PROOF: AaIumt thai. 0 < N,U""'" ill " +00. w " be tlot fun<;tiooo. fram !l into (0. +00) .. hich Ie equal w 1/ (1 + g) 00 1+ E !l : (I + gl(+) > 0) MId w

1 <)o',taK\e th....1.- Si""" I Pdll - I{f + g}'l'- ,J .

".(1-,). I~dll &<>d slnoe

N'{I I_ ,) «(I + 9}'1'-') < +00 &nd N, /,«(lh. h- ')') < +00, "'" have

by HOlder's inllq~ity, or 1 ,,1- , . Idll 2: N, {f )' N,{f + g},- '. Similarly, ,.Ill 2: N,{g)· N~ {f + g},-I. Addi"" tho. Iasl. t_ i"""'ualitiftl. we ~ IU +g}'dll 2: (N,U)+N,{g» ·N,U +gY-', 0IId tbuerore N, U ) +N,(g) N,U+ g). 0

1"'-' .

:s

11m"""

P l"OJ>OIIitlon 5. 1.7 Lei P 2: I k a ,wJ .. and I , 9 II/JO IUnWof18 """' n inlQ {o, +ooI. 111m N,U + g) N, U) + N,{g) (Minkowsl
:s

P ROOF: AM""", thai. P > 1. &nd Itt q ho. 11$ ooojugo.w exponent. To I"""" MiDlIQwakl·. lneqQA!i~. _ may suppooe thai. N,(I + g) > O. N,{!) < +00 and N,(g) < +00. All the fwxtion '" .... ;1., from (0. +oo( into R if! Cl">nw.:.. we h.&.~ U + g)' - "p(ln + gl l },:S "Jrl. (f' + 9"). and"" "',{f + g) Is liniw. SiDoe "',{V + gY-') < +0:1,

Similarly.

N,{g. (I + gr' )

«I

:s N.u )· ( {

V + 9tdll) '{. ,

s:

From \he f_ tbat. N, + g}') (N,(f) + N,(g)) . (r (f + 9}"<4» II • . It ....... ioIkI<>1IlW N.V + g ) N.(I) + N,(g) 0

s:

T heol"em 5.1.1 ( Inequality of Countable Convexity) uf (I~ )~ al bo: A _ 01 jlmetiQfU frcm rl ;n.!o (0. +001. alld leI p 2: 1 bo:" real ....mb.o • . The .. N,n:"~1 I~):;; ~, N,{f.. ). P NX>F: P ut", -

L,g;>,s. 1_for tlYI::ry " 2: l. s: 2..., ,s>,s.. N,(f_) by MinJo:.......i'.

Then N .C",) N,{E .~ , I.) - Iim,,_+oo N,{g. ) is ..... . han

inaqua!ity. Therefore,

L.i1;1 "',(I. ).

[J

DclInlt lon 5.1. 1 Loet F he .. JUJ 8anach space and P ~ I .. ..,a1 number. We define by ppu.) (~ively, .c~(P» the space ol tbase mappillgll I from rl

into F IUd> tW N,{/) .. (r 1/1'<4<) II. is fin;'" (~i""ly, such that I " 1l-"""" wtbIe and N,(I) is li,oM). 10 parbculu ... hen p _ 2. IlDY eJ.:ment of 4 (11) Is called .. ~in~ mapping.

[,,2

eon,,''''..... 1'1>&:4e_

99

Clearly. ?,u.) ADd £'f.u.) .... vector ~pa<:eI. and the function I .... N.(! ) deIiDo:o .. ......morm on PP(p). The ",lation I - 9 If &nd only if I - 9 ,,almost ~"""' ill"" equivalence ll'Lo.tion on ?,u. ), and we shall oflen f..,l w ~;~ ?,u.) from its quotient Sp.M)e ADd £';-u. ) fron, t..'f.u.). Fot.-ry fur><:tion I ,(} - F. !let N;( f) _ 1/ 1·,1,,)" · . We dmote by ~u.) (... peai""ly. ~u. )) the Bp.M)e of functio ... f..,." (} into F (... ~i",,!y, of p-rnew.tn.bIe fur><:tions !) for which N;(! ) iI Anite. For """'Y real n"",her <> > 0, 1" : z _ I z l~-' · z it defined and continUOUll on F - (0\ . Since l
(r

5.2 Convergence Theorems

'e_

Pl"Op08ltkm 6,2.1 kl File .. ...,/ &1IOCh IJ'OOl'. and If!! (!~ )~2: ' lie .. ~ in ?,u.l .1Idl thaI L~2: ' !·... (fa) < +<:0. '€1 L~>, I. (s ) ....... ""'VU tWo/toul_ ".e. If.,. F"'/{:z) - L.>". (S) .1 GImM- ..1I poi"" s ~ tIU$ """'"' ""'....-ga, and ta.te I {s ) E 1" arlJitnlril, ~. tht:n I 1Ie/onglID ?,u.l .. nd N. (! - L,so". 10) 5 L.~. +, N,{!.l 1<Jr ail" ~ o.

The,..,

n-.,tm:.

PItOOP:

the.rnu L.2:' I.

A~

"'' ' ' '".9<'' ID I

in 7;u. ).

o

.. In Pt-oposilion 3.1.7.

Theo...,m ~.:l.l ul (f')' 2: ' lie a Calldly ~ i .. ?,(;l). ~ (b. )a2: 1 ~I (f<)'i!:1 ... th the lollo"'R!J prvpertiu:

(a) L"~' N, (!••• ,

n-. eNU ..

-I,.) .. finite.

(t) n o """"""" (!dl:J) . ~,

"''''''''Y''' "-e.

(ej II.,. Kt fez) '" Um,. _ _ f, _(1:) at oimo.t oil poi"" r D'am! th;, limo"! ..... ts and ta.te I (z ) at'ktntril, "~, tht:n llotiong. 10 ?,{;lJ and the...,..en« (f. ). ~1 ""'''"'VU to I in ?,(;l). (dj F<Jr etIOF)" ~ > O. th..., t:rifts Z C (} .todI IMI p-{Z ) S ""1"""'"' (1'.). 2: ' ..,.,~ u"i/~ ID I on (} - Z. (~)

n..., alU" frmc tiott ~ jn:wn (} and II.. (r ») S 9(r) 1<Jr fill I: E: n .

£

end the

inI<> [O, +<:oJ nu:h thlll N,{~) < +<:0

P ROof". By ind\JC\jon on n , deftM " .ri<:tly increosl", lIt: E !l : 1/ ... " - " .I{>:) 2: 1/'2"}. Now ( 1/ 2") . IY. :5 11<•• , - 1<. 1, and (1 / 2"') - 1Y. :5 1/ •••• - /;.1'_I~ IoIJo"os lhal (1/ 2"h."{Y.)' " :5 N. II..... - 1<. 1 :5 &nd ,,'(Y. )'" :5 2-· , Jien<e ,,' (Y.) :5 Z-· , Now , ~ng II In ~ 3.1.2, """ obt&in t~ .-.suI1_ 0

inucer ..

:r.....

By n.....r-. 5.2.1, ~V<) and C~vn I, S.J!PO>M u,~ ..... ,. " ,.."c/i<m 9 ""'" !l in/.o [0, +0>1 ,.w. IMJ N,(g) < +oc> "nd 1/.1 :5 /I .. e, I~ e-y n 2: I, TIItn / belong, '" 4(si) <1M

olm"",

U. ). 2:'

O)n~ 10

I

"'_fO
in C';o{Ii )-

PROOf': I ;" J"moMUl'abie. Sil>oo 1/ 1:5 II Le., N ,(J) ;" finitt. &nd " 1 bdonp 10 411i). Tho functions 11. - / 1' arewln~ &nd domin&ted Le. by (2g)', Tbo:nn:m 1.2.2 ""'" s~ tlw I/~ - / 1'ds> 00"'''" to 0 II n - +00. 0

J

Theorem 5.2.3 C';o(P) if the cIo8uF'e 01 SI{,R., F ) in P,lIi), .......... R if u,~

"", 0/

". ~ ~,,_

f

It.Dd ~modoen.te, it ;. thol limit ..... of. 8equtnIX 11. ). 2:' of . In,.,n!. of SI(R , F) . Fhr """ry lntese' n 2: t. deline!lo : 11_ F by /1.(>:) - 1.(,.) U" 1/.("11 :5 21f (:011and !Io (,.j _ 0 if 1/.(")1 > 21/ (:0 )[· Tho!> !Io "'longs to SeIR. F) and [g.,1< 21f l. M",oo"", (g., 1.<: . COi"""" ...e. 10 fTboot em 5.2.2 shows tbo.l (g. ) . ~ , con'.I&eI 10 I In C';.(si) 0 PROOf" Ldc / E C';o(si). Si"","

;" 1i'_U~

The,,,. . ,.,

0....- by CP(si, J[) thol tet of functioos 1 from 11 into R with the following propci1y' then: n .... 11 E C;'(P) """" t hai. 9 ... / •.• . A function / from n Into 'It lhue In in O'(p, J[) if and ""Iy If It ;,,/.... , ;urahifl &nd "'.(1) _

N,II/ I) < +ce. Theorem 5.2.4 Let (/ . ). " , 6e "" in.:-Muing ij"''-'''''' in Cft(P)- TIItn 1 _ IUP" ~ 1 fR ~ UJ O'''',lr) if An<)6e that the 1R "'" poeiU .... For all lntqcn' .... n sud> thaI I :5 m :5 n. we haw P ROOr : SIDOO / - "

1:. +(/. - 1.. '1:5 /:

••

~.2

Coo...... .11>000....

101

-.1>eIIot
He!IClt. if euP'2: ' N.(f~J < +00, tllm (f. ).~. iI a c.uduOur tl>oorm3.-- foIlowi i~iately from Tbeoo~m ~.2. 1 .

0

p ...."...ltlon ~ . 2 . 2 lhfine "" orrlt:r OIl t..~(p) .., /<>IImIIo: j '" 9 if ~nd onIW if I "" ..eo, .nd /(1 II ...t-diruLtd nNel 0/ L~(P), anuUIing 0/ _ iliw~. II Nu .. .....,.......""'~ in L~(P) ~!of - .uPw.<;H N,(_) ;, /Utik. I .. tAu _ . the jiJlu 0/ _I ..... 0/ II to ~ in L~ (Ii); tMrt. .....'" an int, jilS' _ ("' )' 2:. i .. II nocII 1IuoI, if". ;, .. ~....u.tiK 0/ .... IIIen 1M ~pper .... t>doJooo! o/IM 90. ;, 1'lqIUlI • . e. to .n~ "" ",'lforn.: . / b. L~ (Ii) ;, .. DoIdind """p/de Ria: 1JiII«.

be.......

"'''oerpu

_,,"';(1,

PROOF: A..~_ tha, M iI finile. Si""", II iI dirt'C\ed uP""l'fd. there exiI~ an lr...... P"ll (~. ).~, In II .ud! tlw M' - 1/ 2'" '" N,(... )" lOr all .. I u. F"n:m> JIo - 90 )· .. IUp(Io. g,,) - g" , il 101;".0.1"", ...... {(II - g. )* )' '" N, { IUI>(Io, ,,.J) - N,(t-l" '" M' - N,(g,."- 'f 1/2"- !cor all i~ .. e 6 '"' IUp II . For eocb t > 0, lhere exil18 " 2: lINd> that ",.(, - g,, ) 'f • . and, if 0 E: II • pater than .... , lhen N,(6 - 0) 'f t. Thereror.., the /ilter of eectionl 01. II mn.~,,,,, to 6. Finally, L;' (li J iI .. Dedelrind complete Ria:l l pece by Propooitioo 1.2.1.

-.1>eIIot
o ~ thaI , in ",,,.. ,0.1. C;'(Il) iI not Dedekind complete. For il>8l.anOe. if _ ....... for p Lebft!gue rneuure on and let E he .. l ube« 01. 10, which

10, II

II

iI no( p-measurlble, lhen the "" II of indicalOr functions of finite JUboeu of E hu 1>0 JUpremum In C;'(P). For every real B!oo ..... spece F . denote by C;:{P) lhespece of p·rneaeursble mappinp from n Into F sud> t.luIt N",,(f ) _ N.. (tlll iI 6nite. and !!quip G'{P ) .,jI b the seminorm / .... N",,(f). Write L f(P ) for lhe quotient .pace of Cf(p ) by the equioalen<:e rel&tiort: / _ g If and ooly if I .. 9 loo:aIly wal_ everywhere.

".,..nd

Lf (li) " . 80nadl qo«. A n . .. ~ amdiIi
PROOf":

~.2.3

Lee. (f. )• .,. be I c.ucby -.l>I:'nce in C;: (p). and put

8 .. (z En , 1/.(z )! 'f

N_{J~ ) for

every" 2: I}.

:s

For ~ " E N , there ailil" .... i""<:g~r k" IIUcl> thAl N"",(I. - I. ) l In for all r ::.: kK and, ~ Ir~ . We tan find alocslly INq:ligibie set Aft . A. ~ 8", IIUdI lbat If .("' ) -1. (z )1 :5 l I n for all :z E A~ and all in~.., r ~ 1:." , ~ 1<... Now put A .. 0..2; 1 A.. and ,. = 1.1,.•. Then tho! ""'I"""'" (g.) .. ;!: I <:<>mtft'lt... uniform.ly 10 .. futldiooo> 9, "'hk:h is t""~fon: ,,· measurahle. Clearly, gill bound..:l.. Finally, N.. (I. - ,) .. N..(g., - gl COD'd gb to 0,.. n _ +Oi>, ..hi<;h p
Provo-Ilion 5.2.4 Ld"l We a """in""", bWn ... r ""'ppirIg """" F" C ill"" H , ~ F , G, H a ... ~ B......ch 'J>Cl'U, and ......"'" ilia.! hA :5 I. LeI p,q II< hi» ..,."jugat< ~ In (0, +001, and I . II t_ de"'~ntl 0/ .c';W) .. nd ~(!<), ~pectiodr. The .. "" lJ. g] 6do"§. to .c}, {P) """ N,h " 11. ,1) :5

N,(f)N,{g).

a

PItOOP; OIMoo.&

In pe.rticular, "" un define tbe in...". product L~v.). TheIl L~u.) "'. Hilbert s~

(i,g) _ / /1$ {P 2: 0) 011

Tt-. .... m .!>.2.5 Ltt" 10:. """,pkt O""i. 1I ~ "" 1t(O.C) . Gm.n a &uu.d<.<pan F , let 1'" joe ito ~i" eel <11<41 'PO<'! "Old.., the ~Ii......r ~

(., "J -

r(,) '""" F " F' inl<> C . F".. ] . Dt....u ... B, (~, ~) II>e d<>.eJ NlI,..:III otn.!... 0 and .... diu I in C~(P) (rupa:tivdy, in C',.{p)). 1'Mn

N.(f)- ,up JlgdjJ ~

..

( I)

J".. ..u J € L~{P), ""d

,... J

N.(g ) '" !Up

1'11"',

I"" oil ,

Igdl'

e....,. ,

( Z)

€ c<.... u.). I".. E C',. u.l. the ""'J;>N>g ..... tin ...'" Ii..... r fo .... en l..~(p), 1IIAo~ """" ;, N.(g).

P AQOF: We

prO'V
i - f Jgd.,

;, ..

( 1)_

SUWOl" lhat 1 ~ P < +00 and N.U) .. \. Fim, _uIM UIs.

L la.r· V,,(A. ) .. 'IS_S _

for find 0
"'Y, "'"

If .....

thai.

and let 0 < ~ < I be IUdllhal

6·

L

la. I' · I,.(A.)I ~ I

- t.

'5"'''. LeI. <> be .. """,pia ..... mb..- of modulus I ~ucb Ih.1 ( , · ,.(A. ) - I,.(A.)I (..... d_ re&I wl>eneveT ,. is r-e&I). ror 0)\.'e1y i<>de>: I !O k !O fI , l bere exisU a, E F' such thai. 1",1" .. 1".1' if p > I (ff'!;per;ti.-.ly. I"~ I .. I if p .. I), .... . G~ ill r-e&I, and Go ' ''~ ~ ~ ·1". I·Ia"I; indeed, whon a• .,. O. tbere exilu :' E F' web lhal 1.:'1.. 1, " • . .:' is r-e&I, and a• . .:' ~ 6. 1". 1; _ , il suffice& to t.alol! a.. -la, I' - '.:'. Put g .. L ,,,. ,,~<>a~I .., . Thtn N. (g) .. 1. On tho ocher hand ,

c:.

f

1"'1' ''

L ..a~ · <> I' (A, ) ~ 6· L 'S 'S~

f 'gdl' :?: ~ '

Ia. !· !"'!' MA, )),

'S ' S.

L

IS'''·

!"!" !I'(A.)! ~ l_ f.

(l ) wben f be\onp to SI{k , Fl· We.,.,... ~ to tbe cue in ... bidt 1 it &rl ...bit....,. elMneDt ..r £'; (1') l uch IhaI. N~(f) .. I. Gl.- 0 < ! S l. tbere exlsU ,., E SI (k , F) web t b.u N~(f - 11') :5 , '. By ... haI. ..... ha"" j ust...n . tbere exiN ~ E J:f,.. (P) l uch l hal N.c.) .. 1 and :?: N.(II') · (l - t) :?: (l - t)2. Then / fgdp. .. + / (f - 'I' )gdp and (j(f :5 N.(f - \0) . N.(g) < e'. HcllCO 1/ Igd,,1~ ( I - t)' -t' , which ~ (l ). Nex!, au"",*, that p .. +oc and N",,(f) > O. LeI. 0 :5 0 < N",,(f ). Tho_ {:. E fI : 1/(:. )1 > o j CIOnt&lns . I'"intqnt.blc!iCt E which io "'" I'"M&!i41bie. f I E it tbe limit , " . &/nx.! ev..rywbere in E, ..r • ""'luenol! (f.),.~, ..r 'Fl.1E· sinlple ""'pp;np, and lbere • • Ii· ln~ ~""'"' Z..r E sud> t haI VI'(Z ) < VI'(E ) &nd 1M' (f.)~~, ""' " ...... unllormly to fi E "" E - Z. bt e, lor fixaI e > 0, ..... """ find n > I .. thai. If. -fl EI < e/2"" E - Z . mappl", I.. ta.Ir:I!o tbe val .... w" ... ,v, on E. and one oftbe I; ' (v.) n( E - Z), M)' A, io __ I'"nt&lidbie. Clearly, 1/(:.) - v. 1 :5 e/2 fo< E A. Gloer> < ~ < I , tbere nio' , ,, finite partitloo P t A) 01 A into l'.in~ ...... 1UdI thai. 6 . V Ii(A) .. 6 · L 8 tP(A) V ,.( 8 ) < L8EP(") B)I. Choosins: 8 e PI A) .. Wt 6 · VI' (8 ) S 11'(8 )1 and lettl", a be one val"" of / "" B , ..... hA"" tal > 0 and 1/(:.) - "I :5 t for all :. E B . T here .."j ot. a' E F' l uch that 1....1.. 1 and 100'1 > 1-1- t. Now lho mappins: 9 .. 18 ' G' I V ,. (8 ) iI intes:rable

Thill

J)rU"o"eI

/11'9$

1/ 11'9$1

'I')gd,.1

n....

o

.u :.

1,.(

n..

and "''' (1) '"' I. On u., other hIwd , f Igdp .. (tI V j>{B ». f 10' I Bdp. Si""" f 10'1Bd;< .. tJ{B) + f(l - a)d 1(1 - a)o' IB I :5 dB . ..... _ that

Si""", t and 6 AI1l arbIt,ary, (I) ill true. Ar&bi,,& III .00.,.,. ..... obWn •.,Iad .... (2).

a

If V ill a dcnot _ _ "'~ of C'j...(P), (t J persili,- whm g extends ...... V n 11,.; indeed. tho interior Jr of 8 .. 11,. ill d<nse in 8 . tu>d B" n V i& den80 In Jr. A similar 01 w .... t;on boltle for t I~ e H + IfIr" all I e H+. n..n, 1M" ClIO" """ &nadI .poa: F. H (O, R)
Ir A is a l'.i~bIe !O!:\, theM "';'u ,. ""'I""""" (1;);.1:' in H + COOVCfliIlS 1(1 IA In the..-n. (g.)'~l .. (Inf(!,h)) •.!:, .«"'''''W'IIO IA In tho _ . Tbua. for given f:> O. I"" ... o:xista all in\.Clln j 2: 1 s,d> t tt.!. f' 11A - " I· dVp :5 f ' lor e\"eI)' i 2: j. But , 01""", II A - g,1 :5 I . ..... _ that N.(i A - ,.) :5. t . New let I .. s. s_.... lA, bol "" element of SI{t. 1'). For Iiud t :> 0, /or IOYOI")' k (with 1 :5 .t :5 n) . 1). Thea N,(J - L,S>S_ ... g. ):5 t. nUl tile cloIureof H (O. RJ
n.en

E,

io ~ _ure on 0 .. r1'&l->-.JlM!d fu~ioM 011 o.

5.3

10,11

and H (O. R )

£::CP) if

IUbe~ of i&
Ot.erve, boac"" , that H (O, R l is not a d""""

p

cooU" __

Convergence in t.·leasure

Defl" it lo" 5.3 .1 Let A be a p-meMUrabie ... beet 01 O. A mappi", I from A inlO ,. metri ...b16 >IpOCe F io ,.aid 10 be ,... ....... urablo> on A if..",. only if

(al I ;' measurable M IA; ( b) /or ,,",a f ,...inlqnblc ouboct SolA , tho= ",,;518. p.rqligib le wbeet N of B rudI tbat I ( E - Nl .. -..able.

*" _

that I ill 1'-'lle\iIun.~ OIl A il &lid only if ~ extell$iw of I which is con. tan! ouuid-ety enlOu~ (or vicinity) V of t~ uniform st.rueture 01 F . e>-ery. I'_in~ sut.t 8 of A. &lid to\"'ry real Dumber 6 > 0, define W (V. 8.6 ) .... tbe WI of ~'" (f.g ) of fw>dioool ;0 £(,(,1': F ) ","vina: u.e1Olk>wina: propc!"ty: titdt (J(I'),g(I'») doftIl>DI. belona: to I' Ntis6.. I'·(U) S 6. Let d boo a distance ... hleh dcfill" the uniform . t .-..:tu", 0( F . •-01" all .. int~ IUbo!et.s »of A and all I, 9 in £(04.1'; F ). denatoe by ds(f,g) t he Thill!

infimwn oft"""" £ > 0 for ...hidll' ( (r E 8 : d(J(z), g{z» > ~ ~.

I) is IcIOI than

If (!')~i!:' io. ""'Iuentt
~

m

~

I and 101" all n

~

m. " .., ha,,,

1'( (... e » : d(J (z ).g{z)) > t~l) S t",: » : d{/(z ), g(z »

>~ ))

s: t ..

1'( (z e B: d(J(z ). g(z »

> 6})

s: 6.

1' ( (z E

For""",y £ > 0, lei. V. boo the set ((~,:) E F . F : d(~.:) s: _). Theil, by the .1 . ] ~. ds{/, 9) s: t If and ""ly If (f, ,) belonp to !V(V•. B. t). The mapplna: dB :II. g) - dB(f,,) io .. poeudumctric 011 £( A.I': F). Dell nit ion 5.3. 2 The ullilOnn OIrucluret, iaI dB (... here B lImO throU&!> the clu! 01 w intog,able ""boet& of A) .. ..-lied tho! uniform atructure of (On'E'",,,,,, In ~...""' on A . A IUboict of £(A.I': F) x £( A. I'; F) is all cntou.n.ge for this uniform 51.ru()tUl"Ioc of t:Onh'8"~ In rn FF"'.... on A. When A _ 0 ..." often omit ",I....,,,,,,, 10 A. If f. 9 I>eion& lO £(04,1': F ). if B is a Wio~ IUboitt of A. &nd II d,. (!.,)., O. then 1-, ~ to\ti) ........ in 8 . n .. ldo,..,. 1M intto_ti"n of all thr. of £(04,1'; F ) Is tlie "'" of pel'" (f. g) sud: that I '"' 9 locally l'"allDOill. 0":YEr)""...... 10 A. We sball d=ote by L(A, 1'. F) tt.. Ha....oo.ff ulli"""'.pace· ; )f l.t.te' 10 £(A. I" £)"''' c.'d». !leCI~~ in measure {reopeeti>"ly. it <:OlIveri's in n...... ure 10 I EO £( A..p: F) if and only il. for~­ ery ~"'C" V 0( F .nd <M!r). p-integrable wbeet of A. the numbet

.:.11.",,,,,,,,,

1" (I", e 8

»

IM",).I.(I'» ;' V )) 8O"'i toO a.t; p and q tend to+OI)liYdy. If And ""ly if 1'. ({ r e 8: (/A (;r), /(;r) t V}) ~ to 0 ... n - +<>0)_ :

,

106

S.

v

Spocoo

E~

""'I""""" {f~)~~, In £(A, ,,; F). eon,..,rging Io<:eJly ,,"&!rI><$ ",",'Y"'lw:on. I<> a mapping / from A imo F al!oo ~ In """"""" to / by £«orov', theoo:cm.

Pl"OpoIIitlon S.3.1 £d {f~)~~ , be a C""",,,y uq""""" in £ (04. '" F) and kt

anII r in B. 1/{f.),.V m_re t<> "" tkmtnl f Q/ £ (A . '" F l. an4 i/(f.,l.. ~, ;, ... abow. rMn {f•• ). / 0/"",,1 ~~ ;. B.

B Ito: a

" .~

(»II"""""'"

(»II""'»"

P kOO': Fir........ P!>(IjIe that B "' intt&n-bll!, and Ie\ d bo! .. dillu,,,,,! compUibic wltb ".,. uniSonn ItlUaU.., of F. By inodllCtion an m 2: O....., can defi.o.e • dno.lblol Rquoenee (f_,.)"~"'~1 in £ (04 ,,,; F) wit b the ioOowin« properties:

(a) 10.• " I. lOr all n 2: 1. (b) (f.... ),.! I .. a MIt,. 1""""" of (f...-, .• ).! , Inr all m (e) For eJl m

> O. the lOOt £...•

> O.

of point. r EB for which

d(l..... (r ). I ........ '(rl) > I /r'''' """ _un: ,,(E.... J .. t han If r · ... For ~ m 2: I. put E.. _ LJ..n ,,(E ... J ::; E.~,I'tE..... ) ::; l I T"· For eJl r of B - E ... _ MV'I! d(l.... (r ).I......(r » ::; I/r ...· - ' br n 2: I and p 2: 0; (I_ .• (:r»~! , "' Ihue. C&.>Chy ~ in F. Put fo .. I•.a Inr all n 2: I. 10 that , for every m 2: 1. (g..) .. ~ .. ill uu~uoe ..... of (f_ .• ),.~ I' Now N .. n..~, E... ill ,,''''''Cli«ibie. If :a: belonp to B - N. t~ Ie "" inde:.: '" 2: I oudo tbat %.x... """ lie in E"., and ~ (fo (%»~>1 ill a Qu>chy ~ in F. fi nally, ""'" that (g..).~ , i!I a IU'-'!umoe ofl/. )"!, . W• ...,.. _ "" tbe ""'"" in which B .. tbe unioD of a R
(a) 9>, . .. /. br all n 2: L (b) (g,., .• ).!. is a IUbMq"",,,," of (g... ... ).
> O.

(e) For eYe<)' m > 0, t~ cxi81&. II-l>t@;ligible.ubget p.. of B .. ouch lbat (p.,"'{%») a~ . 10 a Ct.ud:oy"""l_ For all", In B .. - p ... 10,. _ Iho,.. for eodo. n 2: I . n.m, for ..,..,.,. m ;::: ! , ( ~~ ).> .. is uut-jueocc 01. {~"'.).! " and 10 (10,.( ..»a.2:' 1A a Cauchy ""'I""'''''' ill F fo< &II % of 8 - P. .-hen: P .. U..! I p... Tb .. ,,01'" lbe ~rn • nion. Next. . supp<:lIIIIl that (fa),.,., comeoS'>' In rntaSUre \(0) / € l:{A.,,;F). and thaL (fA,}. » ill "" ~. \\rriU f for the compldioo of tbe uniform ~ ~

!>.3 Coo·... '" Min MOMW'O

107

F . Let N be. " · r>e«licible ....1xIet 018 such lh.ot (I. , (z ) . ;!: l ill. C,'w=h,y

t

~ In F for e>tiy z E 8 - N . and ... lil" ~ on B - N . and {f. ,). ?;. con'."lP in tDeU ..... to I' 011 8 - N . n......fore, I' _ / allDoOllt eo.wywbere in 8 - N , ",hence foIIow1lhe aeoond • lion. 0

t

PropOsition 5.3.1 S~_~ thGt "'" mn find in A G IGmil, ("' );u IJI mllhoAll, dUjlJint. illl~ "II IIlitA u.~ loIm"f p'.pt"" lor etiC1"J ". mlqnl.b/<"' E c A, ~ aillI an at....,., ~ ftlbet J 011 ....,. tAacI E - ( U;o,J E n .4.,) ;, ".~. 11wt / : A .... F" ;, ,,-~ if aM ~ ii, lor ..u i E 1. i!.I rutrit:tiIJn I; Ui A; ;, ".~ The ""'pping / .... (f, ).." ;, an iHmlJrplliml lJl 1M >mil_.,.,.,. L(A ,,,; F) onIIJ 1M ~"i· 1-.,.,.,. a .u L(A; , ,,; F ). WMn F;, .....,pk1<', L(A ,,,; F);, .....,plck. MII:n A ;, " .~. L(A,,,; F ) ;, .... lri
PIWOI': Tbe ~m ilion ilobYiouI. and Ib: j - (i.)It'1 it clurly bijecU-.e. For I':Yeri i E J , tbe rfLlPPI"4j: j _ j, ill unifonoly """tinlIOIlI from L{A, I'i F) into L( A.",,; F) ; heDDt '" ill unifonnly """lin\lOlll. N_1eI. W(V. 8 ,6) be .... _race of ~A, ,,; F) , and let J be • countable ... b8e!. of J ouch lbat 8 - ( U.E J 8 n A,) il l>""",liKibie. Since ,,( 8 ) _ E J ,,( 8 n "' ), there it • finite IUbeet H of J ouch tbat E,EJ_ H ,, (8 n "' ) :s 1/ 2. If / , ~ E £( 04.,1'; F ) ..... ouch lhat (f"g. ) beloop to W (V, B n A;. 6/(2IH ll) for all ; E H , lhen (f, ~) E W {V, 8 , 6). Tbua lb' I ill tloifonnly CI)Dtln\lOUS, and the -.:ond, ". '77

tioo iI

Pf""""I.

When A ill ,,·modcnIte, it ill lhe union of • muntable f.t.m\ly of ~ imocrable _ . By tbe above, to 1""1'" tbat L(A,,,; F) ill rnettbable, tbe>-e it "" rmtriction in..umln« lba& A ill int~. But. if (V~ I-~ l ill, '-ill for the lilter of entou ...... of F. then the W(V. , A, L/ n ) form. '-ill for the unIb1nIty of L( A, ,,; F ), .. hIdt pt"D>...., in f.oct, tbat L(A, ,,; F ) is metli"bIe, Finally. "'WON that F is compiue. By lhe -.:oDd -ruoo., to 1""1'" Ih.ot L(A, I'i F) ill complete, ..., mt.y mllrict our attention to the .:Me in whldt A ill I>"lrotecnble- Then L(A. ,,; Fl ill metrUabie. Let (f.). i!:1 be. Cat>Cby IOqlleOOl! in £(A, ,,; F ). By P ropoaition $.3. 1, there ... 'u~uenoe (f., ). ~ I of (f. I-i!: I ",hIdt coo"(>P al_ ltYeI1""bere in A. The limit I of (f., ). i!: I (arbitrariLy ",,\etlded to the wloole of A ) iI I>"meuurabie, and (J•• ).~ I coo\C' .... to I in £{A. ,,; Fl. TI.e. ero.e. the toequmoe {f. ). i!: 1 'IRtf to / in ~

"""""p

Oboerve t bat the condition of P 'oposilion S.3.2 .. satisfied ",hen A ill

a

jJ-

moderate. We IIhall ,",,0"' Later (Seetion 19.2) th.ot it iI ..... atisDed when" ill • Radon mealllI1!. P~ltlon

.tr.o:no..

5. 3.3 Ld F bo G"'" &nadi~, eqWjip,d IIIilII w~_

kfiMil

~

iU norno..

1011

&. L'S poooo

r.JFor tooerr "-_,,,dle,,1 A, ~ topt'/(u of con.....,.n... in rn ....""';,

_I,,", of C{A, 1" F), OM 1M. ..nifarm

w".,..tiWc oo:W\ the t'I'lCtor.paa: "noch,", .", :. l«f ..th /he "nif""" .Inocl_ Df

lo~ooI

con....,."""

(h) For """ry

_I .... , po« ... obl"'n&! .. in .........,_.

~

u.....

~

.....t n"mbo;r p 2: I, IN. !ol: ':n of '~ (I') iI Ii"""

14, .!.n irw/tl.otrl "" '';.(1') I>J Ihe Iopo~ of qO.p;F).

M For 1 :S p < +00, £';. (11) iI .ten« in PROOF : FQ< ~ ... in~bk

St,,-

qo, II: F).

B of A

t""""

and for "''eTY real number

let T(B. 6) be the ocl or f e ' (A.p;F) for wbl 6)/", _ than 6. Lflt d be the <list&nce (~. 0) -II ~ - .1 00 F . o-iy. rlIlU•• :S 6 if and only if f - g beloop to T(B ,6). Thuo, to PlO>e '1000 (.). II "' c"""&h to oho .. tlU>l l he 61tet generated by the TCB.6 ) "' tbe lid of ....N .... booo <;Qfltai", T {B.6/'ll + T(B.6/2}. HellCl! It Ttmailll ooly to be shown that T(B. 6) io aboorhing: (N. Bourbakl. ~JIOCU """lorieU ropol ;JIt"'" Chaplft' I. §U, M_l. But If:t I be .. l'"_u .... bIot maW", fn>m A Into F a:o:l. for ..... y n 2: I. let ~ be tl... lOOt ('" E B: 1/ ("')1 > n) . The E" deer . to the empty lid , ILO I~ ""," an i.... St' " 2: 1/ 6 tllCh llU>lp(£'. ):S 6. Then ",,- beIOIlp to T {B.6). ",bloc:b Ji- (a). Now 11M 6 > 0 he .. ....J """"-. If f e '';.(1') is.uch lhat f iNd" :S 6"+' and If £' _ (,. E {} : 1/ 1"11 ;> 61 . then 6' . ,,(E) :S f If l"dp :S 6"+'. and p (E) :S 6, ..hIdo ~ (b). Finally, let f he an ."b;\r,.ry element or ' (0,1'; F I and T (B.6) .. ,qhbco-hood of 0 in 1"'- """""" Arguin« .. Boo.., . ..-e _ tbat t~ ";'1>1 a ;t-ln~ ... ~ E Ql B.uch that j<{E):s 5 and such tlU>lf iI bouOOed on B - E. TI>en tl>0.

u

5.4 Unifo rmly Integrable Sets Lflt {} boo .. DOrttmpty ott and p a l'Qil;v" Daniell n>eB/lur-e on a a~ H(fl. C). Lflt F be,...,..J Bant.do Ii*'" and p 2: I a real number. Oefl,,;tion 5.4.1 Suppooe II is a ...boet of £';.(1'). II is said to be uniformly iD1~ of ~r p if and only if the follow;", rondillona hold:

(a)

For every t > O. there"09t8 6;> 1) rmcb that rruPf EH f If l' l"d"

:S t for

all I'"im~ lIeU A .."""" ......\SUre ,,(A) is ic!IIIlhan 6.

••

(b) For ew.y ...."/EH

,> 0,

lhere exisU .. p-inlqn.bIe

let

J1/ 1·lo_B <4< S r.

Whm p . I, H iI aid lin,ply to be uniformly

B JUCh tb!

in~ble.

Ld H IN: .. I1Ikd 01 ~v. ). "nifonnl) illlqi06l. 01 artkr p . The ""if_ -...., 01 """tldgt"'" ... ",eo ... e, "" H , ;, qui fo 1M "nifonn .tn.ch:n! indoooId., IIuu 01 ~v.). P..-opw-ltion 5." . 1

P I\OOF, Let ~ > 0 be .. rsI number , o.od let 6 > 0 and B E -R. be .. in n..ftnition U . l . We -r"""""'" Ilw. p(B ) > O. Put" _ (r/ p(B)'/, and reaoll that H" _) E F " F : I, - _I is written V~. Let (I,, ) be an eietntnt 01 LV(V., B,~) n (H " If ), "" that II - 'I{~) S "for all ~ E l) - E, who:n! E iI • II""ln~ IUboot!. 01 B whoBo meuure iI omaller t han 6. We boe IO_B) S N, (llfl_lI ) + N,{gI Il_ II ) S IlJ>d iii....... N,«(f - 9) 1~) S 21:'/ , ; thertfon:,

s ,,'

N,«(I - , ).

/

U"' ,

II-,I'd". l {ll_1I l/ - gI'<4<+ la { I/ - , I·",,+!.8_a I/ - gl'dp

i I ; " than 20t + 20,

+ «(/ ,, (8 »· ,,(8 _

E ) S (2.... '

+ l )t.

o

P..-opw-lUon 5.".~ Ld H k G .IIb, d oj C'f,(!.) . H;, "ni/ornoJ, illf0. lor ....",.....,.......,. (A.>-~ , 01 ".~ aM "."'oeLato:..", ....idI d , ' " ' '" t . III IhU """'. .... "/~H 1/ 1·1 ... dp """1PefJ<" to 0 .. " - +<xl, for....",. ~ (A. ).~, 0/ ".~ Id.r nell U\4t. (1... >-~, """1PefJ<" to 0

J

J

Almon

t..,,~.

First, ....PI"*' that If iI unifonnly ln~ .... bIe of <>rdf,r p, IlJ>d let (A.>-~ , be a oequenoe 01 p-me...",rable that (l ... ).~, to 0 a l _ e"'<01).bere. For fixed t > a, let B be "lI""int~ _ JUCh that ...."/EN II I''''' S # 2. By the dominated COO>"8"' 1CII theorun. ,,( A. n 8 ) ""'" to 0 .. n _ +00. For n 2: I La.rp enotJ&h, therefore, IUP/~ H 1....... 1/ 1·.". S ( / 2, IlJ>d """lEN 11/ 1'1 ... <4< S t . Con.-.:nely............ that luP/E H 11/ 1"1... d" O"' ''01P to 0 , lor e"'<01y ..... quenoe (A.).~ , of 1I""...... un.ble and II""mode, . 1e """' whith d"u ' 5e, to'. Aalun>e, lor the DlO 0 and .. oeq""""" (8.),.i: ' of lI""i~ """' JUCh thil ,, (8.) S l/'r lor every" 2: I IlJ>d thilllUP/EH 1 1/ 1.18. dp > t. Set A. .. l.I;i:. 8; for ~ imqer n 2: I. and put fJ .. lim sup 8 . .. n..i:' A.. Si""" p( A.) S Ifr-' for tACh n 2: I, fJ iI p-neplcjble. NCM" lhe eequence ( A. - fJ ). i:' Jec. 7 to the emp'y ..... , ben"" lu"H H 11/ 1'1 ... dp coo,,"'p to 0 '" n - +<>0, and we anioe at .. COntradiction. Condition (a) t),ttdore P ROO~·:

oe\8 ."""

"""'"'P

fn_"

......

,.

.

vs_

~.

liD

N.,.. ....III.. lhal coodilion (b) of Definition ~.U if I>I)\ ... tio6li«l. ~ .... ;,,' 1 t ;> 0 JUCh IhM IUP/eHI,,_" l/ r dll > e for all int~ ""'" B. Gi""n n E: N , SUpp<:l!!le _ ha"" obI:&iMd dilQoint int~bII> ""'" 8, •. . . . and .km"',\1 Io .... ,f.-. of H ouch that 111, 1/,\,dj! > t f...- all I :!: i ~ n - I. There ....m/. E H ouch thal. I " _I II,u...ulI• • ,) I/. I'dll > c, &ru! A lI·ln~ 8Ubooeo; 8. of fI - (8, u . . . u 8 __ ,) for which 111011.I'dll > (. W"!IO <:ur>$I:ru<:t &II infinite '"'1<1I:00I: (8~).~, of di$join~ int~!IOU and II. ). ,!:, In H su.ch thM 111"1'1;""'" > t f...- 0.11 n?: \. Now. ror -=b n ?: I, put A.. _ U~. 8;. Tht eequence (A.)~,!:,
B._,

.. '"'I""''''''

3'.

Now lei. H be

&

IUbiet of .c~(P), and <'O<>$XIer the

(c) IUV'EH IIIl~' lIra'1I £OeS

\.0

101100I'"I", condition

0 ... I - +CO.

P n>pOIIl l ion 5 ... .3 C<mdition (e) irnptiu .....din"" (G) <>1 Dtfiniti(m $. ,./.

The, ""' OJIO~

me.. II .. "',~.

o.

PROOF: FiM, SU~ thM te) boIdt. Fo< fixed ( ;> let t > 0 be • re&I D"mI:>er ru<:b. IhM llUP'EH Iln~. I/ rdj! is .... than t/2 If E ill. lI·intqrabio ..I ouch ,balll(£) :!: (./2j.t-" then f6 1/f'dj! S. ~/I>< I/ l"dll + ·II(E) S. !. bonoe cor>dItkln Ca)" ..-lio&ed. Coo. .....ly, ....".. U_ II . dilfuoo and that exonditioa (a ) holds. $uwoee that IUp,(H 11/1/ 1 > f ) doeo noI; SO to 0 as I - +00. There ~ .. > O• (f.).,!:, of OI.ric:tIy j)(IOitJ"" nurnben convera:S"4:\ to +00, and • ~""" "")..~I In ll . ouch thM ,,(i/.1> t.) > a for overy n?: I. >0 be ruch thai II(E) S. fJ lrnplieo IUP'EH 16 1/ I'dp ~ I. For each intoger n ?: I,

,r

.. ""'I'''''''''' _

(S.II

find

Let"

&II

ln~.bIe ",bioi.

I&, 1/. 1""'"

E. of (1/. 1;> I.) such tMIII(S.) ~ inf(a.lJ).

bec._

?: ~. inf(,.,.8). whicb is absurd I. - +00. Tben:1ore, IUP/EH 11(111 > I) 00 O. let ~ ;> 0 be!lUeb t bat SUP/€H fA. 1/ 1"dll :::: e for every lI-in1o:grablo! set. A woo..e ..-.Jure II(A) is smaller lloan 6. lor 1 ~~ , 8Up,,,H II U/ I > .) ill omaIlf.r tloan 6. &I>d .., .... hav" sup ,eH II/I">' I/ rdp ~ t. a N_ I

?:

Tboe".

P r-opollit ion ~ ...... ul

II t..

G tub ..,

01 L~(P)

Dtfinili<m '. 4.1. Then (~) /o<>I
J4lilf~ t>:I1>I!.tl/m M of .......Qdtd Gild" ~ni/onni.

PI'lOO" I'I'-'!IU~ I.h.&t (e) boIo:W. &II!d let a ;> 0 be .. re&I number such tilal auP/~ H "II"" In"..... ~ I. AI9o. let B be &II inl-Pgrablo; .... such llw. IUp'~HIlfl"!n_ .dll I . Then

s:

J. Il!"dp s. JI II

UI1 "")nB

II!" dp + " "II(B n (0 < II I ::: al) ::: I + "pII(B)

5.4 Unifonoly

[~Sou

III

s:

Ie

H . "I"Ilu. [ I/ IP"" 2 + "",,,, (B ). aDd H ;. boo'tI(\ed in C~u. ). Con--ty, IU~ \haI. H iI bounded and that (a) bold&. For fiXf!d f > 0, 10.1 ~ > 0 be suet. thai. ruPf~ H [ I/ IP I .. d" is smaller lhan f for """f)' in~ !eI. A. wJw:>.e ,,(A ) iI Ie. than 6. Chooee [ > 0 to thai. !-P(IUP/~HJ l flPdjJ) 6. Then ,,(1/1 > I) ! - P [ 1/\"dp 6, aDd ~ ~/[". I/IP<4I < f, for all I e H. This J>I'M'S thai coOOilion (e) ilauilfied. []

for evny

_=

s:

s:

s:

n.e1E1\::u, wben " iI .rurU!ll!, every uniformly in~ ouboet H cA £~u.) iI bounded. On the com.....,... .. hen ...., tab £Or I' the 011 R defined by the"".. I at the poInl 0 , aDd , for -.oh n ::.: I. Le1. I~ he the function from R iMO R which iI conIWlt, equal t.o n , t hen II~ : II 2: 1) iI uniformly inte&r&bIe of order i , but i l ;. not bounded in £~ (I')'

_=

II " ~ 1I~ ) ~~ 1 """".,.,.,.,. i .. £~CI' ) «> ".. tkrMnl; I ~I 4C1'), tM:n Il .. II. : n ::.: I ) ;, ""if<>nnl~ m'.., 6k ~I <>nkr p.

TbMtem ro .... l

P IIOOF: For fiXf!d f > 0, there mat. 6 > 0 ouch that [lflPl .. dp < t/? for ... u, i~ IH A ",Jw:>.e moasu ... is 1M! than 6. 'I'ben I' ' I'(I/~ I > I) s: [ 1/. jPdp s: "'P.~ , N,(f• ., for aU .. 2: 1 and aliI > O. Thil impbM lhal .... c.n lind 10 > 0 IIUdI that ruP' ~ I 1'(11.1 > I) illlmaU.,. lhan 6 for aI[ I ::.: 10. Now Le1. N 2: I be an in~ such that N,(f. - f) s: "/' /2 for all n ::.: N. 1"bea. for evny I 2: Ia and eve!")' .. 2: N ,

(1

IM 'dp) II.

s: N, (f. _ f) +

(1

III'dl') ,/, s:

11. 1:>'

il. I:>.

~t'/' + ~t'/p

2

2

.. t ' /·.

s: s:

s: ..

aDd [ [I.[>.I/ .IP"1' t. NO'll. for eve!")' 1 < N . [1/. [:>.I /. IPdp.,.. 10 0 .. I _ +<;0, by the dominated CO", .. ",...,. theorem. For I Iarp enou&h, .... I'!\ [ [/.[>.I /. jPdl' t for aU II 2: 1, Condition (el .. t~ IAtil!5a:l.

Nat., thffl: uilU ... [~bIe IIe\ 8' IIlICh lhat [n _". I/ IP"1' s: ,/2". UK N he an in~ oucb thai. Np(f~ - f) s: t '/' / 2 for all" 2: N . 'I'ben, for evny .. 2: N ,

11.1'''1') 'I. s: ~2 . t l / . + 2~. t'/' " f ( }n_". [n_". II. I' dl' s:

t'/' ,

t . On the Olber hand, for every I S: " < N , there u il'l "" intqn.bIe IH 8'. IIlICh thai f n_... 1I. I'd" t. If .... put B .. • 8' u S; U.,.U 8'N_" then 1n _ 8 1/. I' d" S:' b all .. 2: I , ... hidt pro ..... lhat (b) boIdo. []

and

Now .... pve _ I"

F' II. ...

t.o

COfl'''*''

eul!lcient coOOiliono for a in the ",..., of order p.

s:

""'I""""" ..hidt '""''''''P in

'VI

ale

..,.,."1 . . """,.

...... n C~( ,,) ..... od . , oJ~ 1IUlppi"!I I . A_"", o...t (Ik)>>, Iuu

Propoollt;"n 5.4.5 LeI (I. ). ;!!. k .. _

... re 1(1" " • .,.,.,., W ,."" ,,·.. tAt folit)"';ng ,.""", hu: (. ) For ~ I :> O. tAt"" aUto 6 :> 0

,

",,
ft) Far ......,

~

:>

..w. tm.I. lim lUP. __ IA 1I. l"do<

I'.i~l<:

O. til."., aiM ..

"'P. h._II Ih l·"" ;,. -..Ikr u..... t.

n.e. I (te.•

~ 10

N.V. -

f) -

.. ott A oM.>oe ......."'" .. "'.. til"" " .~

H I 8 .tid tIIGl

4(P) and V.). )!, ..... ~ /() I ,II liLt man 01 i1nkr p

OJ.

PiIOOI" f irst, IIIijIj>iM that (f.).;!!. "'''""'"," to Il.Imoet t\C' 7"'~ For add I:> O. tbm e>;1M.s an lntesn'bIe 9I':t B sudt thal; IlUP. )!. 10_B Ilk l"d" Ie smaller than (f/' )'. For """'Y iolq
B. '"

{r

E B :

~~ Ih (l ) -

1(" )1

:>

~. ,,(B )-II. } .

Tbeo ~ B.. <.10;0; .. ! toe ~ ,,({'\.;!!. B. ) .. d. There e>;ilu 6 :> 0 Juch thaI lim.,.p. ''" IA 1/.1"11" it! tt netlJ' ... than {I /' 'f for all ~ ~ A oatilfyin& ,,(A) ~ 6.

IarJt wo"4l> e<> lbat ,,(o...J ~ 6, and let n, ~ "0 be iIUd, that. .up...,., 18 .. 11. 1'>4> 10 oma!kr l hart (./6'1. T hen. for .011 r • • ~ ",.

r..e.: tie

~ 1 be

I. - I. - 1.·10_/1 - I, ' 10_/1 + (I. - fl · 1/1_ /1 •• +(1 - I.J· 1,, _11.. + 1•. 111 .. - I.' Itt.. ; I .....

N.U. - I.J is smtJler lhan

N.u• • 10_ ") + N.V• . 10_B) + N.{(f. - f) . 1/1 _11.. ) + N.(V - /.) . 1B_II..o) + !V.l.I. - 111..0) + N. U . - 111-.)' .00 80 _ lban • . t' - U' )' ;!!' .. " c.1>Chy ""'I"""" in C~(;o), and It <"h(,,,," 10 I 10 C~v.). We ...,.. ~ to tbe ", '>eo a! .,..,., In which V''':t. eon, eo "'" to I 10 U!' ." ..... From ..-ery ....~ (f.,};;!! . of V, ), )!, .... can e>;UIa" IO!!q\IenCI!I l hat «)<>~ to 1.01_ """'Y"
Theorem 5.4.2 (Doml""u.d Convergence In Measure) Ld (I. ). ~l bo: a ~;" C~ (,.)
. . ."""f

"._nM< t"" !I'' ' """r

P mpoBltioa 5.4.6 Ltc (f. ). "" "" • ~ in '';(si) Ih4l amt'efJ .. in ............, to I E '';('') lINl.uch IMt (N~(f. )) ."" am""'YU to N~(f). :1'ht:II (f.M . .....""'YU 10 I in 1M IIWLn ./ ar4er p.

PlIOOr: First, ruppoee that (f.).;>:, ro".ap to I e.l",,*

~ha-e.

For

fixed t > 0, Jet B be I.D ~ eet....:b. that fn _B 1/ ~<4< iI otrictly Ifeo ,han t16. Sy ~'I ,'-m. we can find. dta oinS oeq.......:e (E,. )."" 01 iDt.ep"abIe aut.u 01 B IIJdI. thai. ,.(E,.) :s l / r b"..--y r 2: 1 aDd IIJdI. that (f. )."" CO<..(' ..... unifonn!y to I OILI B - E,. . If. > 1 II." jn~ IIJdI. tha' f". I/I~<4< :s t16, aDd if we P'" A .. B - E" tbeD fn _A 1/ I'IIp < t/3. Now fn _A1/ .. 1"1Ip .. f l/. I"dl' - fA 1/. I'1Ip co
/

11 - /. I'dl' :S ( I/ - /. I'dl' + (

( 1/. I'dp In_A1I I'dp + In_A

lA

iI ... than t for ell n 2: t . Now ,hoe aa>enl.-.-, In .. hlch (I..). "" ton.e.p" to I In troea8Ul"e, io handled hr the ....... ..-swnmt II thai. in P,op<wi,ioa U .S, obow"i"l thai. (f.. )..",. cor>WfI1I'lI to I in ,hoe D-.n of ....o.r p. c

.&eftite.t for Chapll!f" 5

1. Leu, He._ >KIon;" F _ .hal 1-1.. 1101 .. 1. S - tbat 1- " oad I- - "I 2\<0 - PIlI for aD 0 5 ,, 5 1. Prooo
:s

!4 - !hI' :S 2' . '" - 1""1

(I )

"' - " IoI:S 3p ' ''' - tb( (b) for all 0 5 1:S 1 (...",..Ih&L O:S 1 < 1; "'" '" th&L •

- ~ ..

J.(.

- 1"6) + (I -

JI)(_

- 6),

wb.re JI .. (1 _1)/(1 _ .. ), and tJw.

_ -1"6 .. ,.[a _ 16) + (1 _ I')(a _ 6). wI

e 1' '' (1 - " )/(1- t ) ..

(f.' por'd,,)/(I - I)).

:s I--p/Ij

2.

fn>a> pUt I dod ..... lbal. lor .oJ! ~ •• E ,. •

... 3.

l ie , 1",,11. .., _ llod " ;. • pOOili ..... Oan;dl '_u", Oll • _ 1"((O. C). Co"dude "-! , ,";,y (e) of ~ 2 lhat I"" mowine: I_ I/I "~-' -I from .c).",) ",to .c~"'l io ""i6>< ... ly """"" .......

4.

S u _ lhat 1 < p < +«>. and lot f to. lho Dod ..... - . i_uo.Iity (d) 01 po.
N,(I/ I.... "·1- ~9)!i3p ..... V

~ . _t

~~)·

-.J"Pte .., p _

"'. 111+111»)'" (I

100- .111 aDd 9 of c;.",). aDd .bo.. ~ that tboo ~ 1 - 111'-' . 1 f""" .c~u.) ,...., .c).(P);. urubmly """',."".. on - , - bouadod ",1_. 01 .c~u.). ~.

Condodo, lor I < p < +«>, that .100 m'ppHo, 1 - 111'-' . 1 ""'" co;.",) jl>tO

2

.c).(P) io .. _ _ ..........

Let F boo. ,.,.j a.......::t> _ _u", Oll "'(RC).

. 0 < p < I .. JUI .... _

• .....t .... pooit;"" o-;dI

J

II - JI'
1. o..fi""
tbo _winI; V.II) -

to. +001-Sbo:ow tb.t
2.

U';..., inoqu&Ii'r (e) . - , oI>ow II.... tbe mappi'IC 11_ l/lr-'· /o 10 un!Iormly " " " , i _ " - .c~y.l intO .c).",).

3.

hom {
1111" 0-0 · 1 -I~I '{'-' ' 91!i ~ '1/ - ~I' (111+-1, 1)"'-' b.u I.

]"",

,

01. .c).",). 00:1_ u.... t bo .....1!Pinc 1 -

.c~",)

io woibmlJ

1/1"'-"1 !rom .c).{,.j

cono;"..- "" ev«)f .....,pded lUI 5

,of .c:.v.j.

4. CoadO>do.hat.be mappinr; 1 _ 1/1"-' · 1 ;. .. h<>r b· '.....ptoiom &.;,no 4M""" C).", ) aDd u.... .c~",) if """,pIoI.e. n... C:'(p);'" (»IDp"'~ ~ rioablo V¥*oc .... 1 _ _ ANI SI('R., F ) if ~ iD C'i-",). 3

Let,. be lei: , ... _ .. u",OllI - )O.II. 1'...,<10_ of)O, I ( . """ F .. - , s-dlopoce (F ,. 1011· I.

Let 1

An

e 4 "').

Sbow tM< ,,-,.,..,, ... ~...... ..,.. A. B oucl> tlw B "', .4 U B .. I . o.nd iI/I' IA
J

J

2.

""" bed B > O. PU' B, ...

U

E C}(,, ): N.l f) :5 01 """ write D lot ito ct>DYO>< ...., .'D)< If I E C~t,,) _1II>Ch that N.!I) :5 B ' 2"'-', ohow tILot I IO!o in D (in .ho: -..."'" of pw. I. conaido-r f, ... 2/1A and h ,. 2/1 .. ). Coocl...... bN D - C~(.oo)

3. Dod ..... "",., pu1 2 . bat ~ coati,,,,,,,," w..u Iotm "" C~(.oo) 10 \doni ... callt aero, aod tl r do•• ,b&t C~u.) _ "'" loaUy """>U.

4 1M" be ~ " ' to .. ", "" I .. [O. I[ and lot F be • ..-.. (f" -;. lOll· F.. oJI O. put T (6) ...

(! E C{/.,., F ): ,,(II I > 6) :5 '}.

I. 1M" be • mGtim_ ,0-: Iotm "" C{ / , ,,, f"). Cboooo 6 > o"".bat 1- 0). Dod ...... b.al ,,(ciA) _ 0 lot oJl • E f" aod oJI l>IIocrabIo ....... Cotod ..... 'b.al .. ;" idm'icoJ.ly ......" 2.

t;

Dod..... from pwIl.b.al C(/ ...;F} 10..,. Loc.oJly """--.

L.-,. be. poei' i.. D",,;.u _""'''''. 'pA<>01-t(f1.C). A a ,, __ _ F • ...... ,;.oJ,!o unilot-m 'PI'<"'. I.

urablo!_.

Lot (f.). ~ , be. "'luel><e in C{A.,.,F). and lot I be "" ok",.", of C{A,,,,f"). If (f.).~ , <10M DOt """"""'P to I'n _ _.• t....-e 0* a IV(V, B . 6) aDd • oubo.equenoo U., ).~, of (/. ).~, " """ that . . - of
"'''-'!_

2. Lot (I. ) ..~, be • oeq ........ In CIA . '" f") aDd I an ."._ ••• of C{ A ,,,, F). Sbow .b.al (J.).~ , ----.- to I in _ " " ' if aDd <>DIy if, lot . - , '"'.pablo ... boet BolA . ~ ,u"-,!"",,,," 01 (f.).~, " w ',i'" fun ..... • u"-'! ......... Ih-at «" ...... to I ~ ."').t....-e .. B. 1•• bioI . . .. if, ftuoot.".!tom F lnco. rDetrizabIe W>iJorm..-.. G, ....... tILot C. 0 I')'ll """ ...... to , 0 I in CIA. '" G). S. S u _ lbal A ;" ",'..«Mate. S-.bat.he........., 01..,.._", .. in WP " ,.", 10 the ~_ of._ ,~ T IltI C{A. ,,; F) ..IUd> b-a>'e . be foIlooo-iq ... _ .'" ~ _ ........ (/').zl in C{ A,,,, F) """-M to I ·1_ .... , at , • In A """", _ aIoo 10 I in ,be ...,......,., T .

_. """"n,_

1 1M" be. pooi1i .... o..iell _U'" "" '''-'' 1-t(f1,C) . aDd lot A be. '" _"-"">Ie _. I.

Su_.h-at - " ", ;n~abIo ..._ of A .. bleb io DOt ,,-nqliciblo contai ... an &I.Ota. Lot F be a IDI!IrizabIe u..niloo-m 'PI'<'" &lid 4 • dio'" ... compulblo ..-nh .he W>iJorm ,lnICItIf'O 01 F. Sbow that ~ oeq ........ (f.).~, in C{ A, " , F ) (lOtI . . . " " , i~, P' H'", to I E C{ A.", F) ........... 0100 to IIot:aIIy oJ_ .... "'bote In,t.

'VI

ale

116

6, V S SUP!"'""'bon nioto

2.

iO>tql'ablo out:.et B ct;o., "" ~
Of>

no .... B ...{O S t '" 2" ) M 101_ ........ bOth """ ,I",

Ca) 80.... B. (b)

rOt' ~ ~
I. B..... ...... B. ... ~I form 10 putitio~ of B'_i .' ...... ..( B.on) .. ,
s

'Th. 6>r a111n~ pool .. (~. l) """ lhot h ~ 0 ...... 0 S k '" :t, put / .... .. I •• .• , P ........ th.at (I').~I ""'... ..... to 0 u. £1..( .. ), b<> ••1w. it doeo _ ..... 'u .. toO 10...I1y olmoIO. ~"j . _ in A. 7 Let .. bot. pool,n. """ioon ... · w.... " " • _ 1I:(n , C ). lei U.) . ~, bot ..... _ of /0"" ,,""abIe fu"",_ f""" fl !n", fi ....... dofj"" / .. lim ... p f., r o< - , . ... in~ ~ 8 and _ 0)' 8 > O• .."...noct • /,.i" "$ able aub~ A of B "'. . d : "~ I'I A) ....... t!.an 8 ...... lot _b>::h t ho 1oIIo>wI.. "' ....... ,' 10 .... 1oIiod: b ...... ~ > O. tbent 'b I.(~ )), ...... ,>' li ' n Ih.at lI}') .. I ) ,

f1n,

8

Let I' ""'. pooiti~ DanioII - . .'" "" •• """" 1« 0, C ), and lei U.) .~l b< • i::. £i..(I') - " ,bat (t': ":i! I) io ~"lb'mly _

,,,'epsb!e.

I . ~ tl:&! lim IOIp f 2.

\I

I.d" !

f. (lim_lIP I.)J.. (.... E" . d .. 7).

Similarly, obow tI:&t r (~m;ruf.)dp Po.",... ..... 1.2.1 .

s

Uminff/. <40 Com_

witb

Let (0. f, P) boo • pn>hebility &pOo<:1 ... n , • O',' lybt. f pooitn. _ .. no P "" f ...... lbat p(n) _ I . r ... .-y Int '601 .. Viable ""'" (n,F) i"", R.. .1w. ia,. , .,•• obI!: f tu_ ""'" n im<> R.. _ au_.hat pcs.. _1) .. ("'/ll) . ~ • 6>r 011 ......... 1 ~ O. I. ""boor WOtO&. lho Ia.. of i& ,ho f'o'"y diotribv.I .... .,it/: ~ .. n. f>...a,., ""I Y. " «(5. - n){"r.i)- '"' O
in

n. .......

s..

s..

alb.
Fb< """"y

" ..

I , lot _. .. ( 1/ "!}to·· '''.-' . From

_...,. eJ. .. ••• ,1... > I , ond thM ( ... )..~, Olrictly i..... ( .... Stirli",', Io<m"'-l. $'- that Y. dP "" ~ • .

f

w

10

If ./'ii

2: I ""' ............... boor. p..,... lhat fy, •• V.dP .. 0 lot all ~ S ,,'. _ tl:ot fy,,... Y• .a> - ... . ~' S",liJCI - t /" ) for all .. > Q' ( ... bun 1<>.tiiI- Po .. tho inte:poJ part of Q.,rn).

:t Loot

Q

En,, '

lor Cbapto.- $

117

3. I.e< .. > 0' be an ;N g • """ apply 1100 fA ! . ~I .. t..urim IDmm....... .......... 10 1100 f., lioa z -I<>f(J - 21" ) &om (0, ,, - 11 inIO R. otuIa ,~

......

L 101 (1 - ~) " { "101(1 - ;)oIz +~q (1_ ';;') +Ro(..), os·s.. wt.or.: Ro(,,) s o. """ """"I\>de IW L 101 (I_ ~) +,.,.~+ I )

""S· s.. 4. Dod""," &om 1*10 , """ 3 Ilia> (Yo: " ~ I)

io

aopt.....

io uDibmly Intqrablt wi,h


.-

01 . w, iDcidtDP'I!y, ,1Ia> (5. - ")f,fli -""'$ In law 10 1100"""",,) law IV, (0, I ) (by lhe "....raJ funi\ I " U.), """ ......,. tlo.&t (}")o~. ""...... In

10 "" A be. """,tnuWi"'';1>& (In tbe "VbI , i': _ ) ....t J ... In..,Jul i....... _ p h .... l _ ~ <11 A. Lot E be ... A·_I110....t • • IHzmitiaol loom .... Ex £ . H...... • (" z ) .. • (z ,,.) b: &II z . ,. E E . f"iaalIJ, lot z " ... ,z . be u-\y iad
1.

II

Show tt..l. 1100 __ rIniooo <11. 10 F x F io """" i __ Ie If o.nd 00'IIy If dot (. (z"zJ» ' S.... S. io..,. • .....-
If X io uq.,.". ..... ,;,. 0 1 _ n ~ I mer • """' ..... Wi ... ril>& A """ if Y io 1100 _,;,.
:1;, :

_1_ "' .

I.

Show tllot D •.• .,. 0 b: 0I11 :!i k S n .

2.

Fot - , - I :!i k :!i " , put .. .. E,S-Is' D;':' . D,,, . :l"J' p..,.. thol (e" ...• • • ) ill ... 0<1,,",-"1 _ <11 E. , .... -,- 1 :!i k :!i n, aDd.bat

. (e. ,e. ) ..

D;':' .dot (.(z"z,» ,S" ,S'

(_ thlo ><mark at \be bqinnll>& <1I . he Udolooe). 0... ~ tllol (e, •...• c.,. ) io "",.inod &om (z .. . .. z. ) by C ..... •• O
12 Lel (z ,hs.s . be. b ite ..........,. In on inno< prod"", _ (...... R 8y dofIlO>itioD, C ,o.m'. doot
'VI

CIt

C). ill

ale

6. V Ss--

118 I.

• "" ...., b.....-tr _ • ..H-ni Of ..,01 .,.,!,- if tloM " " G(%" •.. , %~) "I- o ltOd that. i
S-

pooi'i" ,

2.

A",,_ ,hal." .... , """ oubo.po.oo ." E

,poi"

z~

.... 6....ny i .. t po

_,«
"',t. _

It< V boo tho __ (.,,, ... , .,.J. Sbooo- ,hat ,/lot di<, · _ from

V .. (G(%" .•.• %• . z){C(%,..... z~») ',.. Th _ ,bio,..,.,. older ,100 01404 t. (e" ... , ..... . ) """" ..... from (r, ..... z .... ) ~ Crun', % I(>

ortbot!o<>ol~ .....

l.

UO>det.1oo b;ypo:o<'-'" pan 2...... ,bat e. ):.. it MMc1ty pooi\l,.. lor 011

IS 1 5n. 4.

UDo<'- 01 pan 2. lei (~, ..... z.) bo ... orthoo>oo,,'" l3m~y ... ,. .. itb ,boo fotloorUoc ........ , .

la) rOO"..-,. I

5 . :5 ~ .• boo

f.,.. ...••1 i<

_

IOon'ta.I '0 \ 100 _

ou"- of £ _0«<1 by oubopooce of Eo opann.od by

/........ %.1·

(h) For <:Wrf

IS. S ..... Ia". io Mnc.,1,y ....ili... • c. !I<. ) b 011 I :5 t 5 .... w~...,.

S - tbM 8, obtaiDOd lot" OO"H·.. •

,,,oI~iooco

,h.at

fr' .... ,hI

i<

of tho -.q-.oo (.. " . • " ... ).

" - . if (%,),.~, io ... IoJiml'" ""'1_ iat • .". ~ I~" ~ I. (I, ... . • z.. ) io ,1000 _ obu.! ..... b,....u.ono...w. of (z, . ... , z. ) . ............ oar. iR tbll .... \ 100.\ (•• ).~. i< obtai"",,, by ort..... of tbe ""'1_ (z.)Q",

".Ion

U

,!jll,""

If A it • .""""..",,1.,. .Int. 1 • _ ... pqo .... P an '''' '. ,,. of.1ot ...... AIX.!>., of poI,- -,·j·h. 0 ,_diotlRoc, eI .. ." I. o.nd iftbo '''''DC''' 01 AIX.I•• , oI>toImed from P ~ ...... litutiq X~ .". X~ II flqllol \0 0, _.....all lbat P io IIooR d ....ioiblo lot" X. _ X~.

atod"

N_ lei K boo ..

'" tloo

-..", ~ Iat; ... f;dd

.It,,,,,,,. doc ((I /( X. f

~1ooowoI

o.nd • .". - , .

n ~

N . .. rite 4 ( X , •...• X .. )

x•• , )),s .~s.) ." ,ho fotld

K (X , •...• X.. ) ol

&.co .......

~X, .. ..• X,.. ) .. P(X, •... . X... )! n ,s."'s.( X. T X• • ,), o.tOt, P(X ••.... X.. ) It .. pOIy-.i6I 01 t WI 0, , ....... tbM I :5 0 < ',,"'''Iute X# b X.. (" -' Welt. X • • • .". X~ ... ) m ~X, ..... X.. ). S-- th.at tbo lr of II'( X' .•. r.X", ) 00 , tal l io ....... 0-1_ ,w P(X, ..... Xk ) is diyioiblo to,. X .. - X . (ill .. ";....,.. to,. X _... - X ••• ). 00 110M

1. S - Imt

,:S. ".

A(X" .... X ... ) .

Co '

II

,:s....s. .. ~ Co

E 11'.

"u.

r,.

(X ._ X,)f x •• ,_ X •• ,I/

II

IS...,S'

( X,+X~ .. I.

E<e.. . 1.

s..._ "
2 aDd

<.._, eqaal I. F",

lor Choop
ll~

de-.. t". l' aDd • It..

~mplici\y.

X , . A(X, .. ..• X • •O. X• • , •...• X",) aDd 4 (X, . ...• X •. X •• , ....• X,. )

'" K (X , •...• X,.). Exp&Ddi"3 det (I /( X; + X •• , ) obow lhai 1' '' 0 - '"' (-I)' X X~ ~ ,+ ~,.

a.Iont

c.,.

IT (X , - X,)(O_ X••, ) / IT t1..!S'

......

X,(X ,

thot

+ X • • ,).

lS1S_

Th ... ..., obI-ain , _ bpi i<>oo of , . "'pi -' ... aw:iudo.hat c., • I .

.1.

a..l

.+, ·/l,(X •. .... X• . O.X•• " . .. •X ••,·•... . X,.).

wlM:<e .ho -. .. . y
r-

i
Suho
0 lor X , I. _b

oI'~

If . , •. , •. _....... . . .... "" ... ~ ..,," '" K ...... lhat ... +~, dillen " - 0 Ior ..n;. j 01 (I .. .. , nJ .

""-.hat

(CaudIy'. dttetmllWllJ. 14

Lot .. t..~ _ ...... "" 10.11. A .countabio ... _ of )- I f"l., +00(. V 1100 "a l. I _ ... bopN>o of ~(u) ~ by \100 funct ..... I"' (.. E A). Ccx_

u... .

'W

,.

(a J

A t.oo .... a
(\oj

E~~ ... ( ~(o + 1/2) aDd

1- 1/2. +«[

E.>O. ..A 1/ "

.... ftniU.

u.at V dill... from .Q,(u) if ....:I Ii'd. IU>d il> lor oll _;" C _ A"""" ,bat R.e(.) > - 1/ 2.

We iDtead ... _

Lot ...... poin' of 1- 1/2. I.

+ool-A.

wpt1 ft~ .ut.< 01 A aDd If V• • , lie _ ..,btpeM '" Lot",) to ,,,1 t". ,boo j O lor '" E S , t". _ of G,om'* de.albl 0<. It.. dh,,",," d. from '" 10 V... n... _ Ex", e" 13 to ....

If B 10 .. '"

"""""II!

01.. .. (2. + 1)- ' ' ' ' fl. .. 10 -

2.

3.

0IIb- if o:w!itlono 1..) bd<>nc: to V ,

,ha•. ,b. __. /"doot _

I") .... fimu'u_"1y I t

,ha,

..11(. + 0 + I).

I_- 01/(0 +0 + I ) 10 "ric\1y ... IlI&n I . lor ~ '" € A. $,,_ II••, A 11M on """"...,1...... poln< j .. 1- 1/2. _ [. T'- ,bet. bill . ... in J - 1/2, +<»[ ....,.. that A' .. A n I.., oj io infiu.. ~ ,bal p ,"""" ,bat

01.. "'" bo _ _ arbiuarily omaII t". .. b." lor B ..... _ ... , k cardiDaJi'l.

01 A' witb

~

4.

-,-""",.hat A

b.ao no occ .. _,lalion pob" ~ 1 - 1/ 2. + ""1. F<>r ~ <> 0.1/( ' + .. + 1) io (20 + 1)/(. + " + 1) or

... .t, 1 -I_ (2. + 1)/( r + .. + I), .... io omaIler Of 5'.! I.. Iho.o> • • Dtduce from ' hio ,hat .100 !.miIy 1 - I_- <>V(- +" +1) lot <> E A and " S 0 (....pettrmr. lot <> E A and " > 0) .... m moblo if and 0III1y if ,he ...... L:~ ••.~~.(" + 1/ 2) (ni L liwb'. E.. ....,.. 1/ ,,) io a..;",. Coocloode . bat

1. - 01

Il . + <>+1

~.

djllion !>.

r.- 0 if ...:l ....1y If <X>
p,....... l h e . _ . t \1000 bqinni,..; ot."", _< ....

6 Integrable Functions for Measures on Semirings

11. t"~. _

IDe... _

ottentlDoo

on

_0'",,_

dolinod on

~

ODd ..-.

. ... abIe (""".......

,"m_ $.I 1.«" b. . "-,,,,"on 5 ...,uri.,. in 0. '" C nio ......... urablo if ..... ",(1)0 if, ... ......,. E (; S. ,hot ... A n .& io iDlqrU>lo (P>q><JOIt;,., 6.1.3). A fual00 if_....:J,. if, b f;'fn)' E E S, •• , ... 10: ..... '" I '" E il .... ti"" , u . '" oI>npIoo "';0;>''''' (P' ...p .... II.,.. '-1.4). 1'\na.II:r.... "'" ._ ",her Im~ ''1 _ """",i. _ 10< I '" boo ",_lItobk U LoI; F be • &n.do ..,..,. with d ...... If I t CJ.v.). ,,- N,(11 .. _ .... 1 1 .. ~ B io tho cloood .ni< booIl 01 S/{S, 1") (TI ", ' . ... $.~.I)_ U TM. -=t ....... -.Ii-. 1M ioIlowi"l ............ Ld 5 boo tho .....ki... 0( fin~ •..-.. 01. N. T ..... £ _ card E . ..... atd E .. tho 0*'d1,..!11y 01 E." a ....... ~ .. ... 5 , ooJlo poi ... , s.od> • _ t o ill......,." (d . S«tlon 3.8). 11..( 10 lIMo IIIIttkla _ i'tt"", '" 1M otud7 of "'.... ,ot ' , 5 of. m ''R (d . s..... IIoD u ).

oWIi<',,,,

7

f ' .....

7

7

'

r.

....

6.1 Measurability Let

n be .. DODeDIp(y let, s. Berni"", in n, and 1"

(Omple>< n-.re

on S.

For &II .oc.u E 01 fI, "". V"O{E) .. intl"'~" (E.~, V,,(..t;». "", (A,) ... ex~ ~ .1>11 clMI of o . illl I~ br S, VII'CE) .. inrC"' )"', (E"" V I'C A<». ,01><:011 (A.,)..,. ~ """t< the ct.. 01 """nta~

famil~

.uboieu

of 1trecu iud> \hu Uotl A, ::J E . Clearly. V"OCE) :S V,,'(F j for all E, F """h t haL E c F , and V,,"CE) _ V,,(E) for ~ E E ;t.

r

Theorem 6. 1. 1 V,,"CA ) _

I" ,W" J",. ~ '~lnd A (}/fl.

l'

PItC>OF : CItuly. IAdV" :S V ,,'CA )_ To ..cablish the ~...., ;'WI,wily, IIIj>1l(* llIat IAdV/, < +00. For fixed ( > 0 , let I E bf. iud> 1M< r ~ I" and ~V" < IAdV" +~, .nd let (g,.)~l!' he .., ir>c .. "", ""'l'~ in $I· (S) IIdmilti.ng 9 all iU up~r envelope. ChOOfIt 6 eJO. II and put B.. - (% e n : .foe,.) 2: I - 6) £or all " 2: I. ".. B.. Nrm an oeq_ I.b R , and A C u.~, 8... '"

r

r

:r

r

inc:--,,,,

V"O(A):s L V"(B.n ( U

'S,So -!

oll

B", VI'fB..) -

f

B,r) · ·IIPVI'C 8 .1.

1,..,W,, :s I

~6

"l!I

J

g,.dV,, :S I

that V"O( A ) :S 11f" l"tW"l + '1/(1 arbitrwy. we _tude tlW V"OCA ) < l .. dVp.

whenoe .. _

r

~6

I

gdV".

- 6}. Since ( Md 6 ~ 0

P ropOO;ldoD 6.1. . Lei 7 .... ,,·.uUili~ ",ncli<m fro", 5 .. <'(S) ira/<) 10.+00). >riIidI "9'!WI";!A S. Tht-n 7{E) _ f' l .- .w" /",. 011 EE S.

v" ""

PIIOOt': OtwiousIY. T(8):s V,,' (B ) 1o< .oJi 8 e S. "''''''' ,......... that 7( £ ) v,,·C&l b tome E E J, and let (E..J.I!' boo .. ""'1_ "f dlljoinl. w"'-e ankln .....lai,. E. Si""",

<

s-u

reEl _

L ?iE.. n E) < L V,,· ( E.. n E ) _<:'

~1

there c", an 1m., ..

~

I

~

IhM .,(£. n £ )

VI" (f").

< VI'· (£. n 6). n.e..

r(E. n E')+r{E.nE) _ " IE.. ) =

_

VI'{E. ) VI'(E. n E' ) + VI'(E. n E).

,,(E. n E') - VI'(s" n E') + VI'(E.. n E} - rlE,. n E) > VI'(£.. n 6"). which coomadicu , 1>0 fa« thaI .,(B ).5 VI"(B) for..-y B E

A e n;, w~ VI'"( A n E) + VI'"{ A' n E) I~ twrJ S ••d E. P l'Op
~t

S.

if (0114 0<>It

0

II) V #( E) -

PII:OOf: Ld B ~ a subeet 01 0 . We shoo.' lhAl. Vp' (B) ~ Vp' (A n 8 ) + Vp' { A' n B ). We InIo1 .. wen _ume thf,t Vp' (B ) < +00. Fix r > O. Tbere exiN a aequet>Ce ( E. )~ ~, of S·eet.I such thf,t 8 c u.~ , Eo. and E..~,

VII(E. ):5 Vp' (8 ) + r. But V,,{E,, ) _ Vp' (A n E~)

Ior MCb

+ Vp' (A'n E~ )

n, "

VII' (B ) >

L VII' { A n E,, ) + L Vp' (A' n E.. ) - t

."

."

2:

v,,'( U A n E.) + VII ' (UA'n E. ) -

.

~~,

t,

and VII ' (B ) 2: V,, ' (A n B ) + Vp' ( A'n B ) - t. Since f __ &1bilrary, "" 6nd that V,, ' (B) ~ Vp' ( A n B ) + V,, ' (A' n B ), and by Theorem 3.7.2 II follows llou A ill " ·",.,...... ble. Q

",,,,....urabIe

"Ibe fact lhal fur a "'" A C rI to ~ i\ iI".. t 'Y and IOUfIio ' " I I.hI.I It lq)Iil the outer " - , , u n : VI'" an be ~ by COl'''krin,s I"" main proIonpW>n ;. 011' . '-a,ow Vr." _ Vp", Tbo! proof;. Irit to Ibe

• ie< . p ...."...IIlon 6. 1.3 A ..t A C rill ,,·m..........u. ",;""",,Wo/or~....,. E E S.


A n E ..

o Now ..., &:i~ 101"0 ehar""","-ionI 01 """""""",,...bIe functklnl. I.et. "R be lbe "Ill II"fI"raU
Lemma 6 .1. 1 For CWO")' B E il OM CWO")' r > 0, thc...: IIIol VII' ( A 6 B):5 r.

d.,. A e 'R..dI

P R:OOI': ~ exlo~ a~.......e (B. ). >, of disjoint ~ luch t hai B c B.. and £"i!: ' Vp(B. ) < Vp{B):;' r . n......rore. Ie!.Iilll p 2:: 0 ~ an

u.i!:'

inl i " mel< thai VII « U"i!:' B,, ) - B ) + L' i!:"" VII(B. ) :5 t, .... _ A _ U',.; .s. B. baa tbe desirfJd P' OfM'lIy.

llou 0

P....,.-Itlon 6. 1." UtI I ~ " mappi"f fn:nn n ;nl<> ...... ,~"...,.., 1'1Im th~ loillnling ..",.jitiorll B ... """t!Glen!:

F.

(.) I .. P·""FM ... bl .

M

fbrCWO")'Ee S , /IE IIlMli"mil..tm.n1 ~ in E o/ueq,"R«

o/'Rj E·,;mple ....."";, g•.

'VI

ale

(e) F",. 0't'UJ E E S, 1/ £ illhe limir ol..... 01 ... "oui~", in E ", ..

,,' it! E·,;mpk

_ppm".

...,...,na:

PI\OOt': Fix E E S. Let 9 be AD RI E-aimplc rnai>pin& from F; int<> F and ~" .•. , ~. it.! ,1lI."",. The !leta 8 , .. 9- 1( ~J) I~ in 'R. GiV that A J C E and V ,,' (Aj l:J. B j ) 0::: ~/k,

..m

The eetI Ai .. Aj - U'SI~.H A., for I 0::: J 0::: k. btlon,g to 'R and _ ditljolnL For e>'ft)' I :S j S t , Bjn(Aj)< .. ;",,[uded in {Ul<,<;-l (Ai n A, »U( B;n Aj); 1M J , let % E BlneA;)': if % does DOt ""~ AI. then % u.. in 8 , n Aj: "" tho!otber hand , If t; ""~ to AJ. then illioe!l in U'<'SJ-' A" N .... , lor all I 0::: i 0::: j - I. A; n Ai II "",,!.ainl.'d in (A, - Hi ) u f AJ - 81)' beo; .. .... B •. HI ..... diQoint. So Bln (A j )" ill included in U' 5'5. (A. 6. R; l · Fix z € F and lei. h : E _ F be eq,.w to • ",,,Me U':f.Ic. A J • and 10 ~J on eacb A, . Then, of coun;e, h • 1<.1E-timpio. Fl,.rtl>ermort,1> .. (:t E E: A(.,) j. 9(%)) .. lut.o:grable and, liDO! it is cnntaiMd in U!:Sf<. (A,.!l. B; J, _ ........, V I'(D ) 0::: . _ Thuo, for every n ::>: I, ..." <:an find an X I E-$m~ m&ppin« '" rro'" E lnl.(l F..x:h that tbe V,,·..-sure of D~ '" \'" E E : 90 (%) ;. g(r )} is i<:!ll tban Ifl". But I.Oeo N .. Um,upD . .. n..~1 ( U.<:~Dft) is ""S1;g;bI. , and, il % ""Ioacs to £ - N , then! e:dou p > I such that ~ (:E) = ~(:E) Cor tJl n 2: P-

u;:

Thil ptOOfl& thalo tho: f I I l q _ (9..)A>! eomUJI$ to 9 OIl £ - N . &nd be",," 0.1_ ~'"'Y"'ho:re in S . No. IIIJII>OIi'O that I io J"D_ur~ and let E € S. By Prop<)8ition l.~.l, 1/ E Io! the ~mlt tJIDOIIt o:vc<)'Who:re In £ of & fIIlqur:n~ (1M ).." 01 'RJ E-eimp!e mawlnca- By what hal JUII. '-n prov< ew:ry m 2: 1, I/A .. "It.. limit tJn"", t.1'1)'wt..", In A .. of .. fIIlquenot f9-,AlAi!' 01 RI A. ..·oimp!e rnawill9 For ev<:'l' m 2: 1 ""d e\IU)' n 2: I, dcnot
,..r,,,,,,,,,,,.abIo:.

seq",,,,,,,,

S ..

,ho: ,...ill& genen.Uld by S. aDd re<:alllllal the elemenu 01 "tbe 5 1Jor-e11letl".

Mr.e an\. [f

S

E c n is p·"",uurabie and ""TnO 0, for each inkgel' i > I tbeu e>::lscs •

U~J

",,'N·bIe union 8; of S___ 1UCh lhat

B _I.JQ., ~

in

125

E. C 8 , and V,,' (8; - £J)!O ~/,f . 'T'hen

CIOU!lI.abIe union of S_, B cootalll6

E,.oo V ,,' (8_ E) !O

It 1oIkwt' immediatety tllat theN eriflt.I E:' € S, comainlJII E. oucb thal. C" _ E'" "."",U.iWe. Now !bore ui$" F E S whX:b cootal", E" _ E.oo io ".~igiblo:. Then E' .. E" n F" beIonp I<> S. io included in E, BIId E - E' ill ~iJlbk In $bon, theN art E' E S, C" E S Iud> that. E' c E C E" .00 C' - E' 10 J>I!Ili&lWe. Ot '.I" i....,;do,ntaDJ that . br t!:o, ~II,I!;, a "'" N 10 ,,·otel~bIe if and 001, If Ibm! ...... E € l uch lI,at E J N.oo v ,,'(£) .. o.

f .

5

T heore m G. 1.a Lei f be a "'apping '""" 11 ;"/.0 a "",,,....,We IJIGOI' F . TIM: loll =;"9 wnJili..... """... . I"",: (aJ

I

i, " •....",,""'We.

M Far""", E E S, 1IIen: ......1I . . . . .gibl< nbHI N 01 E, N

E

5,

,lid!

tJI41. I I E - N .. "'....... nWoI< SI E-N ""II I{E - H) ~ ..,~

P IIOQP: Fir$!, ....""'" that f ill ,,"_Mill~ .00 lei. E E S. By Propooilion U.2, I I E ill t!:o, Umll of a ""'I1lei>Oo (I.). -t l of iii E-tUnpie mappi",p, out.lide • ntsIlcibie JU*t N of E. Now , by the n:rNrU Just .... ta>dl"( the .... En! tl>eon!m, _ ill&)' au~ that N llos La 5 and that Mdt f. io S I£. "",pie. FiJI: % E F and , for ~ n ~ I , let g,. "" the mappinc from B into F "'1\1&1 to I. <)n E - N and 1<> % on H. T .... '" .... S/ E-oimple r::appIngaar:d coo..... to the mappill,l!;, from E i!110 F w bicb qna wltll lonE - N aDd i:: fIqUOJ 1<> . <)n N. Thlo. , : , .. tbat g io ~ SI B. and that liE - N II ~ SI B - N. MOt......, •• I {E - N ) 10 ~ N_ MCh E E!; 10 cootalned La a c:ount&bIe unioa of diejoIDl (b) II oatlotl.d. Co,:nu , dy. f 10 I'"_urabk .. bH.e~r (b) holdA, """'"' .... , for tACh 11lI.,. &nbk let E, then! allIS E' € 5 !ud> that E' C B and E - E' 10 nqli&ible.

s--.....,

o Le' l be • """vping from rl ....1<> • I'!CI BBMdi '''''" F. .. " .........""w., .n.oI ".mod..-..:.. if .ni <>nl, if I/o.", a ·,t. 9 : rl _ F

T"""""m 6. 1.S

ne.. f

wWI I/of /oI~ proptr1ift: (. , 9 .....iM\u....:oiote..,.,..

M (c)

S•..".

S lor......., Borel .......' B 01 F <1M _,., E E S. The
, +I(8 )I"I E liu in

(~) fI '"'

I 4hM" noe,.....nu..

P JtOOP: T"beo! em 111.3

rou.:.... im.-tiatriy f....... Tbeoieu, 6. 1.2.

0

6.2

Complements on the V Spaces

Ltt n , S, &nd I' bolas in Set'ion 6.\. Then Theorem ~ 2.~ may bol refined ""

"'''''' Theorem 6.2.1 UI Flo(" BIIn.d>. rpw:r, 1'" i4 ncnn<:d d1Ull, aM B tAo clou.t ball ..w. ..nI. T 0 "nd I in SitS. P ) (coMb r:rJ A$ "

01 C~(p»).

l'1l
ne.. N,(/ ) ,. SUP"; II I f 19$1lor 011 I

€ C ~(p).

.w......

PROOF: LH I € SI(S, F) bol ouch thu N,(I ) " I. F..,, " :> O. By tlot_ az-(UmW1 as ,bal in Tbooouo ~. 2.~, there ",,"'\.1 9 € SI(S. P ) ouch tlw "'... (9) " 1 &nd 19 <11'1~ I -c. Si...,. St(S, F) is dcnoe In CJ.(P) • ..., oooc:lude, as in Theorem $. 2.$, tlw N, (f) - ""P .. ,u.i'\ f 19$1for all I E C~ (P). 0

If

"_1>.. '

I

PTOpoooltlon 6.2.1 S~ th.at" .. pMliw, lInd II:! I : n _ (O.+oo( 10( 1" ........""'bi•. AIM>, let P 10( ... tkm""l 01 [I , +oo[ ond q iU co~'" apo,,,,,,t Th .... Pd.,,)' /· ,. ""P 19$, ~ 9 ~ndo ........ tAo ..I of fun",,,,,,,

u·

r

0/ St" ($).,.dt!hat NM ) 'SO

I.

PROOF: When N,(I) ;0 finite, the ptOpOSitlon is true. t,,- tbe ",marl< irnmo>o dWeIy foUo..-i", nldlCWl ~2.~ If,, :> 1. and by T lIfFor
"'*'"''''

r

"',(f. ) _oup aD

"',(/1 .. )

'SO

b-",p

r

1.9$ !i .up

r

Igd",

r 19$. which ~ that IUp r Igdp .. +co.

lbe ~ of a>n""oxeoot in Ir r ..ure. S u _ tbat " ill &nd lee A be. P'ln!lMur&b!e 8et. If F II • melrizabie unlklrm opve. tbe _ W{V. E n A .~), ,,!>Me E ""teoo.. ~ 'R, form • buio for .be unlJormjly nl -'lA, 1'; F ) (notation of !io<:t;on ~.3). Indeed , if B ill a IJ;~ w l-c. 01 A. lbe ... u ..... E € 'R . l1Oh that I'IE t:. B) !i 6/'1, and W(V, 8 , 4} a>nlailll W{V. En A, 6/ 2) Simil.rly. ..hut Fill. Banach ' poooo, tbe _ T(E n A.4} form • basil lor ,be fillet of ".,;,ghl>o.hoo<'&e6 to I € C{ A. 1'; F) if UId ooly "f, for 0.11 E € R &nd al16:> 0, 1' ( € En A : 11ft - 1I{;r) :> 6)) coo,~ to O .. n_+<><>. No..-,

~ an prai>e

0

"'*''''''',

" "'1_

I"

6.3

Measures Defined by

LH n be a oonemlltr _ &
~'I asses

and S a semirilll! in n . Write 5 for tho ".rilll!

P roposit ion (1.3.1 Ld (oz ).o· lot: 0 J"",iJ~ oj MnKn> """,pia "M~, o:t ..e ul oj iruW:a ;, • noIo.d oj n. S"f>I'OU IIlal 1%1 1ot:/ongI k> 5 Jor elle» % E X .M IIlal L.u".\' 1 0.1 ;, fin;u Jor ......,. E E S. TheIl 1M jv1Ictioft 1' : E -

L..,.......,o.

~m.e. oj 1M O. Jor......,.

I'({:'-}) ..

<>liS;' diff~

an atomic

"'_~. MOLIIOIIe"

c/4uu oj alo"", ..

% .... nge.o

1M {:.-}_

....... X . aM

z eX .

PROO" De6ne " .. \be _un: E _ L.un.\' la.! on S. By F,oc..... l·

L...,.......,

liau 6.1.1, " O( E ) _ 10.1 for all E E 5; in p&nicul&r, Ix} iI ..... llIUIP"J.bIe and f 11.1 . J,; _ 10. 1for eedo. z e X. Clearly. V" < ". Fix E E S. CiVftl ~ > 0, Iet %" ... ,z~ be poinl.OofX n Eeuchthal. L'~2ed 10,,1
•

!J.{F. ) - a • . I:S v(F,) - 10••1- v(F. - (%.ll:s: 2" for.......,. I :s: ; :s: n," /I.(F.)I
,-*,_

.. (E - E n X ) .. v(E)-

L

all

EeS. N.... Iilo. x E X . For.......,. t > 0, tba-e .... jo. . that v( E t { 2. Thea

for

" ({%Il - O

I"n:s

an

E

e S c:onWm",,,

L

la.- I'(l z lll .. I"(£) - I'({"H -

rE!InX •• ~·

:s:

10. - I'({z})1:s: .. (E -

1$0( £ -

euch

0.1

{xHI + L !o.!,

•

(x})

+ v(E) -10.1:5 :W(E - (%}) :5 t.

Hence p(lx)) _ o. for aU % € X . If 8 II. p·in~ ~ sucb l hat V,,(8 ) > O. thea it iI net included in tbe JocaIly "'~l&IbIo ~ n - X . Thil means that 8 oontai .... an % E X , which ollOW"I that" ill ou.om.ic. and tbe proof ill complete. []

Of,flnltlo" 1'1 .3. 1 Let. {o. )o<;x boo: a family of 000lp!c,< num~ ... h<>ee let 0{ i~ • a ""beet d. Suppoee l!.al I%} btloop 10 S. for all % e X sud>. lha~ O. ~ O. &nd M1pji(1t!f! lhal L.~(nX \0.1 is finite lor evefY E E S. Then " ,E_ Q. il llIt """"""" on S
n.

L..,,,,,x

For 1~1on witb IllIpett to $UCh a me 0). tho:nl' II defiMooreIn 3,8.1). Mor..,...... lor ""Ht % E X , U>er-e uleU an S_ 8 tOnWnl"l" roch l hal Vp({,,) _ 8)- o. and. V,,((zj} _ 10.1 > O. n,.. !'ri1)o 8 _ I",). rilld>]If'O"eE tlw I"'l l~ in S. Even tbouch ,,"', S\\reI 0: OIetIUrirliln n ~'" of the countable and Iho ~lIt&bIe __ III aDd ~""'" 5 by takirll pIA) _ 0 If " iIz Q)'''JUbie &nd ~ A ) _ 1 If A _ _ _ bioi.

t""

"'*

n.

6.4

Prolongat ions of a Measure

'OOO"'tu~ Itt. S .. ..-:miri", in O. &nd "a oornpleo: n " " i'l! on S. I~ _Ith, pttt 10 lhe cxt.cmi •• 01" and i~ with tUp
Lo:4. 0 bola

10 " &rfI kk"lIItal, U ibowD _ .

Pr0p08ltlo" 6 .4 . 1 Ld 4> ... ......iri"9 ttu'l

E - { lsdI' "" • . S"I'ptI«

(., for

n>
e

liz.,

S, ~ trim 8

01 l'.i"ttv+

e C.

#IOda tlI4I. 8

c

A

!M III ...·

...w A _ H

;,

W'It.li¢W¢

M " n&

ltolonp I<>

i f<w ew.-y A e S

""II ewry E

e•.

PROOf': By P~t"" .U.4 and 3.9.3 , Ch(P1 (0111.&1", Chw. ). and furtlltr {I · oW"" - {I · IIV" and {I'J;.. - { f·Jp for all f E £b(".). [t 101"""" tlw V ".( B ) _ V,,·(B ) for all B E" (Propcwi tion 1'1 . 1.1 ). If F • a locally ",,·Ql'Cliglble .. t , llitn F n A _ " .neglij!lble lor all A e S. by hypothesilz (a). n o,lmpliN thal F iIz locally ,...negli3i~ Now. If B e ~ • ,...necllJlW., tben 8 n ",,""'~lJIbie i «l 10< all E E +_ eed> " .negli&ible """ ......u....:h S E 4> in • ". '''4llpble~. EYio:Ienlly. 1_ remaII>II """ Ior...:ll locaI!y ,...r>egliglble Ht , _hich ttr.emOte • locally " ...,.....ti&ible.

n,..

e ...

Then deli"" 0<\ 51(., C} lbe eemioorm 9 - [!91 . dV ". - f Igl·
At an ttamJllo: tab: lor n an interval of R, and t.ab. !'or " ~t n,'~ measure 0<> n. !Iemlri~ ol..-inte:grable oeu and if i :;J S, thetl 1<.,. p.

J(" ". 'The

~ng

po(l!>
P ropooitlon 8 .... 2 UI .. II
n.

SM_

""'

(a) 0IIdI C••., illoth UHntiG1/J " ·int.,.,..",, ."., UHoUiGlt, .... i~;

M f

ill
f

! .. dI> lur all E E C;

"""/ai,,, S .

(~) Ihe ", •..;.., C ,.""rut<'lll ~ C

1'1I<-a" .... "" 5 . C fI, define £" .. tbe d _ of tboae !leU E IIlICh that tLolly int"&J1l.biI! with reepcd 1.0 both " tu>d ... and IUd> that l """dI>. For e'm")' "E C, e" ill a ~ and oont.j ... C, 110

PJI()()P: For eoclI "

E n " • _ f 1...,,,<1,, -

e".

f

5 c Tberf:fore, f 1£rU.t.<,., f l£rUd" fnr all E E S and.U " E C. Now fix E E 5. e lf" a ~ and cotll.&i ... C, 110 5 c elf and ,,(£,) .. v(E} . a

1

E,

I.e< 0 " .. .• o. be otrictly pooi\i"" n...nbon """'" tbat s.:s;_ G; _ I. Cu:Mdo< tho oomiri"" S l!o () .. {I ... . , "I ...... '0<1,,« c'- tl>o emPOl "'" on
r.....

J._.', s:

~1,

dod""" tluol.

~~'

.. .or:" :SQ, .., + ... +<>.""

lor all., > 0, . •. , "" > 0 (l_ualiIY of tho _ ",.,). 2

u._,

For IIWIl' , " ' _ 1 )! I. IIWIl' (p" . ... ",,) OS'IO, II' •..:1< 1" + ., . + "" _ I, aDd "-"crt l!otqe< r ~ I,.oall tbe _ " ",,!be Bord O'.~.of. R· , ~D<')d.

b:r ...........

rl

,

"1... ",1

~,

_,

'" · .. P.

+", _

ot ....::11 point (0" •.. , G, ) of ( Z' )' oudo that '" +, . . r . 11>0 "'tUPintJmial t.wwil.b _ ( 1 ' 1 , ... •"" ) &Ad r. Om . bootberhud. 1Or..-.,. '8i ""mile<

0<" < I and """'Y 1""?;'Or r ~ I. coJl ,be .......un B (•. ,,) "" , .... &n.l It. ~ 01 R. """" t.,' , .... m ", .. (;)"'0~ """" U of (O. I .. " . • ), , .... to_it! low ...iIh ... _ " ".

"r-' ,....

I.

iotep: ... G .... with n _ _ fl ,be .. , 01 mappings from (I. Z... .. ' ) into G. Let P to. ,be """"""" claoo of 011 ... _ of. fl.s.t1...d by P ({w Il • IIIi'll " n -. fur ..II '" f' (l. N_ let (GJ)'~'S' be .. puti..".. 01 G Old> .hat " I ~ I 10< """'Y I :5 j :5 t . M ... X , .. , .... ful>C.ion w _ ""- ' (G,)l £rom n .111<> R Ii>r GVfty I :5 j :5 t, &lid P'" X ~ [X!hs..H' p""", ,hat Let n

2 I••

~

I .... ,_

"".ho

p(x-'(oJ)_

,(")"'. ,, (~ ) "

r'

0 ,1 . .. 0, .

"

"

"'" """" (0 " ... ,", J of (Z+)t .uch ..... 0 , + .. . + 0 , m r . Show , .... , fur """'Y I :5 j :5 t. Pi X , . %) • B (r , n, / n l ({% J) lor ..u Intqero % 2 0 2.

"

eo.......... t- _

.. jnin5n boIliof tdjllenn, <><>Ion.

For"""'Y I S j S t , 1m., ~ n, bolla 01 , be Irtk roIor. Dr ...... b&Uo £rom ,h • .,.,.. ...... ODd .. itk ,.,1 , '. SIooor ,I>et , Ii>r """'l' (0" . . . , 0 . ) E (Z - )' . ,be proboobilj.y 01" dnt.";"C oJ boJII oI".be j. k roIor Ii>r 011 I S j S t II lP"'n 11,- ,be multi_ i .. l lui ... itk ~ (n o/n, . . . ,n . / n ) &Dd • .

'..v

F<>r "'""" i " , - .. 2 I. lo< p~ be lito m p, 'UOII on , ..... J.. of 011 ......... of N ...... that P. Cm ) .. 1/ " lor """'l' ISm S n""" P.(m ) _ 0 100- -.y m > ~. For mcb r-t .,.j.,.,J "''''''"''' on fl. wri'" £.(1) for /dP~ For.ntesen m 2 I ...". primoo fl. let 0,(") be .be POW" of" in t .... prime ra.c:toriutiooo 01 m . Aloo, Ie!. ~ ( ", ) be I 00" 0 _ "d;"'idoa m or ...,..

I

I.

1

1

£'(<>,) _ 0. - 1'. (0, > ~)
E~("", .... M'g

I<>

"",' ) ~ E., ( ~)o-.

•

L. ( '/(PC" - II)) ""''' .... -

£.(""'") "!oJ" - I -

p.

- 6.) ""',,) +00. and dod_ ,hat

L PIP 1 II""" •

..u ..,

+ 0(" ')'

<>(nO) c, .to s 5 to 0 .. n _ +00 ( .... S.ir~,,« " li>rmulo. ......,imals:l .. n - +<>0). w~

'15

ate

~

£or Cbapt.or 6

III

""" ..... .--1 number %:> 0 , deli... 6(%) .. L.~ .Iosp. Pr-ooo thai

2.

£or_binls[n::!:l .

i1,...-. [2il- 2!il .. pcositi~ for - " poi..... p ond ~

3

If n :!: I • ..., to I wben .. < p

s: :In. flO ;,. L Iosp s: £..(Ios' ) -

£ .(Ios' l - Ios(') +0( .. ' )

·<.s..

From . hio laol i ..... ao.Iily. ded .....1I>.I. 9(" l/n remai .. bouO>ded ... N (In fact, it ~ boo obown .haI 9(n)/ .. CI()ll'"'S to I .. n _ +00). ~ ,~

WIK[e

L .!.Iosp ~ kcHO{z' ). ~. ' 0(%' ) remai ... """tided .. %:!: van... I

Writa 0 .... 'u,oI"""'iril\fl of II, +00/ defined by . 1>0 m-. kc p/ p at - " primo p, ond let F bo .... funclioft % - L .s'losp/ p

4.

)1,+00( intO R. To oimplify -..oioft, put _. -!osp/p io< - " prirDo p. ond .... '"' 0 £or -=II iDl e,,", n :!: 2 .. h.idI io _ • primo. """ _.--I nutnberz :!: 2. '" [..,that

_

-

' l

F (u)

• ~ ' Ios'(u )

""

..

" - dod""" from part 3 .......

4

,,-

""" '"""Y '" :!: 1. let (1( ...) .. I.

L. 6,("') be l be Dumber of diotinct prime divioon

s_ .1t.r.1 £.(g) .. ""ur.u..r. to 1os(losn) .. n _ ex«<:ioe).

+00 (_.1>0 ",a:«Iilll!

" (H I;])}; Show ,!>at

E. • ... tIIOfI

(H I;]) (•. -! I;]))

III.., )+1/ 1"'11. b

OfId~\bat

il.

• ...wler \ Ioan lE.s.l/p. Dtduoe 11iM, lor -=ry e > O. _ .. -

~" ,, -. , "

d;,o;i..::t ........ ,., ~ oudo that " :S ... q :S " .

p.

+00 (Hudy-~

(16/(10& Ioc .., - II :i!: e) """' .... C

. bon .... ).

to> 0

7 Radon Measures

_w. . .

" ..., _ In &O.aI,yoioo _ dilruen. laI.. r ''1 "'."... 1<1 • I>U'" tinIM< ..... of DuIlriI a "''''''" <:ailed Radon ~ 10 on oxaonpIo. In ...... dellned ill IDeally aom~ Hauodoo'lI' II.'" .. __ Ii"""", _ 100.... 011 . he , _ 01 """,,1. _ '''''''1I0000 .ntb "ppcd. 8, ddDilioa. Radon _ _ .... ",Iated 10 ,bt ~ of X : . . . . wlU ...... j ',' hlp "'" l1l&I\)''' 5 r •• Foo _ p l . . ""l' ' • •' ' ' ' 10 m " abio for. _ _ ..... wi i.....".bkd. l't_itIoor. 1.3.1). 11.\00., Rodooa _~ ... _ _ _""'.ntk _~ 1oO .... ...at <>pontlono "" Ii r ",. """,,-.oct, irn';G E &r>d - . . .... ' iOOI of ""',.. .). Maay of t ho _

n.o. .

mU..,,,,

<_,

r..

=

1.1 fo< tho .... wloftoo ohM' "
po.,.,..,,,, oIli1
7.2 IA:t X be alot&ll.r 'x ~ " • .-.;bt l!' _ A I....., b m ~ .... u.. I s-<' 01 O>ap!ex ..... _ 00o. n="'... if, Ior......,...,....pooco K e x , t ho ... I' to \ Iie _ ol <»»ti_ wllllouhOU ift K • E- r pooi<;"" ~""U I".." "", 'H ioa Radon _ ",," (Thooo"D 1.2.1) &r>d....,. R..oo.. _ a.... a o...ioell_~ &.fi". .... ~...w. raptCl l
",Ie,..

.,.,.,i"••

flO""'' ' '

,rict"" '"

n- ...

_

obtai_ in CI<.~ ..- 17.J 'The "'"'" .-.11 .. lhio _'it", (ll- ew 1.3.2) io , . ... lo ' . n.. _ .",:,,,.. / of ... upwanlly dif«qd ... , H . cllntqrablot. loti-. 2 .. . r i _ fu_ .\oat ...... , . . ",p.. s f goW .. < DO io _ i~ _ , COi"~'6 to I;" 1M _ oloq , ,- ftll., of ........ of H . Aloo ... _ _ ".., It .......... la V lac _ if A II _ a..-ob/lo &r>d lIIOOIIo,at<.It.e...11- • > O,lheroo" a n _ ott U ...:I .........bIo __ .,., ol ...... pooet _ It..! V,. ' (U - 1') S c_

1'."""

t.

1. ' I . Ih .. """"""",,, iDtrod_ l be _ion of L.... n ......urabk ....W _ Int ... iti-.ely , ...... fimo;tio .. 0l'0 "!moo' _ -"riubIo _ .. Luoin ...... u ... blt if _ only if it ;0. " ",ablt.

The "r ' ......... dotfuood by I _ 1(",, ) 10< I ..,.oin_ wi.b _' ........ "'wort. III-. "'"" _ Radooo """....... _ 1>0,,'8"""'" _ _ . HQo........., any ""'" 1& r pOUot V"I'. ,<"", 1.5.1). COUftti.,.; ............ _ .........it ft 7S

Le4 ""

~

E' X .

.\alIy.

of.tom.,

=..

Ilodooo _

.'''',...beooy

1.& 0. of . boo pM of I _I<> imP""'" ,be RieInM" I.~ n.. ~ mlf;hl bo. b.&ppy to k..,. that ..II» ..., ' - dooe ... fa< hM ...1IotaDtialIy ' ...... o.ed Ritmann't Glaet: ...... t. "'."""".... _ _ . In~, t bat. bounded fuao:t ..... wh.Iro ___ poco "'PI><"1 • ~" iot.osnble if _ OIIIy if it. _ of t· .... ' Inut\)' 1& t.... I,... ~bIe. 7.7 V.. io..oo 'l'~ of I>&HQW WIIHII"' ''''''' _ _ of ...... Me ..

u_

"''''"'1''_ ..:I""" ........ _ -. _

-e'"

1.8 W~ """,i" .. 0." ~tlt!.b'" of <0<1>"1" ' P' 101 I I . . _ . . by Iooki.,.; . t .!chl ... _olM' . bo. .teaM ... '-;"'''''''PKI_...-I.I.'''''.

TIchI_....,.

7.1 Locally Compact Spaces

_",al

.....e begin by nlCfJllns prope>td of Ioc.uy """'s-t 1I.,o«Iorf( SpA ,11&1. U :> A and V :> B. Obkt .e that a IoaoIly coms-t Ha..oo. If it! DQI. lb: "frily """mat (_ Ex.,cioe 3).

n

"'*'"

n

P r-opo&It lot! 1.1.1 (Ury.,hn', Lem ..... ) Ld be a "",",,"I ~ _ . F " clH 0 o)n U' . P !IOOP': P ut r , _ O..... _ I . and let .... r • • r•... be lUI enumer:alion of . be .-.a1onaJl in 10.1(. We un find Open ICI.S Vo and Ibm. V, $Ocl1lhat F C Vi C ft, C VG C "0 C U. n 2: 2 and V,.", . .• V•• Ilt.~ bee" ,I .. : " 80 t.hat r, < r; im[JI .... ~, C V", Then (mil of t he numbel"& r,,, ..• r~. oay r" will be the ~ OOi: ..hich is 8",.lIer thaD r~+,. and lUIOlber ....y r ;. wilt be tbe....-...l;..... nne Ia:p:r thaa r~+,. Then ..., c..n fir>d V••• • 80) , Ilt.t V" C V,~" C V.~. , C V., . ContilluJ,,&, " " obU.in__ family (V')'E"'''I>'Q of open _ with the IoIlo:Iwl"" po"",,"'- F e V, c Vo C U, and, > r impl .... V. C V•. Defi....

s.._

f()

• J: -

{r if::o:

E V• 0 otherwise

g.

(.). {l,

if J:Eii. otherw"",

and I ., "'P.!.. g" inf.g•. So! is \0.-"" 8emicomin\lOUll &rid 9 is "PP"r -moootillUl)llOlL It "' cle.,. 0 SIS 1. I~ I {s) _ I if J: E F. and

,ha,

that / ,'&IIisbe! ouuide Vo. To oomple~ the proof ~ need only show that ~. Inequality / .('Z ) > 9. (>:) is ~;ble only if r > I, t: E V.. and t: f V" althoucl> r > I lropliel ii, C V._ Hence /, ~ 9. io< all r , I , and / ~ ,. S u _ f er ) < g(r) !'or lOme r. 11> I , r e ti'" and tru. ... oontrw:lietion. FiJ:Wly, / ., g. 0

/ _ n.e

DeIInlt lon 1.1.1 Let n he .. ~ space and / .. mappi,,& from n illl<) an R-_ space « into It The IUppon of / is the clBoure IUPP(Jl of {r , f er ) ~ 0). &!ui.....lelllly. it .. the smalk:oot cloolod I Ut.et. of n outside ..hk/l / vanis'PropwlUon 1.1.2 UI n lie" /ocoI1~ """PGCI Ho1UlJqrl! IJ'OO'. K a Illairtt.d' in U , nICh tAo: f er ) _ I Jar all,.,.-nII t: in K . PROOF : There ....r. an op"n _ V, ..hoooe d ".ure V is ooroJ*'!, IUCh that K e V e V c V, Since V iI comJ*'! . ;t ill normal (in the reWi ... I
>'-{ o

II

g(x) !fre

and obIcn-e lbat / """ the desired

V

'(ren - v .

_"leo!.

c

Theo..,m 1.1.1 SUPPN~ V, ,, , .V. 0,., _01 ~KII of " /ocoI1, ...... PGCI HoouJqrff IJ'OO' O. K " """""'. ,,,,,I K C U,<.<_V•• ~ IIler-t t:ftIt 1lOrIlin_ jt.nctitnu J; (l Si S 01) from n inU> j6, 11 nICh tAo: ra)

Jar eoch

I S ; S n.

sup~/;)

if
,.) E, ;!;.s o /. (t: ) S I /M ~ t: E 0, 011<1 E I,o;' SA/. (r ) ., I 1M m!f'J'

r e K.

PROOF' ~ z E K """ .. neilhbort.:.xl V, with comJ*'! cloomre ti'. C Vi b......,; (depftodi,,& 011 r ). There are pointl r" ... , "_ suc:b that K C V. , U ..• U 1'._' f« I ~ i < n. "" K; he the unioa of t"'- "" included in V, : by Propooiiliou 1. 1.2, there io .. COOlirwouo fUDCtioa g; (rom 0 into (0, II, ..ith DDmJ*'! suppott COIlIalned in vi.•uch thai. ,, (r ) _ I ror all r E K •. Define /, '"' , .. h '"' (1 - " ),,, .. .. /. '"' ( I - g, )( 1 - ,92 ) . .. ( I - ""_, )",,. It io easily ..".;6td , by induc:tioa. that

I , + h + -.. + I. - I - (1 - 9, )(1 - 112) .. · (1 -

g,.).

••

116

1. R.adOII , , - . -

Since K C K,U .. ,u

K~ .

ro..eYffy '"

E K 90 ("' ) _ 1 !'or

8~

leall\ooe i ; to

{/,+ .. ,+ I .)(.f) -I.

0

P roposltloll 1. 1.3 ~I fl io<" IooaIIg rom"",,1 H,,~/J .po«< aM I B «tn. "'ndwm /mm fl .m.. .. ~ Ba~ad>.,...a: F. S"P~ I ;, ltiPp.orkJ.

ti_..,

n.m,

'"' ...",., R ·

,..""tWnJ

(a) ... PP(II;} C K fer _,., I 5: i

~

M

K;

L'9S~ 1I;{.f) ~ I

lor..ll:r

E

> O. u.u. <:lUI am/i""", ... , udtlJuu

'"

(<:) ,,~ ;, ." "rtitra" ,...", <>1 ... pp(h; ) lor 1 50 i 5: n , 1M..

l e:t) -

L

"-(:rll(%,) :$ t

an.
L

1/(%)1-

'5'S.

11,(,,)1 / (%,)1 5: t

LS.S~

I()F oll % e fl. P ROOf': For any • bo'MS; ... to lbo boundary of K . lben uisIB an open ~bood V. 01. O«b lhal \/(: )1 5: t £or &11 Z E V.' Let K' boo lbo "'" 0I11>ooIe pOints in K which book><>« to _ V.' as l' ruM th""'Jh ,boo boundary 01 K . N_ K ' II ~ &IIod (OIIUi ...... In ,ho in~ of K , to 'h<-r<>« to tboo $&mil U" By Th"'dm 7. 1.1, lhere c:ontlllllOUl fuL>ctiono II, from fl 1Il10 10, 11."""

(U.l.

IM1

""* "

Iweupp(h;) C U.. L'5'<. h.(",) ::; 1 on fl, """ For:ll". e eupp(h,) (\ ::; i S n l . t h
L ,S'S. h;(.f) '"' I

1(',

:II" e U,. BUl Ihie i""'lu&ll,y ie .ill '"'" If z 'I- U., \woo' ... . in Ihie h; (%) .. O. ii<:1IU, lor t)\~ % e fl.

for e><:ry c&8C.

on

1(1 - heI(%)) ' / (%1-

L

1I;{:r1/ (zi)l:$ t tl - ho(z )).

IS'S·

I/ tz ) -

L

h; (.fl/(z, )1

~t

for ~"'ry z E K '.

' S;:!;o

Ne:n, Ihot"')/ (z)l:s t , ho("') b

I/ {z) -

eYe<}' '"

L 'S'S·

e K - K ',!IO

11' (" )/ (%; )1

~t

,h.

in cue. Evidemly. this t..5\ inequality ",mal... true ..·hero %;. K . Finally. by. siPllt.r ~umenr.. II /(%l! - L, ~ .S~ 11. (... )1/(.1", )11 S € for all ... E n. 0 [n

....,.do. 1 an be "WlDxhn..ted by dtwm[>OO&ble functiom. LIte.

ott..:.

wise. ""e ha..... tbe folJowiD« prop
I"" 1ocaI1, _p"d "aUldorIJ 'J'II« fl. Id . ... u.. ......n. rint......vtmg 01 u.. K n L' . ...w.: K . L range u.. eta... "'_.mei «II.

Propoalt ion 7. 1.4

I>t'C1"

Ld F ..... I'I'Dl &"",,11.,...,. ,,71<1 1 : n _ F .. mt\I;,....,.,. maPlri"f..w. compod...",.,.rl K . Then !hen: aUII .. ~ (g.,)~>, in 51(. , F). OInIoe.,i'lf .".qomolr to 1• .noch ~ suw(g.,) C K lor <WrY .. :!: I. P ROOF: For .. fixed integer .. 2: l. the .... ..,isu .. finil that 11("') - f(~)1 S 11.. b aU [ m r.nd aU r. , E M •. For """'1 [ j m. put " j" M; n (U'SISJ- ' M,t. and let '" E F be ouch that (1(,,) - 011:s lIn for all ... ENs. If we put g., .. L':5JS" " j. [1<,. Ibm 11 - g,,1 [ I n. 0

:s ; :s

:s

:s

:s

In tbe now.ion of Proposilion 7.1.4, ... ben F .. R r.nd 1 ill ~ti ...... _ may IU_ that o:s g,,:S 1 (lake a, _ infdl<, 1(,,) in the .. boV1! o.rguonent).

7.2

Radon Measures

Let 0 be .. locally compooct HaUl!dorff spaa! and 1(O.C) tbe IJMOI! of a>mi .... ...... compooctly support.ed functions from 0 into C . r .". any oompooct lub>Jet K of C . denote by 1(fl. K : C ) lbe of tlxMe rontinllOlll func:tiono from 0 inlO C ",hich ,_ish ouWdt K. ~ on 1(O.K:C) the norm 1 - ... p..,K ll (,,)1 - suP.~n ll(")I, and. on C(K .C ). consider the norm 11- ,uP.fK(g(,,)I· The rnappi"ll 1 - flK from 1((fl. K : C ) [ntoC(K.C) ill an iIornct'1 of1t(O.K; C ) onto the closed ""btl,*", of C(K .C ) coneiIIl.Inc ollboooe ODtIt,nOOlll functiono ...1Uch ,-anlIh on the boundary of K . Hmo! 1((0 , K ; C ) io " Banach s,*",_

"'*'"

Definition 7.2.1 A c.u,.... form" on 1(R C ) is callM .. Rodoo _ute wbe_ •. lor """'1 compooct ... boIo, K 01 fl. lu _Iricl;"n to 1t{O. K:C ) ill

-"-

:s

If I, E 1(+ . h E 1(". 11 E 1t(fl .C). and!91 I, + h. lhe function 90 . equal to (,11,)1(1, + h ) at pO!ntl ..·heTe (1, + h )(r) -flo 0, o.nd to 0 ebewben:. is ronlinOOlll on n (wilb i _ 1.1). I<>dote + h ,.."io'-. MOIw .... 1,,1 1. r.nd 11 - II, + 111· n", 1(fl. C ) has lhe [>tope> lie! required [n Section u . If " • " R.oon ...... u ... OIl 0. then 1,,(1)1 < g,,' 1t{O. K ; C)I . N I l ( wbe .... K _ .upp(f)) £or ~ 1 E·W and for aU 11 E 1(fl.C) Iud> lhat 1111:S 1 .

:s

~Iore, II

has finl~ ...n..IKln_ Moot<>W:' , every ~og 8eq\>CI""l (f~)~ ~. In JI " ",h;eh poinlWI3e to 0 ..., uniformly 10 0 by Dini'.



IlvJoo em; thus ,,(f.) teods to 0 ... n _ +00. and " "' Tbo.orem 1.2. 1 EIInJ,."nm.. IiIll8T form

I' '"'

&-

Daniell _ , . , .

JI(O,C) II G FWd"" m..,_

h~

P!tOOP: LeI. K be &- ....."p..cuu'-t. 01 n . Tbe"' ....;.u HOlllinLlOO>! funaion I. from 11 into (0. I), whh comf*'l ruppo::n. t <><:/l .hltt 10(') _ I on K . For $\~ g e JlIO, K;R), "" ha... -111' 10 S p::; 190· III and 10 lp(g)( :S 19B'II(MF iI>&Ily, fur e¥ef)' fJ e 11: (0. K ; C ), ,here ex\31$ { e T (t he unit cirele In lbe """,pia P!&M) luch that 1,,(g)I - , ,,(f} ) -I'«g) _ ,,(Re«fJ»). and (P(g)I ;" mWJ..r than I ~,g)H' ,,(fll) < bft· ,,(M CJ

Ii""",

Evoery C U""", form on 11(n, C) ....hid> hao a finite vvialion ill &combination of foor poetll ... 11.-.- tOmw. n."" it ill .. RAdon _ , . , . In 1I>ort. tloe Radon..-.ures on 0 . ... aactly the CH.-r !'or"", of finl" .....tion.,..'tr "H(!l.C), and they "'"" Daniell n ............ Thc,erO'E, ,.., mo.y &ppIy to ~ M","'fOO the 'fOOuI", of SeclIo:m 1 4_ [n l he 1OO!QI>d • ...., ..-iliid. M {I1, C) (resp«tivdy, M (I1, R )) be tbe ,~01 c:omplcx (.....,.,aively, ruI) Radon IMUW'OII, and ,M +(I1) lhe cone of positi"" Radon rneo.ocu,-es on 11. U " e ,M (O. Cl ... e M +(Ol ...., . <><:/1 ,hal IpU lI s v(J) lOr all I io W . tben (P(g)1 villi) b all g in H(O, C). Indood, lOr € > O. lo:t 4 > 0 "" suc:h that (so( ...)1 S € wd 1" (",,, S € for every I>;n '}{ (O.ruw(g): C) _~~ 1...1 S 6: by Plopoosillon 7.1.3 ...-e """ find &- finite family (h;)"" let H+{o'toJPS>(g)) and &- family «(t,)"" of OOtoplel< .. ~ l uch \hal; !9 - E IE , S 6 and 11, 1- E>E, ....I"'! S 6: th...

s:

0,"'1

!»(g)! :s III ( ~ 0,11,) I+ f :s

.

from which

r:, 10,1. (P(II,)\ + c

:s

r:, 10,1. ..(1);) + •

:s ..(Igl) + 'If,

~he

!lquality 11I(f}1I S .. ([gil folkJ'O-s. n..more, f:VelL \.he!tlSl. condllioa of Section L4 bokIe.

Definltiou 1.2.2 Let I "". nonemply inlen.... i .. R.. and Ie\), he Le'-«uo L' ., Ul'e on the aecnlring of I. The line.r 10m> I _ I 1.1.>. on H(I. C ) Ie <:alit
.uoo

"",ural

:s

Obten-e that IIIJ...I.I (IJ - "lUI for.....,.,. J E 'H(l. C). if (0 ,,0] ill &lubinwval of I conLal ning supp(1). !II) that t he Ii""", form I .... IIJ...I. 10 o.ctuaily . Radon '" e"lUte. Hencdonb, ,.., shall do!not~ 1"..1. t he /let of lo..e, setnio:mtin....... fuDCIioDS !rom n mto jO, +001, _!>ere 11 Is a giYUL locaiJy oom~ JlaUlldoo-fl" 'J"'C1'.

~

Lemma 7.::.1 E--r

I E :r

if 1M upper ...." " . 01 11K

m

IgE W;g ~Jl.

P ROO ..; for It\'a)I " e O.uch that I (r ) > 0 and ~ry ~ nwnt- a in jO,/{")!. then: ciata a c:om~ nei&hhorbood Vo'" for wh\dl fill) ~ a 011 V . On tbe otllft hand. tbtte .-xist.s a funcboo 9 e H+ .uch tI.." supp(g) C V , g{,,) ... ... .oo m) S .. for all \I e V . T hUll 0 S 9 S I &rid g{,,) ~ ... which provtII the lemma. 0

for tbe remaindrr oIthil oe<:lion. " "";11 00

~

• poeiti,,, IUodon

meMUn:

n.

.:r.

DefInition 7.::.3 Let I he a function in T he burnt- ,,'{f) .. .uP... .....s/ p(, ) ill called the upptt i"t~ of 1 (... I&ti"" to p). Emmtl,.. ,,' (f) .. ,,(f) for

Tbeo...,m 7.::.2

u! If

Ie ..

It\'a)I

1 E "W .

~~"

-...11I·ctmclc.t ""'..,

01 :r.

TheIl

,,' (RtPrE H') ... RtPrEH ,,' (6).

PROof': Pu, 1 ... RPrEH" We fim PfO"" the lbeoiem .ben the funcdo... 9 E If and lheir upptt ....""Iope I beI.oog to 1( • . TIltn. by tbeoi .,a. tbe fIltK oI.-tiono of J{ conH'&e' uniforml,. to 1 OII.UPP(f) . and .... on fl. S,...,. " io cootin ....... 011 H (O .• upp(f); C ). the I1:l&tion ,,(f) ... IlUPrE H ,,(g) dePr«I . We now,- to lbe "",,,raJ.,.,... C1eari,., ,,' (9) 'f ,,'(f) for all I E H. By lbe definition of II ' (f). ,t suffica to PfO"" lhat. for en!") '" E 1(" .... illfyinc !/I S I , ,,(!/I) ill leoo than IUP,. H ,,' (6). To thill end. Ie\ . , be , be II!I (I" E 1t+ : I" ~ II . fore.(, ) , E H , and let . be lbe unlonoftbe YClope of t be funaiono inf(Ib.I"), when! I" clf:bdo .,....,. • . But Ib .oo lbe function inf("",I") heIon, to 1(+ . . . the flm part of the ~ ptOYftI dw ,,(Ib)'" oup,..;."Pnf(Ib.I"». Now...m '" e . lioo iaa.n • •. Th..

Di"'·.

rouo.... ..

..""""" ... o;OIIClude Immedl&1.ely lhal "(Ib) S luPoEH " · {I ).

Propoooltion 7.2.1

,,'(f, + h ) ... ,,' (f,) + ,,'(12) l<>r all h , 12 E ,;r .

P ROOf' Wben "', (' :specli .... y. '1'1) runs 'h~ {"" e 1(+ : 11'. S hI (, I""'t' ....,.. 1'1'1 E 1t+ , 'P"l S 12)). tbe functions \0, + '1'1 form an upwardd'n:oct.ed.n. w"'-o upper ELhe)Ope ill j, + /o.TIw:..for(,

•

IUPV-(.",)+ "{.... l)

- oup"(,,, ,) + .up ,,(.... ) '" ,,'(!.) + ,,'(h )·

a ,,' {L.u h) -

p .....,.,.ltkm 7.2 .2 Uifirille) m

:r .

Lou ,,' (M

lur n>e1Y lomiJv (finite

ur

PRIOOf': Tbil I'oUo..'1 immediluly from Proposilioa 7.2 1 and Theorem 7_2.1

a

7.3 Regularity of Radon Measures

"'*'"

Let 0 be .. Ioo:-.lly oom~ Jj .".dorlf and,7 " the 11K of pae;t;~. ~ oemlcomi_ fundlool on 0. Sin« the c')Dditione of Secllon 3.1 bold • ...., ....,. ... fredy the resW'- of Chal'l"r 3, and o;dd ""'" ...--.

Ld." be a Radon

III

N ft

on

O.

Denote b,- V the union of -.II p"negll.slblot opel' oet.o. The 101d1c&\pw,;et iI ot.llold the support 01 " , aoo is writ"'" .upp(p) T beQr.,m 7.3 .1 V,,' (A) _ iolu V,, ' (U) lor fW>'J I1IkeI A cl fl . ate""-'.,..... IA<: duI "I,."....," oontaming A_

~

U

w""""

PIIIOO" \ \ 'e may ~ only the caM V,,' ( A) < +00. Fix 0 < f < L There w.t& / E ,7+ ...... thall ~ I .. &lid V,, ' (A) :5 V,,' (/) < V,,' (A) +f_ 'The <>PW tel. G _ !z E rt : I (z) '" I - II roota .... A. On the other Iw>d,

I > (I -I) IG, 10 V,,'(G) :5 (1/ (1 -tj)V,,' (J) :5 ( 1/( 1 -1))(V,,'( A)+I). and the lOIull follcloow.. 0

_icooL.tm_ H,,' k."

T .......... m 7.3.2

U/.

~dil'tl:u4,.t

cl l... nl~.

( rupccli..eq., ~ d<:>a:n_rJ.ofm,;u4 ..,1 al " .inllS' I II, .1'J't" H~"""" I""d~). II WP/£H f I dV" < -tQIl (~"...,ei.d" Inf,tH f IdY" > -00). tII~ Ihl up"," (rup«tit1(J,. r -) CIL"./"., 01 H U "',"rqraWe, "nd I """""'9U 10 9 ,n 1M _ II &loAf 1M /i1kf" 01 _lioN 01 H .

to...

~

P ItOOr, We ..;1l PI"'J'It the stalemen~ ooly ror lowff 8emio:xltlnllOUll fu nctions, o.od ..., m.t.y ""-VS- thai. 1l haoI a """"'''''' elt~ I~ . The func:t.."..

r

(...."""I....,.y. r), ... twre I ~ _ H . Iorn> .... uporard-di~ "'" 01 loom' oemk:ontil\llOUll fundionl ... boIo: upper e."eIope ill g+ (lUIpectiffiy, • Oo:wnwo.n!-diru:led iJeI. 01 upper IemloominOOU6 functloM "'ho8e \ower enl/& Iopo: is g-) . M(Jttol .... , f r.wl' :!O f IrlVl' + f I; rlVl' for ..u I E II. Th.. ,",.......J only Pf'M' lhe n,o lIatnr>enU of Tlw>mn 7.3.2 ... ben H """';-u of pOIili ... runcti(ln$. If H 10 dil"flCUd UJNW and oomp<-.! of poeilivoe, ~ ~in"""" fu"", lons. tben r gdV I' .. "'P' EH r JdV I' .. SUp' EH IdV I' < +0:>, 110 lhe ~r _loominOOUOl fW>d;"" 9 ill i!ll~ and f gdVl''' "'P/EH IdVjJ. Since I :!O 9 lor ev<:ry I E H, lhe fi.1to!r oh..:t iool of }( 0CIn'""'"P «> 9 In lhe

-.

Next, "'_

f

J

thaI }( ill dirccu:d oo...nward and comptJO!ltd of poeili ... , upper

tentinUOUll functi(ln$. We

a"

~

,..ri<:t ...... &Uentioon to doe _

in ... hld>

H has pi"!' demenl ". TMre ...... • an inlECrr.tw. ~ """i!:ooti......... function II Iud> that I, :!O II. for each I E H. c:onsIdt< the func:tlon /' ukln« tbe .-.1"", /I{z) - I (z ) at poinu :r ... t.er-.. I (:r) < +00, and the .-.1"" terO at poin'" 'I. ...benl 1(",) .. +00. When I ~ ~ H. the /' forln &II upward-dirul<:d !IO!t 01 politi ... fuactions, in~bIe o.od lower .. mK:ontinuoua. !.~, f /' dV I' "!i J IIdV I' fur &II I E II . Thull ... long the fillet 01. lIO)dionl ol H, CO.hCOga in lbe lI>CIUI to \be upper tm"dopc g' 01 the N.... " - /' .. f and /1 - g' .. 9 al poin'" is fin;le, III) f coovu gee «> 9 In lhe meLD alone the 611 .. ol 8<>CIiool 01 H. 0

r

r.

..-be...,,,

£ado o:>mpaCI ..I ;, I'.infk, anJ VI'"(U) .. OUPK CU. K.. • '" V 1'( K ) for tl>ef)' ."..,. Id U . Pr-opoo!IltJon 7 .3.1

PIIOOt': Ewry compKt ..... K ;' lnl"V"bie boa._ I K .. InJ'~ H • . ,~ ,.I. N.... , Ii"'" &II ~ "'" U, let ~ < VI'"(U ) he a real nUInbf,t. n.e..., e· ... ' f E 11: + ouch thai I :5 1/1 and VI'( fl > r. Put S .. . uPP(fl. and K• ... (z E n : f (:r) ~ d fore-y t > 0. Then I S I ll<, + t l$ _ II<,. 110 VI'(f ):5 V,.(K. l + cVI'(S). N_ VI'{K.) ~ r "'" t IIm&!I tnI>lqb. a

A ,d A C n iJ ifl.lnIr if. for ~ t > 0 , tM", airt. """..,...:1 0<1 K and an ml,,; 10k Open ..1 U ,~tJu.J. K c A e U . .... VI'{U - K ) :!O t. ~, A if JI"""YO H. if GIld cnIr if, lor tvCrJ' t > 0, _ ..... fi.IUI.. """....".<1 K c A nodi !hat VI'" (A - K ) :!Of . In IlW t:&H, Iht.", aUt. • un;,," 01 diJjoinI 'd x t. A, C A,.uch u....t A - A , if jJ.MgIigibtt. P~;tio" 7.3.2

..,,,,,/4hk

0:>"',

eJI*"

PROOF; If A is inlqJ1o.biI: , l ben, by ~ 7.3.1 , lhen: an Open ~ U :) A ouch that Vp" (U);' &ebil .... ily c~ 10 VjJ(A ). M.,..,.,...... b e....,c > 0. ttwre ....... . pomth..,. upper "'miconlinuouol function I. wlth compKt

,uppon: S, web that / S I .. and /(1 .. - /ldVp < t I l (PfOp(I6ibon 3.7.2). T he .. K _ {z E n : lIz) 2: 6) .. eloooed and contaioed in S for evny 6 :> O. TIm. it isoom ~t and . .. ina! / < I ... K is included;n A. The .et B _ A - K is int"V"bio:, and / S II< +61 ". 1Ienr:e

f

/ dV"

< V,,(K ) +W,,(B ) S V,,(K ) +6V,, (A).

and finally

But. os 6 ..... arbitrary. Vp(A ) S V,,( K)+~ for "su.iubleoompart_ K C A. Come ...!y, if Ior ~· t :> 0 there CXOlUl " """'pod set K C A ouch that V,, "(A - K ) S <. t!>eo 1.. liN in the closure ("";th ""'r>ee' to 01 the eta.. 01 tIM: I I< ( K arbitrary oompoa wbolot of A), to<> A io integrable. N.,... t he !>fOOt. of tIM: tim. t ....., slIlemmlli an: complete. Fioally. if A i9 Int~. 0.6". by Induction" ( K~). >. 01 COInpad oubo!oet.o of A M follooo.T K , C A and VIll A - K. ) S I; K~ C (A - U ,,s.s o- ' K. ) and V,,(A - U' S'So K,) S l i n for " :> 1. 'n>en A K, III negligible. [J

4(p»

""'I"""'"

u.;o"

By Propoolitioon 7.3.2, " ou~ A of n ia Ioea!ly nogliglble if and only if An K Ie rqj;pble for all oompe<:t IietI K . For every . ubeet A om. V". t A) ~ W lll<e " . 1< • , V,,{K ) (Proposition 3.7.9) . A '*' 'Y and $Ul5eioeot o::or>ditOon that" ¥uboott A of n boo: """""",,,uTahle 10 that An K 1M: p.integral>lo: for all compect oct8. Th... clo8cd 8et8, and hcnoe all Borel octo.. an: ""","""",,rabIe. A C n is " . ............bIe if aDd onJy if it . pllts the outer".".....,., oS
P t"OJ><»lt km 7.3.3 Ld A lot A J.'·mou~rnble and J.'·m.qJ~ e let. For ......... • > o. u.c.c enol an o~..,1 U OM G ",,~nu.Mo F 0/ """"'-""" ",r.. , ...h IJo.cI F C A e U aM Vp"(U - P) S •. WIIw n i4d/" 0 cotmlCbk "mon 0/ """'''''''. .. r.., F ma, II< lah .. to II< ew.oed.

""j""

PI\OOF' A""b(c union of diejoim intqtnt.ble 9Ct8 A. (n 2: I ) For cedi integer n 2: I, lb.", aist "" open "'" V. and ..
0

,.

7.4

LUsin Measurable Mappings

Lft 0 ard '" be III in Soldioo 7_3 PI"O~ition

~ G

7.4.1 u l F'

m4 ",,0uid6 ~ ~

,,,,,,,I'g'''''/

J:0 -

( rwl " .......1ri1r "",lrUtlltk), F' . 11= the JoUmving <X>fI
OQ"IJitte!cflt:

(. ) F.... "OO' ;,.,I.... obk 0011 A. lA= oUl "ntPigi"'e nb"t N oj A Gn.d. ~II« (K. ).~l oj dil;iGiJ>l ,.,.,.."...:I'~" 1<J>
"""......:i

~I

1< , th~ e::ciot" .goble ,.,b,oet N

I>J A m4.

"'1"''''''' (K . ). V 01 dUi<>inl """'J'I'Ct "" I<J>"" .." ...... ;, A - N . • \ICII tha.t. cacJ\ fI K • .. """ti""",,",_ (~) F....

""mp"'"

cacJ\ "I I< .n.d cacJ\ € > 0, th~ eN" ......."",,1 ...u.1 H oj K .1OCh /lull VI'CK - if) ~ t Gn.d fill .. "",,",in_.

( I ) For o. th~ erio4. "" '.p,d ,.u.,1 1I 1>J A n.clt. IiIcI VI'CA - If) S r .nd fi ll ..

"""ti""" ....

PROOF": Clearly. Ca) lmp&.. (b). [f K io. com~ sec ..00 ( II.).~, is. ""'Iuen<e 011 in condition (b). lOr t'Vfty f > 0 tbon: ~ P > I such thal. V",( K - u,~ . ~. 1<.. ) :S !. ~ fI Ul"~.SP K. II! c:ontinoous. ThIl'I (b) imp/leo (c)_ If Ie) Io:>kIs and A "' ... in\ep..bIe ro:t, let ( 1<.). >1 be a lLO'! 0 and b- f:VH)" n 2: I , lber-e exi:It.I • """,pacl IU1>!H II. ()( K. ouclL thai. VI'(K. - II. ) S t/'1"+1 and fi ll. is cominuoos T hen lei p 2: I be ... integer .t>do that V",{A - U'S.<,K.) :S f/2 , and pullI - U'li.Sr lI •. Clearly. VI'(A - H) :S t and I I If"IIH,ontlDuL)UG. ~Ior-e, (e) imp!'" \d). FlnaUy, (d) lmpl;... (a), b.!a""". if A is ... integrable ""I , """ can inductively _met a ""'1_ (K. ).,~, of diojoint compact ... ~ ()( A !IUd:. thai. V I'(A - U, s . SP 1<.) :S II p ~ all p 2: 1. and ....,h \.hat flK. is -'Iinoouo..

o

Deflnition 7.4 .1 A mappilq! "' .... I\ed L ...

f $ll.l.W"rin« tbe ~itions of P.-opOOtio<> 7.4.1

in ,"'".""'able.

If f II LLl8in 1"" _urabOo. all ....ppI". wbich ogr<>e with f locally a1!11011; _rywbon: ...., Lualn ,..._urabk A mappI", from !l lnto F that. 11M • Io
,M

PropoIIilion 7.4 .2 Lei (F~)~~, III! 4 ~ of ~ JP4Ct' <1M, ..0. fl. let I . : 0_ F~ III! L.... n "'._~~. FM ~ """'I"'d Id K c !l 4n.d ~t > 0, lilt,." eN4. """"'J'IlC1 n.b(1 L 1>1 K ,..q, tIW. V I'{I< _ L ) ~ € 411<1 MdI J.I L"

"""tin_.

'I'

ate

•

PROOF: For
t-

Tbeo...,m 7.4, 1 Under 1M. hWOlheril of l'ropo.itionT.4 .! , 1>"1 !f.!~l l "nd ,.."po~ 1h4l" ;, .. ""nlin ....... mG".

Then" 0

J .. L ...... I""'~_

a

P ROOF: Obvious.

PropoooiUQn 7.4. 3 LeI M be the ,, ·a/g" u>~.paa ;., Lui.. 1'."'............ ,.,.

he " finit<'> putitio .. of rr inlO M_u such th&t / is <XI ; E 1, ,hffi, n~ • !Ieq..en<:e (K, .. ).,?:. of dis)oint O:;' '''pao:l ""~ of E n A, such that (£ n A. ) - U. a • K •.• io ~igihle_ Si""", /I K •.• ;., continuous for ~ry i E 1 and <m'!fY n 2: I , f io Lusi n ".~. 0 PIIQOO'. Let (A; ).."

P T<>p<>oIition 7.4.4 E.",.., L"';n I' . m ........ rubI.! moppmg / mdri",bk IP><'" F ;, 1'-........ ~rGblc.

from n into ..

PROOF; I,g d be a ..,.,.,pUibie distance on F aDd E lUI illUf;. able !let. ~ uisUl • ""'I""""" (K');l:' 01 disjoint romp&« ""biow;!S of E.uch that EX, is negli&lhk and /l K, is c<m\inuo .... fO<" ~""'Y i 2: 1. ror fiX$! " 2: I &nd -=II 1 i:S: n, ,Iw: ... exisu .. pe.rlilion of K ; into in~1e OIU8 &J (with I :s: j :s: q;) such that the "",,;11&1.,., of f O\'er E; J ia lea! than l/n_ Oefj"" an * / UlIlple mapping g.. from n into F as foIlo:>M: g" .. rt CSIam on """'" £;J (with I :;: i :;: n, I :;: j :;: ,,), ~u.oJ to any IUed ftll>ll of I 00 Ihie tiel, and tqtl8l to a fixed • E F for all % E E ... biclt belong to lIOn< or £' J - Then tbe ~_ (9n).~' oonH'8'" to fiE AI""'"t ~ ... he", in E . []

U",

:s:

en

I""

PropOlitlon 7.4.5 La F Ioc p mt lriUlAk uni/""" 1pOOe. lind kI (J.). ;>, Ioc 0, Ih~ am. 0 ~ ocl K c E .n<e/Ilhol V,,(E - K ) :;: t, I. to ",,","i .._ ........ K 1M ~"''l' " E N .
'VI

ate

l>< G mqping Jrcm 0 inlo " .... Iri....w. .,..".. :nw:n I .. Lpn ,,· .........unobk if """ o.sIJ if p if ,......... ~...61e.

Theorem 1 .... '2 Ltl

I

PIIOOI'" If I ill "-'-"'able, tben it .. Lusin ,,"_urab\c by P,opoe;' t~ 7-".3 and 7.4.5. Coo_Iy. if I is LllI!in ,,"measur&bIc, tben i t . II' w EEurabic by Propooihon 1.4.4.

0

7.4.2. a Lusin ,,"~bIc m..ppinc from n inl.O a ~ opoce ",Ill _ simply be caJlod .. ",w EE' ....bIe mapplDf[;. Now 'Theorem 5.2.5 may be refined .. 1OIt.:.>w.: In view

of 'Theoo~w

Tbeo.em 1 .... 3 Ltl F .. G BGnadI sp
I/

P ROOF": Iu in TlIol<:>oew ~.2.~. _ wa.y "l8trict "'-"""'_ 1.0 1M CAllI: in ...hleb I belonp \.(> SI(* . F) and N,(/ ) .. L Glvet> € > 0, Iberuxisu 9 E St(R . P ) ouch thai 1 , 1S 1 and 1/ / gIl"j 2: 1 -t (by the same arxummt .. \bat at Tbeooun E>. Hi). T here cxieU a finl~ number ol mutually d~m, COIiIj*O/. ~ X , luc:h lhat 9 tabs the COllIIt&lOI. ..JUIl": "" ~ X,. U X .. the uorion 01 the K ,. tben / 1/ ll l(odV" S t. Let U, he an open ~hood 01 Ki ouch 11w!.he U, ..... mutually dlljoint, and lei. It., M" Q)IltinU(IUI fuoctiorl from n inl.O 10, II .,qual to I on K i • ",bolooe IUppott iI OOIIIpoc\ and contelned In Uj . If we ~t " .. E, .. tben II(:r) .. g(:r) on X aDd Ihf:rll S Ion n . H~ / I/lIlll« dV" S t .Dd 1/ IIIdJ'1'$ the tbeooew. 0

:h..

7.5

Atomic Radon Measures

fb. O. alocalJy 00III1'ftCl lhlndorlf "1*"'. P~itlon 1.5.1 Let" .. G RGd"" "' ......... "" o. 11 A .... """""", ~ .... p.oiIU a E A out:A I.W A -(.. , .. ~~

PROO" Let.1" be the ~ oltho!le O)OIpect tell X C A tud> t hai V ,,(Kl _ V,,(A ). The CQ of i"d'cotor IUl>ctO;Q 11( • ....be .... X ""tendo O¥ft" .1" . .. d[.. rcd
V,,(K ,n X. ) '"' Vji(K,)+ V,,(K , ) - V,,( K , u X,) -

VIl l A).

Write P lor lhe inum :lion r:A all elcmo::nte of .1". Then P lie! in.1" by Tb&.....,." 7.1.2. Now. If {
~hood

V. of a 9UCh that VI'(V. J < VI'(A), and then V. n P ill 00&Ijpbt... In thill ~_, oi".,., P ;" "",'ft'Od ~ a finite number of the Vo . V,,(P} _ Q, which COOlradicu t he fact that P be\orl,p to F Th"" t here is & point a of P ouch that VI'({a)) ,. V,,(A ), and P reduce! to the point a. 0

De/I"iU"n

7'. 5. 1 Le1. 0 : n _ C he ....:h that E'Ef( 1"(" )1 is finite for "''CrY o:>mpao;t .el K. The R&don .. PIIIIUA! ,. : I - E 'En o{,, )/{,,) 01> n it coJled .he Radon ........ un: dc~ned by tho: point"""""'" 0 (") (>= in any .el X ouch that 0 ..ani!Iho!oo "n X ") . P l"OpOIIIitloo 7'. ~ . 2 Ld a aM " boo ... in Dl:jinili"", 7'.5. 1. Thot,. ;., alOrr>i<:.

PR(M)r: Let. I EO H· and I: > O. ~ ""..!.II. finite ~bed; Y of n "",h that E~n_ Y 10 (,,)1/ (" ) ::; e . We can lind disjoint nei,hborhoods V. of lbo! poin!.ll ~ E Y , and, for <:ach ~ € Y , a function "'. E 'Heft. C ) $t><:h tl.u 1\10 1 ::; I. supP(
I"ull
L

o (,M, ) -

..,..

Ii -

L !o (I/l! IM - t ~ L ""..

VI'Ul
10(,,)1 1 (,,) - 2£ .

'EU

L 10 (" )1/ (" ) ~o

VI'U) ""

L 10 (:r)1/ (:r)

.,0

1'1_, VI" (h) ::; E.tu 10(:<Jlh(:<) for ~""" h € .J~ . Corn~retly, let r < E~n 10("Jlh(,.) bo! .. rftJ

number, and let Y

boo .. finite oubW. offt "",h tbM r < E ..... IO(l/lllt{l/). rOt each 1/ E Y . 01.,.,.., a rW number ". i" ]- oo . h(I/)[. We..., lind diojoinl,..;phorhoods V, of the pQlnlA V E y . uclt that h(,,) 2. G. for all :r E V.. , and, fur ead:I W€ Y , .. function 1'. E 'H ~ (n) sud> that 'Y. ::; I, . upp(",.) c V., and "...(, ) _ I . n,.,., 9 - E ..,y G. ,>,. I.... in H +- . 9 ::; It, and

V I" (h ) 2. V 1' (9) 2. E .... .. ~ 10 (1I) 1 · Jbr BUitabie numbers a.,. ~ .. a., 10 w)1 can be mad
"" ."'"

VI'(K ) _ V/O(J ) - VI' II - If( ) ''

L 1<>(")1· o

Not« thal lOO (:I:) (:< E ft"""" that 0 (,.) '" 0) ....., rep<e>ientali_ of the dilf= t d ' " , 3 of · ' 0"'"'.

Conveneb'. t".

PJOptCtion 7.5.1 and Theorem 3.$.1, any atomic Radon _ .. O!:/ined by poi'" m E f'OIr~' ,t.omlc Radon "",pm",,, on O. all mappings from 0 into' topo. . , ) IJPf":"! an:

7.6

,,-_urtbIt.

T he Riemann Integral'

Let n bo! , locally tX>mpIId. HAUlidoT/I ~. " a Radon meuure on 0, &nd 'H lhe """ of (»IItinoous functiono from {} int<) R ",Meh b."" ODmpoet support.

o.,flnitl.on 7.6.1 Let I bt poet ~PPO"_ Then

&

bouOOo:l ru]·Y&luod fU7l('tion on O ...itb com-

,," UPll'" Rimz4nn Int"S'al and the \owef ltienwn. (..itb ~ to V"l .

an! called .....poeti""I,.. lhe

I~ of I

Clearly,

/I<11fVl' - - I"( - f)dV " .

t". G the """ of bounded functions from {} into R wbkh 1>1."" (:ODIpocteupport. '111m I"(f, + h )1fV1' :5 I" l ,dVI' + I" loIfIfl' for all h . " Denoc~

C. t'o/dV" - of" 11fV" for

,,,,d r

all I € 0 and Q > 0, 11fV/,- S /" l,w 1'. /.10""0''''', by Tbol<:m::m 7.3.2, IdV I' ~ fl< I dV l' for .,...,.,. upper .. mlcono;!nuous fu nction from {} into [O,+<>o[ which hM compoet IUpport. Lee. F be. rMI Ban ..... ~ and 0, 1M 'pIIC'e of bounded mappinp ..ilh <:Om~ wpport from {} into F. Oefi ~on 0" the ocmilKlrm 1_ II< l/ ldV/,-. in

r

De6nit lon 7.6. 2 Lee. R, ba t l>e closure of 1t(0 , R j® F in 0" . The eIemeou of &recalled the ruemann I',intog.able functlozlS from {} into F. The l i _ mawi.,. V> - f ~ from 'H{O, R) ® F into F (F ro&I or oornplex &n.ctt opo.oe... " lzo _I Qf o:>mplexl has a un;que l i _ <:Q1ItinU
n,

n,.

Since

I"

r il l·
I" II I .
that any Riemann ln~.bIe f"'>tiRUO\lll, it is ~"'" tha, Ildl' _ I" Idp for -'1 lEn,.. N-Iee. 8 be • ftlter t.u;".., G" , and suppoooe there is. oornpoct !let K ....::b tbal, Sor -"" M € 8 , aU fW>CtioDl 9 € M 11&", tbelt suppnrU ior:luded In K. If 8 (»II,tEpI unifnnnl.y to . function f . tben I I... in G" and 8 (»II'e''''' to

'I'

ate

lin G, . P ""I>'Jbiljon 7.1.3 there{"", I'f'D'I"'I that the "1*'1' 1i(0. Fl. consist jill!: o{\.he continuous mappi"9 .... ith compact fUppOrt from 0 into F , is ~

In RF_ Next, lei. I be ... boW>l.lf:Ction from 0 ill\(J R wil h comp8Cl 'lUpport. If I is R"m&DD in"","abie. for eve<)' ~ :> 0 then: ~t.! 9 E 1i such thu 1/ - gi.w" < !/2. and II E 1i+ such that II - 91 :$. II and IlIdV" :$. ~/2; then 9 -II:$. I <9+ 11, '"

r

/ g
/ l.w" -,

LIdV,,:$. /" It/V" ~ /

~ LIN" :$. / " l.w" :$. /

g
IdV/, + ~;

.. I " l·dV" .. I 14V,,_Con"""",ly, i f I"I.IV" .. I'" IdV", lor every £:> 0 there exist op, '" E 1i sach thai op S I < I/> and 1(11) - ... ) ..IV" < f; then 111(1 - op)dV" ~ t; Ihia pcnrl)fl thai I If! Riemann intql'&bI!.. In shnrl." bounded functinn I from 0 jOin R, with compooct SUppo>"l. 10 Riemann Inl~ if &lid only if I" IdY" - III IdV". and in 1m. CMe 111 IdV" .. I IdV".

Ii...... £ ;"arbilrary.......... tW 1",/ ..w"

Thflo."m 7.6. 1 U I I k a bo~1I4td' mapping, orith """"pacl IUpport. from 0 ;..t(I a .,.,., B<mad. _po><:< F . The I~/Qw,ng ..,ndiI..", "re "", .......... ,:

(6) For noe<}I' :> 0, tMre oeriIl .k...... ", a" _... a" fJl F , .Iem...u g" • . . , g". o/1t(0, R), and " /tonditm II E 1t+, nell tha i 1/ - 9,,,, _ . .. - g".a,, 1 s II GndIII · dV,, :$.~.

(e) I .. Ri<""",n

inl~We..tIJJ ....,..,." Ia ".

f/~ ~ ""' MW/i.J., ... ""'~ "'PP"'" in (6) M , 00IIen: M .. 61lj>~n 1/ (0:11.

!hal l E,s;s~ 9,,,< 1s

PIIOOF: First, '''Pi><*> that ((lndit;on (_) ia $&l.isfied. I ill ,...inlopt>ble. 110. b &II € :> O. there aN elem.enta a, . .. .• "~ of F and eJemoma 11 •. _.• g~ of 1t(0, R ) such that :$. , / 4. if k .. 1/ - g,a, _ . . . - g~a" I . Let 9 be equal In !of/lg,a, + _.+g".a,,1 nn the let K .. (o: E n : [g,a, +. ,' + g,.a"I(o:) ::.: !off and In 1 on K . Then 9 ia continuous and

I "
n-

/ II - g{g,a, + ... + g.a~)ldV" <

/ 1/ - {g,a, + .. . + 9~. . )ldV" + / (1- g)lg,,,, + ... + g".". ldV".

Bm I"a, + ... + ,...... I:S M + II - (91a, +.- _+'..... )1. 80 (\ - 9)Ig,a, + ... + g.,a,,1 :S II - {" a, + .. . + ,..a,,)i, ",bleb IeadII to

/ II - ,(g,a, + .. . + ",.... )ldVp :S 2 / II -

(g, .. , + ___

+ ",a")i.w,, < ~.

' l o ... ,'" n:,,, .,,.,....

In ohon, mulli plyil\l!: by the """'" cominuout ful>C\ion " _ may atIppoR that 1 :S M and fbIV ,, :s t12. Hencelo
""wide t he ne&!iglbIe set N . Thut f"ldV" _ f IdVp - f kdV,, :s t / 2, and tbne am It E 1'I +.ueb that It <: I and f hdV" :5 £ . T h .. »f'lMS lhat a>Ddilion (a) impl'" (b). eoo", ... ly, stlppoR lhat """"ilion (b) iI "'Iimed . For """" .z € n , wrile 6(:r) !or till! oeeillalion inr I UP VW(.)( . "}E v . v

1/ (, ) - / (z)1

of I at " . The funCh on 6 : :r ..... 6(.z) .. opper ....... irom;nuout. F ix r > 0, and Iel .. " ... , ..... " , .. .. ,... It be .. in condition (b). Then 6(.z ) .. l he ""'!lotion 01 1 - 9, .. , _ ... - 98.... at :l , 80 f 6dV" :5 f 2hdV ,, :5 2£. Since r iI ..-hitraJ}'. 6 _ 0 a11OO11t ~... here. ~fore. lhe set o f diacoolinult ... of I iI ne&!iglbIe. F inally, """"itOo", (b) and (c) are oI:rviou$ly equi..-nt. 0

We recall ...... 1.<>pOInal 1{.z , :I) : " E n ) iI called an entou.... If V is an entou. . . of n , there am an entount:e IV oudt IIw.

lV o IV _ {(:I , :) e n x n : %e n , (;r ,, ) e IV, (r ,:) E IV } iI included in V. For '"""'Y en~ V and '"""'Y suboe\ A 01 n, we put VeAl _ I, en: 3:r; E A. (.z.,) E V). 1ben, if A ;" • rompooct suboe\ of n , the v eAl, where v uteods o-.-er the set of eotou~. form .. bM. lor the Siler 01 neipborboodo of A. Tn JIIIrt.icuIar, lor all :r € n. the V (.z) _ V((:I}) Iom> .. bMis!or the 6lterofneigbl>or hoods ol z. If an entourqe, uu'-t 0401 n iI aid to he om.aII of order V w!>me"". A " A .. included in V . In whal. follows, n '-;11 he .. oompa« Ht.UBdorif II""'" and " a Radoo measure on n. Let P he .. aet oI 6nite coven"" of n by p·lntegrable seta, oudt that A., n A.. .. II""",liglbIe for all ( A. ). n of P and all diotil>C\ k,.1:, of L. SUppoR that !or """" entour~ V of n there e.:IsU an elernent (A.l. u o f P I ud> tlw. all tbe A • .... lmall of o m.-. v.

V.

P ropOoliUon 7.6.1 Ld I b< II RiM1l4nn jt.in""",w. ""'PJ>ing /rom 0 into II Banadl .~. Then. fo
J

E'

€:> 0 and lee. A Ix tM...t of poinl8 of fI at ..'hich 1M ooclilalion uc: ed. t. S~ A ill c:om~ and jt.ne&!i&ll.ole, there ""isbl an Op<:n

PROOf :

of

I

~'ix

enl.ou''''II~ V such that VII' (V( A)) <~ . ~:< € 0 _ V (A ), 1M <>IIoCillo1ion

r.,.-

of I tbere ""ioto on enlOUrag« V. of 0 for ",hich

... '"

ill strictly leal tb..n €, 10

Then let ("')'" be a &nile Wnlly in 0_ VI A) $UCb that U." VZ , ("',) (Ontain!! o _ VIA), and put W .. fl;EI V. , . If N ... oe\ .maIl of order IV which """'" fI - VIA ), fix i E 1 10 that N n V.,("' ) COilW"" ill l_t OM poitl1 " . N""" since (:<',1') .. (z, . ,,}o(z, 1') lies in V. ,oVz , for all l' in N. SUp(••• M ....... 1/(1') - I(~)I .. _ IW~. N""t , lee. 01 _ (.01. )... £ be Bn e)o,momt of P (O""is\~ oeI8 small of order IV, and, br k € L, lei. 9. be B point of I( A. ). ror ..ch II: E L IIU(:b that A.. _ 0 - V IA), we haw: 1/ - 6.1 'f ! on A•. Tberefo
""'y

1

,".1- 1E.

NOOi we in_ipte tbe pooptlties of noeI·vaIued, Riemann integrBb1e tunc10.. We IIIP1"""" that A • .. l>-
Definition 7.6.3 Let I : 0 - R be bounded. For "''''Y w .. (A. J... of P, m{w) ..

L Vp(A,) '€", inf 1("' )

"'" called tbe DOlbow<

IPJJI'I!!

Obloe~ that m{w) S; ~nn

M (w) ..

.n

IntegrBbkc

'a

relatiw: to

JR 1- dVp

L V,,(A. ) "UP I i")

S;

I

~".

and w.

r

I . .wp S; M(wJ, bec&_ I .. , is

PropOo!itlon 7.6.2 JR IdYp .. inf~,.M(",) PrrciKlJl, for ~ € :> 0, Il>e<-I: mu. .... ~,., W 0/0 ....,;t. tIIllI. M( ... ) ~ JR I· dV" +, lor tadI w E P ......wing cl.-ell .....u cl anhr \1' .

J

Let ~ E }{ boo INch that Ii> ::: I and ¢<jVp ~ JR I . .wp + t/ 2_ Since ~ .. uniforml y contlnUOllS on fI, there .. an entouJ'8«e W of 0 I!UCb. that I~") ~ c/(2 Vp (f1») lor \<¥try (~, z) E W . II ..... (A.). u ie an PRQOF:

- v(%)1

1.&Tho;Ri" ' nnrM, ... o.l

P C'OIlIistillll of IiCtII of oroe.- IV , thcn JUP,(A, vIz) Imo.llcr Ihan t/ (2V~(O)) on "., for all k € r. 1Iena!

d""..:m of

L V~(".) ,....f A.p I!>(z ) - /

~

lSI

mo- ;'"

PtIVp. S ! , 2

t€<

o So, If III Riemann I.<>IlI'&p IV of (J oetI lIMn !Ii

I,,~,

t ben lhe", exd\jl, tor "'UI t > 0, an ".., ,lid> thal 0 S AI(",) - m(",) S t for all ... E P t:OD$IItiD3 of IV, Coo~ly, .... h&"" tbe b1~11III ptOji06ilioa,

oro..

n-

Propooi t ;QlI 1.6,3 Lei I : R k ~ndal. II, I~ """'"" t > 0 , tAcn: trim.n", to P n.clIlIoal M (", )- m(",,):5.~, IIw:n 1 is RicmGn~ ;,,~. P ROOF:

Clearly,

/111/- on L

LA . ' in!

. CA.

j(z )ldVP

S~.

a

Theorem 1.8.2 LeI 1 ... Gil inun..! <>1 R GM" .. JlG40n ..........., "" 1. FWdI~, III I ... a Riemann ~ , inI ..., .We ""'PJPI"ffwm 1 into. &m.a q.:we, n .. I .....iMa ovlftdc .. a>mJl'Cld...mtcrwl J _ (... !oj <>11. F".. tw 0, t/wo! trim a > 0 ""'" mol, I"" <011 • ..w~ ('" )01:. :!>0 <>1 [0>, tj <>1 muIt lui !M"a, _ lu....

1/14<- ~(t..

I)/ (G) -

L

"(leo•• ,, ... I)... 1s r

' 5' 5"

01 tU du>it:e 01 Ii; """'" ~ ( "' _ I }) .. 0).

;".1 .... Mr:nl

in

1(1...." ... 1) (~Iiwlr,

in

Iir",-" ... !)

P lIOOf': For e.cl:, e 1t(J, C ), tbo: function ~: / _ C , that ~ with, 00 J aDd ""0"",,," ouulde J . iI ~jn~. Iv' d, wben II takeo ito val_ in R ', i II upper lmIicoalinlJOOJS with C'Om~ SUJ)l')c. t.. 'The l i _ form, .... 1<4< OIL 1f{J, C) .... R ation - F7, '''' /'J _ J . C\early, for ~ E 1f<-(I }, ...., ha... JII/J.I.V(p,):5. JhdV", Gt....: t > o. let ...... . ,
J

II -

1/14< - /I IJ.dl',1 S

/(J -(f,a, +. ,. +p.,G.)/ r)
+ / (/lJ -

J

J

(g,G!

+ ... + 1h
that Idp /lJ,dl'J. Now, b- evo:ry z € J, "J({ r )) "{(zll, boca .... 1(0) .. infli : g € 1t.{J). g( z ) ~ I ). 'l'ber-rJon;, ~ may repIo.ot 1 b,- J, and it remoJne, or:Iy \.0 apply P l'O\>OOitioa 7.6.1 . 0 which

pr(WeI

7. Radoa M I

162

P' •

"""""'1',

P ropmit ioll 1.11.4 Ld 1 joe .. inttnGl in R , and Id). joe r.drogv m...... ~ ... I. Duig....u b, S 1M .... tbralltmiri"f 01 I . ~

I' I"'"

I d). ,.

~

inf ~S , (g.A ).

"'u

I

'f"d).

'M

I>o1indtJd luflCltlm I from I into R

PItOO" AU funct;"'" t;' E SI(S, R )

l

it

aM

...w. «>"'I"'CI $uppni.

Illimann in«Wlblt.

~ ••,U.(S.Rl·,U inf

Id).. < r -

~rt.

I""

, ~ I , Id 1.. ,61 be .. (Ompo.ct rubin\.01"Yal of / (OnI.Iini"l! IUpp(g). For evffy t > O. there "".. Ill an Inl.tf;tr n ~ I ~ tlw 1r(:I')- g{p)I :S ' l Ib-a) ",".JI.to , It in ja,61_wriD« I,. - )'j < (b- a)fn. Wriu; Co for the oupren''''D of, on [.. + (t - 1)(6 - a)/ ,. ... + t{6 - a)/n). for e.-..y I :S I; < n , and Id '" be the fun<:tioD from / Into R .bleb ,.."Is/w O\I~ [.. ,61, It o:quaIlO C. on Ja + (t - I)(~ - .. )/n , .. + t (b - .. )/nl for I :S • :S n, aDd \I equal 10 c, at a in cue .... Inf(/). 11>en '" I;", In SI (S, R). "'?: 9, :S + f . Thlo pr<MW that r :S and and /(", - ,}II). :S t, ... tb,,,Hoo" tlw r :S /0.. []

F"" find , E Ji(/ , R )

I"

PropoooitloOD 7.11 .5

""t~in&

/,..u I,.v..

Ld A k ..

I,.v..

~

0/ I .,;tII ...... PGd

a........ ...

f.

I" l"d)' .. 1DfE.,.. )'(E.), ~ lAo! ¥"'..,. io I&h:• .,.... /irille /QmiIiu (£').,.1 0/ 5,1& hGh 14! A C ~, A•. S;mU..rl" /" I"d).. _ sup E...:, )'(E,), ........ lAo! _ _ ,,,,. u /.akcn /in;/c /"",mu (£.).." Q/ oIUjomt S· ..."...w. IIo...t U.." E; C A. TIw ..

<>IOU

S... uollUdlthat A C :S L.., I.s,. 110 I'" 1,,4). :S E.u A(E.).

PRQOF:- If (£.)...., Is ~ &nI", family of I .. Con~IJ, for t 'CIJ C

> O.

there aisU '" E Sj ~ (S)

V,,, E; . then

,""h thai <;' ~

I .. and ID tbe form r:.~, 0:.1.s, . _!>ere I \I 6nilO:, the Eo _ dil,ioi.o>l .s-..u. and the 0:. ..... vc-iti~ nu"""' .... n..",

I

I ..

< E..e'.a.M~' I .. :SO <.>, ""

L >E'.&""~' Thill ,,«"M lbe fil"Sl eu.lmll:nl.. n...IItQ)
).(E. )

s

I"

I"d).

1"

t.

o

It F II to rftl s.-::b ~. ~ J E Ji{l , F) may be unilormly awrW' _ted by et.met>'- of St{S. F) which vanllh outside .. 1Utd complId. ..... bI ... ",,",111 of J. T'IruB R, II the closu«; of SI(S, F) in G,. 1"he<ek>re, RIemom>

7.7 ..........

Con.e'....

1:103

inte,rablo: funaiono (with ....pect '" .10) -r be defined thl'OUlh SI(S. F). Ele...etlUry I_ t _ pooc : . J in d ill ""'Y. The followina; eump!elllh.lIt<*te lbe ideM in Ihil oecI.ion. l..t't " be Le~ rneuure on t _ (O, IJ. If H ... """'~ IUbeet 01 J , _hen: denoe Itl t ,.ueb lhat (I < ,,(H ) < l. then IN iI not Rimwln ,...iDtegrahlo:. evetl thou&h it hili. _here der'l tet of dikootitluitir!l. Ne.:t, if A is • subtie\ of the C·m .... _ .,hic:b .. not Bool:liIn (ProposI· tion 4.3.3) . then I .... Rlftnann ".ln~ but DOt. !lord function.

7 .7

Weak Conver ge n ce

l..t't 0 be • locally 0Dffi~ H·"· h,,1( oporoce. Write M (O. C ) lot the "I*'" of Radon meullf'l!l 00 0 . IDd M"(n ) lot the eoo>e of po.1I1~ Radon meuu.... ~n.

Deft .. ltlon 7.7,1 On M (n , C ) the topoIocy of pointwi8c coo' co ""..... in 11(n , C ) is etJled the toI>OIo&Y of ' ..... COO""'ll"''''''. Thill. 611ft bMiI B on M (O.C) coo,uss ~y "''' € M (O. C ) if, foc ~ / € 11(0 , C ), v(1) a)Il~'"

'"

,,(I) alorc B.

Theorem 7.7. 1 wI (,Ii.,,).>, 60: m" - " " " in M (O, C) ,lOCh 1h4t, "" II!WI'J / € H (O, C ), u... KVI'C1OOL' 11'~ (J)) '>1 a>n"".,u limn ,,(I) E C . ~ " ' / .... ,,(I) to a R.>.I<>n ..... '''"', .nd Uo" ).~, ""''''''J'U ...,..." I<> " .

I",

PlIOOf': If K is. """'~ .ubtie\ ofO, {""f1([0 , K : C ), n ?: II is pOInt.... bro,Med. By the Banri-Steinhalll theor<*t;"" DUm· ber MI( Jucb that ""'(I)I:S MKI / i lot / € H (O, K ;C ) 1Dd .. ?: I. '"'"" I'/1{(O.K;C) i11i.-.. and ('.U)I:s MKI / i lot III / E H(O, K : C), .. hJeh pto .... thaI" is. Radon moeuu.... 0

u...

P ' .......ltlon 7,7.1 All c..IIdL,~, in ""fW ~ ..... ~ o:orgvI,. Etlerf c..1ICIo, JiiUr ~ .... M +(O) ""''''''J'U ~.

PlIOOf': 'Tho 6nt ~ followt from Tbeo<*ti"" w-r form OIl1t(O,C) is. Radon mm~ ..., the.....,.,.j o~ment is obviaul. C

7.7.2 Lei B k .. JiiUr kriI .... M '(0 ) ""'''''';119 ""fW1f I<> • R.>.I<>n .........,., " . LeI F boo: • ,...j BoonocA '1""01', ,"":Id / 60: " WurLUo/ m ' ; p illg, """" ......,l1li(. "'ppOi1, /rTIm 0 int<> F . S-.ppoK tMl / ;, " . ... , S,ab'" /..,. II!WI'J A E B , ....: for oil " E A """ iJu.t / "" .. ,,·nz/'giWe «t 0/ --"tinlliku. nw. f /tb> .....""'J'U I<> f /dj. oi<>l19 B. ~ltlon

I~

1. Radon J.k __ w

"··· r

P ROOF: By ~ 7.6.1, 100- - " ! > o. t~ exist a,. .. ...... E F, ,f. E' 11(0, R). and II E' If+, IUCh that If ,o, + ... + "..... 1 s nIl.

I - (g,o, + .. . +", ...11 S II. and /1w4< S t. We WI §nd A e B,ur::b.hAl /1kW < /1w!1' +t and I/{g,o, + ... +,. ... )ch- - /(g,o, + .. _+,..... )dJl I S ~ for all "E A. Then 1/ I
and tho proof Is complet
0

Let C"(O, C) be the fJ*'tI 0( bound«! rontlnUOOi funC1i<M>1 fram 0 LIltQ C . C de hcd with the nom! 1_ II I - ",p.~fl l/ ( ..lt. We Ny \h.u a <»n$.inll""" ftmctkon I from 0 tr.to ~ Infinity wbenever. lor eYer)'! > 0, ~ ~Ist... 00In1*t 0Ile1. K I'Jd> t hat 1/(>:)1 S ~ OlIuide K . Tho 'I*>eCOen.C) of t>JDl1nuooao fw>t\loIlII thM ...a.h &l infinity is, in ~. \he~"reof1'(eO. C) in (:ten ,C ). I(, O
e

'"

. . . m,. _ "'...

DefInition 7.7.2 On M '(n.C). lhe t<>Jl>Olo&T '" point.~ _~'><:e in ~(n. e) (I...-l i""l,r, din. e» "' called tbe topOloc.v of ~ <:on.(I&<".... (.. 'i"" 'lvei,y, '" IW"toW _N " """,,). Thus. RIter ba4is B on M' (0 , CJ .,.,.,. .cISM _kly (mpolCti~1, ...........Iy) to. bouoded fUodOl> -..re I' if l>{J) _·.tlp to I'{/) &lore 8 , lot I:>ftJ' I of COCO, C) (l'tlIpeetlvely. of CO(O. C l)_

"''''paef,

PropWillon 7.7.3 p. - I" E M'{n. e ) : ~ "I s .. } .. .-.tIr lor ..." .. > O. Ld 8 k &fiJl.«r ..... ~ IuT k a .......l aICO(n .C ) ndI IJNtl. u.e _tor Ip(IC.( V ~""ttd.,. T iI mco(n, e). Thfn .. _ _ 'l' ,,1WI.,.jJia.enl """"iIk>n IIlGt 8 _~.....tIr .. lA
p.,..,.., "'>u<

P ROOf': 1\ Is the .~ urUt t.l1 "';t h emtcr 0 in tbo: 1O!>"'nP'0' dual of C"(O. C ). lbue il It .... k'r """pea., t.,. tho R· n.m-Aloa&lu t~. and p. - .P, It -..-I: II E C"{n. e ). Ci ...... ~ > 0, c ' - / in V ... that III - I I < 4 (34). Thea

1..-(11) - ,.(11)1 S 1,,(11 - III+ I,,(f) - ,.(Ill + 11'(11 S 2
III

€

when " lies In. ""itallk A e 8 .

",bl.c:h plOO'eI tlw B CO
0

....-

In putit;uiar, • HIlt\" b&$e on p.

O>O";:0801!

wo:akly If and only if ;t <>Xl'C' &til

'I'

ate

T.T WMIr. Com.",. ""

1M

PtopOItltlo .. 7.7.. A f!"'J and Jt4ficimJ ......titi<m that a ~ (p.),,~ , In M '(O, C) co.I«.,. .....kiW ;, Ihtt.t ,I am....,., '''IgI''"'' .nd that ( IJ<"I I~~, be 6otoJ\lltot. An, -.I; ColldL~.~ am""""'.....dIy. R

PROOr: If (p.)"l ' '"""CL"" vaguely and (l1'oo 1). ~ , is bounded, t.ben (""),,2:' COO',.,' 8"" .......uJy by P ropooition 7.7.3. eon. , ••dy, If v... )~2: ' is • ...,ale Cauchy tbm lup l ",,1 < +co by tbe Ban&ch-$utnh.t.... tbto~",. and (I'~(f»)~~ , hu. limit lor every" / III CO(O, C ). So (p,. )~~ , coo""'P ...,.kI)·. 0

""'I""""".

Propoo;itlon

.........,.

l:t77!I"J ~

S_", 1M topr>Ion 0/0 h4A a r:»L
7 .7.~

PROO": FOI" all a > O. p. _ II' € .M' (O, C ): 11'1 S AI;' compact uod metrisahIe wbnl It • e' .......... d with tbe ~ of "..ak co.... el&<»<"< The propoojlion follows. AIUm&II¥lIy, ..., can giYe a proof thal doe! not uee lhe wcaIr:: OJrnped.... 01 p.: Let rY be tbe comped .pace obtained by toddlq ttl 0 • poinl 00 ( AIeotand ...... '. oompedifical ioo ). rY. and hence C(rY.C ), io .....><1 ..",..table. But CO(rt C I ......y be ldemified wilh tbe IU~ of C(rY . C ) mnsieti.Dc of tbe DDDtin ...... fuDCIion\L ... bidl ....... ish at 00. Therefore, CO (O.Cj . . .~ N.... 16 (f~l. l l he a ""'lueDCe in Ji{(l . C ) such H - (f~ : n 2: I ) .0..-

.ha.

h, C" (fl, C ). By lndDCIioo, "'" constNCI • seque""", (j. )~~ , of otr\c!,ly illU ' " iq fuDCIioni (rom N im.., N l uch .haI. .be J. )(~) (f.)) .2:, in C for -.y t 2: I. Put ,,~ .. I'{o. ....., ,,(.) for all n 2: I. lbe sequentt {".(fll. l l 001"""". for -.y / € H . 10 ("~ ). l ' ,"""u"" WI kly (P n:>po::.it;o" 7.7.3). 0

""'I""""'" U
COl""""

n.en

Now , nanoor r;(I
""''''''''119...,...''

Tbo!o""m 7.7.2 LeI 8 be G fiUer bomr"." M ~ (fl) /I) "" f.lt;iiW!l.l1' 0/ M~ (0 ) nod> that n"l u.w to DI'Udong 8. LeI/be I 6otondtd /rom fl mil> .. n
_119

80"""'"

(a) 1M ",I 0/ Ji' .,,,ilinMI/iu 0/ / ;,

F,

I' .~:

(.) 1 "" · ..........robk 1",-....., A E 8 .. nd /",- 011" E A. ~

J 1
PROOF: Let ~ '> 0 be a .....t number. By 1be ...... "'IUcIlon h OIl fl ....:II thai.

,

7. Radnao M

156

I / -{', ~,

"IoU ,.

n ••

+· ·· +". ... )1 S h S l il OIl.Dd flodl' 5!· P ul M -lId. n..re K f\IodI \hal 1'(0 - K ) 5

Lel K' be ,. compoct ~ of K In O. Il.Dd II!C - h , be ,. cominUOUll fun«ion I'rom 0 into {0, 2M] ..lticb I&«!ICS ..-ItIt 2M - II on K and YIUIi6Me ouWdt; K'. ",.", 10' _ 2M +10, i.e com;"".,.., h' _ II 01' K, &nd 10' _ 2M 0IlWdt; K '. Replacin« h.' ~ .u.p(h.lt'J, ..... may UOiUID8 \hal h' <: II. In Ihis ...... ,
!.

N..... f 1I'/bI _ f h,4,, + 2MH" 1 """"'rg'" to f h,
I",",~'

A E 8 .ud> that

11(9,0, + ... +

g" .... }dv

1 1 1 II4v S

for all" € A.

h'lAo S

-

1(9,0, + ... + g"o~)d!, 'O!

1I'4p+! 5

1h
S "l(M

+ I)t;

n- ;"""IIIalit;'" [",ply that

If /'" -/ 1',1' f """

+1 1eg,o, + ... + ". ... ).w -

1(g,o, + ... + g..a,,)4;>1 + j h4!'

.. to. than "l( M + 2)t; ...hid> p
0

By ~ 7.7.2 • • 1iJ_ t.sis Or> .M~ (n ) mn_gfJ< IiIltI"OWly to "" e;"ment " ol .M ~ (n) if,.nd nnlJ- if It """,.to8"" \"&41",,11 to!' &nd gO', mn""rg'" to npl ...,ng 8. Ot.:rYe tbat. if _ t.l.1a! for 0 the inrerval {O. I J. fur!,~ the Radon meMUleII DO)t tend to ,,( A),., 0 if A .. the ""t of wionalf in O. Themon!, ..... _ that the uadilion tM~ I !'·nqliglbIe toe!; of pOin.. of d*»ntinuity m Tbeo
ba"",.

_net.

P ropoll;tion 7.7.6 Lot (P.)~>1 0.. a in M~{O) aM" an .~I <'l' d<>xJ od ..w. !,,~ boto~J. /11 lIIu 0 . . .. tk i ncp.mfy !,(F) <: Um",p,,~ (F) Ilolb for --.... e/Imd ~I F .

1.I"J'icht Seq..,......

157

PJI:O(IF: F int, IRlppoeo that (.u..}..~ 1 OOII'tt ............... ly to " . Let U be an .open 1Uhae!. of fl. f ill" eY'tI)" I E 1(. mch that I ~ lu . lbe ""'l"""""' (.u..(/)}..~ I Wllieo .... 10 JJ(f); tbus , , ( /)

E

lim lor ",,(f)

:s: lim illl ",, (U),

and lieU ) ~ lim inf ,.,,(U). Now, if Fit., .

,,(11 -

EJ 1Rl'" of fl . ..... ""'" "-,,,

1,,1 - JJ(O - F ) 2: Um 1,.,,1 - Um illl ,.,, (0 -

Fl.

tbal.1a, p(F ) 2: limaup,.,,(fl. eoo",..e]y. I<* 1"-1 1",,1 00II''' .... 10 IJJ I .. n _ +00, and that ,,(11 2: lim IRlp ",, (11 lor aIll>-[-D, and b auobl , ,-1( [1, +000" . l>-
!

!

7.8 Tight Sequences To .udy nartOWly Wi1"t ~t ""'l""""'" in M t (11 , C), it " .-fILl 10 introduce tho! notion of ti&ht 10\. Ddln ltlon 7.8. 1 A au'-t P '" M '(O,C ) io Mid 10 be li&ht wilt"" ....., for "'to) t > 0, there exists . compe.ct lit!; K...ch IMI Vp(fl - K ) ~ t b all "E P. Let (.u.. )' 2: 1 be • ""'luence in M' (Il, C ). and .... ppoeo that {"" : " 2: I} io DOl t,«bt. If , > 0 io IUltahly <.boae1l, lor MCh mmpe.ct _ K ..... can lind .. 2: I ouch lhal V" .. (fl - K j > t. lf"" 2: I if"" intqer&Dd K Ael, Ihttl ..... mUllt DOt ha", V ",, (11 - K ) t lor all " 2: no; <>tilt • .,;.." taki"l L. compe.ct 1O\...ch that K e Land V,.,, (O - L) < t lor every 1 ~ .. < no , ..... have v",, (I1 _ L ) :s: , lor all " 2: I , WIIt.-.dictiDa; tbe cboOoe '" t . n.."""f, for f!IdL int.qfI" no 2: I and eo.ch coml*=! ""'" K, there aM an " > .... ouch thai. V",, (fl - K ) > t .

:s:

."""'ptd

1!oS

1. _ M

UN

Propositkm 7.8.1 Ld C/I~ )~2:' k" _ _ ;~ M' (O, C) ~ to a lUukm ",eo , ... Il, "nd ._IMIIIl~ ' ~ 2: I) if fight- "",,, Il if and (.u,.). ~, CO""'''iiU .... ~ ,<> Il·

..,...1,..,.,ndtd

:> D, t1..,ru xisu " comJMC' oe\ K 0Y0Cb thai V ",,(0 - K ) S ! I<x all .. 2: \. If I E 1(+ ill such l hat I S 10_ ,.. and 5UPP(1) C 0 - K, thea If Jd;>1 - I 6"", _ _ gd;>,.1 S • !of each g E 1i(O, C) aati:dyi", 191 S I ; thcrcfO«!, f IdYll S ~_ Till" ........ ttw. VII"(O- K ) S e . Now fu<" E ct(O,C) .00 e :> D, and Ie\. K be" c:om.,.a oe\ such tlw VII~ (O - K ) S • for all n 2: I. If.., E 1i+ ~ illl ,·"hoee. <>
PROOF: FOf eveT)"

J

J

I" - "'-'ldY!'o.

< ' '' IY 11.(0 - K ) S . P" I

11""'. -1"""I' 4 1" + II'"",· -1""',I,

1/- -1""'I '~I'u

o

P ropoo.itlon 1.8.1 S"PpoM. Ill4l n if" .,....~/.o.bI.o .... icm 01 """'pod ""10, "nd Ia (".).~I in M '(O,C) . II, lo~ MdL • ....,...,...,. o/(p..). ~ " ~"' ;, .. foortJ.c- "" _ _ orIIkh ..........-gc.o .... ' ,oo;lr, ~~ (!'o. ' n <: I) if

-

!Ie"...,.......,.

Suppoo;oe tlw I!'o. : n <: i) iii not tight. C ' - , :> 0 I!O th.o.t, fo< <:Old> MmpN't "'" K , u.."". io an n 2: 1 fo< ... hN:h V,..,.(O - K l :> t. IA:t ( V.l .~ I be an inc,""""",,, """,ucnce of open lieU with COlI'po.ct cloou,..,. such that o _ 0 ' 2:1 V, . In "'"' 01 lbe dioc,..i '" immediately followi", Definition 7.8.1, ~ ean <:o:>ot ruo:t; ... l/.ridly i~",.eq_ (n. ). ~, " f int~ 2: 1 iIUd> that VIl ... (O - V. ) :>. lot all t <:. I. N_ let i _ t eo) be" !LIrictlJ' i'd 7dog funetioa fcom N into N . If L .... c:ompo.cl !let . it Ie lnd~ in VIr(j> lor IIi)mt. j <:. I, and V"",,,, (O - L ) mridlJ' e ' c . k. ThLEI ("",'" : i <:. I) is not tiihl _ "Tberefon:, ll lrom --tL IU'-\I.."IOO of (I'.). ~ 1 ..... can otrloCl. " -rumce ...bich is I~, tlzcn (II. : ~ <:. I ) is tisht. It ",mal"" to be .boom that .. ....1>eC>OO.hid> COlInzgao IWTO'O'ly ill M '(O. C ) is tiihl. In ocher "",rrls, we fUppOOOl llw t be .... uetiOO (.u,.)~ >, """'Ug&I "",,,,,•.-Iy. &od ~ h.o."", \<) p~ tha~ ("" : " ~ I ) io t;PI. Writfl " for ttw. .......,.. li",it of (p...)..~1' Si~ ("" : " 2: I) ill tiibt ... hen (I'. - Il : " ~ I) io liP • .00 !Ii""" lI'~ - Il).z 1 con''''rp 1I&f1"QWly 10 O. "'" P ROOF ;

......d only """"'i 0 and ev' )' l!:l be ...t the becinnil18 of t he proof. For • 6xed integer t 2: I , oul'P'*' we M'" obt.e.iMd com,*,," _ L" . .. , L • . functionl g, . .. . . ,,_ , ill 1f(0 , C ). ""d int.,;tl"l .. " . .. , ... . 00 that (a) 1 !S " ,< . ..
L: '"' 1m.(L.) moIai... V;, !'or"""'Y I !S i !S t ;

(<:)

L:.,:: o L; for eve
(d) V"", (O - L; ) > t for"""'Y I < i !S t ; (e) V"",{O - Lj ) !S t/3 for every (i , j ) witb i < i !S k:

(1/

I" I !S 1 and suPP(,,) c L~+ , - L; lor every I !S i < k;

(S) 1p",{g,)l2: t lor e.-ery 1 !S i < k; (h) 1"", (g; )I!S (l / i )(t I3) fur eve
t.

Since V p"" (0 - L. ) > t, tbt ... u iltll a compa.cl. _ L, .. , oudI that q .. , :::0 L, u V, . , and Vp"" (L:.... - L. ) > t . Choooi", L-.., la.rse (000""" we may ' " ,'me thai. V"". lfl - L ... ,) !S 113. n...n V"", (fl - L j ) !S t/3 kit all j, j .. t~ j < i !S t + I. LeI. g. be an ok""'''t oIfflO, C ) ruch that 1,.1 !S 1, 1Upp{g. ) c q .. , - L. , and 1,.".{g, )I ~ t . Finally, ( ' - "HI > '" 00 that. Ip""... (g;») !S (1/ 2' )(t/3) b every i < k and V P", ... (0 - L-.. ,) > t (which to ~bIt by lhe remark IOlIowin« n..tinition 7.8.1 ). n...n the (OI$f1,lCtion can CODti""" by iDductioo. TIlt _ L~... , - L; are diojoinl . H...,... the IUPP(p;) are diljoiDl. , and 1, 1 S I !:l!: ,g;. MOl"" ..... , ' '"' 9, + "' +" _1 on L; , I'ore&eht ~ I. 009 to conlinllOUl 011 0 '"' q. "",(9) con....... to 0 .. t - +00-

if, _

U'O!. '

"flwo,do.t.

Bo'

and

11'-. (!:.,.. g; )I S VI'-, 10 - L,.. , ) S t 13. ..,bt.

11<..,(9)12: t / 3. wlltooe tilt detir-ed oontradidioo.

, (p,.. (g.)12: t . FinaUy, []

Pl""OpOOIitkm 7.8.3 5"""",," IIuol n .. " a>1
....,'0.

PROO', Let /l be the weak limit 01 (p,, ). l!: " n...n (p" - " ). l!: ' ila rwrow CaLM:hy -rumce. Tbuo it oulIiceo to pro .... t he prnpooition wMil (p,,),.0!.' con•• ,,,,,, w" kly to O. SUppotle that II'- ' n 2: I) '" not tllbt and td.aitI the notaIion 01 p' .......... tion 7.8.2. Multiply;", 9; by a complex. number (; witb ?DOdul.. I ..... may

!60

T. Radon

MN.o~_

...... me lhal ,. .. {g,) -I,.,.,(g.)l if ; itI even and t ht.t ,. ... (gi ) .. - ",-,{g,)! if i is odd. i1>en Re~.(g) > ~/3 if t is e\..... , and Re,.., (g) ~ - e/ 3 if k is odd . But""'" tho ~uenQO (,.,.. {g»),> , (."not bt .. c..llChy 1OlQ""""" and ...., ... ri~ &1 & contradicUon • .-bid! PlO w.. Ihal /". : ,. ~ I) is I~'t &nd IMt (,....)..2:' (IX1~'lIeo1 nanowly \.(> O. 0

I

1.d n be a "'p,,,,,,M'.t _ • ...a I : n _ II: a fOndion. \\-.. defi..... ho: _mo.ion of I ... pOlo ... En .. tho .... mber in II:

wi": I ) .. Ii .. "''' /(6 ) -

......

.-.

b", ",f 1(:£),

fiChl.-hand

side io dofinod (i.~. wben ~m.up . ..... / (6 ) &hd lim h.t._ 1(:£) OM _!>eo botb foqU.O! to +<><> • ...,. bot~ «Iw.! to -ce) .

...!>eo.. . .. the

,.

I . SMor th .. tho:function r _ w(,.: 1l io u"""' ..... i«>rumuo..on i .. domalR 1.

If I . fini'" ""

n. ' - u-..t II _ ...(0. 1) _

0 ud .hat, "" ..-err" E O .

!i",..,p I {z ) - 1M. ( ...)-t..· )

,ho.

(I..«, < wlo ;1) be a ................ _ r :S; ....., • •• lE V' V / (z ) _ 1 (, ) b - , . ..,;p,Ixw1wood V of G . c.o",utel). Itt. :> ",(" :/ ): _ ,hat '""l>i . .. ).u • ., / (z ) - I t.) ~ ,for .....Ilable ~ U of ..). 3. !.d I : {} _ It be .. _ oocaicool i...., ... fu .... \on. Show .hat. if w(",/l .. 6..rte .. _ .. E fl. tlloa !iminf ...... w{z; /l _ 0. For I~io, """" by """,,,,ditto:... oIoowi"l lhat, ..-.. tho: 00DIlU)" IIypotbesio, .1>«0. "" .... poi~" z ..-bitnorily _ \4" ouch.1Iat I (z ) io"larp .. d -' od.

4.

2

r... ..-err ....a..J n"mW,"p/ q I,, "' • . : if . .. 0) . pu. fir) .. , . Show.haI; 1 • _ ,11M ..,(4 : n - 1"00 ..,. an " € Q .

[.g {}

be ... Io<&!l)-

,.... on n. I.

~poc<

Ho, ...k>rfr _ _

........ ( ... i.bq :>O. .ntI'I _ ! .....""",\1..""",.". Q. 1><11

I. -.. """.,.",i_ funD.

If Iim"',,_/(z) _ 1"00 ..... ol! .. € 11.'-

_

in n (10<

,h.a. IN: o.t r ' (+oo) is

V be ... ope!> ....... of n, COftOt."", .. -.q ......... (U~)~~ , of.,.,... wboot. "'- V ~ U~ Iuof """'1*1 _ _ C
"""".nat

w(z,n"

l.

S_ , ........ _ of poi .... of .",..inuill of I io _ ... n ( t l>«o. io "" ... of ....... o.ll1)" in _" ,nl", lbat I .. bounded, b ... may '01:1'00 I by 1/( 1 + 111);
••

3

F

.u_

iC

lor O'P'''' 7

lei

Loot 11 boo tho of 1M plano R > .. _ ~ .. , ow .... lbe ~ of t ho line 0 _ {OJ" R ........ be poiD" (l / .., t / n'). *~" t.ho ... N or otn<:Ily pOOjIm. iDtf:#n &Dd • ronpl """'" .be ... Z or pooili>'& or --'I....

..-.- """'"

'--

""" nery polDI (0, wl in 0 ODd <Wry ;ntqn" n > 0, .... T. {r) be the tot of ."'- poin.. ( ... ~) E 11 • ..cb 1M' U:S I/n ....... Iv-":S u. l'Uo, .. __ of tho ~lt.er of....;at.bo<_ 01_ poi ... (0. , ) E 0 , lloo 010. of l lote T. U), aDd ... ,.:,hboobood. of_k of l ho ~ (., ~) in . ubtoto 01 n a;>nI.IIim..., ( . ,~). &Dd _ , ..... hio _iooo ""I....tty H• ....JorI( '<>;><>kv T 001 n. MOt"""" , _ lhal , i. Ihio T ,_ 01 t.ho nbtoto T. M io """poet ....... metri...bleZ. Dod""".!>at 11. e .cJ with .bol lopoIo:>c T , if ItlocoJly ...... poet. P ...... ll>aI .bol .... "'..,. ind"""" by T OIl io ~, S. Let A \>e l boo tot (OJ ~ Q, c_ br, Il. .. _
ot""

r

°

"

f.

0ed""".1>aI liw:ro O"iou .... intqn" • <: b, oudt.hat .boo"""", D. ol D. , i

p
.··'."'.III. Thtl". ...., <>l ,,,,tM. R- Q - lJ..J,, O•. aodll .b.

;hat at _ ~.

doli_.

n,''''' lOp''''..,.

"... D. ho an im .... I"'i....

Coot.cIoodo IbM _

neiKbborbood of A lal1 ...... U .

• . Dod"", rf'Olll p&tI ) l /\at t_ """" ""btoto of n __ 'iDi", .bol d . . . . cloo
I... boo _ _ ion ol Ent ..... l, Id .. ho t M fut>ctioOll

0 - 0 _

(if",t/n' ) _ II'" Ir.-

IO,-H:Io{.

""" ~ E R. . - ."'" L n (, ) O("" io ·mal .... l han E "",,(1n + I )/N', Oood .... th.... b.-,. ~ ... K,.be"""" E....,.,o(.., io hite 2. Lg" be the Radon _....., "" n dOd. by ,be point no, Cw In 11 - OJ. Let U be &n " " , " , , " ' _ of 11 <>OnI..aitti", D. R-u ;"'" ~ can find an In.... ,. aI U _ I 'i... T. !. ) lOr 011 , of ,,", "I n (R no Q ). 1)00:100<0: .hat ,,'CU) _ +-. I.

..c..)

:I.

Ii

.boo ,.

Ccwodoodo u..; 10 _ ","TZI~bk.

J $uboct 00111 io loGally ","""IliPbIe. l>uI I"'" it

For",,,,,,, iDtes
-lQ.II.

lbe

Ie:!

7. Ro.cioG M:, ..

(1rIttpoJ of II m..M 10 1M "r ·ure" "'" (0. I ....... , doIlned tor \be _ (:ls"{1- z)--' .. _ b poiD:. pl. For ~ ... _ A of (O, I. ... . n ) . ... PII'

!'t A) .. J I .. oj.. I

11 , . ;be fwr>
J

J

2. R>r" > O. put A ., (o !S p !S .. : tIIP)1 > .. ). a:od'- t/o.al I'{A) !S (1/,,') J g....... Coac1_ ...... ,.(A ) S I I(~""·I· 3. i.e< / : 1 _ C bo OOAIIrIoo;>ua. For evay t > O. tbnt ""... 0 > 0 _ IW lI(z) - / (soll < c b ... z , ~ E I NlioIyi ... )z - ,I!S " . Oed""" I..,.,.. ,.... 2 t/o.al. Iot.-,. ~ .. IMp. ''"-9.... """" 1/ (" ) - a..,(" )I!S

(I + 11111« "" 011 z E J. " -

, ' I"" ' (8..,4," ..'&"

lin,. "'l'.."

1/ ''')1. ,.. "''''''" .......... WlII
r

be ;to dmoui ....

.. .. ,,-1/' iD put 2). ,

I. fo< ~ ~ . ~ 0. --.. tor J. ,... """,,ion .. _ r'

from I .. [Ct, II It. S - ,bal B...h. (:o:)" I and 8.. ...,(.. ) .. .. "" all" 2: , ..... 011

., "_

:l. SItoot ....... to< .w
.. 2: I, ""I ........ " '.' ., I _ Po.' .. Pr .• _ ..... poI~iaI n...II7. 100 ......-y '" 2: 2. put .... , "' 0 aDd 1\..(:0:) _ 1". t/o.al. for......,. '" 2: 0 .-1 b" ~" 2: l. Il.. ,.(r) _ I'" 100.. ....... + I\.• (:o:)• • ben p.... io. ~ of ! go .... _ . - I. Mot_, _ ,bat • • •• _ «.. - 1)/ .. ) ' -' ... _••• _. 100 all l 2: I """ all

3. fo<

~

Id" ,bill" equo.I to 0.

-n-..,.

Co

."

With _ ..ioa _ in,....

3,.-"'"

.... _ (1--;;' - ')( '--;;' - ') ... (, -;;')

_./Ie

100 - r "' 2: 0-='_r " 2 I. r.. Ily WI ......", "" t 2: 0. <:II.\eUO! of .....""* CO 2: 0 oudo Ib&I. to.......-y .. 2: l. II.. co l oo, '"..u 0I1'i .• .... liD ~"'~ ,.,.. ...j _oJlor ,!Lan

.,.

7

..... I!'III

I....).,..

be •

ocQ ,,-nO

of """'pit>:

"'1m""''' For alt InIfVn • 2: o.... n 2: 0, do./i,.., I. : !G. II - R by

~ ' .. -.. L...'S' (-II'Clc•• ,. For ~

,-;Ii'.·, ...... ..v.l ..

1.(" ) " .... .

cllIO .... . - t /o.al , _ ulou. Radom "-"'"" " on f .. ~l.Il I/>M eo 100 all n ?: 0 " _ only " I"","" o.dou • ~u""* A ?: 0 -= I""," ~..s . (;)16. -°"01 S A 100 -.y in! Gor .. 2: 0 (·HA....:Io"j.

"~,. < _ b """"' , I.

S-.hat.ho roodi • .,.,

'"

" Y (~that, lor oil .. 2: (I....t o.ll bubo...JlI< of '''''' ,ho poIrw-ia1 .... (1 -"la- Oj. io...

o $ , $ n. to ~-'"" "'... 2. COO .....I;. , ',_ tho rooditioo i. ... ioIiod. u" V be dw _ ou"poooo of C(f, C) _ ..... br ,be I. ( ~ 2: 0). aM let ~ be ,be " - ' lot", "" V whldl """" , . UI Co lot _'Y k 2: (I. U"'" part 5 of Exa ..... e, _ thai, lor OYer)' k 2: 0 , ..(B • •/.) """verp 10 c . .. " _ +<>0, 3. 1'0_, lor _ n 2: I , ...... be .... Ii...... fotfl>

"" C( t , C ). S""- ,bot u. ill comim,o... aDd ,bat, lor ~.t 2: 0, ,be _......., (.... {f. l) . ~l ron,ugtJI '" co ' 0
«In'"''''

4.

8

+,

Loo!. n be. k>calIy """'--' HauodorfI'~. E. _ oubol*'O of c (n , R ). aDd P. _ _ """" in q o, R ) ( i . ~.• • ouboot of C(O, R ) . <>Ch I"" , aDd .J> beIone: to P b oJI J. 9 , h E P aDd all ..mil' pooi. j... .-l """,bon . ). Su _ ' baI, "" NCb h E 11(0 . R l, th P, aDd P: u.. """ of'-, E E lo< .hXb f - h belonp to p , I.

s_ . bat

2.

If Q ' iI.be .upn:mum of the u( f) lor I E p;;, and D" .be infimum oI .be Pi:, , bow tba. 0', D " are finite and .hat 0 ' :!" 0 ·.

3.

11

S_ u..re - . . pOOi.;.., _ lDOIIIIure p 00II I ouch ,hat ,,(f .) _ Go 10< evfty n 2: 0 ir and only if .o.' c" io pool'i"" lor oil in~.t 2: 0 and n 2: O.

~ io ......

"'mpOy.

.(n '" / '"

1;" _

Let." bo .. li...." Io<m "" £ + RA u~":!;"4 "- SIIoor that ., (/ . ) • pooitive lor oJI" e E, n P if and 0<>11 if ,,,{II) lMto;" .be Inton
Let A be a d I ",hoe< 01 R and (e_ ) _ ~ .. 00II ......... in R. W. _ in . ' V wbat IoIlowa: "",.to .. poolt ;'" Ftadoa ='"" ... ,. "" R with .. ,pport ........ t>Od in A, _ that z' io ,..;"tqrablo and :r'40 (~) _ ..... lor all " " - ' ..
u..n.

n.. roaditiott io drariy '...

f

' 'Y. A-.tmo

,be,.ro.-e. in ... bat folk ... tba. it •

...ioIItd.

I. Le. E be'M _ _ ...bo~ ol ClR.R) _~ by.ho runc\«- ~ (i ~ OJ,. 0<1 £ whe _ - . z' to c• • _ P t he ~ in CI a. R l .,.,...;mill 01 t"'- contin"",," ftmctiono . lUd>. .... pooit;." 0<1 A. py".. tbat the<e . .. ~ Iorn>" ... E + li(a. R ) ins " oudt t hat .. pooiti,.. 10< aU / ol p n( E + li(a.R)). Fbr ' hio. oonoideo • ' - ' " (It, ) ... , ol lila. R ), 0Dd0w I wilh WI ........ lot wbleb.....,. _ _ ,[:Ity

(n "

_.n.... <1,__

",1>0« d 1 hao .. (_ &;_8)_ 2.

4,

and

o:mo,"""

~

b!' . ranoIinile h>
s.- tbat -tAl -

w(-'j b aI! -' E J!:(R. Rl. Prow that A OI)fttaiDfl ouPP(P) (oboe .... c.hat 11(11) '"' 0 Ii>< e-y " E "H(R. R I dooot IUI'I"Of' io dio.joi<>t _ A ).

l' ","'<4>(:.1 io <m&Ilor.ban.:,.

b oil ~ .. ~ o.

l

Sbow.hat

,.

r ",.., in.... n 2: o. rot Ji- e:> 0, lot ~:> 0 1><1" rMl nl lln>_ ...... ,hat I S ...,....··, and. lot II : R - (0, II be "OODlI,"_ fun;" : • wiIII .Iot mation to" (1 - ,)1 S "'.... '. ded_ th .. ",- ioo I'-lrMpabioo. FinaIIy,.- ...... f r "i..{",) _ "" .

hocldo:DlAII;y, de li z.. . baI, if .be .r ,.~ --u.ion io .... io6ed • •1Ie """"oeI>I pool>. Ifm (Ie ," 2 tI~'_ w!.eo> A 10 """ ~ 1 _ r.c.t;d.; BHIi g"ij.1'o 'iIi" ..... " .. ' ..... £nmpOolU.p. «lf)_

,,0./

t

10

"C, ••

I . 1.0< E boo ,1>0 ,uboo0 -utiplieal.ioao, ud oXd_ poIy __ ioJ P E RjX11100 La £ wtw. .... .... PI.,) It pooIU... ' " oII:r E R. 2. v.. . A • R . ol>ow ,hat ._ (I>(ldition 01 bt...... II boIdo if &nd. 0<111 if ' • .-Iralic """" Q . : (z ......s. _ ~.• ~. ~'. ~,h ..., pooit!... lot

,hat"

.n .. 2: I) ( _ pan 11- (nolo _ K ............. '. ""","'," pooble .... , S. Lot< P E R(Xl. If PI:.)" pooio~ b ..IL. 2: 0. ........ hat p"... bo_\~ 1": + Pi + X ( pt + p: ). w!.on.ho. P. lie i. RIX1. .. v.-n. A _ (0, _ r...... Lha1 .ho.O>nditJo
I.

E.s... S• •, .... . .,,%0 .... poo;ti... leo

3l·(T1>;o· )_Stielljeo·

2 .....

1"• ., ' .... )

Fi., . -.I ....... be. & :> 0 aIOd • IUUtc Iomily U,l .. , .. x (n , C ). ..... 10< h ..... ..,...1_ f.....:i"* fIom 0 _

(0. I I. .. IUd! ""'I..aI 10 I .... K •

U.., oupp(f. ) ..... 11M """_, '''ppOn, P ...... 111M .......... 511lte 1am11r (o,l,.J of d*1nd po;nlo of K ...... family "',l,.. ito X ·(O) ..... ,hat

2,

I..(/.) -

:t" J /'Uo,)J.(a, )1 S ~ 1ot.oJi

E f.

_,<>p>I.

o..d _ _ put l ..... , .. boto M {O.C) ""'Iuippod with ... """. If beIontIlO <100"", of ,hoo.poooe of IU.dofo meuureo whooo '~f>' poed ia IUPp{jo),

u.. w .....

3,

j

Wboto ,. " I • '" I

.. It< A ..... ,he ..... _ ~ of IUdon _ureo ~ ....a, thai M M"'" wJ.:. ."pp"'U .... hile and amt.ai.....! hl~ ) !lIoooooo ,hu ,. bekN. . 10 t ho. ~ 01 A ;" .11,0 _ _ t, ; q.,:

s

12 Let 0 boo. (OInIpIoCI H ',

I 011' ; OF

I.

If (/' )' 21 ~ (uaifon»ly) ~ 0 in C(fI, C ) and (p"I'2! """....... ~ I<> 0 in M ln,C) . . - u... ,he ....... _ (",,(I.) "~, """-SO\0 0 (01: 2fl .. IbN (ljll)"~ 1 io boo' nded ).

2.

"-wDo that fI 10 mli.ute,. Let V be • neisbbo
s:

q o , C ) b - " ; li: TJ- Wbetl/ it _ . ~_cornbi~"" ofl he h,"t bat (,.(I) : f e VI '" C 1_ lboo Hah" _u.,,.... 1l1000014). 0..:1""" tlw the mapplnf; (P, &.- .M(O, C ) "C(O. C ) into C io "'" """io ...... , ..... boace that tbe _',"",,*0 on ,'-1 (O.C) io "'" __ m o " , " 3. Sbow It..! M ' (0 ) io locally """'1*'1 1_ Ax
n _ ,.(/)

13

LoI. fl be alo
'''I'''kcY. AooWD< that n II _,.1-._. Lon (V").~, b e . _ of ....... ...-'01 O. wilb CUDI'Kl dolo_. ouch that V. C Yo., lor oJl n :i!: 1 aDd n _ U. 2J V. " r",..-,. .. ~ I. let I. be ...:m,io"""," f....:tiod from n itllO fO.I), ..;u. compo« MIPPOrt, 0'1"'" 10 I "" p._ Moo.., ... , Ie\ (J~ .• ),.21 be • ~ ...,._ ill h:(n. ".;e l. ""'" t be ...... ' ..... 10 fi_ , ..... lbe 1<>P'*1L7 T .... Mln,C) cW.Ded by ,be _ i..",a_,. - )1'(1. )1 and tho ..... "'(f~ )1 (n" ... ~ I ... ~ I ), S _ Ilia, , _ . _ 1<>-. .... _ Ind ..... lhe _ ,.,.,.,ano "" M ·(fI). and dod ..... tw M '(n) io -.riMbIe.

01..,

14

..

''''.0.1' -

n

be • \oeo.IIJ ......P"« Ha . - Ir 0$*00 . .. _ copo4oc, """ • couort.MIlo bMit. Let (p,, )' 2' be • 0«1_ in M '(O.C) ......,....;"11 .. "," 17 '" I' E M ' (0 . C ). _ I10t H be OIl OQl'io>dU"""'" . .. _ of C"(O. C ). bOo"""",, ]" CO(O.C). We _ ,ball"I>/O" ""'(I) - "U)I ~ '" 0 .. n _ +00. Let

..... ""..Ibal .UI>,."II'.(f) - .. (Ill do.. '"'" .......... '" 0 .. .. _ +00. Tbett, lor ....i..... I > O.... ""'" find • otrictly i" " -;" 1 " " _ (,.,).~, ill N _ ..... _ (/.)'21 ill H ouch lhat If'.,., I/.) - ,.(/.)I :> 0 lor all ; 2: I. Put

,.,., _,,_ I.

Eq"ip C(O. C) ' ;t h t iloo top<>k>o of unilomt ..... . .... ' 0 " " compooct teII_ l'IMm . .. 10 ....tl-"-'tI. q O. C ) iI _,,;" ... Oed""" ftoon 1<JcoIi'. thee>..... ,hat thmo uboeq""""", (I.. ), ~I of (1' )' 21 .. bIdI ..,., ' UgEZ ;"

u'" ••

C(O. C) '" • fwtoet;"..

2. B, P",,,,,oil;"" 7.8.2.

I

E CO(O. C ),

(~,

_ " : ; 2: l) io

tbo
1(" ).

Sh<,.- t hat

1')[(1, - / HI -

1(", - ,,)[(1., -

4. c.-:lude tbat

/ )
t~

10. Il,

",)[1 ~ i

Oed...,. from to fact ,bal l uch tbal

lor all ; 2: I.

~ e/J 100- ~ l&tp

01I0tI&h.

l(", - "H/,,)I ii_than £ 100- 1: l&tp ~ .IO that we

. - .......,.JicI;"...

yr

ate

8 Regularity

_1*.

A....... Rodooo .............. &r1O _bo;>ooo ....,.,. 1"-'0 _al 0an"]1 _ ........ 04.100. ... 1nd1>OOd t _ d .iDC< _"=- of ~ ... _ - . ~"'. we de~ tho ....t ...... 00IIIiriDc of on i~Ut'YaI aDd later, ... dell....:! .... .,.. thot in' "all. tn u. io ~ ... 3"", . . . l'lo.dooo Ii tel " "" . ' ""," doIi_ .... ....,;ri119 0ZId Radon - ' r·u ....

&_

•

_."' . .

1._.

,~

*

11-1 It. _u", d.lbd "" . . . mlrilOll it oaid to bo otrictly reculat if ilO main "",10 -optloa II a ""*,,,pUoo. of • RadoG """""ure. If - " _ out 5' I of n io • ..... atabk UD"" of cum-, ..... if .. io tho 6cnI ... if - " com~ tot it ,...Int~ ,boa I' \0 Itrictljr nopIat ('Thew _m 8.1_2),

oJ""". . w

8.1 Regular Measures on • ,.".Irin, 4> in fI .. said 10 be stri<:t.ly . . . . (respec\i~y, ~) if tbon "i~" Radon me""".., I' 00 n such tIw i< _ (I.. C... poctiody, sud! , hat p _ p.. aDd if every ODOlpIICI. oec. ill included. in • uolon of .....,10). T btn p. : 1_ 14. is ctJlcd tbo: Deflnidon 8. 1.1 A

Radon

_~ ~

_'""ble

•

f

tl>t : " , .., arising from Pa -

nand . ...

Tbeo .... m 8. 1. 1 ~II' Ioe ~ IUa.d.>n _ ' " '"' mirillf "" .....ring ()J p·i~ IU.I. SM_ 1h4t. 1M" "'""'" """'P
ill RquIAl M.....

1IIc...-... E .... J l~" "" ..

V ~ ,, _i~ 0,.,....,1 PItOO~,

1151

:n.c. fj = il+- M OI Qi(l , ;. - fi.+ if ...&l <mJy

;, ""nlci~ in II .-..tdIt .........

LeI. S he the ...rurillS in

Qf. ·..,".

n oonoist.ll\l! 01 the K n L' to.- all OOttI)"Ict

'Ut.eU K , L of fl. By PropooitioD 7.1.~ . 51( S, C ) lis .w_ ill 4 (,,). ADd, by Propooiliou l.~.-4, 51( • • C ) is oWo OOIllle In !:bv.)- M......:>ver. 4u..) c !.hv.} and J IdY ... - J IdY" for all f E l h(".}. by Propooilion 3.U ."., ~ri<:tIom to i 01 V,, · &ru! V" • .,., .,-...xlili.." and IMy "&f'!f' OIl . , ., they "'" identical. It Kit. a>mpecl "'" and E,. Eo,.,., .. In the ",aumonl 01. ~m S.U , the> Eo, - E, is ~·l>f3liglble, heOCO= K Is ~.In"","abIe and V ... (K } _ V,, (K ). N....., to.- each E E . , thom! il lLII i~ ""'l""OCO= (K~"~, oloompocl. . ul»ctt of E such that E- U.i!:1 K. ;" ".pq!!dble. Then

V ...( E - K~)

_ _ _

V ... (E )- V". (K n ) VIll E ) - V,,(K. ) V,, ( E - K~)

a>&lWflt"" to 0 .. n - 00, .. hlo:h implitol tht.t E -

Next. V~( E)

u..;!; , K• .. jl+.""!tli«lble.

_

... pV~ {K.)

-

aupW". (K. n V) + VJI+ {K. n cr')1

."

" l! I

_

..,p V~(K. n V) +IIUpV~ (K. n lr) '~ I

oi!:' V~ I E n V) + V". (E n U') .

-

lor M<:Slht.I....:h Open 801 the 001 ... meMu ... and ll>o. ... 6ote io ". __ ura!>le. FUrd"'f~.

V.pl''''

V". 01 +-ta,

V,,' IV) _ ,ul'(V,,(K): K c V.KQ)mpACt} oupVJI+(K ) :S V". (U)

_

V". (U ) -

•

oup(V".(B ) :BCU, B jl+, inlegrahlo} "'pV,,(B) :S V,,' (UI,

•

V,,' (Uj - V~(Ul ·

eo.

_

luontly, each ,,'-eJl~""" illlouIly ~-~bk EadlloaIly ~""Cli&ible "'" A rneoets eYer)' E E • In • ,,-''''CUr;l''''", ADd 1>0."", .... ""Cli&ible. lie\. An E_ So A Is loWly 1'+""""C~r;lble. Coo~. let A L

168

8.

~!'Y

be aloeally I'.-'''-'&li«il* ""'" A ,,-. e'<'pOn .. t. U," included in. coumable union of then V ,,' (U)" V,,:' (U) '"' V". (U). Therd<>re, ~..cl> !,-negli«;ible set II "._J>O«IigiI»e, aDd fi. . ... p by P roposition 3.9.3. 0

"'-eeu.

So. gi"",, • Radoo .......,,., I> on R .., """ that"""", metl8uml ~ on lIttnlrinp S&tilfy ;. _ ft+ (01' fl ., jj. ). Now. tbe £UlXwlng f'I'OI'O'itlob 108 theoU......y. PropOSition 8 .1.1 Let .. Wc • ......i.-ing in 0 mn.
(_) FfJ"r ~ <»",,,,,,,11(1 K , tM,.,. erU!.o" 8 E

i -.h thaI

on " . 1kn

...n../itd:

8 A K ;, ". -

n.,tigoblc.

(&)

V". (Jq < +«> ~ ~ wml"'cI .d K.

(e) V" . (U ) _ sup{V"..{K) : K C U, K <»rnI""'lj , ffJ"r

~ c>ptn

rei U.

(4) V,.. (8) - inf (V,...(U) : U ::> E . U opo:n I, /fJ"r .... ry E E .. PROOF, EYido::ncl)', the condltioM &rc _ -or)'. CooVU"80:ly. tuppo!Ie that uch of thern "' Jali$lied. By (a), -=" cornpfoCl $eI it ~-rneu"'abIe, and It it " . [""",able 1>, (bl · E.ch / E H (O. C ) ;. ".-inc"l!l"-bIc by Proposition 7.1.4. o..note lhe liDe*!" "'"'- f .... f f d".. aDd f - f fdV".. (III H{fl, C), ...bicll &rc clearly Radon _ u _, by" and v , 10 that VI> v. l1>eD

s:

/

INdv .

inf

1V<· . f~l~

/fdV,.. ?:V,..( K j

lor all compooct..,.. K,'" V,...(Uj :5 v'(U) for -=II opcn set U. On the 01 .... 1Land. ""(U) _ 0U{>1f f dJ" f E H+'/:5 1,, 1 :5 V,.. (U). Thus V". (U) ~ v· (Ul. It IoIIowf that V". (X j :5 ,, "(X ) for a1llu'-to X of fl. Now V,..( £ j _ lDf(V".(U): U::> E, U opo::n J- inf" v "(U) .. v" (E), f<>r -tI E E ... 10 v' (X ) :5 V". (X ) for each ouh8et X of f1, by defi nition of V". (X ). Finally, v· (XI- V" . ( X I . The ,,!&in ptOloolplions of v . " d V~ "'" t b.. (~ idomtk;al. In pIlrtk:uw, ~ E E: • ;. .,.int. pabLe. Sio<:-e V" :5 v , e¥ef)" complex-""'1Xd .,.intqTabie function is ,,-intqTahlo: and t ht: ~".... £u<m f - f d" ;. <:onU~ __ on 4(") - ei;,(p.). But, ~i",," it &greo'lII .. ilb the U..... , Ionn / _ f / d". on H{O, C). f / dl' .. for all / E eMv). In partkuw, ,..(E),. f l ad;< £uo- all E E " . Nat, SI{. ,C) iI dm8t In and now 9dV" ,. [gdV,.. for all p E H (fl , C) br P'QpOIIilion 3.'U. The ftm,gui"ll; $hmI.-. tlut.t £CU.) ,. £i:lv ) ,. el::CJ>.), and.., f fdj;. .. f fd~ for all / E el::u.). 0

J

/d,..

Lhu.),

1'hI: roilowi"ll; "",-,It ill putleularly

J J

irn~.

•

£n., "

for CI>a;u< 8

169

Theorem 8. 1.2 s."""", tlwt mel! open , ,,",,1 01 n u m ~ ..,04>1 01 ~ Kto. Ld " ... oem;,;"g In n ndI Uw 4> ;, th.~ Ben! .. ··19'1' ... 17Icn e.....,. .......... ~~. "" .. ,lOCh Ih4t V~. ( K) < +o(),."..u a>mpacl ..,u K ".".;a!w ''
PlIIOOP : Ltt" be tbe IUodoD. IDKIUf"IO "H (n , C ) " 1 ....

J I
U V II an opel! M1t.et olO, t~ ""'" an iOCh ,.i~ ({~). ~L In "H+ such that l u _ t UP-.! L1. and that K .. _ M1PP{I. ) c U rOt e,..,ry" ~ I. ~n

""'I"''''''''

v".{U)

Slo.oe 1.. :5

f -"".

_"':P / {..dV P+ ., ",!P / I.w "" ,,' (U).

11<"

I ..
vl'4.{ul .. oup{V" . (K ) : K

for each int.,- ..

> I.

C U, K
N". fix I; IS: • . By PfOP'*tion 7.3.3, tboere ~ , for aU t > C/, an open "" U &I>d a c~ Ii!( F sLrlL tlo.w F e £ c U aod " ' (U - F) :5:. Then V~. (U - E):5 V"..{U - F) - v' (V - F) :5 e, fWd VI'. {UI :5 V,.. (E) H . By P~tion 8.1.1 , II. is M, ictly ~ular . 0

n-

For uampic, tab: tho IMUU"," OIL tbe .....tural.l'!1niri~ of an Int.en-lJ. an: ..U.n.:tly~. P,"""';tion 8. \,2 .....,. be simplified ",hen 4' cootai.. ,he comPOOCI leU. In view 01 probability t heory, _ ....... fix 1IOlI~ DOtalion. LeI: '" be a llemirinll that compact IIH io includai In .. CI"Mlntabie union of ~ ia n Writ<' M ... (. , C ) (' II!Pfl"t1...,ly, M~{ •. e ll £or tbe "'" 01 rtoeuIar meuureo (.Cl .... tn...ly. bounded rtoeuIar II . ... _) on •. TIw:n M ...(. , e ) may boe Idmt\fi.ed ";th a I UboH Cia f..a. a ¥l!C\or 5U""~, ... _ MAIl .... In Chap"

..no

'*'"

Ie< 2l) 01. M (O,C ). Thu!I M ", (. , C) ......y be e<:e. For i alooc: B. In tboe putlc...tar.,..., ",hen: '" io ....mlri~ 01 Bon,I _ ouch tMt .. cont.lI", tbe compooct lIeU. tl>cn M;"'(. ,C ) may be Identified ...-lth the ...-Ide ol M' (n. e ) (11".0" '/0 &.1.1 ), and the study 0( -U coo·.co ........ oimplillcd.

I

!..oK n boo • lo<.oJl,- """"poet " ....dorll" _.ud let S be .. ""';'iac I.a that _ ..,..,poct oct ;. <:OftUirMd in ....."'abIe ....... 01. s..eo.. Lo<

n oodo ~ ~

•

I ro

8. Rqulari' y

"........., "" S deIll>f!d by poi'" m' a . (z E X ). p""", . hat ~ io """La.- if and only 'fE .. z n« I<>,)io lin;" lei< <Wrf comPK' ''' K aDd.hat. bo Ihio __ •• to. RodDoo ' ure ...... (""" " io dofino5l
I :R _ R

L«

I!:t. + r) ~ I (z ) + I t.) b 011 z. , in R.

be....t. l bat

1((P/f)rc) .. (P/, l/!z) fo< 011

E R , q E N . and p E Z.

I.

p....., l bat

2.

S u _ ......100 "apII '" I "' _ del;.oe ;n R '. conti~"""" O. Cood ..... haP. f(z) .. z/{l ) 10< oJl z E R.

%

n- . - Ilia<

ou_ .ha,

I

io

l. LeI ~ be ~ _ . "'.... "" R, and I io , \ o _ S - lhat I it _ _ "0 ( _ P,opoo;. ioro U .I ). Condude .ha,

I(z) " z/( l ) 10< oJJ z S

LeI B "" • _ . . '51 .

...

ER

'" R .......-...:I . . . _

(HameI._l

I.

S - that B it

a.

LeI .., 100. bijoclioro ""'" • po""," .... _

w-.n~

C '" 8 ""to B. Deline • fur>«ion I hom R ,..... R by I(z ) " E ••"l,(.M~ ) '" ~ z .. E-<JI A(.). of It, wbole l,(~) E Q aDd l,(u) .. 0 exoep( lor hileb" ""'''l''' E B . SI (E +r) " I (z)+ I (r) for oJl z, ~ In R . boI, that . 10< oJl. E R, It:. t ot R. p""", , .... / . . . . 011.-1 vol .... " " - " -

,bo, r 'c.)

_.ut

Int.or-'- 01 R "hid> io _ 1.

4

'1** "..". .hI: Iioold Q of .... iooaaI

Ref",-

to

~

2 aood . -

• poln<. ~

/

;0 _

Lob
_""able.

c-.....

be Lt'-c- _"'" (III 1 ., 10. I). u. I . \lie fanilly (I ... ). .. '" diojol'" os- 1nIoe.. 10 deIll>f!d II,- Utd....1001 (III ...... 101_ n., Inl . - .. I ...... oJJ pooi\'" vol,...: lor ...,.. ... ";!: O. p tabt II.. ,.oJ,... I , 2. 3... .• r , and J.. , -11/3. 2/3/. If J. io . be """'" ot tbe I ... ""', " ;.<»diD« to II", """,beoo m OS .., t.be """'I " ,., 01 J. in 1 io ,be un;"" of r H d~ Lot "

""""f*'t ;,.., __ 10 K ... (1:5, S 2'·') ...." ,hal a {K .. ,) < a { K ... ) < . .. < o (K.,7". ')' It K ... - (-.61. ,holt ~ ,0.10;0, b J.+ ,.. l be In~

16 - CI +2" 1('-.)/3),.- 2' · (6 - ..)/3) [ . Now ..-rile E b- 1M ,n

t

"',ho un""" of aD tItc J....

I. S...... . .... ,,(I _ J. ) .. (2/ 3)'" 10< oJJ io "

2.

ID~" ~

"""'0>1«,,,,,,*

0 and deduoo.hat E

gliCiblt..

DeftDO. fuoctioa / on

l ....iIboo ""

10. 11.. 5>11<>0<0;

,; and on Il / l , 2/ 301 ;

for oJJ in.."..... n 2 0 and

IS, S 2"", if ..., pul

K ... ., Ja,61 . 1 tala l be oaI ... (b , . )/2' .' I>t ~ - (t - . )rr" -.I .. ae;,.. "" -.It '" tho In! " , . . I<>(l.+, ..). ~ , It - 0){2··'1 and !b - (t - d)/2"+' ..8(f.~, ..II. Leo Z M . poW. of E - (I) _ io lite ""iCiu 01 "" ""-'tJ """'it:....... 10 E. S~ tlLal ,be riIJt1-lwmd .mi>at ;'" 011 at z .. o.

E>wt[, [ 3.

Let z be .. poiDI. 01 E _ {OJ ..hid> io ,be .~

E. Show

,~

b~8

171

ri&bI """pol", 01110 I.. tenal con-

I

hao DO left·bud don...i .. (!a.I.. (It" iDfiAne) at z. 51""" E 10 _rizablr, llOIlDpact, totally d .......... eeI . ud h.u "" ito "[eel poiDI., it io Itomoc>n~ to ,be Cantor ... b)< •• ' "'i e. , ..... .... ical _Ill •. TL".1ore.1 h.u DO Irft-h&nd don ...;.., at "[' . ,.tobly """'1 poi .... 0110.1). 4.

to

1'...... ,be ............,., 01 .. function 9 from 1 illtO R ouch , hat Itl :5 3. , 11M .. ~ ,'iciblo ... of pol".. 01 diooool.lnuily. Md I (z ) ril )cll _ f g . 1",.•,400 b evay z E I (.... ,be dottUtWec! _ ,.. I<..... 'beot em). Cot>cl!>Cle that , io fUem&nn inlqrobio,
f.

f:

t;

~

_u'" "" 1 _ (0. 1(, &tid. b.U z E I . d Y. b)<.o ,be _lIftOt> tbe .... unJ oomirinz of 1 l(Z. ) _ _ '" to 1!o.>t 4 .. n - +<:0, b I!!I llOIlDpect ..,bin.... .. ",",1>1 01 1. ilMd ""b- '1(l / n)O'(n .... . ) .......... - • . I" ' bio . . ., (z.).~, io O&id to be unifom>l, diotribuWld _ulo 1. 1.0<" be

f

(Il .),!, io Wlilonn/y dinribut
I . Show

\~

.,,'''tI,

""b- if

2. Su_ t~ (1/ .. ) L, s . s . ""pC"'z,.... ) _",",'g~to 0 .. .. _ +<:0. b oil ~ p :i!: I . Show that ( 1/ ..) L'5O so / (z. ) Ieddo \0 f 14 b all " " " , i _ fuDCI_ I from 1 i"", C -=II ,hat / (0) • 1(1), ud _ that ' hio...m pet(IU _ .. bell 1(0) ~ 1(1). S.

,

0ecI.,.,. ftoat pano 1 Md 2 ,hat (Ii. ). !, io u"ibmly diotribuWld _ulo t if Md onlJ< if ( l i n) L ,s. s . exp(2iIrJl'f' ) """'"'''' \0 0 .... _ +00, b all In' S'" p :i!: I.

Rou.in 1.100 -..ion of [,u" eioo S. A .............. (Z' )' ~L iD 1 10 Mid ... t.a.. • limit diotributioa " , ... he« .. io .. " "'''''' "" t be .... unJ ormiriDz of 1, if (1/ ..) L ls . s .... ru......... _uoIy \0 " . ..... , be ...... "limbe< Md PI" z. _ ltI _ [ltIl b all i n! ; '" k :i!: 1, .. he« ItII io \be lllUcrol put oIlt1. I.

WbeII 6 .. irTatioD.al, """""~ (Z' )' ~L io unilomoJy diolribuWld modulo I (&bl'. 11- .m).

2.

When , io ratioDal. find tbe limit diotribu...... 01 (z.).~ ,.

'V'

ale

Copyrighted material

Part II

Operations on Measures Defined on Semirings

•

Copyrighted material

9 Induced Measures and Product Measures

_itl,..

....

n,,,,, ..,,,,,,, Chap" M ~l~ _ ohaIl bo: Ct,ned with m"" g"", " " and t.he!r r-'0'o;\ opel " , W~ obaI! alto _ t- ,,- open.tioat Co_ t 1M ...nb.-ll SUrDmary

Loot " boo,...-.n .... ,. oomirinc S . _ 17b>e; ..... x . 1.00< Y be .. '" 'P'u,-oWo ",bot!. 01 X. Let T be ....IiI ' 1\._"OJOdoer1Jial! _ • Y . .... " ,. , , * I'> Y, _ T ..... aolo od..;o;b odoquaIe po",", utI_, W& dooIiDo . t.. "'" ,.~ Iad-'. 1>,- ,. "" T...:I _ _ to In! ... t e with t'OSpe« 10 ,h.io ind-'- " "m' ~ 8.\

('n :1 ' $.1 .1). 11.2 Let ,.' . I'~ M . ... _

...... _ .....1Ii"'" 5". S" (with 0' , cr _ .bW g ' . b . 6 ,. : A t ".4 ~ _ ,, '( A' '''~( A~) _ ,100 5 I 5 _ ( A' ~..t ", A' £ 9' . .4" E ~ I (.. _ .~_ io n ., 0' >olio ... -.abIe on
inC _ _ peo:\M:Iy). no.

r

.-. t.:r.c, rul)kll'o). t.) w.. doli ... ~

n "

" '' ' ' . . . . . . . . _

r

.,,_

n of R '

( .... /t t

<:

I).

9. 1 Measure Induced on a Measurable Set Lee. X

of X, S ,. ..miri", in X , T a oemlrill& In Y , &DC!" .. a;>rnplIoc _ u n OIl S. We ohaII.,. that 1'(1' .....;,.. boo, a nouemp(y tn,

Y .. not>o:mpt)'

8U~

" •

tal Y!!I

~""'lI3U",bk

MId the T -ae15 are " .illl<:grable;

( b) fur o:..:h A in S. then: aiIIU B E t!lOCb that B C A and A n Y - B io ~nEClI,jble;

(e) A n B belonp to

t,

for all A E S and for all B E T .

Definition 11.1. 1 The llWa8urt " ,T : B Indumd 11)-" "" T .

_ fled" 00 T io called the metISIIJ'<:

The ",los of int<srt.lion wilh,..,.po<:t to PIT ..... &ivm by Ihe Ioilowing;

Tl>eon!m 9.1.1 II

FM ........, .....wing / fro m Y ill1
i

R

(~

l ar 1M. ""'p,tfng from X into> JI: (~ticIdr, inlo F) _icII Qf"bk: .... Y . Q

r

&0",,11 .,"" F ), ..,;u

"""""'g

f'

PROD" Firll. IUI'!'O'" .ha. T .. (A n Y : A e Sj _".J write " ;..1: t or " IT ' The fa<:t t hat Iv(BlI .. lJ 1,,<401~ f h lll,,1 = 1"1,,.(8) for all 8 E T Icadt 10 Iv! :5 11'1tr. let A E S and 8 .. A n Y . Then

eon""nd.\',

II-I/T(B ) .. / 1,,"'""1_

.up .... s ' (S·C)·iool,;'

I/o . III~I·

E""", a E SI (S, C) OI)CI\ that 101:5 1 uti be ,,-,itle!> " • E.H IA. - iii, where I " lifll~ the A, an: diljoint S«u. and 11111 :5 I for oJ! i e f. llul ~ A, n A .,.,. boo puned into S«u £.J (j e J .. IJ,I < +t:c) • ..,

J

o · 1,,<401-

L f IA,"" -!I. d" ~ L I,,(A, n A n V )! IE' :5 L L r" (£.J n Y )I :5 L L Iv(E, ~ n Yll :5 I"I(B). .~ I

Th ... lJolfT( B ) :51v1( B) and . 6naUy, I'" - I" IIT' Clearly, f ~ .. f j $ and f 191d1"I- f 1; ldl,,11Br oJ! g e SI(T . F) {F •• Banadl ' (*:101 If g "' the upper eo",,1Bpe of flO ["""'&lling gequen<:e (g" ),,1:.' of element& of tbm gaM .. SUP..>. f ;"d!.ul .. jdI"l· .., / dl,,1 :5 fdlvl for 0Il1 f : Y _ (0. +0:» . Con~y, if II > io lhe upper cnvdop<: of 811 Iii" ri"l ""'I\IeDCiO (II.. )~~ , w St" (S, R ),

s'''(n.

r

r

r

~~

J1o,,"1Jo1~ ~ f ,." -1.-dl,,1

Iod(pl -

r

i

r

f

-= ":,.p 1I~/yd("J

-r

II/ y d("i

~

r

!d("l.

g.1 M_un 1"""""<1 "" • M_ur&bIoo Set

r

r

r

171

r

idIPl ~ 141"1and finally Ittl"l- 1411'1. Let.A (II...-timy, 8) be tbe ri"ll in X (.... pecti...,ly. Y ) lftI'!I"aud by S (_pectimy, T ). Then

.. t...v.e fo:>Ik>q

!or all f : Y -10,+00]. Next, lei. I be. mappilll! from Y into. Ban.m"PN'" F . I f f" -.Jtially ... lnUp"abIe, it if the limit locally ... almoet t>CI,Where ofa Cauo:by eequeoce (g,.),,;z: , in SteT , F) (~ .... IUbopMl! of .c~(,,)); in tllil cue, It .. obvIouo 111&1. I .. _ w ly ".in~bIo &lid tbu J Idv _ f jd/<. eon...,..,ly, If j II : lhal B c A n Y and A n Y - B io ....... g'js\bI... M.u,)O(r, (.t n Y)n B _ .t n B beloo,p 10 Tfot aU.t e S &lid 8 E T. Hence Prr - Ii by PlOp....ltlol),. 3.9.2 &lid 3.9.4, and lilt dftIimll"/l
'.1.1. AIH, kl Z ioo

P " - ll lon 0.1.1 Ldl', Y. and T ioo ... in Thw,oJO a #tOb«1 01 y .not kl U ioo • -'ring in Z. TMn u./,. )/~

""""itioou Io.oid;

if 1M following f l ) I' /u ::·,t,.

'", if ud onI,

t.

1711

~

ME , ..

""'?rod""'_

(b) B n e bdcng. !~ iJ far All B E T and C E U.

(e) Eodo. C E U

if ~mt.I i h " eo ..""'Mt .. nien o' T· u~.

In tJW ......" (P'r)'u ~ /J,~ (IntIISI!i... ,p ~f ind..a:d ..........ra). PROOF:

Oboe. i t lhal B n C belonp to ii tor III B E f ......J C E ii it i~ doeo

foralIB E TandCEU , FirsI.,""I'J'Ui'" tiw W /r )/u exists. Conditions ( b) and (e) ..... ci...ny ..I.... lied. Z ill WmeMu.abIe and..a. C E U is win~"ble. Next , for fixed A E S, <:hooooe B e f , 8 c A, .0 thai. An Y - B ill ,..,qli&ible, and lei. (Bft)ft~' be" _""'!<" ol d;"jo!nl T -«u w'- unioa conwno 8 ; u.. all " :!: I. t~ ait" C. E U, C~ c 8", sud> thal B,. n Z - Cft '" ,..""IIigible; lben B n(u..~, c.) liN in ii, and A n z - a n (u..~, Cft), as IlIe union of (A n Y - 8 )n Z and 'he 8 n (8ft n Z - Cft). Ie JI·o
~y,

_me tha~ <:nodil"'" (a), (b), and (e) hoid. Z is /J/ r·n r....... bIoc. Eacb C e U ill I'.int.ograbie and conIai""" in .. coun\able UlIion of T--. ~ b(, .. ~ ., /J/r·inl~ For " gi-.en B e T , lei. (A.l-~ , be ....... ...,...,. of diojolnl S-'" whoe!: union CODW", 8. For '"""'Y .. dooop to U, ",1<1 B n z - B n( U C.) _ B n U (A" n z - C.) ft ~ 1

ft~'

o For er.amp\e. if " ill Lebeogue D"liPi\Ire on tbe ""Iural oemirilll! Solan Interval I. """ if T ill lhe _ural ...rutlr>g of .. su.l>inler.....1 J of I , .tooD /J f T """,- and ., ~ nw:o.eure on T .

9.2

Fubini's Theorem

be ,wo '''''''''npl.y ""'" and S , S" I"", ""mitiDgllin rr, rr, •.......,. ~the (t..s _ ( A'l
rr, rr

.i...,.,..

A n B'" _ (( A' n 8"')"

A ~)

U (( A'

n 8') " (An n 8"")).

if It ill """"""'ply,""" be p&nitioo>ed in1O" finite number of S.oete.. Tlierrlon:, S" .. """'""I, callod the prOduct cI. S' ... d 5", ~ by S'" 5".

T boorem 11.2.1 Lei ,,' 4JIlI p" lit ~ "'''''nI'U on S' aM 5", rupoclilldr, _Mid" lit 1M ",ftditm Ufi"N on S bu A _ A' x A~ ... I"(A') · P"(A~) . nom" if • m........ on 5, mtkfl l" <:;} /'" , IdIid> #iUifMI V" .. (V /,') ~ (V " H). P ROOF: LeI.

("' )i~ 1

_

(~x~ ),~,

w"'- union;. ... s.-t

be a.equeneeol disjoint nonempty A ... A' x A" . ""'"

s-c..

b all (z' . ~) ;n n .. 0' x 0" . Foe • Ji~ z'. tbe fWlCtioni ~ _ ~.:a;~, lA· 'A: (z'. ~) we p".;n~ ~ doo-n

..... f lA·'A~(z'.~)d"H(z")

..

~!.~f

[L IA;'A~(r'.z")]


'S ·S~

N_ tlx: fullCtioni

z' ...

f [L1";'''7('<'' z'')]

dpH(z") ...

ISiS.

L

l ..;(z')p...

' S. S ~

(~)

we 1".;mfP*ble. Ilnd tlx:lr .beoiute value. we domin'led by the p' ·;ntqn.bIe function ...... l ..·(r')VIIH( AH) becalH!

:5 :5

. ~f [ L

f [,t.;. 1,,;.": (z'.z"1] f

dV" H(:r")

1.. , .... (:r'.... jdV".. (... j .

l";(z' )" H( A;')] d,,'(r') "

' '' ·S·

f

dso'(",' )

f

1.. ' ..... (r' ,:rHld"H(z" j,

whkft is «Iulvalent t.o

L (II' <:;} " H)(A: x A;') .. (II'

".

<:;}

" H)(A' x AN).

... m:''' ,,,,OlIS, If (A;)q;, _ (A: • Anq;, is .. ""'IU"""'" of di9joint S,ocu all oonlained in till S,eec, A _ A'. A ~, then

.", Ii"""" " is

< (Vp'j S( V pU))( A'

x A")

<+00,

and V (I" s /l" j :5 (V ,,')S (V "N j, It reme,j"" tI> be "",,",n thal V{p' s/l"l?: (VII' ) s (V/l") But, for all limit panni<> ... P(A' j .....d P(A" ) of A' and A" inti> S' ~ 8nd & 1l>MOI"""

S"~ """ptCti""ly,

( L

a"(I'(A'J

1"'8'11) (

L

B " EI'(A~)

L

1,"(8")1) ~

1 (I"s,, ~)(B' >< 8")1:5 V(P' ® p")( A' >< AN),

IY(PI A ') n (('(P\A )

D

DeAnitlon 9,1.1 The II ' U""'" ,,' and 1"',

_II' s

" n is e&l1ed tbe prod\l
If I 10 ... I""",ion I...... fl into [0, +oc[ , ...., shall often ... rilt

/r /r

1("", r')dV" '(,,,')dV p"(",")

I {z', z")dY ,,'(r')dV ,,"("'~l

r

lor IdY". AI8o,Ior ... ,,·lolegI'&bIe mapping, I , from fl inti> a BanllCh 8pooOf:, ,,'" ob&Il ofteto vnilt JJ I (r' , r'}dll'(s')d,," (r') ' - - l of J f dp, U Fill .. Banach opece and a beIGonp tI> Sf(S, Fj, obsen", t .... t o ls' , ,) lief! in SI(S", F ) for all r' E X ' and lhal s' _ J o (s'. ')dpU I;'" io SI(S', F) M~, J djJ Jo (z' , ')dll" - J od" , Heocoeb\h, uru- ot","""ix su.ud, ....un'" lluLt ,,' and II" are pOait ive and put " _ p,' ® ,,", Denote br,J" (.1' '', .1"+, reopecti""ly) the """ of tboooe fuDC\iooo from fl (fl' , fl", ~i""ly) inti> 10, +001 whicb Are upper .... vdoopeo of ioore&ling 00Cj1lft>lW iD Sf"CS) (St~ (5'), St+ (5"), "",poai,,,lyj ,

r

r r

Prop<)llition 9,2, I Iul" _ djJ' lI(z', Jdp" 1M' 0./1 II E .1+, 4nd I djJ?: dp' 1(71. ,)do ... f from fl 'Q/<> [0. +ooJ,

r

r r

PROOf': LK 10 e j+ , o.nd let ("-),.~ I be an iocI pm",""'l.ueote in Sf"($) ..ho.e upper m¥elope .. 10. The funaion z' " (z'.'}d,,'" .. tbe ~ elneiope 0( tbe II\Ci ' 1111 lI!!queote of tbe fuoctio ... z' - f "-(z' .' W . 10

r

rr djJ'

" (z' •. )d,,~

f f s~p f "-II" - F", .. . ~p

d,,'

,,- (z' ,.)dp""

..

a PtopOOlltlon 9.2.2 Ld

f : rr

[0, +oo[ ami

I" : rt' - 10. +00]

D.n.tk., fs/" (z', f (z'W (z") from {} [0,+00[. Then r f * /"djJ .. (f I'dl-f) (r /"djJ'") • ..su. fGClor.' u..

J--c'hN.

-

k I..,

z") -

IM. ft"lCtiarI

ilUO

OM.

rigM-1I4m1 Wk it 0 .,.., 1M. " ther it +00.

PROOF: By P ropoeitk>n 9.2. 1,

r r

(I' ® I"}d" <=

'"'

r r djJ' (z')

f (z')/"(z"JdIi"(z").

f (z'Jr (z")dp"(%") .. rtz')

r

r C?}d" H(z")

lor all z' Mea_, by coow:ntion , o. (0:» .. O. ~Iore,

F""')FI',."....W ,<') - (F 1'''') (F r ..·) . ThuI it ..,1IiotI to allow that r(l' s I")dp $ ( r fdJI ){ r rd,l' ). ThiI iaequality .. clear .. hen tbe ri&!tt. 1w>d oide ~ +00. We nelude tbe CMe in ..1>ld:I one r _ 01 tbe ri&ht--hand lido! .. 0 and tbe ",her +00, and .. it fmIIIi ... oaly to be obown thai. tbe Inequality .. t rue .. hen botb dl-f o.nd

r rd,l' -. finite.

r $" r

rr

LK" e :,... be such thai. and g'r¥ < +00. SlmlWty. let g" in be such llial $ II' and II'd,," < +00."'" coo...-n,k>n 0>«+00) " 0 ;mpl~thal.'Sg" ~ toj" ,lO thac. r g'®g"d" .. (r g'd,,')(r g"djJ'"), o.nd lbe derilw:t laequalilJr IOlIoows. a

.r+

r

r

Propnodtlo" 9.1.3 A' " A" it ,..negligible for <WrY ,,' .....,;bk HI A' .M ""'Y j/' . ...-...u H I A"". Mo"",~" , A' >< 0"" it ~ " . JUgligibk for ~

/ocal/r ,...~

s' ·,...~ od

A'.

yr

ate

P ROOf'", To ~ the first 7 7Iion , .... m.y 7181iUme Aff _ult;' then a ~"'""'"' (If P lOp(I6ibon 9.2 ,~.

;I'.inl~rable:

the 0

Theorem 9.2.2 Lei J' 60:" I"· m...... "'w" ""'","9 from a' ;nlO G m~Irimh'" spa« F . TheN /: (:r', ... ) .... l'(rJ ;. f'.m.,.",rnw".

ring gmt'ratal by S' ($'" and S, l"eIIptrtiveJy). Let A • A' )( A" he .. oonempty S·$Ct. If 0' if; an 'R' IA'· 3implc mappinl from A' into F, then 0 , (:r'. z" ) _ o'er) .. 'RIA.... i.mple. Thm: aiw .. 1"'''''Ili!ible",,~ N' of A' loeb that I'IA' ;. the limil, Ob A' - N', of .. 1Ieq_ (O~)~~ I of 'R' IA' .• imp!/: ""Ppinp (Proposition 6.1.4), 'rbo:a fI.-t "' tM limit, on A - ( N' " A"). of the ""'I""""'" (0.).,., of"RIA...fIirnp!e mappi"., wlLldl pr<M!OF lhat f "' Jl-meuurable. 0

PIIOOf":

Denote~·

'R' ('R- and

~pocti'~ly ) the

In partl
I' if ® f" )dl' - ( I' /'dl" )( f' J" dp") /".,- <>1· Propoait loo 9.2.4

A' io .. jI.n-.urabio and u'·mo
({ 1'1"'<4<') ({ r l ..

-cl, ") s. r

eI' ® /")I ........ dp ~

r

(J' ® r)JI"

( 1' 1'<4/)( I' 1"<4<") ~ 1'(1' ® J")<1p.. It rem.o.ins to he p'p
r

rr

r .,

r-

r

I'®/"dj-< - r'l'®'1"dl'

- ({'W)({.W) • ({ fW) ({ rd,·) o

,.

'.2 Fllbiai '. Tboooem

183

Propo.ltlo n 11.2.5 wi II' "" a ,ub.J.1 "I rI. o.lI(>le., N' doe «t 01 u.r E fl' ndllItal doe M!diom N (r ,·) '"' I r" : (z' , r") E N} 01 N oItkrmineol., r .. rwl !? .. ; ..';61 •. ~ KI II" u ,,' .Mgligibk mel! II' .. " .~, alUl 11" .. Io
s.-u.e

r l",dp :::: r d,,'(r' )!" l.v(z' ..)d,,~ , the fim _n.ioD io

ob.ioua. The - - . i kllion di=tiy from the lim one.

0

wi I "" " ".",
"'til p;OIg from fl ;1ItD " "",tn:.. ..we~, aM" let 11" "" doe ..t 01 u.- r' E fl' If)f" 1IIh>cII/(r. -) " "'" ,,~ ............ ....we. TIle ..t 11" iI /0' · rw>g/igibk ....wn I ........ /4,., ....uik....wn ,," .. ",,,", , ,iltlt.

T beonm 11.2.3

LJ.."

PROOP: Suw- tW I W- the "",,"'ani ,-.Jue ~ outalde A;. when! (A; };", io . finite or ...... wlt. famUy of d~nl nonempCy S.-eu. For..wry i E I. the"' ....u . I'-""Ililibie oubeet. 11', of A; such that I IA; io the limit , 0<1 ..to - N" of .. lequent:e 01 R IA;-silllpie mappinp. Tbm! IlIoo exws • " . t~bIe out.el. NI 01 A: ouch that N,(z', ') io ,,· ·""Ili&lble for all z' in r

A: - N;. If z' j U. E1 A:, tben I (z' .·) -

I (r',·) io ,,~ •..--urable. Now let z' E (lk., A; ) - cu.." 11'1) , and deS"" J _ (i E I : r' E .t:). The A~ , for; in J , "'" disjoin!. and, lor -=II i in J . I (r'.· )I A~ io the limit. 0<1 A~ - N, (%' . .J. of • KqUmCe of R~IAr ";lIIpie mappinp. It foIlooro tw/(r', 'lI A~ io ""'''',..able M " IA~ (.. hen! M " de.r.o' e. the O"·t4ebra of s/'-meuurabIe leU) and Ilw. I(z', -) (A~ - 11'. (%, •. ») io ... parabIe. I (z',·) _ • ouUide lJ..J A;. Therdor-e, I (z',· ) io mouurable M~ and I (r' ,· )(rr - U.ull', (r',· ») io .eparable, to t. to that

1(%',' ) io s/'.IDN&W"&bIe. Th ~ doe --.i .·_aioa. let A' E 5'. The m.o.PJOinIg equal to I 0<1 A' "fl" and equal to • couUide A' " rr io ,,""" , ·"rabIe. ..hid! ohowIthat 11" n A' io ,,'·~bIe. llwee N' is IoeaJIy "'·nqIi&lble and lhe proof io ..,."pIct.e.

0

Tbeonm 11.2.4 (FIIblnl ) Lei I "" a ""'J11iin9 from fl illtD a B"tWJdIlJIII« F, .IUI u" 11" .. 1M KI 01 u.-. z' E n' lor ""'ich I (z' , .) .. not ,.... ... ,,;. .'e. ~ I .. ".~. N' .. ,,'.~ and r' .... f 1 (z' ,·}djJ~ (""'ich .. IkfiMd lor %' ,. /1" ) .. ,,'·inlquubk; ............... ,

J J J Idp -

~ ,,~

.. .....u:rau:

d/O'(e l

I (r' ,·Jdl"'.

and I iI UK>LtMi1) ".in.l~, UI.e.. N ' if Ioaoll~ " ....~, doe .....wing z' - f I (z',·)dp" ......1IliaI1r " .;IIUfr'1l/>k, "nd

f 1dJ< '" f dp'(z') f I(z' . .)r4<~.

PROOf: AlIIIume 111M f ill p.int"!T&l*. Let ('PA). " " . ""'luer..,. in 51(5, Fl, and lee. " E J . be tudI u..t -

ra) (..... )a~1 COIl"'''' to /

<)tI

(bl l"". 1~ h lor oJl n:!: I and

tbe <:<>mpiemem "' .. 1'.rq:1iFble!let N ,

J' 1IdV" < +00.

Theru:xi81.s .. ~ .n<&Ii&ible , uhliel. A' oUt """h that N(T ,) i1 1'~ ·ne&I~hIol lor oJl%' A' . TIdeb" Hm,._ ...... (z', r") _ 1(7/,%") "N.&.~ fot oJl r II A'. On the other hand , heoo' : (r',')dV"" < +w for alb' t 8'. If 7! f. A'U 8', the tequentI! ("".{r.-») • .?:. C'OOvergDI to / (z',-) p".Le. aDd, for n:!: I , 1 ... (r ,·1 1ill dorninat.od by h{r ,-). lt follow!! that/(T . -) ill 1I"·!nlqrabk and f l (r .· }o4IH - li ..... _ .. 1.... (z',·) F . Dy w!\o.t W
t

r 1"/.

1~(7!)1 :!O / f.,o.(r , .lIdVP":5

/o(r. ·)
E f'l'. and dV" " (%', .)d\'#" is 6n;~ A•• oon.q\ICIICC, Hz'..)4J<". defined 011 (A' U 8' )<. ill ,,'.jnleS.able and

for all

J

r

r

r

r

-

-

r _

.-. j ..,.' Ii ...

.-. j •.,. lim

-J

/.1.1'.

Next. a5IIUltlO! that IAN is lIlOden.~ and that I is e!IleIltwly "_"'~ Let g be .. />'-integrable mappitl,ll from 0 IDIO F equaJ. 10 I ""~ • loo::oJly ","",*1i«i1M eet N_ By P ropoeilion 9_2-5 tbe ... is a locally II· negligibie ""beet C 01 rr IUCh lhal. N (r , ·) .. p".~i3lble r." aI!:! f C . Put F'_ (r e fl': g(r.·) ;. "'" ,r.lnI"Ij;.-abIe ). n.en I (z ', ·) iIIl'~. IIIt<-gn.bIe for &lI r f. F U F' and I (r , )dll' - g(t! • -)dp~. Tbo\ IIOC"(>tId ' 55 'lion rollows.

f

f

o

TMon:m 11.1.5 kl I .... p .........1<J"ObI.o: ""'cCi"" frr>m n .. p.m<>krole. ~ /!me""""

i,,'" [0, +<0:>]. II I

z' ... t/(tI, -)dVp~ ...... z" ... j ' /( ..z")dVP'

r r -r r

...., ~ and m<>oleraJc lor I"

j'ldVP

-

...... 1"', .-..po/II:1..",j~; "'~,

dVl"(;Z' }

dVpN(;.-")

1 (z' , _)dVpN

l(-. z~)dVp'.

,

••

9.2 FbbW'. n.w.ew

II j/' ;, modmtIe, t.'Ieft<1Idiqrt r

....

r 1(7!, ·)tlVj/' ;,

r r r IdVjl _

ISS

II·~ and

1(:z', ·JdVjl~.

dYII(:z')

PAOOF: As.!ume tW I If ..-modenott.oo let ( AR)R>' be an ;nc'-;",_ qUCDCt! of jl. iDl.ovab!e lIeU such I ......,:.shes "" (U. A.f. For eact. I.... ttp n ~ I, ""fino! IR _ inf(j.Pll l .... . By Puhini" U....... e"~ \be r~ '1.(r ,.)dVj/' "ddintd jl'· ... lUld is jl'·inlqrab!e. 'll"'tefuoe, g" : r .... 1.(7!,· )4VI'" • jl'.-.urable. M(>!<:o.er, g.,wl" - J'1.dYjl.

r ....

tw e.

r

J'

"The fin\ .-rtion foI~. Suppooe...,.. that j/' • moderate and let A' bt. /"·_ur.1* and 1". rnodcftu let. The mappillf! (r ,:z") .... l (r .:z") I .. ,{:z') " l'"-..tabIt and I'"mo
r


r

1{-r,)dVjlN -

r

I · 1.. , . ....dVjl.:5 J · ldYjl.

J' l {r" ldY j/' ill I"'met.8UnlbIe and lbat

r r r r r r r r r r

ThlI I'«'.ee lhat 7! ....

dV"

f{r, ·}dVjl~.:5

IdV jl .

N_ let A bt a I'"mcesutablt and I'"moderate ouboet of 0. The fwdion r _ J' 1.. (r , -)f(7!,·l dYP" is ".~ and

! ·t .. dYjl _

dVjl' l

1,,(:z',· )/ (¥,·)dYjl" .:5

IdVjl .:5

",hid>

c:om"'r)e~lee.

dVjl'(:z')

dV,.'

1(¥ ,:tdV,.N,

f{r" 'JdVjl~

o

tbt rcoof·

,.''''
Theorem 11.2,6 (ToneJII) Ltt! ~ G,..m(G.I ...... w..n G ttal Bonao\....".. VIc .. , U ,..in~ if and <mI) if eillI~ 01 .w"(r ) III(r,·JdV,." or dY,.~(:zn) 1/ 1(":z"JdV I" ;, /irtm.

r

J'

PIW()f': Th. is • diroc\ _I' Propaoldon 9.2.11

.uenc:e of 11leooem 9. 2.5.

r) into r

(~tiwdJ.

o

ut F' , F", ~n.OI F" ~ Ihroo BaIl4dL IJ'OI<'L'-I, ~n
(.', ,,") ....... lIn ~ • ..... tin ....... h&teo:~

I'

r

J'

.....

ppi..,,..,,,,, F' " F" iJOI<> F . ul

N.n UKJIliallr,.' (~""I., j/'J ·in' ,iLW. mill'" ,;"',.."", rr ( rup<'.di""/r, Jr- rr' in!<> F"). Then I : (:z',:z") ...I'(r ) ,r (T') ;, utmtiGU, W'nI.,.-cWe and J Id.,. - (f I'dj/)(f r dp"). if I' and 0 ...... tegn>bk:, til"" I ;, ,..inl.,.-cWe.

r

,.

g.:u.

P ROOF: By 'I'beo.nm I • I'-~ On llot celie: hand, If b. tlot lOOm> of lhe bilimar mawiD« (. ', ..... ) .... ,.' . ..- ,

by f'ropoeil;oc, 9.Z.~. This ~ IW I iii ei/ientillly ... j~bIo:. Sup' 1>'* that and inlqrable. Then I is ... ~, and tlot,do. t 1'.ID~bIe, and fu binl'l Il>eottttt obooo'S l hat I fdjA '"' U Nil) (f rtlj/') . FlD&!Iy, _ retlll"lllO the t . - w~ and ntlally In~&bIe. Let I' (....p«ti'>'d;y. lot, "",wi", from 0' loco 1'" (' ......... th,,' ly. front rr into F"") equ&I to I' (. tspa:t!¥eIy, Iox:ulIy &.t. Then 9 : (r' .:r"") _ 1'(r') ' 9~(:r"") is "'luailO I locally ..It. (l'IOpOlIillon 9.2.)) and

r_

r

r

n

r __

r )

If.. - I"',

- (f ,W) (f ,,,.) - (f , ..') (f r ..·) o

Let Y ' bill, 1" 'II><. @" ..bIe ... btletolO'. lot y u bill' jAn·_ . abIe ... t.:c of rr. and let r (.-...pa:thdl', 1"") bot a .....1. lag ill Y ' (rtspa:tivdy, Y '"). If " ,... o.nd j/'rr- ""iol, then fJ/ @j/')/cr'1"-) tJtillU""disequalto",...@j/'rr-' AI we thall_ 1100 po.,.,..,Jint; .-suitt ~ 100- RIden..IUfOlJ. Thit, OOwt.(t . ill DQot the tall< ""' P""I'OIilion 9.Z.7. .. hlcb Io! tspetially ..rnI in poObability tl>eoty.

"lei". an

.,.,true

P r-opc>IIlt Loo 9.2. '1" ~I 0' . !WI (Y' .. 1_ ~W H:u. ~I Ii' .. a ~ ... _n: "" • oemiri"f S" ;" rr. Ltc S' ........ iIo 0' , • ...t v.t S' . 5"'. s~1'f'OH. thai rr £<1"(5") ~nJrr E
i....,

s_

r

PIIOOr. Let A • It'" A~ E S b& (iVftl. Denot.!.." C. the d&IlI of tloot
SO_t

.. £ JS,

FiI S E S. and let (A~ II A: )~~ l bot • lIOqu..<>Oe of disjoint S-atU wboooe union oont~ ... S. Put E.. _ s n ( A~ ,. A~) M. =' . ·J _ ~ 1 E~{r'.·J it! an for ~I r' E fI' • ttond lbe function

s:_

r! -

r

1.(%', .)dl'~ ""

r ." L

1• • (%', .~~

•

ill

S" . Bomla.D. Th... If I be'",p

r

1<1

Sr· ($), thea p{r, ') II S" · BoreI;'" and

the funtUoa r _ J'l"r.·W 'II S'·Borclian. FilNllly, if / : rl _ 10. +col it S-&r.:lian. t!>ere ill

(f.)..l!:' in 51"-(5) ..unittil'3l 6i WlI&t _ haw:

hue '''II: 8Cq_ ita ufIPI'r envelope. n... ,,.wI r..Ilowt.. 0 lUI

ox.. 10 far lor the product of t..o",...... .... atends eMily

to /tuite poOdueu ol ...... u ......

n.., dct.o;ls al'lid'l

to the 1'1" ~ .

s..

... , ,.,. _ COIllpll'X WU8....... OD . .mlrinp 5" ... , in fI, • .... fl. , with produd IllCOOIUI'I ,.. lUI
j j ...j 14,~···d,.,.

()t

j j ... j 1(:r,.···. r.}Jp.,(lt,) .. . <4I.(:r.)

J

i"l" ..d cI. 14, and retaia ~ _atioo>l! lor upper in~ with mpo:ct to V p..

9.3 Lebesgue Measure on R k In the E... IidN.D",*", R · . _ ~ tho! vrde<.

_defi""

fOr 0.110, b 1:' R ' "Illfying 0 < b, \0.111 .. t ho! t-d;",en';o""l....unp By ..hat hM bo.en liloom at ,ho! boosinni", "' ....... ..., 11.2 , \be empty _ and ~ Jo. III !or", • .emirl"3.

n,s.s. I-<. "I.

1'-

DefInit ion 11.3.1 Let n be . IIOOmlpty opea out.:t '" R ·. and let S be tM d-. _ 1*;"3 cI. tile ~mpty !/Ct and the rect.",p. A ..I>«< cloo~ A II ..,.,.·;ned in fl. S Is called lhe natural aemiri", in n.

PrDp08ltlon 11.:5.1 hl tile ...,/4.;"" 01 Ih/iroiti,m 9.'.1, rl u .....nt.NIc ....;.... "11Ii.rjoinI. S·.-..:/4"fI'U, mch ~I .' i ct ""')" ... _"""'" t<J "'" of .... e 1- n ,~;~o lp; tr' ,(p.; + I) (m 2: O. p; E Z).

f2"'J

Foe _ \,.. W ' m 2: 0, let q ... be the c .... """,i-U"3 01 all «<1all&1N '" tile form n '5'SO 12'" , (p. + 1)12") (p; E Z lor all I 5 i 5 i ). Put P IlO()~:

JP;

Fb _ {A E Qo : A c rl) p, .. {A € Q,:Ac fl.A .... B _ 8 lor all B E Pol

P.. _ (AEQ .. : Acn, A. .... B _ '

lorallBElJ...s, ,,",_, P; 1

.

,

The ~kille"ta of P _ U..

:r" UA~P.4.,

P... ue C"""1y d;Qoint

~m

S-~18. Let :r E R · ,

IIJIcsI; ~n' IOtt in to .. ,.....} such that ;r,: C O. Then , for every Os:' :S r - I, A. n B .... lor all B e Po. t-a .... A • .u- oot ~ 10 p. and boa' _ the ~kuiC"t. of Q. ~ d~nt_ Now A. C A. 100- ..u 1) r_ 1, and, for SUW'JM' thal

::r. .,

s: , s:

A. !1eI In P" .. hXb oon\.radlcu \.be floCt tbat,. dol!! "'" I:>cIoot to U.. €p A. ",.,.,""', A.:;" n fr< ;. 8 fo,. .-.:b m ~ 0, and ,.". ean <:I>ooeoo .. pOint ",- in ;r,;; n fr<. Sinot Iz" - :'11 I/r for all I i s: ... lhe 8Oeql>CnOe ( ","').. >0 10 :r , and :z belong< 10 the c~ !let fl<. This .bows that UAe " A , and tbe proo( 10 complete. 0

t<:>

s:

a)Il.'"''''

s:

n-..

By P~tlor> 9.:U, -=" ~mpty <>pm IUI:Rt oro io .. counubi<: UDina of diojoint S-rtc<.np,., beliCt the ".';<\11 genenr.ted by 5 O(N1W"" t he optu ,ut:.:te of n, and it '" tbe Borel ".~ in n. NO'O' let )" be Le~ measure on lhe .,..11.1>31 .e
>. . .. >.,*···3>.,. OtofInitloD 0.3.2 Ci...,., fI , .. ~mpty OJ>I'D .u~ of R' . "'" wi Leboeol!!"'" m "'Il/'c on n the "",""ure p induced by .I., on the nMunl 8emirl"" S of fl. 80 ,,{ A} ..

n ,s.,;:.(6, -

1>;) if .be ..01""", of A. ..

n ,s.,;:. )a..6,]. for "''NY

S.~A.

BDm:i&a for Chapter 9 I

Wri'" ~ lot Lo:beot; .... D ,..J - ' io ""'P'~

1.

','" "" R. "'" <> be ..

""o....,.....,pIt>; I>Wnl>o< ...""""

tho.!" _ r I .{i." ""1IO ._ r itllepab!o. .. bt;n Rq,, ) .. ,ttktl1_. ati ... bu• ..,. ........ " .. ~ 18 ........ ~Ml)" uil.. Ho ''''' , '" U" Iou

p",...

I: e"J,/ib .. , _ 0 "'10,+00/

bu .. Iimi • • _ by f:"" ,.., ,/ib . aad. _ +<» (_ ............. - . . val"" Iorm .....) . In ... hal-'>l~ F (o) .. __ lor r l .,fidz.

~, obow ,!oat

2.

For &II 1M • • .. .2: ton00 fu".h"n./~ and th "" 10. +001 ~

.!,. ...(.~.t .u 'I· ",

J. (I) . /.'

Sbow .hal 3.

I.-

I,;: 1.(1)41 -

.

~

J.;. ,.(z) 4

th (z )..

, /.

"e o1 --· t dt •

".

">I (

.

Lot 9 be . hoo fu ...""

._J..• ./i

..!.. ~" G -·)
••

Eu,, "P ppio
189

N , oIoow tlw. a.)0l.:," " " ' _ ~~ibn>ly 10 9 on II/ n , nj, ond doduce drat f,~, f(:r) d.r • '. (1) dr.

from jO , -+«>j iRto C . Ci_ "

2

Pi:.

f.

4.

s-...bat. IM I)) :5 2/(.jj . II - oj) b all n E N ond all I E jO , Oeduce tIw F ( _ 1)F (o ) • I/(.jj . (I _ o J) dr.

~.

Oeduce from part 4 ,bat. F ( - I ) _ .,fi ond F (i ) • .,fie " ".

1.-

+<»I.

I. Lot" be ,be IuDctioor (z, , ) _ . - •• _ , from jO, ~ " I - 1 . +<><>I InIo C . S""'" .bat., io< ->' in! g '" E Z ' ond ~«Ia!j>Kt ... boot Ii of

t.bont.,..... = ..... CK... &l>d>

jO. 2"b ball (::::. , )

I~

/(

~

10. 11.

(DCI.a5io1O of part I). Oeduce drat

4.

s :z

_

,ho.. 5 io iIIRniteb- dirr.". ,I&bIo.

Lot:::: E jO,2'o'[. ~.hat

IIh- (D (:r., )I:5 CK.~

,>I" 1.N_' . \ - ...-'

S. coo,,", ... unifomlly on.be """'~

. u..... of jO . 2.-j 10 .be fuDCIioa ~

,bat.

"""7"

Jr - l'~P'''' . -;:;rdr

• U _ I .'" i.:r •

011 IPIU$1n t ~ I. and dod""".bat. ,ji . .ui.:r_ 1 • 5(:r) -

": ., G) (.a- «3m - L'll &..

•

1

"7. dr .

>I'

I . ,. :5 ::itio<

j. - 7'"

~N

t-m )

Oed.,... from part 4 .hat Sf>:) io ooymptotic 10 .,fi . • ,., .

J; M :r '""""

to 0 ill jO . 1"1. Nooo. ohow ,hat S(2'o' - z ) io Mympto
and COOIcl .... . bat. S io 1'"1.. 3pable (,. • $.

.,." M _

,up{I!.' ~ "'~I: r,f

:5

h'

2~ . ~/III,"I)'

injO.-+«>j} . CiWII n E N . oIoow tlrat

I''';; I '5.(%)1 :5 "* +~, (~) io< all :::: E jO . 2'1(_.he ~.... i_uali.y ill part 4).

1.

Oed""", from part 6 _ I. ond aIoo

.iI<

.ho., S. . 1)0 .•) =·...'i 10 5 · 1)0 .' 1ill 4,(Io} (le., 151 ,ba, s. . ' I• .>.p"""" .. p 10 5 · l p. ..., 151 ,iI<_.

Cooclude .hat S. OO'hE'iM

to

S in 'M ........ aDd ,hal.

'1""'--''' .-

2_ . S.

3

,,~

..... (OI/,fl

Io
p..,.. IIIN
(k E Z ).

Lot I' M ~"" _uro on [O,I( and I 1M Iu"",ion (z . ~) (:E' - ~' ) +- y)' from n ~ la. II ~ )0. II into R.

/(%'

l.

_

Sbow .hat r l/l "'(jo ~ .. ) - +oo.

Z. p......, .1Ia. tbe

f

f

fund"'" " II" , ~) d"I, ) and ~ /1:0:,11) d!>(z ) _ do:fizood on 10, II and ..... inlqlablo. Show that 4(%) l (z.II)d;J(~) ~ ,M,l ! /(:....)
f

f

..

f

Lot n boo an """",,'·ulli ... aDd l' .... ".alP_ in n """, .. ,;", of tho """",ablo azod ~~, on l' bjI ,,(A) .. 0 if A .. rountabIo, and ,.( A ) .. 1 if A ia _-,,,,&blo.

o..ru.. .. _

..

J. Proooelhatll._{ (" . ~)E n~n : r.\I}io_,, ® ~~bIe. 1. Cond""," that Il. io ..,. ,, ® ,.._u~. "",n .hcru,Jb .... ll.(z, ·J and 6 (- ,\1) 0t'0 ,,"-..raMo lot" E n oad , E n.

_'Omo

6

Do,,>!e t>,-.\o ~ N . ,,"' .... fl' . W• ..-I""" .. DOaIinuooo and iIIjoed l~ funet .... 9 (",... [0.1] g((O. '11) io ..,. A,·~isible.

"'to R ' ouch that

...... nambo.- G < I and 1- n . <>(_) - I. II in . .... "I""" Q _ 1_1,1( _ [_ 1,11. n-oot Q(<» ... be un.... ", ito lou, d~ rom_D" Foo _

".I....

Q.(<>, .. [- I , _<>1~ I- I, - <>1 (h(o ) - 10,11 _ {o,11

Q ,(o) _ Q,(,,) -

I_I, - 01 • 10, II

I" , II ~ {-I. - 01·

w_

Let H ...... ~ , .• , ~>.. , H ... boo 1110 """""'botioo of R '. wj,b rot" ( I _ ,, )/~. ,"""""";"" """,\era .... (-I. - I ). ( - I. I ), (I. I), (I, - I). So 1' - '--botioo map Q 0' ''<> Q.,(,,), Q ,(o ), Q, (o ), Q.(o ), ,......,..;-.,!y. O~.bo ot'- 1Iaad, let Ro,.. be .1>o .,.... ...... .,. cl R ' .,:.b _ , to 1110 ~"" .hroqll ( - I, -I) and ( I, I). &ad ....ilariy let R.a ... boo 1110 of R ' witb rdp<'ld to lho line ,'''''''''' ( I , _ I) &nd (_ I , I ). FlrWJr, put J,... .. Ro.... 0 H •.• &ad "'.... n..,. 0 H ... , Fo< intqft 0 5: k 6. ddi"" ....110 fuDCt'oon from [0. 11 int<> [0, 11, lbe of "" oIIi"" fuDCtlon &om R into R, which _ 0 '" k and 1", k + I, .... haI " . mapII 10,71 ""'" r• .. (.t , k + II.

.ymmet.,.

-n

,.,..rIct.ioo \.

:s

be 'bo fuO>o intervaIo t. (O:S k &) and io ouch .haI tho ~ 01 0, I, 1" .. , 1~, Lot

/D

"'I*'i~,

:s

(- I. -I), (_ 1, _ <» , (_ I.,,). (_
olI0SJSa.

Eu .. ·

191

1o<~9

I and ..dI _ . .. (i,. ... . I. ) of .. _ in 10. I. 2. 3. t . ~ fl. 1>\1' v. = b" 0 "', 0'" 0 .... . Sbow .hor. ~.(:r) .. i, + ;. /7 + ··· +;., / 7" -1 +r l7" b......,.:r e [0.7]. ""-Ihor.. pOint r e [0.7] Iiooo in ~. «(O. 7]) if and ""If ifiu _ _ 7up-_ 11M 11M 10m! EO!I ""Ih .t, .. I, •.. .. k. .. I•. Conelude. for bed .. ~ I . Ib&!. the ".(ID. 71) ~ disjoin< and lheir wUoa -{O.1\.

2. For HdI i........

~

"-f1P-' ,

3.

fu .. ~ l. Sbow .b&!. ~ e 10. 71 . . I" .WO 01 .be v, (10. 71) II """ 0Bi, if it IIM.be Ionn ' /7"-' 10< _ I~ 0 < • < 7". II1 'hio~. II r " E ~I (It,./.,..-') """ ~ .. E.~, (1./1' -') ~ r""j«U""'" .be ........

""-"" 01, """ .ho 1m... """. np.IMIon 01 J . """'" u.... the unique .k",elUl' " (i, . . ..• I.) • Iiooo in .·.(IO.7l) ~ (1 ••...• 1.) """

..a.

.ha.,

(A-. •...• .1." . .. Lot (O. ). !I ho • ~uoencoo in jO. l( ouch .b&!. E.!l o . . . +0<>. Tbuo 2 1(l - ",. ) .. D. Dell........""""" (g,.J. !1 of .....U......... funct.,... ftom (0.7] 1<>10 Q .. 1oI1ca .. " .. f.,; 10< ~ I. g,. t.Pe beta obuoI ...... and defi ... g,. . , (u.(I» 10< 0 :5 I :5 7 Md b - n oI.ho 7" "",,' ..... (i, ..... i.); II one at kMt of.be 1.10 odd, put ,..,( .. , (1» .. ... ( .. , (1»; on.ho "" ..... haod. II i, •.... ;. ~ 011 ....... in'""'" _ if j. .. (1/2)i. b 011 I :5 • :5 ... ...... put .... ,(V.(I» .. ... (1__.,(1» . .. bt<e .., .. h" .. , 0" .o h....... J'Tovoo. b bed .. ~ I • • bat. II r e 10. 7] Iioo iB .wo oI.be u. ([o. 7l) """ (k , •...• .1." . (I" ...• 1.) ...... in I*t 3. lbe dofiailbll of 900_1 "" "'I" .. ~.I(IO. 7]) Md "" "". , _..... 71) II" ,. .. (. 1 .be ...... ¥OJ ..... oo.hat lbe defI"illon of 900 .. 10 CO riM ..... '" Lot .... E '21 I./T- ' be. pe;Irt of 10. 7]. Md let ... be . be .. I"'P,"' of (0. I. ...... } ....... hat i, ....• ;_ "",oII...ea. J'Tovoo tbat .... ,<.) .. (It" .. , 0··· .. h........ _) ' 1•• ., (E. 2_., ;./ 1'-1-" ') . .. bt<e j,. .. i .(l b oIII S . S m. f. Lot" ho .be R' . ....<1 wi.h .be - ' " II(G, b)1I .. OI,IbI). Sbow d(9oo (, ). g,. •• <.I) ~ (I - " ,)···(1 - ". ) b 011 ; " ' - ' n ~ I """ , ~ I and ......,. , e 10.7]. Oed""" that \.be AQ • "0 (f.,).!1 " " " _ unifortnly ""'!D. 7] to. com;nl .... fu""'Ion,.

n.

n

,u_ " .....

,«(0.

'ft.

"jet.

h

.ba.

7.

8.

p..-.h.or. p."....

~

_ illj«\i...

that , (IO. 7l) 10 ~-cliciblo if 2' .

_ .. - +0<> ( .......

""'i...

"'"

!of rooIradictiooo).

n.",o",.(I - "0 )

0>11, .....

to 0

·1

1. ID'ho __ ."""of_lt_ 5. b - . in....... 01 {D. 2, 4. ' , ' . _

~

O. b -n ... (i, ..... ;. )

b -n intqrr I S k S 3, 1>\1'

K •.• _ ,. .. (u. «2k -

1.2kJ) ).

So> K ••o • (~" .., o· . '0 ~"' ... ) . / ••• , (12k - I. lIrJ) .. hoeno it _ _ .h.or. j, _ i,f'J •. ... j. _ ;./2. p ...,..,.h.or..be .... K . .. (10< 011 .. i! 0 _ 0II • .and k) ..., dlojoi
yr

ate

2.

IW.... .ho.

*": :.

(K.}.~,

.. ~~ """ I~ R' .. fo!~

u_.,.. of {(- I,-II). (o. - In

•• ' 1(,

.,([l.2I). 91{13.<4I).

_

io

,.(~~): K •.• ( ... her
_ .Il00

fo< """'" d>Iqt< n ctiono ~ 1"''';".; f ', -I. - , I) ~ 0, p , ({l. - I» _ I. _ putti"41 1", o:q ..... 1<1 _/4 .... ,I, (11. - l. 2"'1 ) hI !S l S; 3; for 011 n <: I. 1"0+1 _ F. "" K. , &01<1 F"" if "'I .... 1<1 (1- t j 4)· F.(,..,. ~. (O» + (t / 4) , F. (9-., , ~.(7)) "" _b K •.• (with

«(

IE IO, 2, • . 6)" _ l "i k Sl). Lot G bo. ..... ddI_ .... U.~, K ., "'I"'" 10 F. on - " K • . """""" !.haL, "'" 011 .. <: 0. lor ..n • ~ (i" .. , •i . ) E 10. 2. 4, 6 )' , ..... fo< olI I S l Slo e tUN tl><...! ... 1> / 4 + . -- + j. / 4" + t /C'+' Oft K •., .

f,,,.,,;...,,

3.

E.~ , {4/7'-') be. puint of 10.1) (0 S /, S 6 for 011 . <: I). Set ;. .. 0./1. fo< _ b iDl ES" • 2: I oucl> thai i. ;. ~_ If O ~ of lho I. io odd. ODd if " 10 ' M lOP:: poortioe i_1qIO< oudt \~oJ; i . +, ~ 2t _ 1 II odd, pool F(no) ... ,,/4 + -.. + j., f C" + 0.. tho otbet "-1, If .toe .. _ .... i... S pu< L.~,u./4· ) _ S'- tlu.t dep""d "" O
Lo<

r"

i"

o"

""""- ,hal. F

..

FI,., ...

""'+',

an Fet,) _ "'"

,rr (,

n.... _

en . .. G aDd ,bat F(n!O,1\») - [O. ll

.he.,..,..,,,,

Fb. . E ~«(O. 7)). S - ,bat • h.. in of D. U.» K •. M~ OO'ef, it' "...,..... , ho. &1 ... i""~ I) by .M ~k.er 01 ~bo<>
""

,(10,11) ~.

5<1_ ,Iw " . S 1/3 /no-

.u •

~

I (_

E.

,0• •

.j.oo _

b@.

be), P...... bat. /no- ....itably d>oII
IFCo) _ FI")I

.hat.

""'I""""

s c 'na' _ a~ L+. ,

,. 90>_,1>0., (" .).~. l'*Obeo.cl.'

.... in.,....~. By \\'h\tney·. ~n 'Ir.oo "~" the . . . . . . . COO'II in_1y d ifl'....... iIohIe f""",100 I from R ' into R wbich .,.,..... F _ 10 _ h , bat 01(.) _ 0 lor all. '" , ([0. 71). 'Then

all poOrwo of ,(10,11) ....
1 (,(10, 11)) _ [0, I] iI I)...,.~ u ..

10 Radon-Nikodym Derivatives

,1>

r..

lhio c ~ _ ' . .... 1Il00 _uroo tlw ha...... 1>tlItICri
,we "

obow._ 10.1

w~

en..

".. ...

deli .... oum ..... bk ,.,eiliot 01 _""'" "" a _

"""'i'i", S.

'UI'O "" .. oomIrinf 5 . _ aDdtrIylq ..... n A IUOClioto to bo bcaIIy ~int<wabio ............ ,iA io /,.;Blql bh fo< """y A in S; tbon t ho" ; 'u'" ,I A'dy, ,,,.;~) If .....:l...Jy If I, it ".~ (. f " n.dy, _iOJ!Y jO-ir4ep'.t.Io) (Thee '"'" 10,2,1). W ),o., " ;, ~" 5.re ....... jnt . .-..I, Ie< .. OLr: ..... ,baI 0Yft)' fuDct ... - . 1 _ _ _ Uat IrM .01 _locally .... bI~_ 10.2

Lo:t" bo .. "

,: n _ c

io _

_ _

'1' :. . _ f

r

r

doomI.,. ,

"'''''ia.IlJ.

"'

in'" S.

Lort 1< boo tho rillll _ _ b,- S. A ,,'" "'1"8 " .... S .. oUd to boo ~utdy a;mtinuoua .ith """"'" to " if ew;ry jt-~;pw. ~s..gli, " A_ber oquinIeot «>&ditlon io .100 IoIIowInt; ..-litiooD. F\lr ever, E in S and lor """1 ~ > 0, there enu 6 > (I ...... tbat 10< all Fin 1< "" 'i~ in E and ..aiofyi",.boo i""'llwily IPI(F):;; 6,..,....., I~(F) S a (1'boIo& .. a llU.~)_ T hio aIoo o:-...to to "'" ~ wltb """""" to " (TIwo>o"" 101l.1. Radoa- NiIoodym). M" .",.." pd ., .... S ... Mid to be mutu.o.Uy .ur,pLar ........ h.1 (loll, M) _ O. Tbio " \haI " ..,.j ., .,. "",oce..I,.tod "" diQol ... _ (P ros-h_ 10.3.3..,.j 10.3.4). E.e
L<&,. be .. _ _

(In . . . . . . .

.. ""

,that., "- •

"DO

,

W1iqootl:f... tho ...... ol. _u"",,. at.oIutely """"in"""" .. ith ..".,.. 10 _IU'II ~ oucII .Iw,....:l ~.,., """uaIb- "",,.!u (TIIoof'Om 10.3.3) .

!OJ. Wullo,., tl!at t:,(P) ,+0, .. ...'" 10,) ( 1...........

~

.......

.. tbed....tol ~tp) (wi
0>Qj~

f

_i.'O' .. _ L~tp) (P rop..hloa UU_ IJ. 10.6 w.. "-rib. tbo dual ol t..C'tp) (P. """"' .... 10.e.I) . A.I+ OU" I ... ooadid"•• 'M< .kio du.ol b. -tqlIC1bio ""' (f"r<>ptwilioo 10.4.2).

'P'.,. ...

10.1 Sums of Measures Let

n b. .. ""''''''pl1~ . S .. IOemiri"C in n. and 1<- the " '" gwo....ted by S.

1',. ", be ,wo potitl~

+ ",. Then r I dj, ... r I <4<. t- r Id;., and r I rl". ouPSE"1l 1· l s<4< - I <4<. + I rljJ, b II.IlJ' fuDCCIoa I from n into [0. +co1- Ne:n . let I be • ~ ~

mo:&IIUI'1!II

OIl

S. and

rfrom n

PUt" • ".

r

r

Into. mtdy. _otWly in~bkl "'itlt .dpo::l b«b 10 ". and /". In thlt c_. I JVo. + $I,) _

n

J

J Irl". + f 14,.

P ropoooitiotl 10.1.1

Ld H !If .... _"'·
5 , "'"MaI ~ ... M~ (S) , allrl_K" ;, Itl , ..""""..m'" M ~(S).

from

n iroto [0. +001.

77>( ..

,,' V) -

$UP-
P MOO~'

Firet ........... thAt I ... boonded and ...... blheI ou~ "" R-ge1; £ > O. _ co.n find .. E H lID that v(E) 2: I'C£ ) - t/i / l CLem ..... L3_1 J: - "IOU) 1"U/ I . Is) and "" (f) i!: ,,' (1) - t. e.nd .he

Oi ...... 6 tben

U.

~t lol~

Now

"'~

s u. - ..

u.u I ' .....ion

s'

£ . bu... _.... orily boo.mdtd. Sinft I .. IlIP..!:o !n1U. n ), we "ill 11&." ,, "(I) .. 'UP"", H .. ' (f). Flnally. ~ "O U) .. IUP/tE'R ,,"U - I. ). \be """1t IoIIows in .be gaw:.a!

_

oot.oide an 1t-«l

0

10.1.2 l.d H aM ,. k ... in Pro,."ition II). I.!. arwl ~. I k" ""'J>PirIf tr-< n ;"1<>. mdri'"'' 'p"«. n.-"I q " .m_...bIe if (ollrll>fOlr if) ;. ;. v-mfal.nohI.: /<>r all .. e H

Propoa!~1ou

PIIOO': Oeoote by C ,he dose. of tho!oe S·f!I<.I/C E)"'~, CIoarly. llot unionol .. l'f<'!......ce of C-eeu iI .. C·8eI.

Nexl, Jet K be a ~."'l<'grabLe 5.- and put r • S llw ,,(L) _ r. Su~ to ret a oont. oodic!;ion, thaI ,,(K - L) ill IItrktly poililive Then l!>ere msll ~ Ii: H .1JdI thai. v(K - L) :> O. Si""" / ill ........ ~ caD liD 0 and ,,(L U AI) :> r, ...1Ucb. (:Ontradk:t:l Iht delinition of r . Tl>tr
........we.

DeHnltlon 10. 1.1 A family (,..,)"e .. .,f I>Q6iti..,. P>eIIOIure61)D 5", 6aid to boo ."rom .hIe ... v.G(E)) GU ill ownmab!e!or aU E E 5 .

hoo"""".

£..(8 ",..

v...)~u '" OwnlIilt.b!o: i f and 001)· i f the ul"'...-d·dir«tbt ~ of the • 1.., e B txIa>dI o....11>e clasoi ollinite OUboieli of A . ~ a SUpretlm. In thbI t:aliO:, " : E _ L..E " "",{E) is a !'06ith'" meesun: "" 5 . called the own of Ihe ,... and ...mien Su»p(lote v..)~ i6 a S1; a ""'!>Pili 9 from n 10&.0 a rnetri ... bIoe p-meae" .&ble If and <)J>ly if it "' ,.,.." .. ,,!urable !or All a 10. A; a mawl", / [rom n into a """ B&nId> Ip&O! ill allally p;~ If and ODI)' If it "' tdia1ly ,....im"l'"abie!or aU n E A and tbe lim! L..~ .. r il l d,.,. ill finit~ . in wh;.:h <::Ooee d,. '" dlJ.,.

E...,.. ,.,..

..

,,'(n -

,,:(n

"'*' "'

r-.. f /

f/

10.2

Locally Integrable Functions

Let. n he. II()(W!mpty.ec , 5 ...miri", in n , and" a cornJ>le>:" "I\"re.,n 5. A ""'!>P'II g from n Into ~ """ Banach SpfoOO "' II&id to be 1oc61ly " . illlq... bIe if ,· l s", " .inlqrabie for IlII E E 5 ; iD thill~" 10 j>"meMIIl&bIe. P rop
G..........,..,,, (.,.,. ,.

PIIOOI'c 'J.<" oo..1ou$IJo E E 5, WOl haw V (gJ.<) :$ V(gJ.<).

&

rneo<SUn:. Slnl:e I('I')(E)I

1, 1. V" , and

i.

:!O

,.,.,w,.. <0 be

(1, 1· V,,)( E ) for 1lI1

ohow~

t hat

191· VI'

E""

:$

Fl.rst, ouwoee Ihat 11 _ C< . 1 ... , ",!>ere / is finite, the "" "'" o>ml'l"" " ..mbe.... and the A, aN: di6>int S--. Ci ..... E E $ , 10. ~ i E / t~ ""... a Iinlte p&Riti<)" (B.... ), u. of E f"l A. into Now, kIc IlIl I E / and k>< All j E J;. Id. ptB,.;) ",n through the claBo offlnite peni.1ono of. B;J into

s.....u.

s_ _

(1, I· V"j(E) _ ~>. I· VII(E n A. ) ;(,'

,.

- D" -L N, "EI"("' L ..1W)I '101 iV . 1'( ...... )

.. L L SUp L Igl'(f') 1 «;1 ;EJ, "'EP(II."I .. L L V{gIi)(8 ,.J) :s V (g,,)(£). pt8, ... )

«;.1

,E',

le,>ef;&! <:Me, let E be M S-llel. Gk-en ~ > 0, \tw:re ni:!!tII a ill SI(S, C ) ...m S __lhal. I.. - jI . 1. l otVI' ,/2. For .u fini te ~itlo.,. (E.)"" of E Into I~

In

J

:s

L J<> ' I~dl' .f./

:s L lf~'IK,dl' ~I

1a,4V" +E/ Io 9 1' iii.

J

:s VUI')(F:) + 10 - rI' i.dV". So

"""'(lit,. V,,)(£ ) ..

f j,1-

l.tiV"

:s

f lol l. eN,. +f la - f l·

l"dVI'

:s 11 (<>,,)(£) + f/'l:s V {gp)(E) + f. and. ojncel .... arbitnzy. "'" condOOe thu (191' V,, )(£1 :s V{g,,)( E). 0 ~CIo) oflocally ".!~ funo;tioro:o from 11 b>lo C if_ UInlpie>: motor fIll'Ol'. and the JIl.I.pplft,c 9 - 91' from ~M Into M (S, C ) if Unear.

The let

"" .. 1121' If and 00.1,. it lf' - 91 1- V" ... O. Eq,uivakntJ,y, "" " rnl' If and only if " .. Ih kIaoJly '"'to j wbt:n:. When I' is pOIIl ",,",

".0.1_

Inl{g, /, .9\lI') ... int{g,,. h I ' j.<

for all locally ".intqr.bIe functlon8 from fI into R: ,nde<'!d. sup{g,.,.) ill locally J>-iMqrtIbie toDd

.. ..

,ZU'" + , ""J ,,u,+rn)"+ 21, ',-9JI'1' ,'2U, + + III - " 1"",, "1') ... i ll ig,/, -

.. /12 .. "'PUll .,..) ·,,·

10'21 ....... lIy

I~

F\I"",loot

197

_ute"

o..lInltlon 10.2.1 A on S • oo.kI to be a _u'" ";tb ' - I' if tben! ill a locally p-intqrablo! fune\ion g from n into C such that" . 91'. Then ~. which • limned UI> 10 • locally #'nrsliliblo! let. it called • dcnoily of" ",1&lh... to 1', or a R~Nikodym deri_i"" of" with , apeet to 1' , _i""", wril1w .w/dp.

T'->n!m 10.2.1 tel g : n _ C lc /oc:allr p •• nl'5,allk an
IJ! poillU Z E

ra)

fll1Idt IAIJt g{z)"" O.

.4 "",.., E 0/ p.~

n ;, 1oaJ" gp.~ if annIr if E n X

(II) .4 m.o.pl'in9! frgrn

<mI, if /tx

n

;, IoaJI,

in./o a "",,,,,,,w. II"'« ;, 9p·m""~",w.

;, p .......... n:.bJ.o: on X .

if.n
(c) A mq p;,og! ".",. fI ,...... Banado 11"'«" gp .............w.: if.M """ if J, ;, 1""'= , ....101.; it ;, .......,.,...,Ir 11"""4' .w. i/ an
J

J

P IWOF: Lo,t E be. locally p .J>ef:liJihle .uboH of n. Fix A E S . G;""D r > 0, t,,"",cxiot8~ > 0 0IICb dlDl J" I, l dVp S r forever)' ,,·in~ ... beet 8 of A .... too. ""'''u''' Vp( 8 ) it Ie8I tliM ~ (Section U ). Lo,t ( A. ). 2:1 be a Iotq\IeDCe of dioijoill1 S·~ ;,..,luded ;n A Ai5dllhat E n A c U.2:1 A. and E..~, V"(Il,, l ~ 6. Then

."

- L. j ll l·I A. N"

." .. j 191· !UA. N" S.

0,

and bo,><e V(/,,)"(E n A ) S. r. ThlI "" ...... thai. E • locally I#,,,.,pi&ible. Now . If ! if. p.me&8urable .....1'I'in« from (} into. _ri"hIe BJ'IIO". II • 1!,-lMUIIrabIe by Pros-imn 6.1.4. NeJcI . let E be. locally 9#'nrslilible Il0)l.. We abow tha,

• !'-U gliglblo! for all A E S and all 0 > O. For """" intqer m ~ I/o, let (.4.....). >. be a - t _ of S·oeu contained in A.uch that

E- -

.,.

(U .4...... ) n {,, : liJ{zll ~ r} :J E n A n {z: Ig(zli?: r }

,.

.

V,,(E-) <

L V"(A",. . n(% : Ig{%II~~ )) "?:'

m

.. bid> pro",," thal E n A n 1% , (g(%)1 ~ ! I i:s " •..ec!i&i!k. Tben:IOre, E n X ill ~ly j.I·..ec!igible. , - '(0) Is ".roeuur&ble and tIC) II'"D>t/OOurable. lei. A E S. Sino. An , - ' (0) ~ 'I'"integr&.bie, """ (An find a Ol'qucnte (O' )"l l in SI (5, C) ..IUd> con~ w 11"""',.(0)..(1) outside a II'"~bIoe aM 8. n..,., (""')"~1 CO<>.aga! to l(.U,,.(.)oo(I) ., _ 0 O1J\.Side U>t aet 8 n (% , g{%) -! 0). wlUd> Is j.I.oq!i,sibl/:. Bec"_ J I"., - o.II,w" - J I". - ".l dU, IV,,) br aU p ~ I and q 2: I, (O"I ),,;:' II a ~""by teql>tnl:<'. ThUll

,,,.c.ucilJr

j 1{..., ..,,,. j..(l1 d(I, IV,,)

j 1" .. 1d(lgiV,,) - .!:!.'!.oj lo.., ldV" -

U;x>

_ 0, which _)'II thaI An, - '(o) II '1'"I>tI1igibie, and therd"on: that , - '(0) Is locally gl'"~~ So • ~ E ~ locally ,,,·neglWbIoe E n X II loooJIy I'" nqli,sible. Nooi Iel I be a m.t.ppinl from fl loto • metrizahOe Space. Fim, ""PP""" tbat I", IIl'"meMurabie. For every B E $, ...., (An find • IU*, A E S of B n X tIC) lhat B n X - A Is I'"ne&lisiblo: . T l>ert exis\$. I'"ncg!igiblo: ruhle\ N E Sal A """" thai IJ. _,. .. 5,._,. and I (A - H ) io ~ (Theor-cm 11.1.2). !>iDoI! N U (B - A) .. 'I'"nqligiblo, ...., ..... IW I II g".

,.""never

_ .able

l1>te8l1Tt\bIoe.

c...,u&tly, oupj)o8 ruboott B al X , Ie! A E S be u ..booll<>l B Iud> U,., B - A iIIl'"u J&ligibk Then! exi!lu a '1'"1III&li«ibloe ~"""'" N E 5 <>l A rucIt llw f , .. _N ir measurtiNe S' A_N and I(A - H ) io ~~. Then. sino. N u 18 - A ) ill ,,·nqligible, It, ill lb I .. ...hle ii,., and lOCI I f" iIIl'"meMu. able. In sbort, f II ,,, •....,...,,,.ble if and only if I I ~ iej.l·meuun.bIc. :0 poonic\llar, a ""'wi .... I from fI into a B&D.Kh '1'""",..uabIe If and only if Ig ill ,..... ",,"'abIe. Now Ie! I be a mappi", from fI into a Banaclt 8~ F. Flr8t , tbat I II bltiallJ' ,j.I.iI>tqrabIe. ",.".., exist.o • ""'I ........ ) . ~ L in 51(S, C ) ..1Dch ~ 10 /loooJIy 11'"a11OC1St ~'ben:. Then (0.,-)">,

op&ee"

,,,.c.uchJr

' 'pp
• a C' !I"t,y .... uenoe in

£bIP) and it
.~

.-ntLally

Tberdon:,

19 •

! I,d,.

s.o.intqrable and

...

Ii'!,..! a.,ds.o

.. •!i.'r'...j a. d(gs.o)

- J''U,) Coovenely, SIIP1""'" tlw I, is _iall1 ,..lntqrable. n.e ... ex;w • !>-=able aDd j.l-b..,dt, .te 0lH ..t ....:h that II van ....... bt:aIly ,... . ; _ .....erywhMe 111 O- A If / . tho II1.I.JlPin« from n into F .hid> au- .ith I 011 A and ."",w.:. outtide ..t. tben I, .. /g oooep4 OIl A' n 1>= En: Ig{r. ) 01_ Thill I, .. 111 JocaILy j>-";""""~ tTCiYWbcre. and / .. / locally 9"''';''''' nc.-Iog .... .,.,.,.,.of1t-.u......:b tb&t / .-I8beo ouWOt and, '" all n ;:: I , put F• .. (>= E E" : ]/ {r.)] S n). ~ tho I ·1 , . U"II ,,..inlqnbio, t.bo _!'Pilip / ·I,.· g ~ ,..in~.

+

u.2:' e...

Morea ..." ,

j l/ l· l,. 4(!,!Vs.o) ... j IJI,I,• . !, ldV,.

s j !JlldVs.o ,

r

and ...... I! II,. loCi 5 to III .., n leDds to +00. _ _ lhat 11I4(lilVs.o ) .. ftllhe. Tlw:.do.... i io ,j>-lmq:rablc, and I io .....,iall)' g!>-Ln~abIe. C

Tlreonm 10.2. 2 LlI .: n _ C i.e Ioooilt/ "';nllf'

r

/
r

~_

Th<-n

lliloNs.o

I.". .tl"'~ I ".",. n ...'" {O, .oo{. 1'1I:00I': W.. mo,y auppOM: that s.o and 9 oan pOSItI~ Fnulmpiicity, put v ... ,,..

,....

If " io tho upper o:n~

J.

(If

J

an iDcri'lPlSiq: l1l
J

J'

Ir" .Iv .. IUp h.,4,. .. h,dl'_ ' 2:' II 5)l1aws IIw I,dj.i s Idv for all fuDCtioni / from 0 iruo {O.+oo]. If l,w .. /lnltl:, there eUru a COIIn\abIc union E of S .... ....:h ,hat / """bbN locally "..~ .......,..,..hMe outside £ ; lben I, .. 0 JocaIIy ,...Lmol5l _ '/"t..... on n - E. and

h
r

r

r

r I,ds.o" r 1,- I"d;J, SrI - r lrr:,w ..

l,w,

Cooverwly, let I be • function from fI into (0, +001. We prove lhal

f' l,w s:

f'1,dl"

We ~ ..... ri<:\ oor attenliotl to ,be....e f'1,dp < +00. lMo tt..~ exw. a counuhio IInion E 01 S.oeto such thal If vanisbooo ~ jJ-al....~ OIl fl _ E, and It..."","e ouch ,hal. I ,"" .. iU lnctJIy ,,·al....everywt..reon fl - E. ~ f' Id" "" 1· 1.,wand I,dI' 11.,dl'. In view of the o.booo:. """ may imppOlIO': thai. I.....,,~ ouUidc. countable IInloo of S.oets. Ddl"" <;> to be It.. function .. hich i8 <:t>no from fl into +on]1IUc:h tloat " 2: I g. Then " . Ix - /I,pg Is I'.i~. "" is ... intqr&ble and 1 ~"" - I"· Ix dp.. But I -Ix S ~, ... hen<:ol I· Ix "" '5 1" · I.~do< < and I"dl' (.,.....".. o - X is locally ... DtIliCI'*')· We oondade thal I doIgdp, .. detire::l. a

r

r

10.

1"""" -

1"<4<·

r I"" s: r s: r

r

'3

r

"v>

Theore m 10.2.3 f.,tf" , fl _ C bot loo:>oIl~ Wmf"""Wc A ~mIIotd f~..mo .. " <>PI fl i. /ooaJ/JI hS'-mttgrubk 1/ AM onI!I 1/ "g, " /ooaJ1~ 1" inkynlWc. In !h" """"', g,(g,I') '"' (g,9:lll'-

o

10.3 The Radon- Nikodym Theorem

Dellnltlon 10.3.1 Ld.1' be a _u~ "" S A ". ,u~" 0<1 5 is,...;d to be abooIulelr -,I!OODIII .. lib. p""" to p (and _ write !hen " < I<J if c,",ry 1o<:alJ)' 1'-<>e:«!i«ibLoo ""' ill locally .... 'gliglble. Equivalently. u < I< If ew:ry I<-nesli«ibLoo S·Od Ie ..... nt&lipble.. AI\)"_~ "wlt.b bMo I< iI at.JIulely """"inllOUl with ....1>«\ to 1< , br Tboon:m 10.2.2. Com(>:""\" _ ha... tho: foiloori,,« leIItllt.

Theorem 10.3.1 (~n-Nikodym Theorem) f.,tf 1' , " k .....
~

C·..et.o '""' mlOhHli1y di-ljo
cutnti<>U~ ".inl~.

•

lG.3 The RadoD- Niloodym Theooa"

2(11

PROOf': Th ~D. IIU~ lbat " and V "'" positi"" and that V io bowx\cd, Le\ F bo lbo elNa cJ all ,,-lntepabk fUI>Clions I from fl into 10. +oolluch lbat I" ~ v . Then M - IlUP/E"- f lOtOeorettJ . I io ,...lnU«t&bIe and {I"I(E) - IUP.U~,,)( E) ~ vI E ) fot all E E S. Tb.. I" ~ v. Now _~ br mrur..tiction . Su~ thai. (f,, ){E ) < v{E) for"""", E E S . and "" ~ > G bo 51>Cb tbt v(E ) - (f,,)1 E)-~I'( E) > O. In PIOpoiltion 3,3.\, ~"by v - I" - (" and ....Ul"" the F... are S-au. We """"lude tlt&t lhere • .IJl $..eI B C £ ouch tlt&t v{ B ) - (f,, )(B ) - .,,(B ) _

IlUp (,,(A) - (f,,)( .4) Ad ... ~ "

.,,(A») > G.

v{A ) - U,,)( A) - . ,,(04 ) 2: 0 for all S....t.:u A of B . Put ~ _

J.",<1" - J."f

f + (. l B. 'Thm . for all FE S.


J.,

I dp + v{ B n F ) -

-J.,-.

J.8""

I"

1
10 IeoI than v{ F - 8 ) + v{8 n F ) - I'{F), which ......... IMt 9 1;'" in F. BUI

Jgdt< - Jf
+ . ,,(8 ) - M + (,,( B ) > M .

and .. "" ha"" • """,,·&diction. The,"'or" . f" _ "and the t!>eorettJ • J>f'O""lL. Y"" E io contalDfJd in the unlon of • teQlImOI! (A.)~ <: I of C·..u and cJ. ,,·negli«ible.el N . To OM'e noc.alion. put,. _ g... lor all " 2: 1.

Tho>

,

r

(g.

1. 1d\iil ..

L: j I", . 1, 1d\iil s I" I(E). ."

n.c dominated oon~ '""",,,II> oN .... that ,· 1,. is I'.in~and that J,. III"" - L Q I J,., . llldl' .. L.~I v(A. n E). Since N 10 .... ~bIe, E./! I ",,t,, n E) .. v(E }. lleo>oe '1'''

...

0

Notioo that tM modlt .... 01 TbeoiEm 10.3.1 AlE ""t¥>ed if fl bck>np to 5 or...... $hall _later, If I' io rqu\o.r. Theorem 10.3.2 Ld,. k

....... " S.

"""Iitibol ..........~ on S. " ... Id" k" """ mOl·

no.. foJ./t)lIJi~ ~

0",

qooi""'~

(a) F.,.. coery UKIIliAllr ... i .. l¥"ble jwodion f fnnn fl mID {O, -too{ ell 0, III""" -rI. 6 ;> 0 ...dI Uwol the m..tio:>... 0 f 011<1 /I"" < 6 impIr ""l,,1Sf .

r

:s: " s

r

(t) " io!/o..,. /0 ~ ..... ,..,... .,..... .M(S. RI.

red ~

I' i ..

/Joe DMWn4 (:G1IIpkrc /Nfu

(e) F",. c..., E E S And ""'l' f ;> 0, u.c.... tAolI 6 ;> 0 ....,11 tIwoI., f()T <11.1 F E 1t a»
....

0~1,

(I01IcIm_

.,;tJt. rupeel to p.

PJWOr, We.." I't'JIItnet 0Ul atlol!ntion to the cue in which .. io poeid .... By f'l Oj>Olitlon 1-H. " bdoo&t to 1M band ~ by I' If and only Jf, (or .,..,ry I € Sf· ,S) and .. elJ f ;> 0, l hen! 6 > 0 IUd! that thoe ",1M1ooo /I E S I+(9), 0 S I> S I , and /ldl' S 6 imply I>d.! S f. TIn.. ~ition (_j imp/iN OODdilion (b j . ..hlch La tlll1l imp/leo <::OnIdl, and lei. B be • P'nqUJi'* S--. G lwn E E Sand t> 0, .... 6 be .. in condilion (~). 'Thete io ........ iDe oeq~ (s..)~~ , of "It l .u tueh thaI E f"I 8 C U. ~ I8" C E and 1'1u..:t..8,,) S 6. Thon ..-ru.~, B" } S t, and 0() .-( £ n B) S t. Sinco • wall arbitrary, E n B 10 .... negllciblo lor all E E S . 1l000e B .. .... neg~Jibl.e and <::On 0) &lid write 1/1 (or the function t~t it equal 10 IlIt:z) at <:W!ry ~ E X and "",iBhOW 011 n - X_ 'There cxIIu,: n _ (0,+<:01. _tlalIy jrlnt .... bIe. wc:h that I~ _ II'. Si".,. Ix "' lo<:allJ- .... intecr&bW:. Il/ielo<:aUy 1 ... lntqno.blo. ThUl k _ (111-" iii locally p.lnl-l!grable and

J

"""te J

,U.

G.J.-e)<", - (;) <M -,," ,'

10.3 The Rtodt.- Nilood1n>

n..o.-

N_ . for 8Wf)' l"tp~, II 2: I, put A. _ (>= Iii 11: k(>=) 2: II}. Since k · I .... is 1l--..abIo, I .... is .t.lHDNMIf'&bIe and!lO A~;s .,.."..aew:ablt. The funclklnil / . LA. dec .... 10 0 M II _ +00, and they ~m.o.in bounded by / : roo: fW:d , > 0, 'MS"'" t~~lo<e lind an inlq;er N 2: I IIUCh thai. f / ' lA,. tW S e/'l. Then, if h : fI _ I NJ.io&B O!S Io !S / and h
r

to.

,,"(h ) !i

" · Ch · I .... ) + ~· (Io( 1 - I ..",))

S ,,' U' IA.~) + ". (I x' h i 1 <

i + (Ix' ...

I .. ~)

)*(10(1 - I ..... ))

:S ~ +J.I"(Ir.(l- I ..... )k)

s

~ + N.II· (Ir.)

:S t ,

o Deflnltlon 10.3. 2 Two 0D
II" o;rni ... ..,... diljoillt.

PropoAltlo" 10.:U

P1lOOr: The relatloo

0_

iDt(Vp. V ... ) _

VI' ;

V(j.

+ ...) _ V" + V ....

V ... _ IV... ; V"I

lmpt;eo

(Vp - V vj _ Vp +V".

0.. the oU>er hand , V" '"' V(j. + .. -

"I s VII' + ... )+ V~

V .. - V" sVIl' + v).

o

u....J, i/O io IM.~ mo..

Propoeltion 10.S.2 ... dll.ol .. ...., ofPjoi'" i/ .. fttI!

tJo..t .. o.b>l~t..J, """""_ ...tIl ~qod to lot4.11 .Il.0l ...

are p<:IOiilj~, I' and ~ are d~nt If aDd OD/y II BU.) n B M .. (0), ",!>ere BU.) and Bfu) are the bando in M (S. R ) Ietpectj~ ", ..uat.l>d b, I' and b, u. The n:tull IOIIowtc on IoCCOI!nl QfTheorm> 10.3.2. 0 P ItOOV: W$ rn.y

~upp<108

th&t I' and

~

Tl>eoretn 10.3.3 (Lebeooa;ue'. Oe<:<>mpo".ilion Theorem)

"be

"'...,u.... ""

I"'" """"""" S . 1'IIcI I' ... " k ...u..... u. + 1'., """.. < I' .. nJ u• .i l'. Mool">\lu , ~. crnd~. ~ .. ~niqwlg del~ed ~ tit~..,

--

U.

U I I' <mel

By the RirR d«oinpOllilio)n t~ Re~ '" ..;, + .,here u;. ~ M (S. R), V. ~ M (S. R ). V. c: I' and V. J.p. Similarly. 1m ~ _ ~ +..:'. Then " .. u. + .... _here~... + iv.' it ahooIuu:ly conti ......... ';111 , ~ 10 I' and ... ~ + ill,' It disjoint rmm " . N"", lei. " ~ I . +8. b$ a"odoer <wwtpo:$llon '" " ..·ilb reptoCllo 1'. n.en", - " '" 6. _ ". II abeoIul tly _lmtOUtl ";111 ,et ..... t :.0 " and 10 " . - 11._ T hUll ". - II. .. 0 by P' <>pOfiIIon 10.3.2, wheoce I ... ". and So .. ". , 0 PIlOOI": ,,_

v. .

thaI I' It

pClSili~_

v.

v.

A _ MY and ouffidenl condit ion thai 1_ I'" · 7'u ..... " and v Oil disjoint;' lbat they be 0CID
S be

pm\oitkm 10.3.3 A . . -.. " on S is,...;a 10 be ~Iraled on £. C 0 (e.- carried by E) wh."."...." 0 - S Itlnnlly JI-""8I~bIe. P ropoooitlon 10.3.3 U'" "nJ v lot h
I" · ,,

to<" ~ A ~ S,

a,," I" ·,, .... "'""""'wlttti' .... 'W;O;1II ot" E "nJ F, til".. " ..n.:! v ...

~.

P ROO~ ........"., that "and " are int(l " . p , 1" .1') It alD(f:tItrst.l>d OIl

11>erdo. " lnlll' ." ) '" o.

poali_ For ~ A E: S. I,, · 1..1(".v ) '" En F, ..'hlc:b PfOI-"" WI I" ,1n1U., 1') '" o. (J

v...).""

Pr-opwitlon 10.3.4 Ld k c ""I........ 0/ """"".... "" S .1Itdt thai ~ "n.:! "" .... m... -.l1J "npIa~ lor oj! diltirKl p E N .. "J q ~ N . T1toL, /0<" _.-, A E S. ~ ·d,.~ ..... ( £..).. ~I 0/ diq>inl S·otll 0>IIIain A.. nodi
til",

PIlOOt': Aailume ~hat 1'. II pOIUi",,Pw. I' _ '" 1'" By ~be Rt>dno- Nikodym lheor-em, the", "";111 ".intecr&l)h, fuoctloom I, and h from n InlO (0. +0:[ IUd> thai 1"·,,, '"' h" for alli S; S 2,

+

We .....y.uW* that /, .."ilihel

Oil

0_ A &nd tbal h / A II" : w,rsblI,

S,,, (Tbawew 6.1.2). N_

inf(J"h) _ 0 p-&Imost eoerywbert, to 1,, · II , ..

avv-nlrated on

...... 1" . II. iI a",cerura,ed on

Ft -

{>: e n : It (>:) .. 0, 12(>: ):> OJ .

f or eto:h inU(ef j:> 1. ,bert exis1; disjoint S_ E:[. Fl, incloo.d In A. auch that 1" ' 11 , isOlnied by Ef and 1" ' 11; by Ff. Then 1,, · II ," carried by E:, - nJi:' E{ MId tbe 1,,", ...... <:ODCf1llrated 00 F, ~~~, Ft. In the _ _ ""'Y. ..-e can 6nd E:, c F, and F. c F, that 1" . II. .. carrio:I by t:;, ...... 1,, · II; '- 0Inied by F2 lor 0.11 j:> 2.

P. ox ee jl", ... ilh the COII$Iruct\orI step-by-Jtep, .." obtain the .......""" ( £,. ).~ " which hu tbe deo:i.l"fJd ~y. 0

In I*rticuw, if fI beIonp to twed on disjoilll.

lOA

S...u.

_ u .... "" Ibtn_l_ ............. ~n­

S, tbe

Combination of Operations on h'leasures

The foIlowI", !..:II ...... Immediau:, MId require linlo: comment. Let X be a _ ,ply 1Ie't . S a -run", in X, Y a ,,,,,.."'pty ... t.t of X , and T a .....iri'" in Y . If " ' and II, ...... t _ me&II\lJ"ej on S such t hn and " ' IT exist . tben (p, + II, ) IT alto exilw and iI equal to (PL , . ) + ""' IT)'

""t

P ropo:.lt ion 10 •• , 1 Ld H .. Gn

...... t 0/ M ~(S), ~ ~, """ Ittll .. U. ~"'" in M (S,R ). Then. ·b, . owI n/!icKrol IMt " 'T e:riIl ;, IMt " I ......t J.... 01111 E H aM tA4.t IUP"' H u( B ) .. /in;u. for""'" B e T . In tIIU ...., II, . - sUP~H " Ir' ~"....rcI-dimc:ttd

.......ntion

p ROOf': .u.ume that "" exiIu for 0.11 " e H and that ouP"" H v{ B ) II Snit.! b eto:h BeT. TIle .. Y • II, nu",abIe. the T _ ...... lI""im"l"bIe, and II( B ) .. "'P~ H u(B) lor &II B e T . F""LX A E S. for every .. 2: I, tbere t::ri8I. ". e H . uch t hat II(A n Y ) - II.( A n Y ) S l I n, and B. E t ouch that B,.

CAn Y

and A n Y _ B.

"( An y - U 8.) -

"".-~bIe.

1I.(A n y -

. ~,

U B.) S(P- ". )( A n Y )S IIp, .~,

10 II( An Y - U.i:L8. ) S IIp, for &II p 2: it p' ''''Il!&lble and that II" exio\.t.

N."..

I. ThilL

ptO'IflL

t hAt A n Y - u. ~, B. 0

,.

.

Propoooltion 10.4.2 Ld H Le" nommpl~ ,ublet oj M+ (S), Ioo>unded a.Iooor, And kl # Le it. ' """""IIm in M {S, R ) . 1/ # 17 en..t., Ihf:n I'h ~ OUP.., H " h '

P ROOF: For all ... " .., In 1l , !u(>("" ..,) _ (v, + '" + l"l - ",1)/ 2, """""

,11(>(""

"")/:r ., 9UP(V'/T' "'/T I·

It IollowI tlw (aup~J ")/7 ., auP.€AvI T) for ~ry nonempty HniU sub6et J of H. Sirxe # iI the auprmlUm of ""P.." "' , ",he.., J ext<:nda """" lbo. clt.8s of I>ODemply finiU IJUboeta of 11 , the prOOf i. oompleu. 0 Nert, ~

(j>,),u be" ! "bUMbie family of po8ilive M""""" 00 S ",ji b sum f'. Then f' /. exloU if and (lilly if ~fT exU;1Il fu< a U i E 1. and lhe family ()..-( B)"" .. aummabio! for Nd> BET. In tbil caoe, f' h - E"" "'/,. N"", let {l be" no<>oempty lIIlt and S a !lemiling in {l. If I" and I', ~ two 'rp·n .... on S and if g : {l _ C is Ioc&Ily inu"n.bIc. lor botb I" and #1. then g 10 Ioc&Ily W, + ",,}-;otO'gnl.bIe and + f'1) - 9f', +

gw,

g"".

P ropoooltloD 10.4.3 Lei H Le A RCnemply ~t 01 M +(S) , II<>uruled a"'-. ondleC # boo it. n prem""'. Theng : {l _ (0. +co( ;, """'"'ill f" i"~" (and ""'" if) it ;, """'"'I, v.intoy, d k '''' 411 I' E 11 and (g ... ; " E H) is II<>und
'v.

PIIOOP:

.u in P ropOSition 10.4. 2. ".., m.y IJUppOote .hat H iI di~ u~.

By Proposition 10.1.2, 9 is f"lIW:Murable.

I

e I' l.rd" - sup ~"

I"

MOf'e(l\",.

.

9' l"di.o - ou(>(g ... )( £ )

Nert.1et (,..),u be ",ummablo! f&n\ily ofp08itive me&OU«'OI on S with Sum I' _ Then, : {l _ [0.+001 is Ioc&Ily ... inV'p"bIt; if and only if it illoo:.oIly ",.integrable for all i E f """ the family (f I ' 1 B d,,;),u is oummabie ror each E E S. 10 this C_ .,,, - u j",. P""I"'Iitloos 10.4.2 and 10.'.3 an: partiru[..dy UgefuJ in the !;tudy 01 _ IIIl'UIlfOI; _ 9hal1. not pumoc this topic . .".. folbo:ing two _y t t . )I tt". an: ""'Y useful.

E. howe""".

Theorem 10.4.1 ut X Le a noMmply "'I, S a kmiring in X , And" a ......."'" "" S. La Y lie a SlOb.., of X and T a , .....iring in Y , Iud! IMt " 17 ailtJ. Fin4J.I" 101 9 ' X .... C 10< 1ocoJ1~ I'. inl.tyrubk. 1'Mn" n_'Y"1I and ~ wrufilion I/w;I (g"J/ 7 aill;, Ih4I g . Ill' Le ",inky. ,b'e ,'" all B Iii T . In Ill" 0:»<. It. is """,",l~ I'h .intb'e w d (910)/ 7 - (f,y ) . (j> ,. J. PlIOOf': Each B Iii T is ",;ntcgr.bIe. and h..,.,. iI <»nt&io'ld in a countAble "nino 01 5-""",. A.. ume t bat 9·1B is " .;"t.egrable for all B E T . 'Then lB is,... imegnble. Mo, eooe" for every A. e S. t here exisu BEt such tbat A. n Y - B

1(1.5 DlltJity of L' Spwd

is ,...negl.ipble. aDd tbomore gi'""negI.ip~, which peo,,"" thai. (g/')/r 0!Xi0I.a. f inally, for .U B E T , (g/r ) ' ]a is /'/r .i~.bIe aDd / ] a . ' " lIIu./ r )" / g . I a III/, .. (g/')/r ( 8 ). Q

T'->rem 10.... 2 Ld 0' , 0" boo tW>o> """"""t~ ,d, ,,"" " , I'~ I.... _ r e i .... H"IIIiringl S' OM S" in 0' Galli ra,..,li",,/). if g' , 0' _ C ;, 1«o11r /,' ·inl" ' M V' : 0" _ C ;, 1«o11~ sI' ·inl~. IMfI g' 0 ,.. ;, 1«011, /" 0 /'~ . inttgJdbl. olld (g' 0 g") . 0 P") .. (g' ") 0 (i' I'~ ).

rr.

..w.

u.'

P ROOf : 0bri0uI.

10.5

Q

Duality of V Spaces

Let n be .. nonernpty ett, S .. toerniriotl in n. and I' .. comllla .......ure on S . Let p E [I. +00] be Ii....., and IH q be it.
J

n->rem 10.5.1 WIoC1I 1 < p <

+00,

I ""'I" Lh(p.) "nto ( L~u.»'.

PROOF: Let T be .. """,;n"""" lin ..... rorm on L~u. ) . and put O«A) '" T(l .. ) "" NCb A E S . If (A. )."" is. of disjoint S_t. ..... · itwl in "" A e S, t he K i a r:. ~, 1.., con"",,", W IU .. in L~u. ): hence

.

"""I"""""

..'

'

..bleb pow•• thu" is .. _un: on S . Gl ..... / E S I(S.C), 1'-': aiIt. d~ot S-u 04 ... , 04, and complex " aclt 1 ~ i ~ n. let numben VI , ... ,»- such that / .. LI :!>.<, 1.. ,· ~ . f or (B..t) ,~~, he "" S·panitioo of A, aM, for .,. • • y I ~ j ~ j ;, IH C;.J he .. complex number 11Ich dw \e,J [ - I &Dd C;J · v( B,... )· V, - 1 v(B. ... )·" I. Then

,.

.

is 1M! than

m · (JI .~e

.fss.

Co.}' 1"'4 . 11-1' oilpl) 1/,

'"' IITI'

(,~~ Ipl(A,) ' I",r ) 1/,

_

"',(1).

IIT~·

ThUll L.$'S' IvI(A.) · 1tJ,1 - f il l dl>-I ill smaller lhan 1711. N,(I).

In ,*""",Iar. for all t > 0, "'" have "'I(A) ~ ~ lor all A E R. """" that Ipl(A):S (t/ ITIl' , which POO<el lhat " is.oh6olmely cont;nUOUllOl'i\.h rUlpo:ct (Tho"H~n

10.3.2). NOOI let F I:wl tile cl_ of ,..."., I'O'litive fUnelionoi 9 E !hv.) lor .. hid! 911'1<. 1"1. n..:n 1.0) ji

N.{so ) ..

sup

f l glip

. . .. , .. C)

".v!i·

~ ""pf l/l d(gll'l) ~ ,upf l/ldlvI~ 11'\I f f

for all 9 E F . P ut M - 1U1>",FN.(9), aDd I. .,..,.,....,. w F aud> that N.(g. / . Tho... is " - 'Ii",, fuoctloa 1/ E 4u.) aud> thai (g.)~~, COO""'P to 9 1...1 _ ~""re. ()t",~y, ~I 1>1· ~ .... 0 ... that IvI(e) - I.9IPIICE) - tll'!( ~ :> O. By IIIe arne argument OIl thai in 1boon:m 10.3. 1. "'" eIIn 600 an B ind uded io £ lor ..·bich IIo I( B ) :> 0 and h _ 1/+& ·Js l;.,s In F BUI II' +(c· Ie)" ~ ht (Lem ma 5.1.2) yioeldi f go dlPl + c< . 1I.1( B) :S f ht oill'l and "'.1.9) <. Ni h). Ttui'J "'" anive U a O>IItradictlon; "'" QOOClude that Ivi ., 9 ·1 1'1. FOl" tbill, . "'" can find .. ~ ( E.. ). > , of disjoint that 9 van ..ht:a on n - U'~ I E". Then fl - u.~ I Eft is locally v-' >("gl;g;bl<-. By the ~-f'likodym tl ..... eu., " is • mtMure with hBIIe fiji, and .., then! "",isla h E !hv.l _ h that .. _ hI" Sino! T and tho! Ii....... fonn / - f Jhdp are rootin...,.. on L~v.) ..:od agree 011 SI (S. C ). t hq are i
s:

s..:u.ucb

Tbeo...,m 10,5 .2 S""'* ... (aN finJ " clau C ql d:ir.i<"~t UKJU.olI, p' m' ;' !.Ie .." ndE that cod! £ E S ill a>fII""'..r In c .nio~ qJ ..... nlal-Ig ",cn, C·..,,, aM 01 " p.nqo/iJiblCtion A .... T(l .. ) OIl S. Glvtn A. E S. If ( A . )ft ~1 Is a 9C
L ' :$fS·

TIl ..,)1s: ITI · N, ( 1.. -

L I S ' :$~

1... )

10.:; o.../itJ' 01 V'

S-

209

coo>e.&&, to 0 "" R _ +CO, and """ is a -additi''e. Mon!O'I'ei", since I,,(AII < JTI' !PI(A) for all A E S, " is . "",asun:, Siuce 1"1 :S ITI' !PI, eKb -..tially p·inttJn-bie !Itt is .utl.ally ... !~. By the Rodnn- Nikodym theomn, .!>ere a locally p·iLlle&f&bIe fUDoCT.lon, . ud:llhat .. ... ,p. We have "I nTi locally ,...1""", eWi1*'he~. /uthe Ii...,.... formoITand ' ,. DLL Lb(P) .... contin\KlUlland qrel!oo Sf(S, C ), they .... idemical. c

.,xi.n..

:s

Now auPI>OM t Mt p '" +CO , Write M for the elMs of

,.._utabIe

leU.

Lemma 10.U Sf(M , C ) .. .w- in ':; (p ).

PROOI': Let f E £j)(P ) &ad N ,. {J: E (} : If (J:)I > N",(f l}. Given an j~, .. 2: 0, 1M I he Lbe _ oflhooe i '"' (p,q) E Z " Z sucb that the ... U&M R, ... J"/:r' , (p+ 1)/2" J " (q + l )n~J i n _ U 1(0. - 1'0'). For eKb ; E I , PIIt E, '" r '( R. ) and chooooe any Ci E Ro. l1>eo 0 '" r:~ II£, · Co beIonp to St(M , C) and N",,(f - o j :S 2-· . J2, "'Mdt prooes the ~

J,/2" ,

o

Propooltlon 10.5.1 Ld T lit .. II ..... ' /orm"" St(M , C ). In <>nkr tJW tMrt. ...... , E 4;(p ) I1Idl UwII T (h ) ~ / ,hdp I", allh E St(M,C), iI .. ~ ,, " .... ~ w.t w.1<>l1ot.J;"!I cond.tiMu MId:

r.) T (I ,,) ... 0 I'" """'1' I'.Mgligo~ S· .., E. (~) T (I "" ) """wYJU 10 0 10 w. ....Ptr ..t.

I'" a.rt)' - " " " ( E.. ). ~ ,

In

S roNdo Uti

(c) II E E M andT(I,. ) .. 0/", all S -..II F .ncilldtd in E ,

'Y'

w.n T(I ,,) '" O.

P ItOO+': First, I UPf>OI'II' that T ( I,,) .. / /I ' I"dp for all E E M , for. l uit.t>Iy chooen II E 'b(P). COI>dition (b) holds by the dominated (On.e' KtI>Ce lheorem. On lhe od",. haOO, let E E M be l ud:I that T ( I,) .. 0 for all S· lIN F ineluded io E; if F is .. BoN!I IUt.ed of E n {" : lit.,) '" 0) sucb that E n I" : lit,,) ,. 0) n F' is p·""SIi&ible, lhen g]" '" g], p-almoot e.",,,.1w:re; thUII T ( I,,) ... / /l1"dp. '" J gl,dp '" T ( I, ) '" O. 'y, .... ume that conditions (a ), (b), and (e) .'" "",i5Iled. l1>eo T ( I,.) '" 0 lor every locaI]y p·necJi&ible _ N. Tbe function ~ ; E _ T{ I ,,) from S into Cis .. ",hW;h is bounded ( P ropooition 2.2.3 ). If" ill it. rMriction to S, tben ~ ... ~ ( P tOJl(llSitlon 6.U ). Since ~ It bounded , tbe", n\aU, E 4;(p ) 1ud:L lhat ~ .. 91' . For all E E S, 7(1,,) '"' ii( E ) .. / ]".w ..

ea:.",

_U"',

/, · 1,,4.

NOOI' 1M E E M . Then T ( ] anl",o<~)-nl) .. 0 t.,. rendition (e). On the other haOO, if F is .. Borel out-! of E n {:r : g(:r) O} 1ud:L tha~ E n {., : lit.,) '" O} n F' It p-uegli&ible, tben

r

T ( ] ""1.,0<. )....1) '" T ( I, ) ..

f, ·

1, dp .. / II ' I" dp.

,

••

o Corollary Ld T N G <>0<11;",""" ,"-'" /""" .,..

Ci;tu) IIi.:h

tlW T .. " tion JtU.l Aold.

if """

L~(P). Th~

<»II, if o::>ndm.-

aUt. 9 in (6) I1nd fe} Qf Pro~

PItO()F; If t'-o) coodillot,. hold. t~ ex~ 9 E .c::{P) >iU
identK:aJ.

0

On L':(VI' ), coo>Oio:iet the <>«Ier relMion; i~ "" j , .. ~..,. 10 ., /. locally aI"... every"'1\en:. ThllOl" lIIIqutrte:e Ci.).» In ~ (V ,,) ha$ II oWl it>! in6~", if and only if inf. ;!:! f." 0 ioo
i/.""

P ropa.ldoll 10.:;.2 Ld T N

4 (P)

,,.oft

IIuat T""

Ii........ <>nIy if g

/o.m ""

L~(PI.

n-t "';"U

g E

(a) T(I.) ..... t'e'JU /Q 0 /or _ FY d..,......;"g ...,........ (i" ).~ I i .. L,:(V 1') umitrirlg 0 ... iu mjim~m. E € M ..... Uwt T (lp) ~ 0 T( Ia) .. 0.

(' ) for

aoer}'

/0<" all 5·..", F eE, ..... /wi""

P ROOro If fjctiaw, OODditioa (a) boIdo by the dominated oon'''''X....,. lbeor=. COI",,,.-..I1. 1UW'*' that ooodit~ (a) &nd (b) hold, &nd Let 9 E 'i:v.J boo - " that T {A) .. 9"<4< Joo- all II E $t{M ,C) ( P f'OpOCitio)o 10.5.1 ). If f ill pOIhl~ and bd"",p to C:(,, ). il it , "" upper ....-.Iope 01 an incror.sing """lUCDOl! (f.). ~l of domenu of 5/+(.04 ) (PropOOtloo 3.3.3), and T(J - I.)

J

COOh( 'st'! to 0 ... n -

+0;),

Therefo",.

TU1- ..2f'!'... TU.) - lim / f .gdp. -

j I ,d!., o

and T '"' ,,_

10.6 The Yosida-Hewitt Decompooition Theorem' w~

DOW

""""~

the 10000tntd dual of ~(P). Obeer<e tlw.

f

".;'E~(p.J}-{,- ,h.w,,: heLC(I'l}.

n", .. tlJre. in what followl . ..., may suppooe that" is positive. Wrile Ljl'(P)' for the l!*)e of all continuous linear forms 0<1 the lUI Bao..-b ,pea: L;'(P). It is the order dwl of the Rieso: spe.ce L:(j.). Simllarly, Iel L~u.l' he tbe ..... wed dual of 1M &Mch SI*'"' L~u.). 'Ibe:II L,/:,(Pj' C&D be idenlffied .nth tho ...aI vector oubopace of L~u.l' ""IIiP;", of tboee T in ~(P)' such thou T(1l is real for all f E Ll:{P). For ~ T E L~(Pl', the .....ppi'" t : I - TO> from L'{;'(P) into C belonp to L1l'(P)'. Si...,., ReT _ (T + 1')/ 2 and 1m T _ {T - t )/2; are real, tbey lie in L;'u.y ; WO<eD"",, T_ ReT+HmT. UIi"lN'l"mcm.limiw to 10000;n Sodion L~ ..... find in Ll\'u.J' . smallm poeItlve el'''''''1 L sueh thal. (TIIlI 'S: l..(lf ll for aU I e q'l'{p.). Thlo ,'ement is written

17'1. and

IT I(f) _

sup

IT (II)I

/o.E 'e(~)·I~IS I

lor o.U p 0, to 0 ,Itew~. Then Ir ill'"~bIe, It I !£ I. and II _II<. Hmce 1TI(f) ... IUP... ~(~). Nw(o):s, IT(fkll. In particular. "'MIl there m.tI g E 't1J<) ouch that. T - ' •• we _ th&t ITII/) - I lf l <1.1' klr ell I E L~IJ<).

J

Definit ion 10.8.1 T,. T, (n Le'IJ<)' are said to be disjoim, or mutually sin-

«War. if inf(t7" I,IT. 1) -

0 in

t..;1J<),.

Then. from tbe relation

,

"' ( ,." ~ II", 'I_IT'I+IT. I_l lTd-2 IT.1! ' 2

it foIJowo that IIT,( - IT.II - IT, I + IT.I. But. since IT,I - IT, + T. - T.I :5 IT, + T.I + IT.I

llT, I - IT.1! SIT, + T. I :5 IT. I + IT.I, and we cooclude th&t IT, + T. I _ IT,I + IT.I. 10.6.1 ( Yn.lI....- H ewluj kl G ... Ik ~ 01 all '., lor f in 't{pj, lind Id H "" Ik "",1M ~ 01 L~IJ<)' ......wing 0/ thDK T E L~(P)' -.:Jo>dt. ..,.., c/i,joiu'! from G (i.~., from <>11 ekmemo "/ G). ~ ~Itlon

""= T,

"""lI T in Lg'(p)' ...n be ,..,tI~ T, + 1:" belong. to H ; IN. d.=mp>lili<>n .. un;q.,.,
belong. t<> C OM T.

IiTI .. ITdl + IT.n.

FmaJly.

PRooP: F<>. aUy E £1.. (P), define by "'" .he mApping / - J /9 dI' from L'J::(P) [nUl R. TIt"" C .. = (",. : 9 E C:'(P)j is .. """"" $ubip&ee ~muu preoeo:jing [)"6nilion 10.6.1. We PfO"8 OW C .. 10 .. band In Ll:(P)'. For this. Ie\ 9 E Ci..(p j and. T E Ll\'{P )' be ouch IW III < h",1 in L':{p )'. Tbe!lt(. fu nction i>: E _ T( l.: ) is finitf:ly add't''''' on S. and

IP(ElI-IT{IJ!')1

~ ITI(lIf) ~ I'P,I(I ~) '" /

191· ledp

for all E E S. The """Relion p or i> 10 S ill a bounde.J _u....,. and P(E ) .. J I" dp for all E E S. Sin«: p 10 bouDd!!d. t""~ e>rel ..,beet of P.n{ >: ; g{>:) yO 0) ...:h tht.t E n ("': g{,,) yO 0) n f"< ill ,...noegIi&ible, Ih....

J

T (l en( ... .c.~I) .. T(I ... ) ~ / klyl' I ... d;< _ / Jelf l· 11ld;<,

J

and T (l.) "" Jelf l· Ilf Itj •. llwoo T and "'ti.I' ... hich 1t,p'(le On Sl IM. R ), ~ kkntkal. This ~ that G R is M idcttJ in L;'(p)'. Ne> 0, I~ ~xi>;U 'i'. E D 8ud> I"'" "".(Is,)
",,"

TI I .. ) = IT - "".ll l s., ) + ... ( IE..)

~

(T - "".)(1... ,) + ,/2

'" L.:lIl than t , and 80 Ttl .. , COO'Clgts 1.0 I) 88 n _ +<:0>. On t~ Ol~r hand. If E E M .. ...:h that 1"(1,) .. I) fot.o.ll S-eet.9 F eE. \ben "".(1 ... ) _ I) fot &II our:b F . 80 ""( I.) _ o . ~ all 'P. E D. and Tt I d ~ . UP... EO 'P,(I,, ) _ O. By I"" coroIla. y to PropositJob 10.5. 1, T kiongs 1II GR, and...., conclude IbM G R ", actually .. band in L1: (I')" By the ru- lhal. T .. T, + To' N(>W T E L~ (I' )' is di$>im fro", G io L~{P)' if and. only If 11'1 is disjoint (tom C .. in L':(P)', tht.l. is, if and ooly if ReT and 1m T are dioijoiDi. from GR in L;'(P)'. Each T _ ReT +; ImT of L~(P)' eon \""",Iore N "";tl
ITI ~ fTI( 1,.,) =

IT,)( I) + IT.I( I) .. IT, I + RT. I.

LeI. L:CJ<J~ ~ 1M

L!:'(pl' sucb lhat T (i. ) COlu);}. An element ttl of tbo6e T E

{i

i

of L:lu) ~Ionp \0 thil annihilator if....:l only if N... U.) OD.","&& \0 0 .. n ~ +00. rot ev(p) lildmitling I) ... infimum ....:l oucb that i. :s: Ii i rot all n ~ I (Ihd.. Tbo!ormI 11.4. P. 70). From this. "'" dtd""", the klIlowinc. P f"<>PWIUOll 10.8.2 , rroop. Lblul onto L2'lu )' "ond .".". if fl '" 0/. jimu ..limber 0/ """'" 4nd 0/" 1Dool1~ p.~igihk "".

P ROOF: S,,_ that G - L2'lu )', TIIea G R - Lj\'(p)'.10

L:lu )~

-

to"""" L\i'lu)'

L;>luJ: _ {OJ. Then (N.. U. )) • .!, <:OI'I~'g<'lI to 0 for ~ ~nc "'"I'''''''' U.) ..~, in Llflu)' admitting I) .. infimum. becauae \.he annihllal.or of L!:'lul; io t~ ... hoIe of L!:,(p). Suppoee there II allequence (A. ).. » of inequivalent .torno. lteplacing A. by ....:l

A. n ( A, u· .. U A._ I )". if ...

"y.-"'" may ....1ImO that the A .. are dioJoint. FOO' all .. ~ I. put E., _ Uv.. A •. Then n..~, E., - e. DUI p' (£..) ;> 0 2: I. bono! "'.. (111,1 ~ I ....:l "'" ani~ at. contradictioa. ThUll there are fini~ """'3 clegs '? of . _ Let A " ...• A. he ~tat~ of tboao differmt ch ' . ... hid> "'" may """"""" \0 ~ diojoint. N.... let. E ~ • p-i~ subeet of fl - (A, u· .. u A.); E contains 00 ,\Om. If (£..). 0, lhen! exi'" n ~ I au.ch IIw cr .• :5 € for all I :s: I :S: k • ....:lti>en N ... (/.) :5 €. Thedore. ,\'''''(/. ) eooheiP:S to 0 .. n ~ +00. Finally. w""" Llflul'. '" (OJ ....:l CR '" L!:, luJ~ _ L!:,lu)'. "'" _ lhat L~lul' '" CT a

for..m ..

for ..m

"'"I"""""

For ....,.bft p;vol of Pn:>poeitloo 10.6.2, _

,7.,

E"u cioc \3.

••

10. Radoo-Niloodym o""i.... iwII

214

Ext/Lilu fo r Chapter 10 1

c;."". .. aDd b in fi o.od:t tlw B < b, deoou by d'z Leb.o.ie:"" ....... ure "" f '" )<0. lit ' Lot .. bo a ' IL , ',,,, fOnd;""" "" r, ,Iw. io,. 0( o.od:t.ba a' io ",b-intqrol>le for.u i~.qen " ~ O. Deoou: by E ,bo Hill:>et\ _ Q,1",u) .. ith in11ft' p:rod\lC'l (g,h) .. 9/10,, <1% &nd by II I_ (f,I )'" ,,,".,.,.., 01.1 E E. W~ will idemify a poIy""",;"1 (wi,b """,pie>. welf»:ioM.) wl,h Ito to f.

f

tat_

be 1110 _ _ obuiDlJd from la· )."" by C,,,,,,', <>rthosonoli~iou (~5, E>:etcioo II),

t:Ot

(p.),.~

I. p....., Iho. Po ... onitary pOIynom .... of I weIf»:"",t •. 2.

For - "

)." .. !P--,(-'. Po -

2: 2, obow lha. Po .. (r - ).,,)Po_ , - ,,",,"_., ..1Ieu (ZPo- " ,..-o) aDd " . .. 1,..-,1'" )p._.1-· I'~ '" ,ha,

0_ """'_. . .,_ '"' " , . .

!~_

n

. " rp,, _, ••

bj'-'

,Po_' ) '

3.

i~'_ n ~ I, let '. be ,be codIicitM 01. ,,'-' iD Po· SJ>o,o \ho.t )p.r' . (Z}'o, Po) .. " - •• ~,. (Ol a , •• ~ .Iw "Po ""'Y be .. , .. IOn ..."" + ... + .... ,,...,, aDd «>mpu\
4.

" by a. , ... , a. Iwi,b z . < '" < ... < z, ) ,be real I'OOU of,.. .. bid> ti:o m ).0,10{ &nd .. _ «don ollDllltipli
F"" _

D.

"""• ..diet;.,...

!o. t:Ot n E N

15

~

5 n,

d<~

by q. , .... poIy""",ia] ~.<.,.~.(:o:- a,)f(a. - a. ) aDd write c, '" 9O ",dz. If Q io • poIy""",iaI wi,6 <:omple> ..... fli:cio_ dover io Ie. .han 2 .. - I, obow ,Iw Q . ",dz .. E,<, ~, 'Q(n), (Ob . n:e .bat Q .. R + ,.S, ..her. S 10. poly..,... ......itli dqr... _ than n _ I , booce on.~ to Po, ODd ...her. R .. E, ~ . ~ Q(a.) io ..... ~ In"",:<''';'''' pcoIynom .... -=!a.od wi.b 11>0 a. &nd .ho Q(z, I.)

0, f

6.

2

F.,.- ..... ry in .......

'" . .,. ·

F\:om..,," ... dod,..,. .ha,

w"'-

f

Ior."""" q. la,) = 0 lor j i- k &nd .,. Ir. ) = f I, .. b

.. c. ; t"""

Let fbe. ,.,.j·valued fur>eCioo'I.

r.,u, Coo

d:eIi~

I, &nd In>m put S,

io otrictly pooiti"",

.... at' int
d"' in
of J ooch ,bat f '-U(o " O. I.

F\l< oil 1 5 q 5; n - I, I
2.

Ciwt> on int 8

! 5; q S .. _2, ... _ i. hall ' - " -..blio'-l tbat /"-' 1 van"",," .. " + ... + 't - (, - I) poiDto 01. J .. l
3

f

.).

S",,", tbal.

l.d

I, "' •. ... , z.

O

""" point.t _ t In tllo iIIterio< el J.

-1) . . . . . . . . .t

_ n. , .... ... bo .. in Exotcioo ( ... - I) doeriwti_....,. ...... Ix arbi'rat)' al Z,.

7.'~

,bal.I

and I.. fi ....

I . S",,", u..t 11>0 uniQ ... polynomial of ~ _./tan n _ 1 ,bal ........ al r ; ......11 .. i.. fi .... (... _ 1) dOli ...;.... (10< all I :'f j ~ p) ... W ~O.

2. l.d .f(X) _ L.s' so _,U/ Irl)a..X' bo • pOIyDomiaI el dqreo _ t/tan n - I . Shooo- tloM r.net itt firat (n, - 1) oX, ioatl_ an "'I""i at:<, 10<

s: / s:

-" I P 10 I aDd Ito 6rat ( ... - I) doeri .......i - . ,"""",",i~,. if and ""Iy if (
,bat.l>e«o.,....

1.

2) aDd ... _ lioat I ..... n doeri .......i _ OIl I . ~ by ~ 11>0 ",,'.quo p;>Iy-uaI ell,,, ........ h.at. n _ 1 SIIdt thai , ' )(:,) .. P )(z.) b all I i p aDd all 0 5 t n, - I.
~


1 (1_

~

s: s:

~

1_/(1) _ ,(1) _ !. (I _ :<, j"' .. '(I - r~) "

" { (deptA
" 0 " , at z .

lbt

•

s:

=,11.,. """'I*" in,"",,*, oo"toi";", ., and lho z, . 10< .hid>

hn><>tbooio and.- ,ho -.1.100
Afp._

I.

D '4~ by II 1100 unique ~ ... IfIOO ........ 2n - 1 o..dt IbaI. ; '1(". ) _ f' l(.,; ) for alii :'f i:'f" and all 0 t :5 1 SIIow that lhen! io.
s:

/(/ - ' )"<1'% 1.

/hp!"'
Dtd_ fn>m pan. 3 01 Ex«cioe 3 tha.

/ (/ -

"""~-

f

1"'<1'%.,

(M ... ..,. ... Oqu.alily).

t~

ens.. { E 1 lor .. hidr.

~) ..
L '~ 'S·

C. '

1(.0. ) + ),.,1' .

(,!,)•.t'"IW

~ ... 4 P-' method to appro.;im&I
f I.

h .n • •

ott ...

L!< '" be tbe ...npt 1I,,""loa. "'1>(-1'/2) on R . ond
H.II) .. I-Ir . up (r'lll . D" ( "'I' ( -

,,1'2»).

po!y._

I. SIooor thai. H.(I I 10 • .........,. j.1.
*

,<"" «, I H.'« '" OS'SJ./'II '- - I • 2' . ~lln n!_ 2.1:)1 . (. _ .. • 2.

ww..o frof21 io u.. ......oJ - ' of n/ 2. GIYoD R E N, ohow.1o&t f Ho(I)!' ''''' ) oII ,. 0 lot oJl i .. •• "OS t < .. ...d l loat J H. {I ),· ..(,jolt .. Ih)'J'lI· n,. (UM in...... ioco by _ ) 000cI ..... (H.)"z- .. ..... -!

£ f......

(f").~ by

Or...,'. or'bc>pmali""ioco. 3-. P'roooo that H.II) .. ,H••• I' ) _( i - 1)H._. (f) lor oil j n! ., ,, . n ~ 2. CU ... - ' l of F cI"" I .) '- $boor lhat H'. (,} .. nH ••• (I) lOr" E N. tloat

&. Or... ..

E N, .... ,t..- JD(np( - " (2) ' H~(I))

w r, _z> 0 S ..,11&bIo 1flIOl "

t ...

itU_. _

. ,' .

..

0 Ior.u

.t < ... Dod..... u...t H:(I)-

tH~('} .. GH,{I ) w...... G io • holl, II••• HZ(' ) _ 1It:' (1) + nH.{,) .. O.

Oi_ S E R, 'be r........... I ' • _ ",p(- ..' f2) . exp(uz) lull _ R on "' _ _ _ in ;>o CI ....... of. 1ooo 8 '" L..l;O Q. (,,) . ( ..",nI). S - thai. Q •• ,(sl ,. ~. (s) - ..Q. _,(s) 8 oJl 1~ .. <: I. Corod'.to .Io.oi "'11( - . Ill· *"PI.... ) .. Eol;O H.(s ) . (. "'.01) lot oil s .-.I " i~ R.

;eo n 2: I. t" II • . tioco' _ " . c'" f-. R inloO C io bounded (io ot:.>Iu ... nloe) ..,, : , _ 111".,.1' (' .11J(2) .... runo _ 8. p ...,." tloa& g lwIc_p to Q,( .. dI). (Otl .. 110M 0\1\- I'll 5 2.. ' - . \tI.) .. Lo:t I E Ebe,,"~ •• I, ;D E, 100,1000 ruO>tt;...r (n 2: 0 ). 1 (1) . ."p(u..). Gj>{ - I' Ill '" Ii """-pNc I" C . _ .bot oil II< ,- "'" I" io E it.tdf. (t!.. T1>oortm 1604.1.) i . Foo- «Yer)I1 i: E, Itt " be tlwo runo:tloo>

1.

1Art .. >

0....:1 8 .. (. E C ; 11t"'1 S "(2). 0;-,..,. ... ro,

,,-.1oM

•_ J

• _ 2'J'lI . .. ' " ' /(~~'" . • ) . ( _ u ')

.. R. Show .... 1 - 11/ 10. ~ 1ioatooU)'" or G,(", dI) _ IG. CGrocIu
L~{,* ),

.... ; , _ (- 1)· · 2' / ' · (nI)-II' . 2-·· .. - /' . UP(d')· D~( ",P( - h-,')

Iom.. \LItoi extl>oo<>r..w , , _ ;B ~(dI). ob<~ I>,- .... _ I ...... \"

Lh{4I) of tho...."........ (I" · o.PI - :O:I' I).zt'

·b.

E
,, - H. ("v3)~ -,,' /1)
R. Sb.ow .Iw.

and

Ie ' "

b"Cbap«rIO

,,·.('ur'''I

21T

.-'~ . ,. (~) ~~

.,t!i. " ,,(,,) _ r / .(,,) - fo e,,) and "" . ... . (,,) _

" .. (,,) - ,~ (,,). Oed ...... hat /. ~ g" . 11

Defi........ """"'" "anob"m F~" of ". by

f ~-""' .

FIl.,,(l) '"'

h. f,,)u .

,...., " '"' t.th

;D ,. (,,) _ g,, (,,). and o:w:I .......haI .T h.!I) ( - i)". 10.(1 ) b" I f: R.

,

Ci_ <> > - I and 8 :> - I. ..., "'DO""" ... ho functioD

p,..,.. thai. fw - " i..... ,,

,

;m .D" (...-{I) - (t'

L

Ii- by

",, 1 .. ]- 1.11.

t ... ( I_I)'(I+I) " I.

a

;!; O.

- I)") ..

H )' · " I· (:

~ ~) ,

e;") .

( 1 - 1)' . (1 + 1)'- ' .

~ ' S'

",bon D" (...-{I) . (t' - I )') ......... h dori.... "'" of Kl(1) . (I ' - I)'.

1

!:/diDo ~ J..,.,bi paiy""""'*l 1':-" of doll .... " (wi.k ~ D . M by r.-"(I) - 2"1"1 .

~,)

,0' ( ,.,(1) · (I' -

I)')

b- I f: J - I. 1(_ p,..,.. . hat lbe _inK _1IIc ..... of P• ..

1":'" ..

~. (" T~+2>'). s.

ld. V. boo tho _ ......1»« "'- E _ i{(I ... dr) _ _ oed by 'M fII .... tiona 1". 1' . , ... 1", S _ ....... P.(t) . ,' . oo(. ) dl _ 0 b oll lDt g •

f,

OS t < " . CU.. inlql'''''''' by J>6tU or ordot n;

[ r'Ij:..M:a-)u_

L

(-I)'" · ",-,- ' 1(,,).

,(o)(rll: • H

J"

o,s;rS' - '

if /. , .... "".............. """,_I ~ 1-.1\ __ b.o... " COD!;. .... Ded_t"".II.. P, ., , . . P. focm ....................! '-io 01. V• .

~)

4.

Shoo< that

1_', P.(I) _j" . tu(1) 41 .. 2".~.".' . na +- n" I )· r {8 + n" I)

'-

r (a +tI+ :lr!+l)

.

........ r .. tho E..... pmma """".,., Conclllde II..... ... all .. ~ o. " _[ "

r +'"

, r«I+" + I ) . r(t.l+ " + I)I- I" . ~~

Q+ I' +lIo+ 1 I'("i- l), r {a" " +n + l )

ob<.oiDOd by ort""-", olizo..

• tho poIy...--lol 01 ....... n i .. tho .... _ Uoo. ol ( t ·)..~ , !..

Lot ..

2': 1\>0 . . - . -. 51-- IhK

1. . C<> _ 8) . (" .. S .. :z.. 2"

io tho ...,...... ;,,1

6.

•

n- !

I) _I""

01 P:~ 01 ~ .. _ I.

Lot ~ ~ 2 boo ... LIlt'SO'- Sbow 14<

.. . P. (t )_

I a" lI. "' -1 [(O+8 +",>C' 0 - 8 , .... IJ + tn " l po.,CI) 2' .. .. tl. " 4 ft - I) a+ lh:m _ 2 +2n)(ai-n -l)(" ... . +- C<> +Co.II+8 .. n)(a +8 .:m _ 2) II ~i<:oJIy -... ( UM F

I ) ..

I)

rO t ' I

:ciooa I , po.n 2.)

1. Ci ...... E' Z· , tboNailt "'O E R, . . . • " "

!

~ . (-
eR

j') P';' (I J) -

~ lil&l

L ..,P,(I) ~, .

fI;w oil I E J - I , I ~ Show lbat. lor all 0 p
....!... . !.( w(I)' ( I 00(1)
( I - I'), P::'{I) "

k S

n, n, J P: .. dl ill ,he tcaLor

1' 11'0 (1»

Ded_ that G• • 0 far O.:s t < .. and .~

:i

(I •

•

-..{a

+"

p• .

+- .. T

I ).

eo..:_

(I' -" - Ca +fJ + 2)1) . P. (I ) +n(<> +{J + n +- 1) -P. !I ) _ O.

.. S_ tbat """"l' po;>I, ....... w.I 01 doqp'" "- c.ha;, .... 1Iich oaIloIleo tho dif. it<eDt1oJ oquatiooo 01 pu'I 7 10 5 ' rily ~ \6 ~.

(Notat .... al E:uo - 5 l.)

'15

ate

E<e.< ." h'~lO

I. Show

21i

.Me.. kit ~ Il\IqIef .. ~ 2, n · C:(t) - 2(,.+ n - I )· t· c:i_,(t) + (2,0 + n - a) ·0"'..... (1)

10 -icalJ,r ...... l.

... , '1 R. Show ...... boo functio<> h , .. _ (I - :b" . . ' + "''-' _ I _ J - 1. 11 .,.... be apended i~ .. _ _ _ la-ideo" (1 - ..... OD.) p""", tba. h")(O)/n ) _ Q.I_'), ..-bon Q. 10 the

r'

3.

E 1- 1. I I aDd 101; VI be . be functio<> .. - (1 - ~ -+ ..' )-"> Oft I. 8y u.. ... 3 eli .... ..,(.. ) . E.~ Q.(%)w· kit ~ .. E T. 5"""" tbat

Let

%

(1 - 2>0% -+ ~'J
_'MI

»Q. fr ) - :l("f -+ n -

~.

I)~._ I (% )

+ (2,. + n -

2)Q._,(%) _ 0

kit all iatqent n ~ 2. Coow:lodo . .... Q. (r ) _ c.:(%) kit all .. <: O. n.er-. "'"' (1- 2kl< +.')-' _ E. ~ C".l(:r)b· kit ~ " E I-I, II (Geboo>bo..,.n .,.panoIoc of (I - :ha -+ "')-'). Lc<,€R.S'- .....

,,-h~".')~ ·D-.,..·-." · ( L (7)( :.:)'-) _20

OS. S_

loI-o.lI"'1 1- 1.I1· !..

SuppoM ..,,. O. Oed .... from - ' ~ .ha' IC".l (" JI < CZ(111Ore>ft)' I n< n 2 l aDdewry r

,

"- J- I.I{.

_"'''P'Y ...

LoIo S be .. "''''irine in .. n. I' .. <:O<'tIpIe. - . n "" S. and 1 Jl < +«> .. .--1 Dum' . . . - «IUJupIoo e o," ' n' 10 , . !At. : fI _ C be oudo tlt.r.t Is 10 l>"itt1os:rablo lor all / E £~i;o),"" tha., '" 11d,, " locally

.s.

--. _.be,. I.

_

that" / _

J /g ilI' ''

contln....,... "" li:i;o ). (U.. "Tho< .w. S.l. '

I~tlooooew.)

:I.. o..d .... _)JUt 1 tila/., i l I <

Jl

<

-kD•• Ioent .. ~ E

, _ h locally ....

l. ,

."

Let

f1 _

Q,(JJ1-"

I hi

If, _l,olo:wtbatglioooin~(jo).

f1_lo, 1,:1, . . .)

be I ... _

or: pooll;'" iD~ let S

-pot, __ ....

1i... 1". of.1to "" Swelt ..... I'((n}) _

I'" -,.

n

~

be u. oemiri", In

Finally, lot I' b& .... ,

, ...

~o.

'VI

ate

I.

o.-~I'(I

: ,i...!y,t"" J tbo """ otd,..."oropo<:$QOrloni",olt"'....... _ r .. (r.)_~ 01 oon>pIu nu",ben"""" Ur i " t:_~\Z.1 < (lospt",h.,U', n~ p..>o\Z~1 < +co) . St- . hat tt... ........ 1111 f _ (!(n)).~ .... Ii.....,. ..;...",.,. of Li::(P) ont.o I' (I " ...,cti""ly. of

.00

.h.o.

. .. ...

~(P) 0<'110 '- I·

2.

F<>r """'l' i........ n ;;: o. put ~ . .. (6.." )"lOI .. .....,~. . . I or 0 .. ... .. .. ..- ... ,. ~. S_ thot ..,., " .. (z.),.... e I' • lbe limn;" I' of ~.~~z.~ +co.

... ,. _

).

fbr <*:10 ~ ., ( .. ) .... e ,... lot ... 110 ,110 "" ... in"""" Ii.....,. loom> (".),,~. _ t:. ,... ".!/o 011 I' . S_ .b.a1 .110 m&ppillll' _ ..... Ii.....,. i., >tll)' of t""';:"".bo ,o aot.
4.

0 •• Ok ~ l' u..'lo ld . -............ 01 1'"' _;"";1\101 , ' - ........... t.o O. fbr _ " Ii t mil"'__ ... U:ft <011 . . . . .

f>.

F<>r -=h We" . Itt ... M ,1000 contin__ U""", form (~.).~ - t:.~~!/o Oil l'. Show that u.. IIUlpplDc , _ ..... ~.....,. ........ .,.,. 011' ""'" ,ho _,DOd dual

(l'y of l'_

0 . - ~ t;' (I LIi_i""ly. & ) l be _ 0111........' , . (".J. ,.. of 1"OOil ...",ben .. hid!. .,., ... _ (I't' ;:ocll...ty, .. biclt «>D'''~ to 0). \hi~ (1;:)' lor 0 _ of.",T Ii (1;')' ... hldo ...... 0<'1 4. • an idoooIln (t;'y . F<>r <:very rei:'. Itt '. be;.ho; 1_ fonD (r.),.~ _ t:.~.".!/o 0<'1 &IId.....,.,J1 ,b.ot {_, ' , Ii ,:.} 10 .. _ 8 In (t;' )'. If T e (I~)' io pooIl ;"", W .,.." tho.!! .... ' . (, e I:'. r ;;: 0). Md if T ..... w... on <'10. • . - . Ihr., T .. 0, c-Ju:io !hot ......,. T e IIAY _ bkIt _ .... 4 io dlojoint Irnm 8.

6

r: .

7.

!I.

If T 10 ..

~ I ;""

...._ .... 4,

c.-Judt.b.a1 T

.... 9" if T ..... _

to

!At " &lid .. ~C,, +

.. y ..

ho; , _

t ltmtn. of (/A)' , d.jo>ini from all _•• , ....,... .hr., T

e cI""r io d iojoin. Oft

l'.

from .ho; 081>0__ I' of ('- Y if &lid

dioJoi"' '' ." .... on

• ..",,1",,« S in

n.

A_lilt lbot.

4,(P + ~). Lot. ~ be; .. """Un,....., ~_ 10m> on ~(P ). &lid .. for 4(p+v j b aII.,e a;'(P+,,) , ~ B M""""ntahlo u aiQa. d! ,bot. f ,u.I •• "".... B. and ~ E, F too dio,ic>int S...u _~ <&1"')' and I .... _ _ i""U'_ Show lli.r.t 9(oio) .. f J . b . ~ J" lor ail ~ e .tZ1J'l. &lid ~ ,hat ~ (,,)' .. Li::(P )_

,itt"

I."

II

Lot. S too .. _

itt ... In .. _

"" S. Aooumo ,b.a1 t..C(,.)' _ In

Li::M 10 ~r"""'pact ·

ply lOOt 0 and " ,. 0 .. ~ti'" dllr_

Li::(P). 00 lbat

'ho do"

t

......."'"

uni. ball .. i
'I'

ate

" '""•.~ 50< Ch'p''' 10 ..... £ be. ~~obIe aDd non'"'ll~bJe oet, and ~ ( E" l "~ , be. de" 7 dill ............ 01 ",~ ... bocto 0/ £ .uch tha. p(E. ) .. (1/ ,,),,(£ ) for - " .. ;>: 1. By S""" lj"D'. t bwo<m, from ( n/,,(E» 1«, L '" t:aII. _net .. ""'I ........ • b;(:b , e ... .... wMkIyl<> ~E 4,(,,). Show that fg ' \ s, 4 - I lot 011 1' 2: I • ... ..... ,i·. c:t>DtndiaioD.

l.

12

..... S bot a....uri"" In .. """""'P'Y"" nand" .. pooitl .. atomi< " 5:_

S . AIo..-.haI B .. u..~ A • . I.

\~

.................,. (A" ).. ~ 01 iD.eq\livalenl. ....... . _

1"'"

funct""",

For - " 1.u--_.....t.Ie O'JG'lpIex·..,Jued on 0 _ - " in,*, .. 2: 0, oil, >Ie b,. c..t.l lbe uruquo """'pie>; Dumbet ouch ''''''' ''' <.oCt) ~eo .. ,.~m A.., ~,hal ,botma.ppln&jo _ ( ... II),..{A.»)"~

io ... ioorne,ry 01 L.i: (h,,) 00.1<> I' and that .he rnappinc io

"

on

an ......... ry of ~ ( 1.1' 1

p....., that G:t!oJ 7-

i-

(c.,(/) t~

0<1'" ''". c.-l_ thai. Lh( l ap) 7- ~(1.,,)'.

L~{j.)' (_

£n,t.. 10).

13 ..... 5 be • """iri", In 0 oonempty ..,. nand" • _ure 01\ S. If ~ ("l' ... !.l::u.), _ ..... from ~ioM 10. II . o.nd 12 IbM n io • union of. bile nurnbet 0/...,.... &lid 01. locally I'""'gl'Cible_. l. eor. ... oeIy. if nit. union ot. 6DM Dumb« of , , _ ...... cl. Io
....... Hciblo 14

[n I . . ....1<... , if

lei . ....... ,.,..

. . . .,."..,

L<;'!PJ' '' L~u.J.

_,:T. "" 's

.hat. if n ill .

P, •. .. , /'0 .... pooiI.i... difr... _

""""'P'1 " ... on T, ......

be. iD

n, and

{(I', (8 ) •.... ",,(8») : B E F} •• ..,.,,~ """_ IUboot of R " (L)'apoIInoY'I'boof ... ,).

Gi_ ... In........ ~ 2, " " _ .too lhooo ........ beon ,,"01'«1 10< ,ho " " ,, - I. We obow lhal it io 'rue lot .. i ....Lf. Pu, " .. ~1 + ... + "". I.

~

ltoo dtlaJ",*", Ll: (P) of L:'(,,) wilb I.. -.II \opoIoc:, . p..,.., 1"-< IV .. II E L;t b. ) : 0 S '" Odd - . - ;~ L;, (" j. I.d • be ,Il00 m·W"" 1 - (f 1 dp.. ,~,s" "".., L;'(.oo) ifIIO R". at>d let 0 _ (a' •.... o .) be an .le"..... "'_(IV }. S,-- .bat V .. w n . -'{a ) • <XlI""""'boot of L;:(p.). heoce bao aD "",rem.oJ polnt, (whK:lt _ . . . . , Ok -,w"oblo F wilh val ... In 10. Ill.

s "\ • """''-'

• rom_,

2.

> O. The .. ",Ct EIO,I D > 010< ...... .,.. l S i S " . utd. 100- oimplici O. CbooooeO <: t <: L{2lOlbatp..{Z) > 0. ........ z .. I- ' I~. L -tl ). "'""- ,,(% : r(%) Eto. I I})

Odd let A. E F too _\>CIt 1"-< A. C 2 Odd 0 <: p.. ( A.) <: .... 2 ). By .ho induction ~heo ......... ""' F _ B C A.. C C 2 - A. tudt ,,,-< I'«B ) .. ( 1/2)", ( A. ) aDd p, (C) .. (1/2)", (2 - A.) b- l Si S .. - I . For ''''labIo.--l nua>bon •• I. _.hat ond .+11 V ......... It .. 0(2 · I . - LA ) + I(lZ_ A - 2 · Ie). aDd ol,.i». O>n\ndiction.

, - it

_'4'"

II

~ (,,, (,-'(1», . . . , "'" (~- '( I J) ) .

3.

From part 2,
~.

p....., lhal L _ ·, U.. tela"""'; .. nlid lor rnI dill... _ureo.

I.e< S \wi .. oeonirin« in .. w """.xy

.,..ttl ... bounded _~""' .... fl.

~.

!let

fl, iUid let (p..)'>1 be • ""'I"""" of -

If,..,.' ....... In part 1,'_ Uoa••here ""iota "" _ i a n)' ,..intql'ablo /uno. ion

/fOOI n intc> 10,+00\ oucb ,hal ,.' .. t ,..

~

IS 1.00< E be. Rl! iUid H" p<JOIt ive Ii.,.,... fcrnl .... E tudo ,.,.. " (""II " 0 implWo. ~ .. o. TIoe ful>Ction ~ ... . (lzl ) deII_ • -... .... E . Su_ E io «M>I.~ no ,m. _m. T1oen • .., ~ _ tbal E io o..ddohod oompiote (Cb&ptoo- I , Exercioo 13). 1ft ,be __ ' ion of FY Id"" 12, I.

s- , ha"

Exe,,'"

x.

""~ito,. lor .100 IUodon _~'" Ji " f _ ~ (.- I(IJ) on For _ r E E+, ...,.n that . . .. . (z-l .... """\.in ........ IuACtiooo Irom X imo IO. -tool· """",.bat tJ;. :;: . (z ) .... t bat •• io J'"intep1obio. S_. ito f..,., lUi If ,. .. toI %) b 011 z E E .

r •.

J• •

3.

p """, tbat

% ... .

r..:o

•.,. 01 E ORI<> l,.:"!.u) and on iooomorpIoiom ("' .... "' ••>1 .. ,.... . .. \&liM 11oo<>rem).

(zl io ... _

of o<de.od _ _ . _ ~.

I.

lor <:W;ry:< E E'. tboo fi ll... , ....... _ 00Ii''' 0 '' I<> z In.be """",,«1 OJ*>! E C _ 13 of Chapt .. I), ... lha• • -1 (1f ") .. _Inir .

2.

C~

ofJi(X . R ), _ . r o w , in . (E). W~ g lor ,ho ... p<emUm of H In . (E) iUid f bt ,be upper _Iopo of H . Recoil thM r io lbe u_ ..... ioon. int>OUO """Iari ... tion of , . Show lhat u( . -'(fJ) .. tul'lo~H"( . - . ( ~», iUid ronch.de ' bat 9 .. I ,.." """" H be .... upward-dIT
....,..,... 5.

c_

FiI<(I E A, ond let F ho. ckl I and _ ...... _.u_<>f K • . o..no... by of 011 doput ,"-...co of K. cont " pint F azod, far 011 U E ....... I" b ..... it>die.tor of U .... X . ThllO 1,. .. ,1>0 ........ on,,,,1opo 01 . 100 I" (w...... U E e ). lbat 0 io.be infimWD of.be 1" in . (E ), iUid <»t>clo>de .ha. F io ,..."",Ii«\blo.

e ....

,,,,,,,,;0,,

e,

s""""

e " , ond let M

<_

be . I""....i&lbie.ul.ot of K • • Denote bye tbe <>foil
&. Fill (I

•

m

" ,., .._: lor Ch.apwr 10 7.

II

F'n>m patIO ~ ..,Q 6, dod""", tbat tbe locally l'-0 _""" ......... _S'- t bo.t .upp(,.) .. x.

n.

IOto ate

u.cd,.

CI

Lot S be • _ I I I i•• ''''''''I~r let Equip I.bo 0pM>I M '{S. of t..,,'''''od .. ' W. _ fI -.rilh I ... -... P _ lui .. N_ ,kN M (5. R ) if on orde.ideooI oft ... JUra .M (5, R ).

f ,""".

"*'"

Lot" E M' (S, R ). ~ """tho baBel. ill. M' (S, H ) Ct.." . ... by ,, <0"'0,,0£ ~ ~ E M' (S. R ) which "'" abooIuw,. .;t h "'pod 10 1'.

""",i.....,..

11 Lct S bt a .. miri.oc; ....."".mpty ... ...,.. .bat • ;, on ioot ..... l]' and AD" M '{S. R) _ r.I.!I<).

n. Lot X , ,., aDd • •

be .. ;" Eaac" Ie,

"h"" of ",d..,,><1 _ _ _ _ from

I.

p,..,... th&l • ' - • URiqut C-U ......... _Ion, .till , f t ed by •• _ M' {S ,C) ill.., 4:{,.) ood thai thi< edE,ui", io • li _ ' " pry of M '(S,Cj octo Ll:, (pj. (0 1 1 \$ '''''', if T .. 10 E C , I
2.

" . ...."' t b. H will _ . ~1 CWDpoocl...- of M '(S,C). A d En' cal ~ H' cl Li.v.) has waIdy '-'PK< ~ if &lid only if it io bounded aad =ibmly lD1OI!JabIo. AIoo, it oo.yo tbat 1111 , g E W ) boo!ben 4l klr oomPK< <100 ...... TIl.., ti .... ~:> (I, t~ if. OOI'npoo
propooit..,....,..

N<w put Po •. •

" .. • - ' (1 - b

...._

-'0.)· Lot, E

_.). S, -

tlw

H.

i .. • (9).',

.. • - '(! :-'IL ), and

If,. ::: ' 0\bat 1'.1 be!oop 10 1M bud

_ _ by Po i" M' eS, R ), a»d lhal "" . d~m ""'" po. iDM'(S, R ),

3.

""Sod -

Let, be • pooil .... b ... - - ..... .. lIb _ I ' (£""'1' '

I., + , . ,

w""'" /. ;"

" 'tift " " n, 0I>Ch thai tho "'I~ .... oil 15). By put 2, - " , E H """ be wril_

I.PI ill ... tl1aa 1/ ... We put 2, of Ch&ptot I, ..... band

p-;"t~ a»d ;. _

"- 1I 1I1-1I~1'I!I S 1/ ,,- By Elon'Cioe IS, by" lit M 'lS, R) ill . . I 1m M ' lS, R I. p,.,.. that

...... at"" "

"

' un with _

~

ill •

,.

Coach ..... h.a!. I.... 6 E H "".. . ~ t-.

'I'

ate

11 Images of Measures

I be R ' II (~. ,) _ (~ _ J,,' + ,..1). ......... , ... _ _ oftbo ~I 0( ,. + .. 110 10.::.(. w. 10M--19< ••• ) 41< .......... , _. rOl' ....,. • ~""int
'5,...

to 1M "'" !map oIl.Jt~ ...dot / . t{,o a , _ . 1M. """"'pio ..... 1<1 "" ... _1"11;- A • .. oloaIl _ . the toct u..t ,,( A) .. ,I.{J- '( A ) do~_ • I' p~", <10M _ d' 0( p
po_'.

r

7

"'"11.1

r

C~." r

""" .... Sian_.-· a ·

,._nblloO'._ .... uyt<>

deft. . . _art; on. ~rIoc SO ill rY bJ ,"'(A) .. ,,(.-'I A)). Thio";O 6t6• • [ ' ''", if 0Irtt0I~
-r- f

Iv

or",... .......

W. owlY tIM _i00i 01 ; _ 01 • _u,," to ~... _ " ' " on R. 1 U n.iI If • $bort iat,oot.,.ioot to 0f\I0'dif: lboooy. no. ....... r$I.l. '" t~iII * 1.... • 1Jirtld('o.....,.t>c ~In; U I : n _ R io _ially ..-lDIcnbk_/. " fo ~' (whore. , n - n q.,t"'- -VOl" "J. 1_ ( I I .. ) ~.S4-' , . ............. ", c.iaIly "" .... tF~ fw>«I!y _~ &ad o . lo
r .. r

'0_

11.1

Jl-Suited Pairs

Definition 11.1. 1 LeI. fI be a DOn~mpty !let. " a romplex mN!I\ln! on a IOtnIlri'lS S in f1, ,. • mappi", from fI int.o a .. t n' . o.nd 5' a """iring in rr. The pUr (1',5') Ie sald to be j.I"Iuiw! if (a) "'· ' (A') is_ntially ,..intqablt ror all A' E 5';

fb)

for evtr)' A E S, there WIlt. ""'lUI""'" (A~)Al!' in S' and a " . ntgjipbie

"" N such

ttuu N u

IU.~ I ,,-I( A ~ ll
CMt.

tlot fuQCtion A'

~

,,( .. -I( A' )) is. measUJt on 5' called tbe 1~ of " UI>Oer ,., cIeno!.td by .. (JoJ. Ip Ihie

In the roI\oofIns , h",:.. ,,,, . ...., _me lha, I' .. posit ive.

"'. lIwa.t (...S' ) II " ....iW, GM p": 1" "

".(Jo). n,,,,,

Theorem H . I . I o4 ....

(a) 1"(1 0 .)dl' :5:

1" f dJl

r f <41JI f

toll"" .......

fr>r 0>11 f : n' is .m~;

JI · m"", w

(&) , ,, I' is I' ...........r'1Ibk, fr>r .... ry " ",.lri
iO. +cel]. ~!,

oM 1"(1 0 ,,)<4< ..

_ppmg , fn:m<

(Y into

(0) f 0 1' .. cuentiallr I'·.nttvrd/c. fo~......., eucntW/1y 1" .'nt.,...,1Ik m"J" pmg f In>m (Y into G - ' &!1Odt opts«; m _ _• 1(1-1<4< - 1 f dJl.

E.....,.,. ,

P ROO" , E 51+(5') CIlII be "";\1.... I:." n. · 1.0:. where I ;. tillilt. !he Ai are disjoint 5'4I!U, and lbe (I , &n! positi", nt1Ir1ber'f. N"...

I

gd,,' ..

L

(I; '

I"{A: ) -

;';1

L

(I • .

,,(,.-'( 04: ) ..

.( 1

r

(,go I') dl'.

If II Is l be upper ...""Iope of an increuins ""'l'0fll0! (g. ). ZI of St*{5'), ,bet>

r

/ldj.' .. . ~p

I

g. dj.' .. "',!p

r

r

(g. 0 .. ) dl' -

~1emenlII


r

(/la) dl'.

ro,

lIeDOe 1" U " .. ) dj. :5: f dJl fOT &.II fUDctions f from (Y inl<> +cel]. Now ~ 'R be th.c ring c-on.U!d by 5, For .......ry A E 'R. tbfte "", .. I • oeq_ (..t;.).. ZI of 5'..." and. I"ntgj!pble £lei N such that N u (UAl ' ,,-'{A!,) oontaI .. A i flO

rU OI')' I.odj.

:5: r U " I')· I __ ' ll!.t:. l dj. of

rf-IU ..~ d",

<

I·fdl",.

IU O ff) dp ~ {fd/-l' If E' iI a ".i~ te\, there aliIts A' E S such that A' C E' and E' - A! is "-nesl~bIe. Sino:>e ff~' (E' - A.') ill \oeaIly jl-l>fIlipblt. the "'" ..- · (C ) "' p-mt.. urtb!e. Now fi~ .. "_rmuurable iot\ E' _The lIeU A. E S tor which A. n .. - '( £') iI ".~ folD>" P-""" C. II A. IiflI in 5 , tbere exlsl a 8eq~ ( A:')_~, of S'-eeu and .. J>-ntsJjp!Aete\ N such d'M N U( u..~L ,,- I ( A.~» tonUi ns A• . ..d \.ben MI1r-'( E') .~ __ \lI'abkbeo_ k li lhe WlioDof AnNn.--'( E' ) and the A n " -'(A:' n E'). H"""" C .. S Wt ()Q<>Clude .h&t E n If-I (E') iii I'"Lntecrabk lor all wl~bIe feU E. So ,,-'(C ) iI Nat, let p boo .. p'. ~ m.o.pplJIf; from 0' into ~ rnetri ...bIe..-Gi...... I'-inucral>le IO!t E, ,,,-, en.1 .. ~~ (A!.)_.l: ' of $'- u at>Iaioed In LJ..O!: ' g(A:' - 1'1;), it~. Thil provtII ,hIu g " "

,.._11IbI8.

""-~ Now the f\ll>Ctioa

,,· C" - ' (A')

" . ( ,,-I(A'» II ,,-additive on 5<. and - ,,'{A') lor all A' E 5'. IIwce ""("-'(,01'» ... ",, (A') fo,A'

all A' E ~ (Propoeilioa 6.1, I ). Gi_ the ,,'_ m'P ""able IH £' . ..~ M .-e ,,"(E') _ WP"'t ... p'"(A' n E' ); bu~ A' n E' ill the Wlion of an and •

.t.""'

...'-""&Il&iblt Rt • .o

and finally,

.....(C ) _ .... (1'-'(&')). Each ...'.~ fuDct.1oa J fron, n' Into 10,+001is the 01. an ;ne
Uppc1' ~n~

Finally, Itl F be. n:aI Banadt ~ Tho: functlonall1 J ~ J J d¢ and J .... J(! 0 .. Jd... from l~V') InlO F ..., con';nUOUl v.d .hey J«rI'Ie 01> S t(S', F') . lleooe .bey ..., IdeotiocW0 If ( .. . S') II _Md and, II . COUIpIu. ,'Il1MId funttion on n' lor .. hid> , 0 1' II ~~, thenlios 1>1> [ ' on 10 SUPP
.....

I!.I ,..&i .... Po.i ..

11.1.1 ( n-allalt lvlty of Irn.ages o f M ..... .....,.) 1/{.. ,5') ;, " •• virM and ("',5") ;, "O"Il·..ntm, tMn ( ... 0 .., 5") ;, ,,_it/!d and C'" 0 .. ){p) .. ,,' (,,{p)). Pro~lt\on

o

"Ii

If 1'1 and 1" an: t"", complex 1'1>elW;ure. the ...."., ~iri", S, and 1£ (..,5') • both 1'1..w~ and p,....ui~ , Ibm ,5') io 1'1 + p~led and 1t{1" + 1',) .. " (iiI) + " (Ii.)·

e..

P ropoIIltloo 11.1.1 J..:I S lie G ..miring in .. ~",pt,.d n, .. M Id If lao: ." MJI_..t..t;...,ta/ ",b.., oj M .j-(S) hoving .. 6IIJI"'f'I~m Ii. Firl.JI,. Id .. lao: • """"""9' from n inlo ....I 0' , ,,"" ld S' toe G ....... ri"9' in rr. Tht-n " "«a"" ..wi n,/l'icienl OI)I>di/ iolo II>~I (1r . 5') ... 1'· ...ilCif ;, IAoI it ... " · ...>ICif Jar oJl " E If .. nd WI ruP... H ,,(.. - '(A' )) ... fi";l, JlK" 011 A' E 5'. I n IhU OIIH. "(Ii )" ruP""H l1'(,,1 ..... ume lnat ( ... S') "' "-,..,,; H" {"-'( A' )) ill 6nite rQlr 011 A' E 5'. Then ..-I( A') io _i&l1)' ,..1ntqn.bIo: for &11 A' E 5' . ....1lI A E 5 and, for each n > 1. thoooe ". E If flO that ,,( A) - "~{A) S 1Jn. ~ & teq~ ( A:.. .• )... ~1 Qf 5 ' _ ,uc:h P JIOOF:

n-:

that

A n (u..:;tl .. -'CA:.. .• ))' ill ".·nesligible. 1£,..., put B ..

U .. - 'CA:....).

""...

W:n, lOr allp2:. 1. An B" io ",. nqlitible- So Ii( A n 8") .. {p - ",)(An B"):S: {p - .,.I{A) 10 loris t he.n 1/ 1'. ",bid> pM'" that A n B" 10 I'·ooclilible. 0

Noo,. 1et (~ ).tl be. lummable r"",Uy of positive measures 00 S (S "'mlri"i In OJ. Deli"" 1' '' !:;.;, ~, Let 1r be a m&ppill,fl: from 0 IDle> a ... n' . and Itt 5' be a semirins in n' Then (... S') ill ji"6uilb:! if and ooly if (... 5') '" ~..w~ br.U j E I and tho family ( ..(p ,))""

• (p) -

.. ,"'limahle. In Ih. . . . .

L.u "(1'<).

PropoIIitkm 11.1.3 Lei n lao: " """""'pt~ ..I. S 0 .~irirog in 0 , 11M I' .. ......]OIa ...... IIn: "" S. Lt.t Y ..... tubld % .. nd Id T ..... ~rin9 in Y . Fi....II,. k l ( .., 5' ) ... .. " ....il. IAen (."" S') i4 IifT·'Wl.Cif ..lid (..I Y )(pfT) - .. (1 Y . Ii)·

o P rop
0"

.110\ Ih4t , i. I«»Ily p.mtqrcWe. TM", .. 1«»11, p' .irlt
if . ...

•

P ROOP: A_me tb&*, is Ioxally "'·;n~bIe_ Then. lOr toll A' E 5', g' I", is "";111",",,*. 110 (g- \ A' )~ is t$lMially 1'·lnto::grable fI.nd .-'(A') ._twiy (g 0 .)I'"intecrabie. ~\Qo:W',",

....mt tt.at (..,5') is (g<>..-}I-'-CU itfld. For f!&<:h A' E S', (g. I", l_ ''''WlJ l'"in~bIe. N_. """"' ,-1", iB'". n . 5·" .... bIe, "' _ l hal " I ,,· .olitJly ,.. ·lnl~ Htnee 9 10 locally "'·int~ 0

Corn~ly,

iB is i

Proposit ion 11. 1.5 FliT 1lI1, E {1,21, I'"

n.

k" ...........'1' MI. S. " ......,. ring in 0., IUIllI-'< " _ . " , "" 5,. leI,"; k .. "",ppm, from rl; inl<> .. HI n;, al>lf a..........., in 0:.. fJ (."S;1 is 1'; . .... th.
I'" s: k

'1"".

o

11.2

Infinite Product of Measures

Thlo action.", parti<:ularl)' UIIlduI in proI:oAbility theory. Dellnitlon 11.2. 1 Loot. n bot .. '1OO1I!1hPCY tel_ A 0:1_ C ~ , nI suboleu 0( n ill taId 10 be ""mIlK' if r\;! , 11', ;. • for e-y ""'l1lt:llOt (K . ).;!, of C-. satio!fyina: S'S~ K , ;., for 0.11 n ~ I .

n,

Pro_ltlon 11.2. 1 L
c·..

n,

PROOf'; Ltc eH;),,:!, be .. ""'l""""" nI C' ___ """" tlw <.c. H. ill IVlP«RPCY lor toll Int.... P 2: 1_ For ..... , ~ 1. H i 10 .. union of linlto:ly l1WIy CfIeU K.,.j (whoere I :S j :S ... ), Denote by Z tbe III!t n ,2:,{ 1,2.. .. .... 1.110 thai ~, H, Uo..,(~, For ..... inlql!r' ~ I. I~ Bet Z• .. (n E Z : n'IlII lbe vaI_ of q lOr which ,,! _ "" tllm: are in1ini~ly n'W1y b: ", hidl n1 .. 0" By 1t>d\iCtLoa ~ .... , define a ""'Iumoe .. = ("" _. , , .... . .. J, "'hid> belo::>np to Z ...-.d ",hidl satlsflooJ the 5oIlowi..,; "'''I*.ty: lor .......,. lntqotr P ~ I, tben! .... iII!. lnfinil~ly """'y in~

..

K,...,)_

Z. ,

K....,)

,-..1_

q ~ 1 .ucb that oT '"' 0 ,.0; ., 0: ....• o~ ., 0" Since

IOR'It

of lbeet

in~

an:~lhanp.OEZ•.

Now

n,Yt, K;.o. iI ;'IOJ•••,"",pe.y lor .all. P:=: 1. ..t>(, ... C Ie .. ooml*!l clut.

n....fun!,

~ ,K, ....

f.'. and alOftIOn n. ~, H, f.'.

[]

.<1.

Pro.-IUon 11.2.2 ulO k" ~pl~ 5 .. _iring in 0. ami" on ~ ""'PI'i"f fn>m 5 into i1 11
dmoJced by " . lei A E SO. Tbtte an: mutu.tJly disjoint 5..-. B, . ...• B • ..booe unino iI A. Ci""" • > O. " -e call lind K. E C ";Ih K. c B; II\ICII thal ,,(A - K; ) < ./n. Heooo ,,(A - U,s.s. K, ) and

:s '.

,,(A) _ sup{,,(K l: K E C. K c Al. to 0\>001 ,hAl, il ( A; ).~, to •• ,,(A,) «>tI,~ to 0 as i _

is a ~net"I<:e of 5'..-. .. hid! de(' +00. II DOt. idnce (,,{ A, ») ;~, deb .. : I , ! _ in! ,,(A.) "' 81tktly p
:s

K.c(n K .)U( u ".-K,'). IS'!i_

IS'S ~ -'

Tbio implies thAt I'lfl, s .s _ K;l 2: ./2 £Or all n :=: I. Noor it IoUows thal n, S' S ' K, ~'for all I , and thAI K. ;" beo::a"", ella CO
n:=:

n.2:' '" _

I.d (0.).." be .... infinite r.ntlly of ' ..... IIL~ """', For ~ i E I . Ie< S. he a 1Ie~ In 0., thAI if. a 1Iemlring in 0. contain!", 0.. ~ by F the claM of all finite "",..,apty out.M of f . For eecb J E F. write 5 J for the 8eDliri"4l 5; in flJ ., n ",J fl. and denote by qJ the projection (>:; lot" - {"",)oEI from fI ., n'E,fI, O , hAt J. c J, . denot<: by ,he ~ projection from OJ, onto OJ,. Tbo: d ... S of aU the (AJ ). lor aU J E F and all AJ E 5}, ill a .. miaJ&ebn of IUt.e\.o 01 fl. '"dE Ei, If q; '( A}1aDd qir' (BK 1 belMS to 5. Ilten q;'( A} ) • q~/«A N/<) and qir' (B/<) - q~K { BNK ) • ........ A.NK " ..;)..,/«A }) and BN /< _ "ir~/«BK) bt-Ion& to SJu/< .

n.•}

"1.J, q;

Delini,lon 11 .2.2 5 "' called the product 0( the tem~bras 5; and "' de--...d I:.,. niEIs, .

NOla that the ,,;'(AJ ), b J E F aDd AJ E !h .. ®~JS;' form an aJ.,d>nlA and ,h.at 0 0(;1 ~ i!I lhe " . riI>g 8"'etatM I:.,. S (aDd by A ). Definition 11.2.3 For...:b J E Y , Itt I'J be. ~jl;"" II\CaIiIlle on 5J "';Ib !nUl I. If I'J, _ %J,.1o"'J, ) £or aU J , aDd J, E F ruclo. lhal J, C J, . then "'J )J~F Ie said w be • projed.i"" .)"Iitem of meu\I~.

Tboorem 11. 2.1 (K oJmocorov) kl U<J)JV /It G pooj«fillC

"oj

iii

<>/

~.

fa) The", ;, an UJili"" /-(1iD .. " .... A nodi IIuot ft>r.J1 J E F aM.!1 AJ E !h .

vlq; ' (AJ»

_ I'J (A J )

(6) Write I'; - I' ( , j For.J1i E I. I/. Jvr _ry i e I , ~"';, a (mapld do.u C; oj S,-MtI nd IIuot 1',(04;) - !lUp{J'; ( K,) : K ; E K ; C A,) ft>r .!I A, E 5" 1/1"" I' _ "I' ;, a ........"".

c..

be tWO A _ and let J , K EF. AJ E SJ , aDd BK E SK be"""" Ih.at E; _ ,t{A J ) -=I. F _ ",..' (8K)' Then, \n our prcoiou8 noutloa . ,; ' (AJ) - ".Mr{AN "), '" A ........ _ q~,,(E;) and •• imilarly, 8 N " _ {F}.

P *X>I'": Let E . F

'.IUIot.>too .. ...·,

1' .. (8,, )

~ I'JU,,{BN K ).

If S _ F . l ben AN .. • B N .. · Thilsayo , hat I'J (A J ),. I',, (B,,) , and.....,..... thM. II 10 wei! defined . ~ E IlOd F are dilljoiol., ... are AN " aD
"" Th,.

~

ill oddil;"" 00 A .

~ i E J, then! ~ • oomptlC\ dUll C. of S;-uo Neb tbat 1';(04, ) ...... p~,( KI) : K , E C" K , C A ;}. Dmote by C tbe cia. N"...

_"me tW, for

_llot !Dgo(lhe~,;

m ,tJ A,) .. here J E F IlOd A, E

and let (1t1-) ~i!: ' be • ..equeooe of

Kj-) E

Co £or

c-u.. fOr e""'"Y n 2: I , K t- ) -

&II I E J.

n,u Ai-)

C, 'or aU i In _ ... beet J. E F and K!0l _ fI.; for aI\ i E (/ - J.). If n, Sos,. 1;1-) ia dO, ..",pty for &1\ P 2: I, Ibm s • S, K~·l io

... here

n,

11..>,

nonempty lor .JI p > I and all i E f , so iQ~) is ,*,"illply and, finally, KI-) ;. e. This sbows that C is .. comPact eta..

f"l..z,

Let

.,.te,

,;'(04 / )E n ;,sl S" and ~t n _ card(J ). Given f

> 0, t ' - allu, for

J , K; E C; tud> that K; C A; and jJ, ( A; - K. ) S tJn. Put KJ _ n ~J K ,. S;""", U, O ( A.i - K i) )( n ;,s/-,/I 0 ;) <:ON.ai1ll , ; ' (A I ) - , ; ' (KJ), _ ha~ .. (q;'(AJ)- q;'(K I » E ,EJ I',{ AJ - K,) S t, ... bleb $hoon that j E

s

.. (, ;'(A J ») _ . up {l'{ K ) , K E C. K C q;'(AJ) }. FUrthe""'Ae, by P /op08i1ion 11.2.2 ... is ,,·addltl~ on

n", S•.

c

o..ll n il ion 1l.'U In ,he previous n()(.alion, if I' _ v/ 6 ill ,,·addlti~, it .. ..n.d the projec\i~ limit of tbe /JJ. Then 1'1 - ' J(JJ) for all J E:F, l'{E )I. djllor all E E A, and " is ,,·addlth-e.

J

'fht,otMI 11 .2. 1 l"vol....:! """'I*" d '

T

7 Thio ..

DOt

tbe cae with the

IolJoori.. ,...w,. T bec.. .,m 11.2.2 Far"'""l'; E I , kll'. !Ie G poM"'" .........,.. will ...... 1 "" 5,. Far...dl J E:F, JhI.I 1" _ <8>." , 1'.' n..... (I' J)Jel'" Io.tu G p.oj«li"", limit.

PI\OO', l.eI. ( E. )~l:, be. dK::t,osi . . ~ .... """' in A ouch tIw. (l'{E., »)." , doef not """~ to n. It IUIIioN to.oo.o tbot E~ is """""'ply. For ~ n ?: I, t ' - ...... III J. E :F and AI . E 51. l uch that ,;.'(04/.) _ E". Let (;~),.l:' be • ~1lI':fI()t of dietlnd dements of t ruth , bat ( j~ , p ?: I) :J J... !-lISt. ...., Introd""" lOme notaUon. For ~ intq1:r p ?: 1, .. hat ...., ha~ done 10 f.... for 0 can be done £Or OtJol _ n .el/.F•.....•• )) 0.. Let fol be t be eta. of rJllFOI>Cmply finl~ I Ut.eu of I - ti,.... . i~'. For e>-ery J E p I, de,oou by qlJl the canonkrJ pm';"'tion fmm !p) OI1to OJ . 'ThMe ....II ... additive function.,/Jo) on tbe rJgebra A( ~l ronsiotin« of tbe (, 1J1 )- '( AI ) for J E p I and AI E 5 1. M"......"", .,(P) _~ ,.t.l (q~l )- '( AI » - I'/( AJ)' For e¥CIY in~ p > L, every (:t:; p ... ,:r •• ) in 0. , )( ... )( 0.., and e¥CIY iD~r n ?: 1, denote by E..{:r;" ... . :r.. ) the lJI!CIioxI of E.. de\ermined by (:t:; " ...• ll" •• ). By de/lnl'lon. E.(ll";, •. .. • r ,.l ' 00 ., al of lhoooo (:r, )..e(l_l;, •... " )) In (l
n.>,

u..»

'* """,..In

beloogo to A lol. In __ (i, .... ,i,1 a>IlU.illS J~. E. (z" •. . . , z'.) is d. b- tw"ry ,, 2: 1, dtfu>e

B. _ {Z', EO" : .J')(E~ (z, ,)) 2: t / 2} .

.J'I{E,,{z ,,)) - /J J. (A,. 1 = ,,(E,.) 2: ! for all z,' E: 11;" wben<:e B" ,. 0". Now , if J~ - (il'. then B~ - A J • &:>d I'<,(B.) - /J/.( A,. ) _ >'(E.) 2: •. Fin.ally, if;, E J. e.nd J. -I- Ii,). P l"Op(Ieit;,;,n 9.2.7, applied 10 1'<, ® " ,. _ [;,), ~ thM the fu nct;,;,n

z,' ... 1'1.-1' ,) (A,. (z" )) ill S,,·iloreli&n; thus

• =:: v( E,, ) ,.

B" E S"

,,(I) (E,,(z;,))

and

"J.(A,.) - f "J._[" I (A,. (z;,») 4"" (z;, ) _ / .JU(E.(z,, »)d;I;,(%,, ),

which y>.:ido

, , f.

"(1){E.(z;,)d;I;,(z,, I+

S.

f

10., -so

,)1)( I>,,(z,, ))41'<,(z,, 1

=:: /J.,(B. I +tn· TIlcreIore, ( B. I~~, ill .. dtu --;11,(1 ""'Iuenoo in $., ouclt ,hAt I'<, (B. ) 2: ./2 lor all n 2: 1. SinCe J.I" .. ",.additil'e, B~ is oonem",)'. and there Cltisu

0..,.,

z' , E 11;, SIlliosfyill,(l "c')(E~ (:r" I) ~.n "ry n 2: t . BIIt , as (£~(z,,)).>, ill docreui,,« in A flI, _ ma,y apply IItt argument used abo>" for E.. 10 fu;.j %" E: 11;, with " I') (E.(%" ,z" I) 2: tl4. By illduction, ~ , define .. """!,,,,ncc 01 pointl {z .. I.~1 E n.~l n" IIOthat .,{.)(E. (",,, . .. ," .. I) 2: 2-'t lor a1lp 2: and all n 2: I. Let, E ('I be .. poinlwhol;e <:aDOn;.:..! proJe<:tx;In(ln n~l is (%,.). 2;" Ci-. an lnte«'" n 2: I, let P 2: I be oud> that (i .. . , i,l Cinnw... " So ~ E E. , &nd J•. As E. (z". ,".. 1is nootmpty. ("')'0. beloo9 to AI.' o..~ l E,."" t. 0

r;, ....

n..

I

OefInition 11 .2.S UDder the bypotlta!cs ofTboorem ] ] .2.2, " _ " I S is called lbe Infinite product '" lhe J.I' and is do!-ooted. by ® 'EII'<.

/1

ate

"""'"u'" ...

be • posith'" it.h n.- I 00 .. ~.. 5;, &nd (I;),.,) ill .. partition of 1 Into """"mpty IRIb1ets. Put I> .. ®;V(®"," 14). For all j in 10m! finite iuhilet K of / lei. A, € 5;• .00 let A. .. n, III E (I - K J. Otnott by h< 1M finite tet U € J, l, nK" Then fbi" ew!'l' i € I . let

"'''''''*'

II;

t,.

, (11 A.) ":1

• 11 ( 11 ",,,.) .. n 1>,(.4,) , o~

. ~ /,,, ,,

"' '"' n

."

~, I> ..

I>.{ A. ) .

® of,I 1>. ( te,ociotivity of prOdua 11-"""), ThillMl. """,1\ »UsioU 10< &nite I w'- lbe 1>, an! CDIIlpiex mo:eoIu~ defined 00 Iimlirlnp.

11.3 Change of Variable Let I ... Ca,.) be .. ~PCY inl..-val of R, ...IUI Jo.f\ endpoint: a aod rilIbI. in it. L« .I. b4 ~ _un: on I and let F be .. n:aI Ban...:h

"""point.

."""

No--, li""" .. IoxaIly .I..intt&t"&bio: fune!,;on , from I

I:

intO R, _

G be an

indefinite intt&ral 0( 1> " ,.1.. So G(p) - G(o) '"' g(1)dI for all Q, fJ € I . S""", G;" cumin...... (Corollary to ~ 3.2.2). G(I ) 10 an lot........! of R, and ... let" be ~ " w " " , on the M\urai ~Iri,,« S' of G(/ ). ~!Or tn., n_at that g ill continuous. If I: G(/) _ F io t:ontin_ thea J:(f <> GJ, dI ... ~ 1(.. ) dto lOr all 0 , IJ in I . Indeed, for eoocoy " € I , tilt fullCtioooll: ~ ... J~(~) /( .. ) du from G(/) into F Is .. primiti ... of I; .. (Il <> C)'e.,) '"' I (G.,) , (.,,) for all '" E 1, from which _ obtain (/I o G )(P) - (Il oG)(o) ..

The formula

[(f

<> G)gJI.

I: (f oGndl .. ~~ f{~)du <:an . , . be written f1f" I{..)du. J.~~(f <> G)(I)G'(I) JI- JG("1

"The latter fOnn It ' .....'IIII!!(\ .. a ctww:: of oariabIt; !ormula. In fact, ..e <:an ~ mud> """'" ~...J cI>an#I of -'"hIe f~ ~b< thi&, ... ~UTn to tbe _

In .. hid>, II JoeaIly

Propoalt icm 11.:U n~ limit.

0(.... ) ..

~illl~bIe.

..tUM"'" thai, ;, .I..~. 11l~n (G,S') ;, 1> " Mittd. ~m '_

."

G(.,) and

G(~_)

ui.ot. .. nd il J;' ... in' • • :J

G(p) _ - I J

· ".

PI\OOF: Si~"

...w.

G(..·) and G(~_), IMn G(p) .. IJ ." or tuG( .... ) < G(b_) orG(a") 2 G(b_).

indtuUd in G(l)

~ndporinu

.Ix>undo>d.

(G. S') III ,,""sulU:d. By tbe domina-told coo·.u ", '1O! theorem, G(a" ) &Dd G(~_) o:xioI. in R. and it ",mai... I<> be proven that G(p) _ :I: 11"" ~ / E ~ (G(I) . R ). It ou!licao I<> ohow lhal. f (/oC), dJ.. - :I: f / · I J Ib; (Theorem 8.1.2). ~.., bot; .. 1>ow>ded cootlnlJOlll!l function from R Into R tha$.
I")

(g,, ). ;!cL cnnvergs to, J....a1mooot ~~

(b) (g., ] !! h 10. all" :!: 1-

f:'

19.),.2;1 wn ..... gooL 10 , In CliP,). If we put O~(",) ~ 0("'0) + g,, (t ) dI. wbel t 2'0 . . . !hed point 011 , the fUI>Ctioo"" G. (Om~ unilo,.ruy to G, 10 O~ I""), O. I~_) wn",,'I" 10 0(.... ), G(~_) , ""'pectimy. N_ 1(10 0 G.Ig.1 • omalIer tban 11'1 . h , for all n E N . ThUll f(", 0 G. )g,. dJ.. COO"",!", 10 f(""'G)gd). ... n _ +00. F(J< ~ R E N ,Iet (e. ,d..1(..ith c.,. S d..) he .. compooct. Iubinl.en'al 01 1 CORt";ning lbe oupport 01 g., 1"hcn

j (", OCo )g,.d).

-

[('POCo)(IIg,.(c)dl 'I"- )

,. {

'l'{Od{, C, lo.)

boc., ...

whtrt c.(",,),. G.I"" ) and G. (d..) '"' c. (~_ ) G. is """"I&Ilt "" -.ch oitbe lotenU:o (.. ,e..1and (d..,~). L<:tting n ........ '" 10 +00, "'" """Ihal f(l oO)gd). ~()d{. Nal, 1Uppc-. 0 (.. +) ~ G(~_) and let I",~l be & a>m1*'C ... binterval 01 O(1), Includ«! in [G("" ),G(~_)I, CKlt.sldt .. hioh f · l J ..... nilhea. Then f f· IJ Ib; _ f: 'PI.()~, Bul , iii""" ... vanisb.o.I "" 11>0 com~"","1 of 1" ,...1 wil h .... p O(b-).

t:(:;]

o

Propooillon 11.3.2 l.d z., E / .

,.)

u.~ li""iU

G{.... ) . n.d G{~_)

n.. .... (G. S') ;, II"aud if and <>nIJI i/ <:nil in

fl.,.

(&) ,. l (e ...l i. ).-;"fLi'''WC cIlcllt:t'Otf" G{a ")~ I" G(I);

(c) ,. I I..M ;,

).-~ 1M~~ G( ~_) 6d0-n9'

t.o 0(1).

••

11.3 0 ..... of. V..iooblo

2M

l/{C, 5') ;, 1'-... iuJ and if J ;, .....&Int"""<JI 0/ G( I) II1ith -tpoilLto G{,,"' ) .. nd G(~_ ). thrn GIp) ... 1J ' ~ or G(j<) ... - I J · V . . Gj,,"') :S GIL ) or

G( .... ) > G(b_) .

P ROOF: ~ that tbose mnditiono are w l$6ed, and lei. K be a oompact ",bid of G( / ). Then B .. e- '( K)n[zo,~) is cbd ill (..... , ~). IfG(b_l ""~ 10 K , (" o, b) io I'""in~, and the WI>/: is true of B. On the odler hand . If G(~_ ) ~

"'" b 2"0,« to K . b ~ DOt lie ill the t\')iWlO of B (.. Ith I"OIpooct to R ). hillO: B ill c:ompod. Similarly. A .. ("."olne-'(K ) is I'"i~. Thus (G,S') ill ~ted. COD>uidy, -..mo tMI (G.S') 1& I'""wited. We arsue br o;Jatrad and thai. G(z) doa; "'" """""'"I" to AI Ill" con'"l'll"!"" to b ill I....., ~) . W~ C&II find r <: AI "';tb tbe loIlmoill« propot\y for all " E (zo, b). there em', V E ('" b) .......n th.t.t G(,) :S r. t..ttill« • be an ... bitrary ruI number in /t". MI. there exid l • Rrlcl.ly hoI I'ill« ~~ ("~)~,,, lI> [z.,b) rud> that CH" .. _" zaol) i& ind"drd In (r, .), G(" ... ,) " r , and Gl" .... ) .. .. SII>OF tbe act G -L(fr,.])

u..",J:z.... ,."...

ill ,..lntqp"abie br hypocho:oDo, E .. 1ill I'.ln~ .. well, and /I . 1" Ie ~.integrable. Then I>ri . I • .t~ .. E~l' 1/1) • 1,-_, ._ I.v.. ill fiLIi"". But thio ill D UrOI€ that At and 171 "'" ",,,,!ned hr G,t....). n.:.. (zo,6) 101'" ill~, bec& ..... It ill lDduded in G-' (lm, MI). So 9' 1(0.,10} ill ~.integrabIe and G{b_ l ....... . Na\. 1eI. ! € 1t(G(I). C). If G(b_)bt:Ionp to G{I), then g . 1(.. ,10} ill A-

r

I~ and

r

raf::\)!,w - fU o G )~. I ,""..)
od"", harId, If 0(6_ doeo not belonr; to G(I), tbotn: eritto z. <: fJ <: thA~ jp, b) doa; DOt meet 8 ,. C -'(",W n (" G. 6); then

n

On tbe ~

ouch

('<61 / dv _ JU oG)g. l"o,8) d..\

JG1 .. )

(Pt .........tlol> 11.3.1), th.t.t is.

.

t:..,'~' /,w .. JVo O )g ·l .....

) 'O' .

'" both CUQ;, tbe IlIIIt equality 10 \rule. Likcwioe .

{'t..) / dv

JG(o*'

_

JU GG)/I' 1(.....1<1">..

.... 'Y"

ate

t:.0(.*,'-' J !iW _

(JoGlgd),

a Theorem 11.3.1 S""~ 9 ;1 pMtlw. Thm (G.S) ;., ".",Ueo/ Gn
J

f(f"G)~d;>·

PROO" G ilK'" " '" G(,,·) &Dod G(~_ ) exiU iI> R.. U1. Z(I E I. HGeL ) ~ to 0(/). IbonI ex-. ~ in (~,b) !I\OdJ 1hat Gle) ~ G(l _l n..n , ~ .l-.J_e-uywherein (c, b), benot , ·IIao"';' ~.in~ Similarly. •. 1(....111 )"illtqrablo ItCC" - ) ~U1G(n. T'bettfon=, (G, 5") _ ,Huited. Now I~ Lo8t "nttiootl ~ from Soldioo 11. 1. a ~

_ tliall _

11.4

in Chapter 22, tb/:8e n'!$ultl CM be refi!ll!d.

Elements of Ergodic Theory

I" ttu. _Ion. _

~

Birkholh up>rtm.

~rnma 11.<1. 1 Ld (I,~.:s~ m . . .11 j..u,.~ tJW 1)

; taitII the

I, + ...

""'*

f<>Il.o"i~,

+ 1, ...

60< • ji.we _ " ' " of r<eI n.Mb.tn, aU /d :S. m :s; n. ~ b., L ... ~ .ttl 0/ ,,..,..,.. p,up..,.l.., /lien: ~tr "" 0 :S. , < m ta.co\ tIwot

~ I). ~

Eot-t- I i ;,,.,...,.....

in.,.

PIU)C)Pc Let i E L... and In , boo tt.. smalitet imoser in (O" ". m) ouch thall; + .r. + t .... ~ 0. For 1 S q :5. p, _ h.t.~ I, + .r. + /..... _ 1 < I) and "+ --' +1.... 2: 0, + .. - + I.... 2:0, "hJcl, .... O0:' :"1 tQ L ... If L .. .. """""')", _ can ~ J, •. or J. IT 2: I) 0( ........ ""'ptr inten1l.l8 01 (0. nl n z + oueh IMI

be...,.,_

(a)

P(J.~ , )

_ruct.

< o(J.) 10< all 2 'f J: 'f .....

(b) J , u" ,u J•• jO,8(J.lIn 1.- for all I < k 'f r;

(e) E",I. I.,
l ri: e~ ••~ "" ha.... obt..h,.,j noroo.mpty in~"'" J , .... , J._ I (.. ith • <2RiooI (a). (b). &J>d (e) bold for all "# L".. U i . .. lbol $m&I1ootit inlep 0( t 'f • - I, and I"'" J , u··, U L". - (I, U"' U J._,) &rid p, II lhe ..,...:Jl$ of lhoe. ID~ 0 'f , S rn """"

J._,

II,~

that

t. . ... ...... t ; • •• ;>: 0, ~ ul<e J. ,.

may !I"X" j .

Now " _

thai. E,~ t... I, ..

Flomonu of Fzp:Iic Theory

231

{i" ... , i .... p. I. Tb.. the lnductio<1

E,,.,.,.,. (E."J, t.) ill pOIJili~.

a

I" .. hat foIlow.. Ie\ 0 be • ..,."""Jlt7 _ , 5 • ..",Irin& in O. I' ,. 0 • ~liYe WE nre on 5, and ... znawinc from 11 inlO O. Aaume lhat ( ... S) Is I' ..uiud aDd .. (1') ,. 1'. Then (..·.5) III'.ruit«l and .. 1', for &11 t E Z ... For u.d> maPJllnl: 1 from 11 InlO • _ rt' . put 1. .. 1 .. .. ' b all t E Z+.

'v.) ,.

PropoolUon 11.~ . 1 (Maximal E'&<>dic Tbeo""n, ) Ld 1 : 11 ~ R Ioe oe-tiel/, 1'.;'11.,....we. and,1Il

A .. (r E n : 1o(r) ... ···

+ M r l ;>: 0 1M GI.

U-

It.u.I """ p;>: 0).

n- f" 1 dp.;>: o. P ROOF: For &IJ k E z·, 1.", e!IIIenUally ".ln~bIe. Fix m E Z·, and Ie\ A_ be the -'" of thoee:z: E l1 .uch that 10(:z:) "" "'" 1.(:z:) ;>: 0 tor at Icu\ ..... 0 :5 , :5 m . Let ... ~. n ;>: 0 be j1iYeD. For -.II 0 ::5 t ::5 m + .. the let B. , eI t hoee :z: E 0 rueh that (1. + ...... 1••• )(:z:) > 0 for &1; bH one 0 < , :5 Inf(m, .. + m - k), Is I'-",c·urable. Moos,.., •. B• .. ,, - · (A .. ) lor 0 ::5 Ie :5 II. Now, for """" z E 11 , Jg 1... (.0:) be t he !lei eI thoee IndW. o < Ie ::5 m + n for .. bich l he ... exillu 0 :5 P < inf(m. m + .. - Ie) ouch that (1. + ... + 1. ... )(r ) 2: O. Then, for all :z: E 11 and for oJl 0 ::5 Ie :5 m + n , " """" 1".(.0:) .. 1 or I",(:z:) .. 0 .. k beIonp 10 L- (:z:) or 110\. Hence Eo:s.~_ ~ 1.(:z:)- 1".( :z:) .. E.u _(. ) 1. (.0:) for oJl:z: E 11; howe""" the ri&hl-)WId oide is ~UYe by Il .U , and .. Eo:s.~_ ~ f ! • . I".
..

~t;""_

I.,"""".

..

Nort,

Since E... '~'S:~"_ f 1• . III. 40 II 10M t ban m f Jl 140. __ • finally, tlw. (11 + I) fl ' 1,,_ dp. +m J il l II" b! positl..,. Letti..., n Lend III +0<;>, _ cono:h.le ~ J1·1,,_ II" II positi ..... N_ A .. iI included in ,t... " lor &11 m ;>: 0, and the dominat«l DDEI""" IItIWX theorem IhowI that fl ' 1,,_ II" (Om(.1t'I1O J1 - I" II" .. m _ +00. 'l'boerdore. J1·1"11,, iI JIOOiti~. a

Tbeon!m 11 .•. 1 (G.D. Blrkhoff'. Ergodic Theorem) Ld 1 : 0 _ R Ioe ~iAU, ".in~. ~ e:n.tI An .J eo eM'/» ".;"':j.o.&le jtmctitm l' jnmI 0 into R nocIz !hal (I/n) Eo:s'S~-L ,. ""'.....,... ID l' 1«>sIt, 1'~I ~

e ... 'ph ......

..... -

+00. Mo. I\O"" ..,

l' .. l' 0 u 1«>sII,

I'-~t

f'IIOOF: Lee b be. ral number and Jet C.., 'I>Iially ".~~ eel-.b 111&1 b < limsuPo__ (I/ Il) IAs' s.-, h (~) to<.U ~ E C. Denote by A \1>0 let of 11>00II: ~ Ie n for which \.hom ........ P 2: 0 ouch lhal (10 + .. , + 1,)(..:, -" .. ~. atd(O:S J: :S p: "'(~) E C). For each Z E C and each 'I(J 2: I , ~ caD find II 2: 'I(J .., \ha\ ( 1/ 11 ) LoS'SO_1 I. (~) 2: b. Thll'l A <:OIltains C, By 11>0 mpimal ~Ixodie IbototClI', 1...(1 - ~ . Ic)d" .. I ... I dl' - ~(C) iI p()8!tive, wbkb leW 10 bJ«C)

:S 11I14j.·

Now Jet ", b E R be , uch lhal " < b, and df:rxKe by E !he r E fl ..xt. that

, <"' 4.."

Um tnf 0 __ "

.

, <"' 4.."

I.(z) < a < ~ < ~m"", • __ "

let

of \boeIe

I. (r ).

O5_,so_1

~'S. - I

Tho- tfICh w inl.tpablc aubloo:!. ColE,

-J1/ 14j.:5

"I'(C) S ~(C) S

/ l/ l4j.

(to,- what ~ ha",,}usI. ..........), and hen<:ol (& -II)I'(C) :S 21 1/ 14j.. Thill imp/ieo lhal ,,"{E ) 10 finite and E Is _ ntWty />"lntqrable. Let z E fl. n... fojlowllll I)I)(I<jiliooo .... cw.any eqw..u.ut.

(.)

~

<

~lIIaup._ ..... ( I / Il) !:os'SO-1

I.(..z) .

(b) • < ~m..,p._...,(1/") Eo-s' so /. (z). (c) There ....itu /I > ~ ouch lhat 11 (.. / (.. + I» !or tnfinilo'I1 many inti;.,.. Il. (d) Tbe--ol .... iIou /I' > b ouch ,hill /I' finitely "...,y iDd;"" Il .

:5 (1/(1> + I»

~.s o h (z )

:5 (1/(1l + I) !:OS. SOt.(z) for

in-

...d " (E) illDcluded In E , In rod , ,,'(E) c E for all J: 2: O. WrlU! A for the eel of tJx.e z E n ...m lhal there ex~ " 2: 0 wilh IV- 6)· 1.1.(:1)+ ' . . + ((1 - b), I.I,.(z) 2: Then An E is lhe let of tboele z Ie E tum lhat. / o{r ) + ··· +/p(z J2: {p+l)6 for at Jout _ " 2: o. n... A n E .. E In ...... of \1>o detlnilioo of E. Now , aJ1P11in& tbe maximal dliQdio: the"'_ to (! - 6) · I. , ~ ~ /!Jo(El :S fl· I II 81'. Similarly, 1 I · I. djl < ~( E ). So I'(E) " 0 and E Is k>caIly I'"necligible. Foo- uch (a , b) E Q )( Q Judo l bat " <~. put

o.

Fl.")

..

( '" E {}: lim lof;;

L OS . So- 1

t.(z) <" < b <

limrup~ L OS . ::;~-'

t.(:I:) ) .

F' _ f% EfI: !iminf~ L J.(%)< lim.Up ~ L ~'SR-'

l

~.~.-,

/. (Z)}'

.. the union 01 tbe 1",...). ill Io<:tJly I"-oepilible. \Vbton / ;. p. fon:, thtre eIisU an eMentlllJly j>-tntevable fw>cUoo from r:t Into R .uet> that ((I;'. ) ~. s.- ' roovergeo 10 ioc&Ity 1',a1moot everywbere. Writf,. I .. r - /-. ..., .... that tbe ptooediJ>« III ,NIl ~ ...hen I it DO(

Los><--.

Eos.<--.

/0 '."

r

r

poolIl¥e. Tho: lao! -.lion ill obv......

C

ThIo theorem exl"'" to a>mplex.""'....:1 !unet"""'.

Theorem 11 ....1 S1IJI1I<>I( i' ;. bo~n.ded. Let J : 11 - C 10: ClocnIioI/J w ~. end let n - C 10: .lOdIlhct ( I/ n) !.).~ I """""'iC' to Iooolir ,.·almcot ..."",,-<- Th~ J tI" .. f I .11'_

r

r:

r

Eos.,s __,

PI\OOI":

o u ,. is bounded , 01 5.' ¥e that (( l I n) Eos.~.- , ").~ I con, "'Ad 10

4v.} by P 'OpOIition ~U

r in

J""

Dfl'fInlt ion IUI. l I : n - c ;. aid 10 be ,,"In_iant if f .. IoeaIly /""&I ....., e~ .. here- " II said 10 be ~ic .. boone ..... ...m ;>-meuurable

and l'"invariaPt funetioo from 0 inlO C '"

OO<>Itant Ioc.o.lly

1".&....

Pl"OpOIOltlo" 11. ... 2 " ;. ,.· .. lIoJi<: if (anJ ani, iI) " O{A ) .. 0 Of' ,,·(AO) ., 0 lor..a I' -.........".u. ",/ A Bdllhat 1.. ., I .. G" /..".IIW J.<-IU.

P lIOOr: Let I: fl _ 10, +0:>[ boo ... m"o .. ""b06 and I"-invo.riant. Or:/l"" I . . . in Pmp<.:aitioa 3.3.3. n..n I. """Nllont louIIy ,,·a.,,- A_I. -I."'" _ thac. tbe __ .. true {)( I. and tbe poOpOSitkl<> k>l1owa. [J

•

:uo

I!. Jmng.. 01.~

Ert:rcUu Jqr Ch(Jpl(f' 11 I

Let. I' b'> I.or:I>
_=on

I. S...... hM "

. _ .... ( .. Iii T ). S..... l bat .hoI ' - 01 ~ ~Dder .... ~"" • _ • - , is" ita
1. 'l

'="" ... ino.wiant

Lot T .. h Iii C ; 1. 1-

,_..?";'_ T _

I}. 0 1_

&I>

""""'"

&
9. lot

inalional .... _

~

be t ...

~

T.

bo ... i""",.w.. uod put !. _ 1 0 . ' for 011 i"' ,&21o k ~ 0. B1 Blrtldr'. ~
lAoI

! :T _ R

on T """" ..... « I/~) E.~. s. _I /.l.O! I COt>,...~.. M OM

r

to

r am-t ~~.

b . Jr~ - JldM

I . S
r l' )- I

4.

3

1"(·) ~II.w ........ ~_ thM. lo . _ pic.

Wri",). for \..obaJ- "liP"'" "" R. Let I ~ .c~ l l) ... tw:r. F;,. Bon",*, _ , ADd III!. g Iii .c;;'(,\) be .... , Iudic ... ith period I. Filially, lot Ie. )"~I be &
.~'!'.. / Wim. ... _

>
p""""*,,,,,

/{z)g(ou

l bat

I

t- G. )"'" -

(f I IZ)U) . (1' g(Z)U) .

boIoop to SI(S.F ), ... ""'" S io t ... nMurol....urinl! 01

Lot "boM ln ~d.. l . Oi _ k e Z -(Il) . _to .. lor T ~._ . '.

I. S..... u>&!. uo( ..) '" " . 1. Let I Iii .ek(..) and g Iii .c;;'(" j. SI>o>w , ..... ;J.

I / (.j, I.") .w(.) """,...... to

(I ! ({v )· (/ ,,,,,) _ ~ Iii N tendo to -toe>. (Appi)· F*,·. Io.-muIo.. ) Le<, E .c:(,,) be . ..... tI>&!. , OB _ , ...... .,., ...... put c _ J, .w. Let

I

I be

,100 1pcI;"",,,. ~, g 01 II Iii T : gflj ~ c) "" T . By 1'",. 2. J(g ou" )4o> ........ .... to (J / .w) .(J g J~) .. , - -toe>. o..J_ t b&l 116-<)40> = ADd I"",,eb_ tb&t, .. <: . 1 _ ~~ ;0.1' Iii T ; 9(')
J

o.

4.

c.-Iude


lor all iDtqrnI n

1

~

all z EO R.

o . 8 ill F,.~_ tbat tbe fu _ _ ,_/(z +,') -0

GMu z E R _ _

0_

I _ I (z - I') - 8 hom 10. l/21 imo F ..... A/lO .•",.inIepabIe. Deduce

hom I'oja-" ......1Ilo .bat oimilNly

'" ( J. o

1.''

( / (Z + ' ) - 0) .... (:(~t) 011

lIz - I) _ 8 )oiII( 12n+l)1rt) . 011) ..- to 0 _ n ""'( ~')

~

1-

+<>D.

_ conclude lbat 5..lz) OW .......... 10 + (1)/' · 4. Gi_ z E R. ... _ '''''' f "- al z. "",,"1IaDd limll l Iz .) _ .

101\.._ lilllil/lz_j. Aloo, ..._

I"'"

(l(z

+ I) - / (z. »/I

_

(t(z - I) - / ("1: _»/(-1) COD""" to _ _ ~miU .. I :> 0..- to 0. StK.

.ho< 5..(z) ..... ,_:b ·

~ (l(z . ) + I(z _) _ " -

+<>0.

'VI

ate

12 Change of Variables

[~tM. ~ ... otudy thot

i..,..., -, ' T'" R' . By makinc _ _ mp' · of ~

In ... .",.... ""bot< V of R ' about." dillete_ "'>k; ... 1Ind ,.,. ' ..... e . plO:lUy 1m ....... 01 L<:~""

u"""," a fu""' .... T, V _ '1ooI:ii~", ofT, " ~ _ po _un and t ho
7

u.. d;"~tl.oJ ofT. """'->

n"

I _' I<> L ot " .. _.,.. in V. Nul., ... onoJroe tho ....... Ionohipo bot ......... ho 11.1

I

I

]n u.;"

a_J r

5

"loa . . .

E... ! . . . .

oI,,_u..po"'""t .. of\"dori>'ati..,""~iflJl'

10 tIaiuo " '"I.'horo,,, 10 iW>ooIuteb' coati=_ with _,...:t I<> ~ _~<e11,2 FIno, ..... ood.~ ' bon..,<Ji.). _ . ~ _ _ ......... Tho modal ... 01 ... _ _ ,", I h;,m of R' 10 tbo abooht", ftI ... 01.1 ..

11.3 In 11010.:1;.", .... _

"<" .....

-..u....nc (, ......... 12.2.1).

'Ioo ~ol~

""""w'" l i T 10. <&fIemoaWlk

,,,,,ptuom f . - V ""... W , .I>
Jm:. .... J -cobi.. oI T

( 1 Hml2.3.I). 12A nu. .:1-ion It de, .. ed to polar ..,.,..,Ji ........ in

12.1

R~ .

Differentiation in R k

OJ,,,,,, an i!lt.,. k ~ ! ~ • nWlpty opo:n subeet n QI R ' Let ). be le~ Hit·... "T"' .... on the nalU~ .....1rin« S orO. 7

IHHnltlon n . I . 1 Equip R' with t be E",,\ido6n """ric. A ............,. (&).l!' 01. Boord leta in n is Aid to ""rink to z E R ' nk'ely if ,~ 10 • number 0) > 0

aDd • .oq""""'" (~. ).;>:, 01 lirictit paoiti,-e numben cooverp", \0 O. which alilfy the foIlowi", conditions: (a) FOE ..I

~I

i ~ I. fI contaillll the c~ ball 8'(z. r ;) of l'Iodi ... ~, oen~

z.

(b) For ~l i ~ 1, £; • included in the cloeed tJ.l1 8' (z. T,I .nd ,1, (£;) a · .I.(8'(Z.T; ».

~

Dell nltlon n .1.2 Lot p be,. mmpJa ""''''''''"' on the naturallt:llliri", 01 fl. aDd let z e R l . lf A e C . ouch tlw li""_ ..... p{&)/ .I.(E,) .. A lot e>'ery lflq\Wenot (E,),;>:, that "'nnb \0 z nicely. _ call A the dm_ive 01 I' at. z. aDd we wrile (Dp )(z) .. A.

The princll*! rtotult 01 th • ..ction .. n...or..m 12.1.1. The foIlowi", Len>..... 12.1.1 aDd 12.1.2 win be r.eedtd lOr tho: prOof.

..no... •

Lemma 12.1.1 II C if • t:Ollt..otibn 01 J - lt . P ROOf: C'-,. compoct!OK K"" that K C IV aDd .I.(K) > I. Sinot K • ODmpKI. . It 10 00, ...,,;1 by 6ni~ly tnalI,)" cLemen" of C. MY U, . ... ,U., which we

....y order 10 lhat their radii .(U;) .... mfy r (U;) ?: I'{Uj .. .) for I :S ; :S p - I. Put B, .. U, . Diac&rd ~I UJ ouch that j > 1 and Ui tn\e.--:\l B, . Lot II, .. U.. be lho: ft"'t of the mnailli..n« Uj ( if lhere are ""'Y) . Diac&rd aU UJ auch that j > i . aDd Uj inu-me" ~ ; let B, .. U" be the ftm 01 the mnainlOC Ui' Rep et1 th. ~ .. often .. """,tllle. Thio ,h~ tho: di diao::ard«l U, 10 • ou'-t 01. bo.Il C. ~Iric with _ 8" _baR radi ... II th ..... ti..- that 01 8;. Thill K C U' ~ ':5;R C•. It rouo.... tlw t < .I.( K ):5 3l . E' :5;.:5;_ .1.(8 , ). C

Lernrna n . 1.2 S _ I' it " "".,1;"" .......... "" "" S, ,,"" /d A ... . 1'_ nz,t,il>k nhM:t 01 fl. TM!o /hen! .. " ...1 A' C A ....... U\at.l.(A - A') .. 0 ",,", (DI' ){z) " 0 for ft
0I..u z e A for_hich

.

p(B' {Z,T)) .. 0. - 0 .I. (8' (z . r)) Itm

or, tqui~lly,

. p (B{z .• )) __ o .l.(B(z •• » .. O. Itm

yr

ate

For """y ; E N , Ie! P, be tbe lie! 01 &U Z E A for which

,..{H{z ,' » 1 I"~!p )'(H(r ,r» :> j .

We show that .I.(P;) _ 0 for all ; E N . SillOl! Ii - A' ~ Up, P, . il ... ilI I'ollol> that ).(A - A') _ O. Fix ;, and Ie! f :> 0 be gi ....... Si""" ,,(A) _ 0, the", is All open . lIhot!. V 01 fI COOtalni,,& A . with ,,(V) S c. E~' z E P, is.he """t~r of All open bo.U 8. C V oudt thai; ).(H.) ,,(H.). Let IV be the union of t ' - B• . For tJI t < .I: (W ), Lelll_ 11.1.1 Jhooq that ot",,.., is .. diojoilll euhcollcction lB." ... , H., ) of l B. ,r E P; ) .t><:b IMt

s ;.

1 <3'·

L

).(B.,) S 3·; ·

L

,,(B.,l-=3· j·,,{B.. U .. ·U B,. l

S 3'j · ,,(V) S 3' #. Th", .l.{W ) S 3'j~. Sinot P, c W and € it arbitrary. ~ ..... IM t )'{P; l ,. O. NOlO" Ii:. rEA' U>d let {E,)./t . be a ~,,_ IMt lIhtiru.. 10" nk:o:lJr. Let (I ::> 0 and (" ).-~' be ... In ~ftn!tlon 12.1.1. ~n

14£.) I " ~{~B'd('~'~"~)) .1.(£.) S ;;:. ).(8"(or ,T,))

Tbe",rore. ,,(£.)/ ).(£.) D,,(s).O.

CQIIW' gallO

0 ... ; tends 10 +00, wbicb J>I'OV"" lhat

0

Theorem 12.1. 1 Let I' k a ""np/a: "'.......... "" S. D,,(or) II dtji>ltld Ir>r ). • ..mu..c "'I .I tilt- ft<m/i<m z _ D,,(or ) (.uji>ltld ).·"""...1 pwk:r.:) "Ioaa.llp ).·inl 9' M . JliQ'W '(" tilt- LdeIg'Me mp»Uion 01" ....lGli"" '" ). " " _ (D,,).I. + ,"" .mil DI'. (r) _ 0 )..",....,., ~m!.

"'fI ......,

"w'

PItOO" It ru!lid->i 10 j)rO'o"OI the lheoo"em !IePI'",1dy for" 1. ). and lor I' < ).. AIoo, we t>«)(\ only obuln the ruu1l for ~ ," . I f " 1. )., then ,.... .1 )., and thern" a ""' A C fI "';th ,,'( A) _ 0 and .l.(A' )" 0 (P'Oj:O($tion 10.3A). 8 )' Lemma 11. 12. (01'+)(" ) ., 0 .I.-&!mI)6t ...--ywheft. Tbe .. me arglitnelll . . . . ."""" that (D,.. -)(r ) _ 0 ).·a ." . lit .....

(DI')(r ) _ 0

).. a ....

N"", ........... me IMII' < ).

and ,.. ..

,.".1.

;" a Ioo:aIly .l.-int~ funchon I from fl lb\O R ,uch that I' ,. The: theon:m will k>l1ow 00<:e _ show that DI.(r ) ~ I (r ) (.I.-..:mo.,

~

1.1..

~"""') .

A...,.; ·'.. with cedi rational numbo:r r the oeu A. _ {r E f1 : I (r ) <: ri o B. _ Ir E ll : I (r ) ~ rl , and define , , _ 9, by 9, _ 1",(1 - r )).. Sinot 9, (A.) _ 0, Lemma 12.1 .2 sbowlo thaL.hen: ..... e68 A.~ C A. JUCh thaL .\.(04, - A~) _ 0 800 (Di. )(:z) _ 0 if r E A~ .

11.1 DilreRotiatioa ill R '

2~

Put Y .. U (A,. - .4;.). T'bm .\{Y} = O. Pick 'lOme % E n ouWOe Y. lei. (Sd, shrink \(I % niedy. and cboooe r ;> I (z; ). Then % E A•. Sinoo I"' - r ~ .. (f - r)~ < , ....~ ba...e

/'tE,) < 9.(E,) + r ~ (£;)

-

~(E;)

Sinee % t Y , ..... ha~ % E A~ . Hence li"uup; _ _ v.(£;)/~(E;ll:s r. Thill II truo: lor.,...,.,. ruloo&l r ;> / (%1. n..",fure.

~.'-+00 m,uP~i:• :s I {%)· We h.a~ ....... ~ the follo.>wilJ&: "hn':MI~ """'>'

%E

n lAtis6ell

~~!;!,P ~~i~ :s 1(%) for I!YO!f)' _ (E;)."i!: ' that shrinks to % oioely. If...., ~ I' by - I' . and I by - I , It f<>l~ tbat &lmoM ..tidlo:& ~

e¥O!r)" %

E

n

1(%)

"" - r ~ (E;)Q. that iobrinks to:z nio!:ly.

o

The proof of the \.I'...-.[U .. tblll complete. AJ>I)I.het lmpOrt.anl. ..,...It ill the IoIlowlng theorem.

Tbeo...,m 12. 1.2 1..:1 I Ie q koa;Il~ ~.;"kgm6U: mapping from &.nacII 'J'II
,E!:.. A(~I . 16, 11(>:) -

1 (%oll dA .. 0

n ;"1<> •

...-1

,.,

PRoOP: no..., art! a '\'~isihle oubec<. N of 11 and • """"' ..... ou'-t D ol Fax:h thf,t I(n _ N) III Included In tIM: do"un: of D. f or r E D. put 1'. . . If - r l ·.I.· TI...,.e", 12.1.1 shoo<s that ( D",)(>:o) .. 11 (%0) - r l for .1,.,.,.. all 7'0. I.cI. Y. be tho: u .....:.00.1 ""' . and put Y .. N U (U. Y. ). Then ~(Y) '"' O. If 7'0 rf. Y , If (E;).~ . obrinu to ..., DIoeiy, and If ( ;> 0, t~ cxist.s T ED . uo:b that 1 f(~1 - r l < e. s.noe, 1/ (%) - /(%0)1 < 1/(%) - rI + f for aU :z E n,

...

A(~.I . E

.I I(%I- 1(%o) ldA(%) <

("

Si,...,. r o ~ Y. D".(~) '" I/ (ro ) - r l, and I~[o'" lbe !l.ft-Iw>d side of (2) 18 _ IbM 2€ r.". allsuffidentlr ~ i. Thla ~ ell r.". every r o 'I Y. &nd <:OmpleLf:$ the proof.

a

We !>OW r
'""I"""""" that shrink

n~lr·

Lem m ll 12. 1.3

ut E .. nIS' S. Ja" tIoI be ~ .......emp(~ ...:t<>ngie .,. a ". 1'IKn

E;.o dUjo-int ..niono/rtd4ngie. p; E

z· lor ~1Jn')

].B.- P,2: 1,8;- ;!j

IT

(n E z ·,

ISIS' lSi S k) .

PIIO<W : For e""b .. E Z·, let Q. be ,hoe

(I ....

<»nSistlng of all the _tAn&I ..

(Po E Z..

for .. III S ; S t l·

p, " IAE Q,: A C E, A n B ~ '

for """,y

B EIM

P. _ {A E Q.:AC E , A n B _ '

Iorevery

BEl..io5.lS"_' P,)

l\ _{ A E Qo , A C E )

Clearly, tbe rectang"" of p .. lI..,o p. "'" diojoint. Let " be .. point of «I ,AI wlli<:b does DOt beloo,g to U"u' A For ~ Inteser n ~ 0, if A. Ie tbe unique e~t of Q. which contlll ... " , then A. Ie not included in E. Otherwise, A. WO\IId beJoog to p. , or would be ;""tuded In one ,,!I._tit of l..Jos;S"_' P" wbicb is iulpoosIbie. Now, for each .. E Z·, ~ r l-I in A. n (R· - E). Since 11';") - "i l 2- · [0, all 1 S ; S J:, the _uencc (r l. I,.>, COIl' e' g<>i to z . But tbe Bet n ,s'<. ) - «1,,'1.1- E Ie (~ In n '5'5. 1~ "" ,A), the.efo> e % ~ E , and

n,s.s.J-

s

E =O,,!pA, Mdeslred .

a

We St.y that .. nonempty ~k 0 .<<< . 10, •.8.1 is,,!lquan': ... he~ .8. - a, 00... oot dcpmd on ;. - Let I' be " ...61 _ ' " on S. Fo<every % E and every n E N , !l.t c.. (r ) be t hed_oft",*"S«ta wlli<:b ...., "'I ........, with edge leng\hiI "'rictly l.- thrt.o l In, and ... hMfl ""nuln %. Put .0..1")'" ~UPIi«.($) (I'( El!~(£l). 1fT < ~.(,,). the", existe E E C.lr) such thal. 1'(£lJM E ) > r, and ..... CIUl fi t>d E' E c,,(z), E' j E, ouch that ,,(E')JA{E') > T whidl io " nel,ghhorbood of r. The",fo-re, A~ ( r) is al90 the BUpnmum of tbo! I'(El/ A( EI , " E ran,ges ""'lI' lhoee eJnn.ulB of C. (z) whidt

n

12.1D;!Ie..... ia>.ioa Uo. R'

2-&7

are ne\sbborhoodl of". HtllOt. lo, all liE N , tlot functioo >: .... ~ (s) II lower ~mlo;:ontill\lOllf on fl. Finally, >: .... inf~>l .0. .. (,,) .... Bard function from Ilimoltand _ _ lb __ n( _,.)· 0 ...... tlwlb{s) _ !i,.(s ) _ D,.(s ) &I. every " E 11 for which O,.(s) uIst, From now m , _ _ume that ,. .. positive.

n,. ;

PropoAltion 12.1.1 UI A /100 ~ ~I "fn ~""Q > 0 ~ rmJ ............ $""",Ie t1uJI DJ« s ) 2: Q for ail s E A. 171m Q)'- (Al ::; ,.-( A). P IIOOf"; Theft II no ""'rietioa In ...... ming that ,.-(A ) iI 6nit
i.....

A.

n

CI_ .. e N . Given 6 > diljoint

0-,

,,'

.,

have ,.· (A. ) 2: (Q -~). ),· (An ). no,.,,,,~, ,.· (A) 2: Si""" € iI arbitrary, .., moclndt tha~ ,." ( A) 2: Q • .I. " (A). [J

5;...,. 6 iI ... bitrary, _

(0 - () . ~" ( Al.

Pl""'>PWilion 12.1.2 Ld A /100 .. ,..u.1 ofn 4"" <> > 0 4 rmJ n~. $"PfIOIe t1uJI !ip(s ) ::; <> /1><" ,./1 ;r EA. Th~ ,."(A) ::; 0 . .I. "(A). P IIOO" Wt ""'Y ""PI"'"'" that )" (Al iI finit 0 be 1P>"tfI. For every n E N . let il" be tbe _ oftt....;r e A NCb that p(E:)1),(E:) ::; <> + ~ for all E E c.. (,,). Thuo tbe _ A. eo to A. a _ n E N . Gi..,.. ~ > O. tben: c:riN .. colm hhle f.amiIJ (E; MI of diljoiDt S. _1IICb tlw A~ C U.U E; aDd E.." ),(E. ):5 )'" (A. )+6. AfKUlna: Oil III P •...,.....i Liod 12.1.1, .., oonclndt that ,." (A ):5 <> . ),· (A). C

ioc."

Theorem 12.1.' If,. 2: 0 U ,;ngoola~ (Lt., if,. .J.. .I.),

u.- !i,.(,,) ...

+00

,. "~I ("'i"~'

P IIOOr: Let N be .. >.-"",Iisible_ .. hich C&IT"ifs,.. For-=h i~n 2: I, put Eo _ (" E N , n~,,) :5 n ~ . """'" ,."( Eo) ::; n· )'"(E.) Je.ds to ,."( Eo ) _ o. ~, Is EN : !i~,,) ... +00) bOil ,..~bIo compl ' '''''nt . [J

"

.

Propoo;ltJ.o" 12.1.3 /j1J,.(Z) p~

+ 1'.

Let ,. .. fA

"""

< +<0

be the

(O......

~

~~ ~n,.

c: A.

McompOsitioo of ,. rel.lti,'e to A.

11,.. ::;: n,. +1)(- IA) ... D,. -12(/A )::;: D,. < TOO, a

and,. . ... 0 b)' TloeGiW, 12.Ll. Wbw t _ I , we lui... a "ron&"" ....,,It than Pmpoo;lt lon 12.1.4

n .... ( ' n 11.1.3.

WMu k ... I _""''' if po>41'tkre oiAgoJar. QI<{:z) ..

+00

P !tOOI': ,. It co.."enu..1«i on _ A.roS!I~ble Borel "" N. For ...,h lilt. . n ~ I . define E" M tho> "" ol.1! zEN ... " 'hicb l2Io(:z) < n . Gi~ & >- 0, Itt V boo .... open ...hoe\. 01 rl a:>ntaI.I>i,,« N outb tlw A(V ) s~. Lee K boo • (:()nIpid sublet 01 E~ . EedI :z E K liC11 in .... S-«t Eo c V S\IclJ thai 1>. ill. ~bood ol :z and "(ErllA(E. ) S .... Bei.n& <:Om!*' . K iI w>'e.od b7 finitd)' maDY ol u- S.tcU E. .. If IIOinC pOi'" at rl !ite in th_ S-«u, ""'" 011'-<>1 .... In lhe union of the 01"'" 1_ and can boo ,t" ."td wit hout ~nc the union. In Ihifo .....y. _ ,,,, •.,,,, lhe ... periI\lOtlOl S.....u e., and ....,. _u_ that no poIm. IJoo in more than 1_ d. the S - E••.

n.....

J<{K )

s ,.(U E •• ) ,

S

L,,(E •.l S n ~, .1.( £ • .1 S 2n.1.(U, e.,) , a

12.2 The Modulus of an Automorphism Let A. be ~ meMU!1l on It'. If" is a p.. ...1Im! c .. ,.()O. Ij· ). h_~ d.

for

evtcy t

E Z" ,

b- all (p. , .... "") E (O,I , .. . , 2' - I)' , and "" for all (p" .... ". ) E Z". Therefore, ,.'(U) .. d ' (U) lOr e=y opctI stLbet< U of R ' ( Pf'lIpOIIilion U. I), which """'rtIf lhat " ... c.I... If" iI. in-r . uloOi' .... pI>illm ol It' . the poUr (,,- '.5) II A..... iu,d and _-I (A.) ill Invarlant. W>dtt InMl ........ lienee lhere ~ .. """iii"" number callfld the .-lulU!! of II. sudo that ,,-' (>..) ... mod(a)A..

-«_),

••

f lo(~ 4 (tI- '(A.» _ f

lo(E) () v- '4A. =

f

111 "("°," )- ' 4...

for all E € S, ",hftooe ""'..., tllA' mod(!>" u) .. modltl) · mod(.. ). fOr all i. ; E {1. 2.. . .. tl . 1e! E' J dmote the t " t nwrir: tbat ball tho: ekment in tbe (i .j ) placeequal 10 I and all Mher eimoenUi equal to O. If ; ~ j and " E R, put B..; (ot) ,. I. + (l E•.j, ",here f. Is tbe unit mIItrix at (N'der Ic. For any murix X of order k. 8;J «I )X ill obWned by addl.r>g (I timoo tbe j t.h row of X to tbe ;.h row of X . FUrthermore. B;.;(ot )- ' .. B;J( - (I ). PI"1)J>mI,1oII 12.::U Ewry i,,~ibk t • t .....mz " A prod ... , of "",/ri€a of 1M form B, J« I) And G "",/..u of 1M form I, + (a - I )E ... .

P IIOOP: Consider in....n ibk maIne... of tbe form

, , 0

0

x_

.

0 0

0

0

,

0

0

0

.

( , ,, _h ~U_ h

,,,,,,

..

,..

.

{.-.-,

('-' -,.'-" ... ( ' ''- 0

.. ~ 0 S " S k - 1; i f " .. k - I . tlw:n X "' an ..-bit ...". in....nible "",.fix. T be proof Is by induction on " . If " .. 0 ... m .... have { ... " O. tI~ ..... if ... multiply 0<\ tbe IefI. ",," , ",vely by tbe ",une... B; .. (-{,.. ' (.; ) for l S i S k - 1, ~ shaI1 obt&in tho: matrix I. + {( ... - I) . E ... , and tho: pt os-it io)o io truol In tbe ca8O: " .. O. N_ IUJlP080 !.hat t be.......:tt hai beeD pr<JI'I!d lor ",. O. I ... . . n - 1. andl)l)nO!der t he aee " .. " (1 < " < k - I).

5...

:

,.-...).

::

. ..

{

...

"j

tbe,.., ~ exisUi & noo:w:c-o element {... _ ~ , fur oome i ouch lbat " - " S t . Pmnultiplyi", X by 8, .•« 1 - (; .. _.){;~ __) lOt _ !nde.o:j ouch that I< - n S j S 1: • ..., rna)' ....ume l hat (;"_ft .. I. Mulllplyin& __ 'voeIy by the mu~ 8,.JI- {,-.• - . ) lOt r" j . "'" end up wltll. matrix fur ..IIid! (." _ft .. 0 if r f- i . and (;.. __ .. I. Filially, In t he eMf: j f- " - n, ",ul, ip/yillll' . hlo """rix by Bj •• _ ft ( - I) . B' __ j ( l ). _ <>bteJD • matrix of tbe .....,., form but with j .. Ie - n. Thio o:>mpjeteo tbe induc:tioo and ... Ol)mpletes tbe proof of tbe poopOllilion. 0

is

""P.

Tbewem 12.2. 1 mod (..) .. I del(" )1 for ........,- """""""Pi>;"m " of R · .

PIIOOP; C\Mrt:y. if thut .... jot... E R' such U>al IItr" . , , . :r. ) .. (r ', ... . :r'_I, or. ) lot all (:r , .... , :r.) £ R ", the~ !nI)<j(u) .. )a). Now J\lp~ thaI u takee the form (:r, .. ... :r.) _ (:r, .. . . , :r. +a:r, •. .. • z; .. ... :r. ). ",here i, j ~ tM) dllfen:nl ln~ in (I . __. •.t I. ,. it a ..,..) number, and :r, + Q:r; III the Wlrm ol rank i in (:r" .. . , :r, + 1U, .. .. , ro). W~ wish 10 ptOot that !nI)<j[_) .. I. But , if Q ~ 0, tben u ... .,- ' """ '"', ",here '"' is the automorphlom (r' ..... :I:. ) (ilh . .. • il" ... , orj • .. . ,:r. ) of R" and" is the automor· pItiIm (il ,." _. :r, ) _ (:r" _. . , :r, + :rJ .. ..• z, .... .:r. ). So .... "".,. ""!>POIN' lhu 0 .. L Write (~ , •. . . ,hI lot the o;:aD(I<1ital baeis of R " . L6tins

A .. ({.., ... . , ... ) 00 )0. 1)' : y;:'

,; }

_ha~"""

A. ..

(11(., ......... ), (:r, . .. " r. ) E)O, I)·, z, + zJ::: !}

8 ... t . ..

t..t",•." , :r.): (:r io'"

.:r. ) E )O .1)' .:r. + :r, :> 1) .

__ henc:t Iff> ded~ thac. A. (IIt )O , I)' » .. A. I )O , I)' ) and !nI)<j(II) .. I. Finally, ",bee> II. arbitrary, "beoo~m IZ.2. 1 rono,.." from Pf'OptlIiIlon 12.U.

0

l"<w. if II if a Uneat ~Iim ol R' ",hkh" nc>t bijt:rti,'" &nd if P iI tbe rank olll, tbore .....,. &II (l ..(R' ), 10 that . (R· ) is Arrqllpble.

12.3 Change of Variables Gi~

.t E N . let A be LobeIcue masu", on R·. For

I! WIE' U of R ' , denou by

>...,

~ m ' !PI'"

"'ftY IIOrIempey

opal

on lhI: tatuntJ...rurin(

s.,

oW. P~ll lon 12.3. 1 Gi........ .".... • ..but V 0/ R ' . kl T be .. 0II'IIli..""", .".... ""''';''' from V inlo R ' , alUl • .."...~ IMI T u djff."WN0"k a l _

I

J.

"","U E V . Po4 6.(r ) ..
.lITE) _ AI ) , :r - ' 1 .l(E )

ll.J

Chance of Variableo

2S1

PJtOOP: We IDIoY IU~ IIw r .. 0 and T (z ) .. O. Ftm, 1m""," dial OT(O) .. id R, . and let 0 < ~ < I /~ be a 1ft.! number luch IIw I -r < (1 _ U) t < (I +26)t OS I + r . We can liDt all ~ e V ..tiosfyin& III :S l I n (.. he~ 1,.1.. 1(" ,·· ·,,. )1.. "'Pl ,.;<~o (l)l; Il)· La E e c,, (0) with edr;e len(tb I, and let E,. &. be IWO IQ~ ~tric with E ..bo!oe edr ba~ ~bo (J - U}I and ( I + U}I, _pectlwly. Then

OJ If ,. beIonp to E, ( I) .howI that T( ~) t;." in &.. Hence T (E) c &.. Ir. belonp to tbe bouDdary of E, (I) olIQW1; tMt T (, ) <:toe. DOl. lie ill tbe

.

.

interior E, of E ,. H..,.,. E, n T(E) .." two diqlim.

<>pftL lIeU, ..bo!oe

•

. . , . E,

.

n T(E ) and E, - T(E) .. E, - T~

union •

•

• Since 6 < E,.

I /~ ,

(\) 1*0_ that

•

T mapo the.,...,teI" of E into E, . B UI E, • ..,.,noc\.ed, IO E,C T(E) . Now E, c T(l' ). Since, for .11 V e 1: - E, T(V) dooM DOl. lie III E" ..... ODDclude tlw E, c T{E). ln 1Ihort, E, c T(E) c &.. who""" it foIlowI tMt 1- f S (I - U )· S

A~~»

5 (I

+ 26)· 'i I + f,

.. dosIrtd. NI!XI . _ume lbat II .. OT(O) • • Ii".,... IUlomorphiarn of R " . Then, D{.~' 0 T)(O) .. Id R•. B1 .. Iw .., haw: ~

an imep:r n

illSt

1!bow1l. to "'di f

> 0 IIIne

> I oucb that

, I"--'(T E» - I OS ,'(TE) A(E) - 0>0<1(11) .. 0>0<1 (11)' A(E)

f

for all E e C_(O). fi nalli• ........,., tbat II .. OT(O) io singuI.or (IO llw de1;(II) .. 0). Then II(R O) ill ),.nqlicible. Fix f > O. The ... exlo", , > 0 Iud! that A(E, ) S r, ..line E. ill tho let. of all pOIn", wboee diotana: to ..([_1 .1]0) • Ieee than 6. La n e N beauch that IT (,.) - II(~)I OS ~ . III for all , ..t;,,(yiq III OS l I n. If E ill an elca.ent of c. (O) with ed&e length /, then, for all , e E, ..... ha~

IT(.) - 1I(.1I:5 /6. Th..

d(T~V>. II~·» ):56

and

d(T~V), U ([_I. I ]·)) OS ~

1'ben!m. T(E)/ I ill included in E• . and A(T(E») .. A(T(E» < /. A(E) _ r.

o

m

12.

C~

of V&ri&bIm

.... mark that

P roposition !l.3.1 is true ewn if the hypOlbe6 .. tlw. T Ie open is deteted; li>e proof Ie moce difficult .. might be expected. The foIIa..;ng ch.n~e of variablffl formula .. . orntral _ult of this book. w~

,\IbM"

Theorem 12.3 .1 Ld V , IV lo< II... """n 0/ R ' "lid T "hom..,....,..... )Mum 01 V .,.,.to w. A ..""", lA4t T io tli/J.,..,nI.~ III ...a. po>inl. z E V , G..... p1II J(~) _ det[DT( ~)I. n.. .. IJ I ' % ,... IJ (% )[ io 1«oI1~ ~ v ·;"ltgn:Ib-Ic ""d .l.w _ T (lJ I ' .I.., ). PJWQP: Put I' ., r-' (.l. w ). By Proposition 12. 3. 1, n,, (z ) .. Allile for all % E V. Hence I' < .l. v (Proposit ion 12.1.3). Now IJ (%) I ;:0 Dp(r ) for .I.". allDOllt all %. by Proposition 11.3.1 and Theon:m 12.1. 1. T his ~ that IJ I "' Ioeally .l. v·i~ and " _ IJ I· .I.." .. I>o:nce ..., deduce IIw. T(IJ I.I..,) .. .l.w (P ropooition !I .U ). 0 Obee..~ that

12.4

f / J),w _ fu oTjIJld..\..,

£or.-..ch /: I\' _ 10.+<01.

Polar Coordinates

Fi>< an i~ n ~ I. Writf: S~ 10< It... unit.phere It E R H ' , Yz i - I) in R~'" (...to.re 1·1 Ittt... Euclid..." Il0l"'''), and 8 for.he Borcl " . algt:bnt. of

S·

A=

AI ...

(b: ; I £ JO. 1),,, E &,..,1 <;et in R~"". Denote by .....,., ~ ""","un: 00 R~~I . fundion A ... ( n + 1)....... (.04.) '" " · oddi!;",, on 8. and. bo:no< a P<"'it;"" ""","un: dS~ 'Ill B.

For """'l' A E 8 ,

n..

Dell";tlo,, 12••• 1 dS" (0' tbe R.3d<m" ~u,.., on S~ arising f"""
(I(dS-» (A)

-•

for all A £ 8. Then:br-e, ! (dS- ) _

(dS" )(r '(A» (n + I).... ~,

(I-IfA»)

(n + 1),\".,. , ( .. - '( A») (n + 1)>.,. .. ,( ,,) _
&ru! dS

ft

;,;

invariant undtt ortho@:onal

tran!lfQrnatLoo$,

Let g be tbe ~ism (1, %) >--+ Ix from jO, +r LI:~ D>ea>;un: on ]0, +00[. Now, f<>r aU reel

n"rnMrI r , • alisfyi", 0 < r < • and for all A E 8 .

(t· dt 0 01S- lOr ' ' l xA) ..

•"+, _ r··' "+

I

oISft (A)

=

(.ft. ' - rft.'Pft.L/A)

..

...... , (p{]r , . ] "A »).

So p{tft dt 0 oIS NUl . lei " be l be .... ppi'" from

ft )

...

~" 'fR' " -IG I'

a" into Sft wl!id1 ...ndI I ... (I" ... . 'ft ) '"

("" • .. .... 8, " " , ,"" . ... ... 8, sin', . ... , ... 1_ sin 9. _, ,sin 9. ), where lbe U + 1)11 coordl~ " ",,_ x .. · )( " sin'; for """"y 1 :s, j :s, n.

"" J.'

P~hlon 13.• . 1 Sd P "']- " ."I

" ]- " /2,"I'lj"-'

and....u,./<>rllot. ...........e .... - ' (.ft) " ...- - ' (1. _,) " ... "ODI{8, ) drI, ••. drI• • ~ Ji, .. ~ m "I and dIJ. if Le,",- mMFMro "" )- ,,/2. "I'll lor .,/ 2 :s: i < n. 17Im~) .. oIS- . M

'

_

""

]-

"

.

Denote by (e' •...• c,,) , ho canonical bMio '" a · . LeI J(I I be tbe matrix '" ( ~I~, .... . ~I~_. I!>('J) ';\b respeel '" lbe o:anoolcal bull of a·"'. Th.. !be mappi", (1. 9) _ I ·I!>(I ) from a " a· into a··' bas J..:obi&n !leiftmilWlt ( - W . t· , det (J (' l) M (t . ' I. If .. ~ 2. wrile K (I l lor !he: n" n matrix wJ.... firM row by multiplyinc!be lim. row '" J (... .... , ft) by "''''" and wJ.... itl! row . for all '1 :S j :s, n. ill tqU&I '" l be itl! row of J (B" . . . •' .). Similarly. wrlle l.(I ) ror !be .. )( n nwrix wJ.... first row ;" obilli""" by multiply;", !be firM row of J (I, •...• 9. ) by ... 9" and ... hooie ilb row. for all 2 :s: j :S: n. iI equal '" the itIL row of J (I" . . .. I.). Now up&Ddi", del. (J(9)) .Jonc tbe firM column 01 J(I ). " lee tbat PII:OOf';

"obW"""

det (J (' l) ... - -
By IlIdaction 011 n. del.

(J('ll ... (- I )· . ...0-' (10 ) .
Nut . PILI {} ... a·'"

-

{Ol- Then

S· _I!r( P )_ l(z, .... ,z •• ,) E S·:Z, :s: O and z. _ O} . So a · ·' - II ·I!r(I ) : I E (0. +<»[ and 9 E P) " ~ .. ,.r",ll«ible. By the <:h.t."... ol...n..ble formula, ~. ,,,, ill tbe im.or;oe _ure 01 t"dt 0,. under

12.

~

01 Voriabko

the mappi", (f, 8) _ I · -';(8). 00 t he other ha.od , we 1rn0>\. tbat t" dI @dS" Is the lrllqfJ "'o:,.u"' of .I.."'ffl under lhe ",""Wing: - (lzl ,4U' II). This imp~efI that f~ dt 0 c!S " Is the !1D!Ige measure of I"
(I, B) .... (I, 16(B)). The pro(l(lIIitiDn followo.

0

Theo .... m 12.4.1 .1.... , ;, 1M mwge meafart 0/ j"dl 0 ,.. und.:r (t. B) _ I·,,(B) from )0. +oo[ x P irI:.o R"'" . PROOf"". This £01"""" from P mpooit lon 12.4.1, becIt....,. ,~ of

I"th 0 dS" under (I .z ) _ tz from

.1.... , is the i"""", mel>-

10 ,+001x S"

inl<> R "·'.

0

If / .... A,. .... ;o~ £unetion, ....., <:&ll, IIlX(IrdiTli I<> l'lteon:m 11.4.\ , oompuU f / d.l.. .. , in polar ooordilWell. All I<> the signif;conoe nl we let (z " ... , Zft",) be .. point of S" sudl that tbe oorxIition z , ::; 0 and z . .. 0 i. _ !l&Iis6ed. Fm....:h in'<,#"

e" ... ,e",

2 :f j ::; n , the point (1 - (":. , + ... + z! .,l)-"" and there is .. un;quo Bj E J - .. /2 , .. / 2[ sat;"fying

sinB, - (I -

(",.. .. . ", +I) lia in Si,

(4•• + ... + z! .. ,))- '{lI. Zj.. "

Finally, there .... uniquo: I , E [ - .. , ..[ such that

(<'<119" rid, ) .. ( I _ (~ + ... + Z! .. ,)) - '{1 . (l: " z.), and we haW,. '" ~I).

Who:o " .. 2, " III tho>: "U,titU! (D.!>(B)c, .. .. . 1)¢(B)"". ,,(B» ) is 8 bes;s of R"'" , the WJcton W (9)c" ... . W(B)c ft form 8 free system.

AlM. let (e" .. . . 0• • , ) bo! lhe CIIllOlli.coJ '-is of R · +I . Writo. ,'.1(1 ) for tM JacobiM mum. nl " at I , aDd M. (B) lor tM.ubmlloU;X of M (I ) obtained by deleting tho>: ltb,.".. of M(I) (1::; k ::; 71 + I). It 11 easily't>own that del. (M. (8)) .. (- 1)" - ' (¢(Illh )«(>$"- ' 9.)(0::06"- ' '. - ,) . , , (coo 9, )•

.. bere (*18)1• • ) is the .... 1• • product of 16(B) and • • '

0

borneomorphism from P (IJJIO ob( P l, and tbe inwrse ~'" II, from ,,(PI in\.<> R " , iI 8 cha:t of IDe manifold S· . tbe canon>cal be.Fix 6 E P ....:l put,. .. ,/>(B), Deoo~ by M of R O. Si""" de(.(J(I ») hM ei&n ( _ I)·, "'" .,., that the (" + 1).tu!lle ('" Dt(B)e' •... , D¢(9)c. j is direc\ ill. R H"'. Lc! j be lbe c&IlOnico.l im"",,· lion of SO lll.to R " '" and. for &II 1 ::; k ::; ", lei be the uniQ.UO: _ 1m in By P ropooition 12.4.2, '" indUCftl

&

«(,.... ,(.. )

v.

12.4 Polu Coordl_

tbe ~t l pece T. (S· ) l uc:b that d.i{~. ) • D"Ol-(8}t!• . n..-n tt, 11···11 u" ;. in tbe orit:nI.allon oCT. (S· ) (if S~ ;. ~tM "toward tbe outside"). For eacb n e N. doeoot.e by V" tbe volume ~ ( B'(O, Il) of the unit t.J1 In R " , .. hoeD R O ;. Ii...... ita E uclidean oorm. and by the ou.rt.ce ..... I dS"- 1 of S· -I. Siraoe A" " lhe imqe meNU", oC 10 - ' dt 0 dS o- 1 under

n..

f

(1, %) _ tz.

n.. _..Vo . Now

V.

-

[ ', u"

_

V._ I

I··· I I B·(O.I)L"'~}d.I:" " .u-._,

j ' (\ _:r! ){. - L)" dz.

-.

/;(2

/:(2

lor eva)"" 2: 2. Hence V. _ 2V. _ 1 COI"8d11, and ctUo"8dIJ ;. ~ to compute. Fbr all Il > 0, put r (ll) _ / ~- • . r"-' u "o ._ I' "The function r : Il _ r {_) from 10. +o:o( into jO . +<>o( ill called the EuIotr pmma fUDClion. Clearly, r (_ + \) _ .. . r (.. ). Siooo r ill _ I, ~"'"' that r(n) _ (n _ I)! for all .. E N. On t he «her band, r(.. ) - / lap( - :r'j . :r;1o- ' u /lO.+ 0. N_ Jet .. > 0 and ~ > 0 be t"", rNJ numben. Md lot / be the fuoctlon (:r;. ~) <XI

_ 2 ap( - :r;'J . ",10 _ ' . l a p( - v') .,a-I

10 , +o:o[ " JO . +0:0[. n..-n / / f u /lO ._ 1d~/lO._1 - r(a)r (6).

Buo., Ii"""

T : (r . 8) _ (rcoo8 . r lin8) ;. a diffeomorphism from jO . + cc[ x jO , "'/l [ ooto jO. +oc[ >< jO . +oc[,

II

f dz/lO.+<»I "r/lO ...... j

-

r (a + ~). 2 / (coo1o- ' 8)(lin"'- '8) d8/lO .• n:I

by Lbo chanp <>l "";"b!M formula. Th ...

'J(~ -' 8)(sin-' 8) di

110 .• /21

_ r (a)r (6)

r (Il+6)'

'I"lkinI "" b - 1/ 2 in the last hle<juality, .... obtain r(1 / 2) - -Ii.

{ (2 rooO di _ ~ (r(i) r (n;\) / for ~ry n e Z .. , ..heooe

_'e _

Next ,

r(n;2))

that

,.

.

11.

Chant< of VorIabI.

lQr~n E N. Th ...

...

lOr~yn E N.

,"

"' --"'

E:ern:ites lOT Chapter It I

Loot C be ......... &IIk ....... . ap of t ho odditi.., SJOUP R " . /I, .... _ p (_pee. 'iveIY. C) of R" .. o.id ... Iou 0.1'....<1'" (""'t>f!di>tV.• C~ if. Ior_ • #- 0 .. C . oM be... (. + p) n p _, (..... pOd i ~Jy. if R " _ U .-(!>(. + C)). A .. _ P .bid> .. botb • G·podd", &ad • C_inf: .. ealIed • (p •• : ' . . \oil. IA:t ~ be I.c,-,," . 'ra OR R '.

I.

c·..,'";''' &ad P it. ~ ._~ C-poacti..,.

II Cit. ~.~ ""- ,Iw ~(C):!: ~( P).

1. Ld d (C)

be tIM; iert ~(C). "t.e<-o C run< 'ru.o...ch all ~.irUp_ of R ' If A io. ~.~ ott ..... that ~ (A):> d (C). ""- tbM tI..... - . . . " 0 i~ G n (A - AI·

l.

C-u..

SMPI' l ,J.or. it an imqtablo G·~l&tiool p . !At Go be. ouboJrvup of bit. _ ~ 100 C . ....... .... . •.• • • .,. ' .... ' ' ''''''~of 'hoi" " to 01 Go ill O. st- that ~ • + P) ... C ...• .,' ,Ila, ...... &Dd """"Iude lbI. d (Go ) _ • . d IG)·

U,s.s.("

2

...

1m .... of ,be

Mq .........

..-.II ."",.

.k _~'~ ' 'Y

IM_ ';" 01 """""",,.li~

~

A """""",... i~ P""P -.r be rqatoc f_ oyot<m in G """ at ....... " .kiD< " '. &ad..-ory ... I~N .. of io f..... of rank IoN , ........ Fi..ally, if C io . d!oc...... o..t..,_p of . ....... 100 It', , . _ to.io of G """'" Z io £roo in R' raprdod. . . . _ _ _ R.

a

.m

a

_ 10< C to. • •r I " .. ~ 3' _ oI...,k .. in R · ..... 10< A be • oubtot 01 R ', . ,."unotrio U>d ......... . _ ,bat ~ ( A ) ~ 'l' .
.-r.pootl

I . Prooooe t bat
2.

,ha, ........... 1. Ana . _..." .......", (i.... J>f"O"'O ~!iDlIo::owoki .. , .........). AI< '1>10. oil" •• thai. A _ A .. .. bon A, ~ (l + I / , )A.

~

n... ..

"' Ld IS: '" S "

be an Intqtr and . lor - " I S:' S m, Ie< ... be. ~_ Iorm .... R ' , {:r, ), S-So - :t ,S-~. c. ... :r, . .. loon ,1>0 .....tro
,ho,

,:>

u.,,1<22 1orCbap\« 12 I.

m

Loot I boo ......... _icalloomoono<J>lo. .. ol.boo """" z ~ onto z~ IpZ~ , ao>d ~ 1... 101's~ boo.boo boto""",phiom i - (... lij) '~ ' 1~ 01. Z· Into Z~ . If Iooernd 01. lol-ol l s,s ~ , .ha. ~IGoj" " dl ..... 01. ,-. Dod...,. from M~ '• • lMoor<:m .h.. , ...... ~ .. - - . . d" Iknt i ;a A n Z· .1Id> , ......... Ii ). 0 (mod,) lot oil l Si S ....

a."lhe

2.

•

u-

1m .b.. .... e.... " prove lha• ...,.. (lAsr_·ollieGIUD ).

I.

257

Fix .. poi ..... , '" 2. Let

.01 , .0• • .00. z •• _

...

Q

;nl~'"

n
E Z and 6 E Z. PTove.haI. ....... eorirt

l~

all O. oudo .hat

u , + .... . .... {mod pj.

bz , _

=••

z_ (mod p) .

4 z,'+ z,'+'+ z, z.'<\1'2 _ ;p

(io &" .... 3, ....... lot A .boo e _ boJl Uo. R ' .n.b on 1- 0 ao>d radio.

(....,(4/ .. jp)"'). 1.

01. 0 and • iD (O. L. .... II' _ 1)/2 ) .1Id> .bao o' +".. + Coodudo tbat. if z" z,. z •. z, are .. im put I, ti,eD

p~ ,,,," "" .."'...

1. 0

(mod

pl.

zI +rJ +z1 +:or! ", J>. 3.

Fo<""" qu.atenOOoo z _ (z,. z, . z., z. j. "". N (z ) .. zl

+ zJ + z1 + zl .

!Ibaw.1ui< N(ZlI) .. N (z )N (V) lot all qil&l<.Dioonoo z". fIoom ,Ioio fort, dod ..... li

~ ••

tl>
Let F boo .. IIonodt '1*>0 (om- R Of e). E; , _ ...... _ . A &II _ ... _ 01 E . and / ' A x E _ F .. ...... imiOClUOly dill"""""iobIo moppi~ s.._ tloot r .. {(z,r ) e A x F, / (r .r) " 01 .. _"pty and ........ Ior _ (;t.rl E r . tloo poottiol de
I. Sbow .Ioat , b all (zo .... ) e r • •bete ...... Nt ope .. lW\s....... bood Vol (". .... ) mat;,.. to r ,1Id>.hat ..... ","ridloa 01. '" to V .... too'··_I"· pltiom from V onto an _ ball cecner.d .. "" aDd """tain
2. Oed...,. f ...... put 1 l/tat - , . " " " _ and PI(G)" opaI;n A.

.boo'

o

Let / ao>d

r

be .. ill


G of

r .. _

i.

r

Exfte... 5.

A path I. A 10, by do.Iinitioa, .. """"in"""," m·W'll 1 : fo .bI _ A ([o.~ .... """'-' ... biD....... 01. R dep z..,tiD& ..... boo pat h~ ; a < ~ ); .. Il!\i.. 01. 'I' it> r .... COfIIinUO\>S moppi". Ii : fo,!oI _ r .1Id> .hat PI 0 .. .. .,.. A homopy bo A 10 .. """Un"""," .... Will: IP ""'" fo.~. le.oII illlO A (0 < b ao>d" < d). &lid .. lift. . 01. IP i. r ... """'-ill ....... mooW"t!I ~ fran 1e .1oI " le.tiI iroto r .1Id> tbat IP - " 0';.

'VI

ale

12. ('!o"B' '" Vatiaw..

H._IoftI> . ..., _ .bore it ."",,-"'<1 """'por>1. _ ...... ;" G n ,,'("!f.), tbore e><1ou .. !iltlne; ~ 01. l in r _"II t ho ...."" '" .. t•.

n ~,. 10 f.

~, ......., Iia~ i~ r "' .. _ palh l : [0 .61- A ..-.l ;( .be ... """" 10. 6111Udt .Iw - ,(Io) .. ~.(Io), ....n. oleatly. Q, _ . ,.

N_1ot ~ be a)w M
G."""

If _Inooouo OIl

e

w.

Co .61 " Ie. 41.

- A be t l><: pa.1> t - I'Ic.(). fOr _ , E 1<0.61. t .......... t-.l.......t>en rll):' 0 _ .(I) ;.. 0 ,uch.haI..M mt..--'lion I~ of r _ B{1(I).rll ) "B("' .... {I). o{I) io «>lloalned im G. _ ....1> ,hal.

e;,."" (E (c. oI! ....... : Jo.~

(_)

(b) I.

'" ll>(! ........ honv ,"" ph ;'", " - V; ..... " 8hlll.r{I) : Oo/la.,)" 0 0, 1( •• , ) io _ _ .... V, .

If ~ II ,Il00 u,.;q ...

mawi..

&om B("1(I ).rl C) into 8 ("''« I).0(1) .... h

~ (a." Ca)) bdonp I<> r "" 011 " E 1.I (1(I).rll) • . - , hat; " ..,.,.in...".,w,. dilJonntlo.blc (_ u.. impllej. flloo:. ioa .boo<>rom). 2.

~ ."... • lini", w.,1lJ" (l, ).., In

io

10.61 .""" u..t tM

B (1(1.), ir(l,J) ,, 8 ("' ... (1.).

~o(c,))

_

,, oiaIpIiQo ,100 ........ ioo>. 1>'1' « I.) _ " _ .C'.) _ ••. I..ot /of b.\.., u _ bound b I D,J (".,)"o V,l(r.p)R.. Cr.,) ru ... . hrcruP

r n U (8 (1(1,). ••) " 8(", ••«l,).0;) ) .

."

Ci_ 0 < ~ S "', «(1/2)0;) . loll 0 < • :!: ;.1 «(1/2), ,) be ouch .hau M S <. ~ ~ . .... 6 :. 0 ..... t hai..

100- evt<J" { E [c.d) oaaiofYi.. Ie - " S 6. _ _ .....100 _ _ • ""<>(0 - I'><'f(ll :5 < ...... 011 ."" oo ll.t-r _ . h>(1. U - '11(1. ()I 1011 ! E C- •61. eo.ooi6ei ouch • ( . ......."''" ~ 10 ,Il00 ... - 1'>",1 S ~ .... Io .11. Ot " .. ~ ,hal ,,«10) boI<:>o>p to 8(111,). ( If'l}r,) " B(I'>"« (I.). ( 1/1)0;) b_ i E I. _ ..... ,hal u.. bn>ot"-io 10 < 6 _ '" • cont~"".

:s. .....

l. Concl .... 'bot (r. () _ ", (I) io d lot .... E Gn p, (11'11•. (II). Oed"", f..... 3 .haI. .~ .,.;0. . . Uftl ... ~ 0I. 1P if, C .. bld>.w.. t ho val ....... (10.(0). 4.• "l::1ootd ~h in A' io • pot~ ~ : 10 .61 _ A _ h ~ "1(0) - -n~): if "1(1) - 1(0) b- .. II E 10.61 ....... ... it .. pUl>" . A ~<w IP ' 10 .&1" (c.d) - A io <: <:1< 7 d poth I _ , - '11(1.4)".

T 1ft the - . . . '" F ,.:

"'.
_ ,&lit

-=-.....

,ha.

ElIti'

to. Chapt.r 12

Let 'I be • "' ..... op)" of c~ paths in ,4, d./lDOd "" 1.. . /11 X Ie.ell. from • ."...... path 10. ct.-l poolh ""1. o.nd lei " be. lift"" of 'P i n G. ~ by , tile .upremwn of II.".. ~ E Ie, dJ welt that ~.... ) .. ~~ .) "" Ie.{]. S - that 1110 bypot""-io , <; d Ioado 10 .......'rodktiDoo (_ port I of ElI.. cioo ~). 2. In ,ho -.tioG ol port 1•• _ l u i e _ "( ' .d) 2 I pooIlI. Old .... l lou , if ""1 io., ~ . I poIh in ,4. bomotot>ic in ,4 10. conotanl pooIh. I _ ~ lifIiq of ""1 In G io a d· d poo,lI. I.

io .·.

3.

._pIy . . .

FiDoIly, " " _ ,hal ,4 io <:ODDo<:W o.nd lha< ~ <10& 2 j poIb in ,4 io hoIn ,4 10 • f)M1 . . . . .. patb (i .... ... _ ,hit ,4 '*'«1).

mot"""

From port 2, dod .... ,1uoI ,be ..-rictiDoo of '" 10 G • a l' • .... ' ..... m. &om G 0
Let E be. _"""" _ (....... Cj, ,4 a col1DOClCIiDoo. ODd I ,he moppi~ ( .. . , l -.? - g(~l &om,4. C _

c.

I.

lei G be a "",_loll ...... ""'''''. of r. Oi_ mo.i.. 10 ,4. and iP- 100 E' C to. .h.id> • > 0 and 0 < . :!i . . ..... lluol '"

["

... ."PlIOO)'" 1'1... ),

boa_' ..."".... from

Sbooo lhal G

'il.

V _ (B( .... . ) . B("".») n r ""'"

iIId_.

u .. B{.... . ).

roat.u.o V +(0. 2bi) lor ...... sui,",*, k E' Z . C;"..d1ide ,hat

,., (G) " ,4. Let ""1 : (a .1II _ ,4 he • path ond Ie\ ~(a) E' G n " · ha). \\"ri\
...""""u.m

""'''' ..ca ).

p....., l hal

<:

>

a (_

Exetcioe S. port I ). S. . , finaU,!'.

,bo,

<: .. . .

\I

3.

S u _ ,4 . _ ply rom'*'«l. G iYCD ... E ,4 ODd 100 E' C . ..... lhat ."PI",) " II ... ). ohow that ,bon io a un;q"'" ...... in ....... l ullC"l""' '' : .4 _ C rud> llou " (z. ) ... 100 ono:l.l' .. , . Fwtbu""" • . _ ,hat " . ...... in .... 0IIIIy dilr=atioblo.

4.

Su_ .4 iI limply """'*'«I. GiYCD ... E A ono:l II E C oud:I. ,hit .: .. II ... ). dod""" &om J IbN ....... iI • uniq................ 'W>CI""", k : A _ C to. .MeI> k (zG) " r ODd It' .. , .

Let. he aD .klDlDl ol IO. +<><>{ ono:l g, " ,_ ;",., C and It. ......... ';...ty. S " _ ,Iw. (a ) (b)

r IIf" JIc""} cIz.-- ... r. Jg(,,)Ic"'" cIz iI 6n~

_ " _ O. ,(z ) odmn. ,be 1O)'DI_", ."panIioa

1'1 ,,) ...

(<:)

",able "'nctioao from )IUI

L

A,. z"

+ o(z·')

'S,S· ... bo<e _ I < 0, < ..., < ... < Q. ono:l ,ho A, ...., a>mplu aunlben, IhIn ..... " > 0 ODd ~ > 0 Iud> ,Iou II{.. ) .. -c· ~ + o(~" ' -") OI z _ 0:

'VI

ale

(d) .boor<> ~ 0 < 6o :!i b iOUclI tha, II'~ _,,_ OIl 1>(60-) for olI "'" (60."' .

w. P'''' ..., ..ympl.Olic

npOm.Ioa of

method )_

J:~(,, ).'·('I .u ... _

I

St-- thot I> It IItrictJ.y- ""SA'ive "" 10 ,bI .

2.

Let6 O:iO,60(. [f 6 < b, ot-tb&t

I.

1O,.s.,[ ...,. 1>(:I) :!i +«> [ .... pI..:e ..

,("~'l(') "1 :!i ,¥ .•(1-110(' ). [ 19'("),,,"(./ "

"j ' . [

.. -... .. I 2: 1_ Pro.. tIu.l . be Ieh_baod IIi.

3. Pro.. , hat

iDr 011 I

4.

~

j

~ T,

wlwn!" " , .... E..1eo- s ....ma [ullCtion.

ktl 2: 1be~ . S_tb&t

, ........ [

g(%)c" ") "

IT .

-

.. boor<> F.M .. II<)W to ""l' ( .... (~.-'/~» It 0 otberwloe. Ded"""

,hor,

G.(~) - <xp(II>(JIl - "-)) .

1.- F,(,) d~,

. vljIl- '/' ) If 0 < ,

< 6,' 1'

aDd

[,w,-,,·) - L . . ,(~ -l/.)., I

, cv") . Jexp (,1>(,,-" ' ) -+- .,...) - :; . [ L ~,~.

-+- up ( -

. . ,W,-',·)··[

' SJ~·

11 0

< , < 6 "'/', and.here G,(, ) _ -<.q>( - ""' ).

L . . ,.(pf-I")" , , ~,,, ,

11 , 2: 6 _,'1'. ~_

AM 011, > 0, ' - ' thai. . "" • . G, f, )

eon_

.<:>0 .. I _-+-cc.

8. Gi_ .hat III(r )1 2:
- I m exp(
'0,,,-".) e'
/ -",

ll>on owIY.be mi~ con,,,,",",,,,,, .I-b».", 10 ohow IW

,"," .[[ n*'O(.,,u _ L •

t·(tW · '·"

.r(
IS,S'


10 Ld b E: [0. +<»I. and lot g. II boo ."'" _W"abIo fundiooo _ R, ._'ively. Su_

10,10[ illt<> C and

(_) J' Inr)!e"'· ' ,u/lO .O( io finite; (b) .bore .... e > 0 and (J > 0 oud> .hal. 11(,,) _ -~~ + o(~) 00 r _ 0; Ie) .btft _ 0 < 60 S b _ .ha. II bao. otricl.ly oqu;.... do:rivati ..... in [0, 6o( and .udt. .bat h i" ) S 11160-) for 1111 " E 1"'- •~. fu"". i. Show tha •. rr.. 0"W1
I. Tbe

,.

" - _ that ...

~

_ O.

,,~) .

10.60( ""'" )O .<,o(A,-)/ hao "" 2: I.

,vIr) admi ... be ooynq>toti< ~

g(~). t/>'I~) _

L

A"

rO, + o(p")

'i,5'

.ntb _ 1 < (I , < ... <
["",,'''I'lb_

L

t (dr"'i' . r(
I:S>~'

• .. t _ +«t.

11

I . Sbt.w that k.'~I -'HoI b r l... I . .. . r(t + I) alii > O.

rr..

+

J.-.,.:>oodlHI-' l u

(I-'l:'

io

«tUIII

-4 '"""

to

, . Co ·, i"lor .be fund .... 9 , (I 1'-. _ -1, 0 C i ..... C . Sbow .haI 1(.) it .be ...... 01 • ~ _is .hooe totW _ - Ioc{i-.)-., - D· l" , - .' • - 2''~ """" """ c..- DO _ , w ...... = ...........

,'--I.' ..

1 _ 01" ~I>iq ... ../1/2 """,.'

r e.),.

!( , - IocCI

...... I (z ) ..

J.

C lor ",IUd> 1(0) .. -1olc(1 -.J - • 10<
boI<>morphic , .......... , fn>aI D int(>

0<" < I

and t lw f(z) ..

zl'" lot - I < :t < O.

-!( , -

10&(1 - ..) -

Le\ \PI

~ ,,,- funeti",.." -

.." u.. f.......... "

- :t) - ,,)"" lor

\"c(l- rJ - z)'I> f""" [0. II ......, 10. ""' ( - 1oc(1 + z) + "J'" f..- 10. ....10 (0.+00(. ( -

_ &Ad .... Il1o .Iorir "" .... m..o.ppiq:l. Writ 0. ,ben! 10 • ~ionI v 01 D{O.r) into D • ...t. t hai ,,(0(.» ....... 011 • E D(O•• ). S_ that ~"" .. o(r ) o.nd . .(II .. -v( -,) lor a.II 0 .s , < r . 0 -'I>l10 t hat w., ...... l..tlllitely d ilref_iabIo ....... It..< 04" (0) .. (-1)"- ' - *l" (O) for an k E N . 4. Put .. .. ." . 0..:\1loI from F a ioo 10 thai; , lor ~" e N .

(" [C"" " ' .'.rClj_

L ....!.... .•(Jhll{O). '-·. r(HD] O::S'S~ ('lk)1

... 10 0 ... I _+«>. COlIC ..... that

L

r(1) .. ff. ,,, -1 - ' " '. -'. [

~ ...1>*· "(0)_ 2" ,I (~ ) .,-' +oC'-·lj.

._il,

"':!O· s· 6.

r , pu.'''' .. ""~(0l. By pu<3, 0(.) .. E.Z'I <>,... F'tom 11>00 'C1.hon "

1'0...11 k E _

L

_ _
DeiPbo
..

- lot: (I - 0(.1) - 0«).

j - " , ' .. ~_,., .. -2a . _• .... all ;.' G"" "

~ , . Si...,.

's>t_

0 , ..

8.

0/'1. ,ho abcJooo Iormulo. ""'y b< uoed '" oompute ,be numbon 0 •.

Ood_

from - ' 5 tim ",(»(0) ..

ADd ",!OlIO)"

V;.c.....ludo

-3' ..

that

r(I)".n; , ,' - '- "'.~· '·

.a....I· . 0(1") _ 0" t _ ..! • .. .. -

+00>

+00.. In

...-'lit, ... . .1. - c-.

1lI(0) .. .;; . . .«1(0)" : .

,-, ,-. [

I

1+12+ 188+ 0('-'),

-,;cw.. .

-II + \;" + ~.., + ,,( ~. ) 1

ISllr\io,s'. fonmoIo).

12 Gi ........ ,ws ' .. ":! 2, Ie< ( •• ~) - (~M be tloo 1U.w i~_ pr<>dl>d OIL R " . ...:I Ie< ~ -1-1 " (-too)"" b< .Ito! ... uaJ """".,.. R". Write tS" lor to. _ u " ,
b
n.. fitno;l"" (r .u) -

from B{O. I) " S··' !nco /0.+00( ;0 called .100 PC

&~'Zf. .. p(r ,u)

- W ...1 relali ... 10 B(O. I)

I

S _ n ?: 3. For ..t. In '7 _ (n - 2)/2 (ChapI.er 10. E_ciot 1). For _ k E Z~ ...:I._ ~ E S"-' . ,be fu nctiooo ~_ 2k -+- n - 2 ,C. (IIl.)

z: :

"-,

from S" - ' i _ R 100 called tbo .. t .ic&I ~ic .. il b dtJ> ... k ODd pole ,bat 100 "onot..... "" OKlo. p......JIeI wllb pe or'looI<>rW t<> ~.

r.

•. s_

E S"-'. S_ IMI lbe """,,"ical ~ia z: LMM ). 3. I.d u E S" - ' and v E S · - · . Write f fD
Fb; ~

.... on~

In

, _ (I _ 2r(u)o) -+- , ')- ' !nom

(0

P_ t bat ,.(,..,u) .. I (r) -+- .. ~ 2f{r ) for ~, 1). Dod""" ,ba, J'( .... , .. ) _ L ZOC ujr' . For &=I 0 s: r < 1 and

10.11 into

s:' <

R.

.~

u E S"-' , oIomr that..boo ...... on "'" . .1tt.haD
"""""#0 '"

, . Oed""" from pot' l ,bat

U

fA'"

0)0/.:0 (..) _ I lor olI :.: E 8 (0, I ).

S' _ ( .. E ft' : I~l ~ I) io...,. eqwppod witb.boo p-oup otnoctuno lor _bid> S ' 100. oubs_p of . be mul\ipj",*,;.o, VO<\p C· . I.d " E SI. W~ .. ri", ior , ....... " on' f."",1oa I 0Jit S '. aDd r. , ior ~.....,. k E N . ior tbo u _ (u/.) ' -+- (v/ .. fn:>m!ll ;...., R. io e&Ilod ,1>0 opbrricoJ h ....,..;c wl.~

z:

fwon.,..

z:

r'

dqp ec k aDd polo: •. 1.

S"",,".bat pC" ... ) io , .... fU! - ' of .. u -+z _ 1 -+- 2 "L. (')', fot oJl _ ~ Ii

".

" E 8(0. 1) _ v E S ' . Oed""" .h.. pC ..... . ) _ E .~ Z, C ..,,' lor oJl " , U of S'ondoJlOS:r
S"""".bat .....

12 ,..", .. n ..... ior n _ 2.

from n Inl<> C (w"",,, n io an """" ... _ of R ') io I";"" dilf."",ti.ab!o":.: E n, _ call Lap""""" 01 ~"* z . on.OabIo in n _ 6~(,, ) _ 0 10< olI z En. J.g

n ?:

p«I9O
2 boo lUI ~. If

Ii

fu_

~

I . 01_ .. E S"- ', let "' ..... be fu nctiooo" _ p(% , u) from B(O,I) into R. Coml"'~ D'op(:.:)(:I , :f'"), fD< ~ z E B(O, I ) and oJl %' , z" of R ". Dod""" ,bat '" is harmonic 8(0 , I) .

in

2. I.d f be • <:OnIinuo....... pIe>; •..JLioO
J

l.

~ lbat

S" - ' • 10

11>0 fufld.ion F • .. hiclo io "'lu.!

I<>

P(J) on B(O.I) and to /

""",in"""" on , .... olotood b.olI. 8'(0. I ).

Oft

11. (:!r.·-l"olVw' h'zz 4.

Let G bOI .. " " " , i _ ru..ctioor. _ U (o.l) late> C whieb io llanaooUc ill 0 (0,1) ..... • witb I on S~-' . S~_ ,bal , uod G .,.. f'<*l>1II1IlaI t ...... ..,io..... "'Iqft ! :; p !> n io< wloido.bOI pu.1&I de,i ...... (d'h/~:j(61 io ... ioctJ1 l'<'Oill.., f'ro,.o , ....

" '" .."""., .. ",,.,.ud",,;"'. &)

w. """'" bade

I<>.be

s<""noI


in wbido

I io complu;_nl ...... o.d_

f...... poOrt t tlo&l G .. F.

15

Let .. ?; 2 bOI ... h" G _O ... op
,*

I.

Let lbo_ "_Ie _pi02:·,-!..od '_ioor. ill fi Ded_ " - e-... 14 ,10&, 1(0) _ J / (.+""I _(e) lor aJlo E 0 WId aJI, > 0 -" .10&1 0 __;' ,bOI.k, ! boJ:I 8'(0 .•).

,.

Let / be ...... ioo _ _ """,·nIuod "-':0000 OIl fl. tlw.. lor ~. E u ;oo .m.U '", ........ > 0 ...... d!al 1(6) • ,(~ )oIcr{.). p."... II,,", f io ","'10' i" in n. (S 0 """" d!al fI .....at.. 8'(0 •• ). doli.. , ...M

J

1).,_

...

_ 'rwil r

s.._

.-"""'-1,

--.ai_ f......,., " - 8'(•.• ) 1_ R wtoiclo iI h . """,It;,, O(o. r ) ...d .. " .m.~ Jon""' ....... +rS~ - ', ""I~illll by".." .....""..... . - t bo.t , _I 01> 8'(0., . ).) ... Let Ha{n.CI lot tbOl _ _ 01 \. ,_,It fu .... _ f""" I) Imo C , S __ II. AIto:r F on H,,(fI.C) ........""' 1<> . fu ..... .,.. 1 ~ ..!6onnb' on ,100 ' : 2M . ...tooot. of 0. f • b.annot>lt

.!>aI.

.Iot.

13 Stieltjes Integral

not"""

Ai tbt oDd ollaot OO"'WY Stieh';" introduced 'M of "d"lflbuho" 01 T" • on an i~ . 'The ..,..ol " - ol 1<1. z l io an incrM&ins f~""'ioD; 11.1 d _ _ ti .... 1tiot .... 12 I' --t to mET'll WI .... "U..~ at """ pOi~t. Gi", ......... ru..ctloo. ~, S.1tkjoo eN ... ,hat, "" "'tr <XIft.im,.,.,.. fuocUoo / . t he "IUtmann ""mo' E/(O(,!(:o:,.,) - 'II" ')} b.r..., .. ~mi. *-"" by f. / (I ) dl)(l). T he i,"" ' '';", flo""doll 'I io _iI:r ,... " .. I by ""1 dill",. :0 of ;DC" fu...:t ...... . hat 10, by "'IY fuDdIoll with ..... ,nded ...!aI.Ion. n.;,. ci_ an uplicjt
"r"'4

IAtonai. !I I_ Let P C_ In\OCl'obloo.,.... [a ,lI! and. Fez ) '"' J(I) III , .beo ,100 ~abilicJ' 01 111'''' " , \ha.t F it .. t......;.... ol _ _ ..o.' ioa. Ik if F it .. f,,1ICtloa 01 bounded >Wi&tion wtoo... den ... i"" C...hidt ...... . ~ ~ .[ ).bc oquoJity ~ if and. GaI:r if F io "p -h/dJ """"'_ *hidI _ that tbt ..nation 01 F Oftf " " " ' " opdo to 0 .. i\~ . ho .... , , ,,'" 01 J.

r. a_.

OD. ...,

r

'!PM.

S.. ",muy 13.1 I.ri I he aD imerwJ in R and. J .. fu .....,.. from / jill<> .. IDetrio ~ (£ .4. If J io .....bi-...;J of I, I" .. MId ... be 01 6"io. _latloll if.bo eo< d(/ Co, • •), / (o.)}. whono lJ. nInO """,".he ... of hi~ portlu.... 01 J . iI bow>ded. Whoa E _ R, .be _;',ion 01 1 It b:&IIy 10".",,10<1 If and. ",,), If I io tho dilr<_ ol t_ u.u d na fw>ctiool (P'op;aitioo> 13.U). 13.2 Da>ote by S l ho natural ...... ;,;",; ol l .ubintca!!). bou"d . d _ (n..on... 13.1.1). Tho Io<mulo.b ~ poot\O I>oIdt IlIIoO .... "'"DOt.....
Cr:..

<_

inlqr.''''''

cJ..hOI .. ...,... .... 13.4 W. po..- the I.eboopoo doo:uapOOi' ioo 'booo .... , If M 10. funct;"" of locally boooDdtd ....... ,;.,.,. 'ha. &Dd AI, 10 ~b' OobtoIntdy coot;' >_ (n...:..,a 11.UJ. "100, if M 10 I," " ' ia,,.bet. M 10 dillet ' Iabio 111_ om) .bon! &Dd itt deti.....i .. 10 locally int..,.-..blo In--

.....,.1a<

"""'- I3,U). 11& La '1Mo ........... Ii. .
_~d

_

be _ il ted ....... \Ody ,"" uppct &Dd Io<woer doi ....

to .....1oa

13. 1 FUnct ions of Bounded Variation If J is a ftOMmpty Lnlcn1tJ in R. ..., call • deo:ompc!l.ilion or J U\Y ttqo.toer>Ot ("G,a,.,,, ..... ) In J o.uclt l ha ..... < '" < ." < ..... W hen J II ~I*t .• tubdi'#Oo 0 -.h of.be IUbdiyilion. UntU furtber nocice. will deoow an Ln~ Ln R. which I!t noc a pOInt, LeI E be a metrk "I'*'" wi.b metric Po .00 Ie! !If be a mspplns from I iIM E.

r

DefInit ion 13.1 .1 If J 10 • oonempty ... binterv1ll or 1, l ho lou! varlalion cJ. M ~r J, wrill~n V(J) '" VI M , J), ill 11>0 ouJK"'IIIUm of

L '" L A

I'(MI... _,), MI... )l

l~l S .

wbm! t:. _ 1l1li, ·· ...... ) <:X\etldri over l be 8e\ of dt<:oml'O'litlOr>l; of J, M ill eald ro ~ or bounded variation oon:r J .......... «r V(J ) it linj~. M ill aid ro be of loWly bounded varir4ion if it is of bounded ..........ion _ .u com~

... blntervab of I. 'k'ben J ill com~, V CJ ) it d twty , ... euP"""um or 1:;;\, whe", t:. e-tA:ndo ~r ,1>0 _ of eubdlvlliorw of J AIIIo, VI"'. c::j) ., VI'" .111) + V(~. cjl 100- aiL pOin.. _ . b, c in I .. tWring a < b < t:, bta._ V( [.. •cjl ill .ho ""fftmmn of lhe numbm lor allaubdivi$i<ma t:. of (a ,cI ((ICIwnilll ~,

LA

Lemma IS. I .1

Jo. /J)

iro I. V ([r . .8l1 "'"'rtrpt,I I<> ia 1<> •.8). S;milan" I"'" an, ~I, (a .,8( in I,

F".,. .... _ P I )

V(!a . .8) ., :r - a vIla ,rl) ..... wrru lei V«(a. 6() ... :r. - 0 "' (0 ,81.

PROOf"; We will pI"O\'e
V((r ,m )

Jo ..8l into [0, +0:>1 d.:oe"'"1 I , I<> il baoi a limil .. r - a . .,hI •.8) such l hat E", 2: ~ , Then

from

V(1r ,OJ) > V([110 . OJ) 2:

L 2:

•

T

u.rnrna 13.1.2 A ....".., IAc.I AI " o/..,..."okd ........ I:i<m -'" .. o:mIJlClCI ftMlII~ «a ,BI ~f 1 (a < fj) . Then V([a, til) ,. V(]a . BI ) if.".j emir if AI "r1ght.amU"""1II a.I Q . PROOF: First, OlIvvo.- tll&1 M "ri&ht.-amlinOOU!l &I Q. FOr &Q)' (> O. I~re exiN 0 < 'I :5 fj - Q lucb 111&1 p( M(a), M {z )) ::; ~/2 !or all z E1a ,a + '11.

Eo>.

> VUo ,BI) - c/2 for .. Suitable m.bdivisioo t:.. _ (.... ... , .... ) of In ,6]. and..., ~ IUp»Ot': 0, ::; It + 'I. Then Na.

O!:

•• • V((a ,.B])- " - ,,.

SinDt ~ .. arbillW}", V(ja , BI) - V((a ,.BI). .,.,l" IlUpp .fII) - c. T~. If <> < :r :5 ....

eon....

V( [a ,.8l) - ~:5

L :5 p(M (a ), M (:r» + L :5

•

•

V([a

,mI.

p(M (Q). M(z) ) :5~.

c Propooltion 13. 1.1 S _ lIIal M ;, 0/1«:011. :r. .. m I . lHfiM Q fo=tion V... "" 1 ... /M.~: V (z)'" ( - V([Z, :r. 1J V([~ . :rt)

...

I

Ioot,,,•.u,, .......;.n.",•• no! Id

forz:5~: /"rz i!:.~.

n 1- oo ....p/[

if . M <m4t if M ,, ",M.amI;.........

AI ~.

PlIOQf": Lee. 6 E: I , 6 > e. By ! em_ 13.l.l, VUe, " I) _ V(1e . /ioj) _ V(!" , "I) """'ttP to V (Ie. "' ) - V(jc,bJ) .. :r ~ ~ in /c,&!. NOOI V.. _ V.. {e) + V.. 10 V.. (:r) """ 'tip to V.. {<:J + V(/c, bj l - V(jc::,"!l .. " ~ < iIIlc.iol. By Lemma 13.1.2 . the last I " tion oIlbee SlUm>cn1 follon. a Slmil&rly. V.. .. Iot\-coot.in UOUl 0' .. point ,, 01. 1 n J lnf 1 , +oo( if and only If M iI lefk:oD!.inUOllll U ""

'I'

ate

Propoo.itlon 13.\.2 S.~ E u _plett..nd At .. DII<>CtJlI,.......,,,,, tWi"tiD ... Then M iI4I a HmU from til< righl (.-upt.ctit'd" from 1M kfl ) "I ~~ "..w c <>1 rn J - DO .!nIp I I (r"Jlf.Cti""IJ, DI I n I inr I ,+<»[).

- Ie, oup I I, V(Ie, . ]) hili " [imll. Tben coo>(z"u; to 0 ..bm :z: and ~ t.eod to c in J .

PROOf': Wben ~ , ..... to ., in J

rlJl

V(Iz, III) -IVlle. ~)) - Vile ,

A fortiori, p{M (r ), M(r)) coo.'~ t.o O.

0

P ropoAl t lon 13.1.1I S!q>P("" l1l4I E .. R . """" M u .. ","ctitm DI lo
P ROOf': Tbc cooditlon Oil clearly sufficient Now "'PIl"* tblot M .. cJ.lo<:aJly boundtd ""';al.loo, and let %0 E f, [f V.. .. tbe [uncWn ~md [n Prop& oitlon 13.1.1. then At _ V.. - (V.. - M ). and V.. - /II in<:' ! 9 I t.o.:a. .

v.. (,) br all r,

~

V..{r ) _ V{ lr

E I oatlolfybljf; Z

.111) ~ M (, ) -

M{r }

o

<~.

Finally, when B .. C , 01 .e thaI M Oil ()i Io<:aJIy ...",ndtod....-iatioo If and oaIy if ibl ft*I and It''.~no')" p&rU~,

13.2 Stieltjes Measures ~ I .. (... ~) (..

< ~ In Til bo! &

IUbi~",,1 of R,

aDd lot S be tM 'Ia' uno.!

oomirillf; In I.

DefInition 111.2.1 I.d. I' be " ~ on S. At : I _ C .. callfld an indefinite i""'Fal 011' M (/1l - M (.. ) .. 1'( I" ,~]) lor all <> . fl E I Mtisfyi", .. < fl. For all <> , fl E I , ..., put

.,be,,,,_

t

<11' '' ''( I<> ·flI)

[<11' - - 1'(1,8 ... ])

At(P) - M (.. ) ..

If .. !i fl

if fJ5 n.

L~
PropoAltlon 13.2.1 LeI I' k " m_ ....... S AM At <>II .,fJ)) lor .. ,,~,~ C .. . 6) 01 1.

;"1". ".

PROOP: 8 )' ddinltion of (PI. IIIIC ja,/JI) " V(M.Io./JI) ro.. all o . pe l ..u.(yI", 0 < p. N_ let J .. (0 ,/1) be • ..,bintenoo! 01.1 witb endpoiUI.l 0 ..... fJ Crt f- 6I. lf J - J
.-.

VI M . J) .. lim VCM . Iz . /JI) .. Um IIII( jr ,fJ)) .. 11I1' ()o ,fJI); ")0'"

i(J _(rt ,tJ!.

.-.. .....

V(M,J) _ lim V{M. (0 . z]) .. lim 1" 1"( )0, zl) .. (1I1"( )o ,/JI).

o By P f
ro..

.cd) -

'fbeoonm 13.2. 1 ut" /Ie" _ ' " "" S. :nw. ...... ;n4finilc inUfnll M oJ" Y oJ 1«»Jt, IIotoIWltli' ....rMItio1I a"" Y r1ght--COtIti~_ "" (a , i{ . Mo,w"", .

,,(Ie)" M (c)_

M(e~ )

lar ..nce)4 , ~), ~ M (e-)" Hm. _ M (z ) .

•••

PROOF: We CII.....:I)' know lhat M .. of IoaIly baund.od vari&tion. Le\ e be In (" ,i{, ..... 1et (z; )'1::' IIe.de<;. ' '''''Iaeq_ln Jc,~) con>'tfll"l to e. ~ II("', Z~]) oo... ug to 0 .. n _ +<:c, ..... 10 M ;. oiI;bw:ontinuouo '" e. Next. , let e E)o . 6). CIIId let (z ;).1::' be an iuti oj", ""'1_ In (n, c( conver&I-"I to e. Then ,,( ""~ ,c() .. M (c) - M(z~) to M (c) - M (e-)

con,u"...

Mn_+oo, IOII«(C}) _ M (c)_ M(c-).

a

T ....... m 13.2.2 UI M : f _ C /Ie ollaadt, 1oo~1WItIi' ...riotion, coruI ~ """""',..,.,. on (" , i{ . '"""'" y a ~"""" -....... " on S 4IdmiUing M ... an irnkjinito inl.,..,J co"" ICIfYfying ,,({a)) .. 0 ~ a e I . PROOF: Thio foIlocrn from P'<>tXMitiono \3. 1.1 and 13. 1.3, and from P fction M .. : ., __ M .(z) from , Into C

a.., _

:no

13. S,;"I,;" Im,,,,:,...1

io obYiou4Iy ti.sht.-cootiOUOUS QO ( .. ,/II. Beller ye" OV V{M. {c , lI1) + IM... (4) - M(4)I. Indeed, let (:ro. ·· .... ) be .... bdiviAM of Ic. d). For eAch O:s k :S n - I, let 1/. be A pOl~t
IM.(v.)- M .. (II.. ,)I < IM.(,.) - M()I. )I + IM. (I/•• ,) - M(I/ .... , )( + IM{II' ) - M(II. +. )I

£or all O :S k :s

n -

I. '"

+ IM.. (4) - M(4)1+ V (M, Ie ,41). Now. lettlng II' ~ to z . in

~

1>:. ,..... ,{ (f<>r eodo 0 ~ k :S n _

IM.. (:r. ) - M... (:r. .... )( :5 V (M. le,d))

I), "" obtain

+ IM. (4) - M(4) I,

OS':S;~ - '

...

~red.

III Iohort , !of... \Ii A function of locally bounden,in\lOlOl on (... 61 · DefInition 13.2.2 1be unique meMu", I' on S at!miUi"t! M... ... an Indefinitt: inlqral, ~ M.(a) - M(a ) ... iu..,.... u a ..1><0 " I><1oop 10 I. is called II>< Slielljos mtUW'e dto/l""" by M , &Dd \Ii written dM. Now .... descri l>< II>< Radon

!IIOUIlre

......:>ciated wjlh dM.

P l"OJ>OIIl tlon 13.2.2 u.1 1/10 • <»ntin""'" mapping from I inl» " &uw.do IJ'CI« F , riJA rompcld "'PP<"1, hm« ...nilhing <>vlricIt G IX>mpCI<:I ",hrIknIaJ '" ,1:11 oI l (0 < (J). Fo.- ~""'l' ( :> 0, IN..... ....w. ~ :> 0 ...,," IMt

If I

dM -

~

(Aft:r, ) - M{:r,_tl) ' / (1,)1 :5 !

' S'S"

10.- ""II .....",,;.,;.'" (:ro•.... :r. ) 0/ 10 .1:11 "';ou rnuh iI . ",aller IMn 6. on,d 10.- on, .,.,.... (I ... .. , I~) 01"",,16 .,uil!lling z,_. :s I; :5 :r, lor 0111 :5 i :5 ". PlIOOf' : Oi"",n t :> O. then: o:xWt5 ~ :> 0 oucb lhal I/ (z ) - / (,1)( :5 ! for all :r" in "'.PI S&tiofyin.g I" - 111 :5 U. Let (:r", ... , z .. ) be & subdivision of

..,..J1e< than 6, and let (I, .... . 1.1 be • f)'Item of poIQtI .... Wyinc z,_, :S t, :S z, for oJJ I :S i :S n. Then

10 ,.8\ ..bole msh •

1! /dM - "HQ» · / (QI-

L

(M.(z,I- M.(z,_,I)· / (I.II:s:

1$' $ _

~·I"I( \o ,.8I I·

,,,.. ,,({o )) · / (0 1 +

C"

L

(M. (z,) - M. (z,_,»· I (t, )

- L

(M (z,1 - M(z._d)· I (t. ),

, s·:!;_

e .. ,,({ol) · / (0 ) - (M.. (ze) - M(zol) -I (t,) + L (M .(z, ) -

,,({oil· 1(0 ) -

(M. (Zo)-

1 (0 )" o;r Simlilrly. beeo,

M (z.»· (/(1,) - I (t, .. ,))

M(zoJ)· I (ft ) .. (M.("o) -

0 > 0.ben '

M (z. » · (/(0 ) - I (t,»,

M .. (Zo) _ M (Zo ) ..

,,«0» if 0 .. ...

(M.. ("') - M(z.» . f(t.) .. (M . ("') - M(". » - (/(t..1- I (ft). iweo'. I (JJ) .. 0 if (J < ~ .. h.c>: :/ 1 M. {". )- M(z .)" 0 if p .. ..

"'"c ..

n . Lmpla

(M .(Zo ) - M{"o» . (/ (0) - I {t tl )

+

L

(M.. (z, ) - M (z,))· (J{I; ) - / (I,+, ))

+ (M.. (".) - 011(" .. ». (/(t .. ) - I Ifl», and tbe...Jo",: dLat 1eI:S ~. V(M.. - M . fo ,fJ)I. f inally.

I!

I dM -

L

( M (,,;) - M (,,>-,»· f (t,)1S

'S' S' ~.

r" I([o .(Jj) + €. V(M.. -

M. [0 . .8\1.

o

'VI

ate

11. g.iol.p

In! pal

Tbeo,..,m 1S.2.3 ( Integratk>n by Paru) U I u. ~ /It I_lund><>", ~f k>""'lIr 6.>ttndtd t'Grialion""", I into C . .4........., /hal It "nJ ~ h""" "" com"""" poinl (]of ditoonlin";l. OJ 1oaIJ/Ji dv·;n~~. v ;, IoonJIy du·;nl~, ,.,.J d{....) .. udu + ."...

n...n ..

-=

PRIlOII': On o::xnpoo:t ,ubin~ cl I, u ilI l be uniform limit (lEa !lequence of mp funaiono (PropOIIilion 13.1.2). n.e .. be, It is locally .w..~. I.)efi", II>/, fun<:tiono .... and It .. In tbe tIIfUIJ way. If" E f U! luch lhal v{%) ~ v.. (%), then., is. poinl of ((IQIinuily cl u, &IX! d~,, ({"II" O. Sinco: v ha6, at ""*-. "," .. otably many d;."ootinuiliel, ..... c:onclude lhal vdl< • .. v... du .... When a e I , writ S defined br pIad"ll: tIM: unit ......

••

if .. dnG! not belong to I. and .odu

_

"Idu. . + (..... (..) - ..(..1)..1

~

v.. du .. + ( ..... {,,) - u{,,»)tt{,,)c.

if a I;", In I. Similarly, if " "'- ....:

beIon« to I . &lid 1Od.. '" .... 4 .. + ul")(.... (,,)· o(,,))r.,

If a Iitolin /. But , whe<> a bo:io>r>&lI to T.';nee ci
to{,, )(v .. (,,) - v(" I) + (..... ("1

V

io contin...,....t ".

...(" J)g(,,) .. .... ("Iv.. (,,> - ..I"M")·

d(ug) .. dlu .. v.. ) when.. "'- not belong to I . and

dIu) - <1("+,,.. ) + (.... ~ ..(,,) when " E I , it iI!

""""'I&h

"v(an,.

to . ' - ' that

In other ...".,.. woe h.~ to J>f'O"" the theon:m ... hen u ""d v .... rightcontin"""" on (",,,. In th., ~, let <> . {J in 1 be given, ou.ch that <> < {J.

,,'

6, 6,

... ..

Hr,~)' <>
6~

..

{(z,~)'<>< r"J/ S/l)_

{( r, w)' <> <~< r S/l),and

'VI

ate

f

f 11~ .~(r) · f It.)(,)cI~-{,) .. f llo~( r)dv{lrlldu(r), f

14;(du e dit) =

d.,(r )·

l a.(d., ® du) .. 0

W-' ,... d.. and dv ha,,,, no COnlIDO<> point rn_ and eu:ept poomaPl'

f

&I

a)UDlahIy !ll.I>IlY poillUl or 1 . On

f .. f

la ,(d., ® eluj .,

du(:z)

· 11~ .P1(:Z) '

~

du{{:z )) .. 0 lbe other band,

f

l lo.P1(, }du(,)

l lo.P1( :Z) · MPl - u(z) )du{r )

..

( ..(,8) - .. (0 ))"(,8) - (~..n.1O'" ,lJ!).

=

f

Simil ... ly,

f

l ... , (du ®
elu(,) · ! ra ./ll(,j·

f

1r-...oI(%)cI.,(r )

.. ..(,8) ( u(,8) - v(o) - (udu)()o . lJ!).

So

(.(p) - ..(o )( v(P) - <>(0 )) .. (el.. «:> elv)(jo •til) " Oo .tIl) .. ( .. (,8) - u(o ) <>(,8) + ..(,8)(,,(,8) - "(0 ») - ( udu + trd.. )()a .lI)) Tbio ......... that

d(orul(j .. . OlJ .. ('''')(0) - ( " v)(o ) .. (udv + trd.,)(]" . ,8!) , N",,", Olio«! cI{..,,) and ...." + t>
!IWIII

at .. , ... hen .. E:

t, _

may 0

WriU: >'/<>1" !.eM"", ....... ....., "" t . Lei ( ..... ) _ "" bo " fI.ilill(tlJ" .....",,"9 from F" G iftlD H, ..m.. F , G. and H """ thrtx r-...I 8o>w>eIo 4 ...... AiN, leI t - F .nd g' , 1 - G bo k>t:altr >..~ /o<-..ctimu.•nd k l / aM , bo "iMeJi..u~ inuyroto· of " nd g' , _ t i..I~_ T~ru 13.2 .....

.....nil......

r'

r

"'~ [ (r'I{I)dt .. U,)(,8) - U ,)(o ) - [Ug' )(I ) dI

J<>I""I 0. 0 E 1.

'I'

ate

PIIOOP' Gi,'OO o. 8 in I. let " be ~!",-"I' For all I' and g' in £~u.) and ChfJo). ....pettiftly, denote by 4' (1',g') l he qlWltlt)'

[(1")(1)'* + [(fg')(t)dI - (U,)(6) - (fg ){o )).

.so. DOl de~D
tho cboioo of the indefini~ inlograk / ...IId fI of !' and g. The bili....... mapping 4' ; (f',I ) .... "'(1' .g') from IP) Inl.O H '"' (OOl in1>O""_ Since. it '1lIlioshet On (C:" (P) ~ F) " (e Rlli) ® G by Th.eon!m U'U, it .. Identically zero. 0

which

'ro.) "'I'

Propwition 13. '2.3 (Second Mea" Value Formula) St.p_ 1 ;, """. l"'oto "'" I. F ;, Ih<: "'qdj,o~ % .... 1(l l d! , .. Old 0 if .. clo,! ' _ _ I'd in £ O".,UII'nmg F (/ l , ""en tMre t~ V .-.. D

"

nd<

t.V.1

t. J{", )*),u - g(..

f:

}tt.

PIIOOF: Let • be a Itq> funo<:tioon 00 I , pocitift Md de<:rtAlIl~ Suppoee • iii ~li!lOOUJ on '" ,I{ and nm idcm.icaU)" ........ Finally. let (.... ,. ,' ..... ) be ... subdlYloloo of 1 IUd:> I.Iw • is coruin ....... on ....:II [";_1 .... r ( I i :5 oj.

:s

Tho>

[!.d>. -

' (<>oll F (o,) - F(.... )) + ... +

.(n~_,)IF(.... ) - F("~_,lI

F (", )[. (4o) - ' (0,)[ + ... + F("~_L)['(""_ ') - '(Q~-tl)

+ F (.. ~)6{"R_ ')_ 51""" lbe codlid.,,11 ohlle F(",, ) on lbe l'i&ht-haDdIide am pt8iti.., will> own

*), ( 1/ .e.. ) f I,d), lio:oI in D. Now ~, br the Jtq> fuDCtioa

..m (... _1 . ... 1( I :5 ; :5 ..), .. tv:re 0; -

.~

o:quaIl<> g(6) at 6 and I<> g(O;_ I) 011 .. + (;/.. )(6 - .. ). ~ .... uroce ( •• ). 2; 1

10. at OoI~ ~here. Mot tle "'., 1/ ... 1:5 g{..)lfI. By the domilUl\OO~..::e tMocWI. I ... 01.1,

<:om~

c"."..,.,.. 10 eomploele.

13.3

f /gd),. Tborefon:,

(1/ 9(" »

f Igd.l,!;.,. in

f

D . .oo the proof II 0

Line Integralsf

Let I _ ' .. ,6) he a tIlbin""'.....! of R, S the IlII.tur-a! .. miring of I, and lei. M : f _ C he a function of Ioct.lty bounded Yllliation.

'n.. OOOdioa ~ be omitled. 'I'

ate

OeflnitiOS> 13.3.1 If / ill .. D>8ppi"i fu)m X :> M ( I) into a 811nod> Sp6OI: such that / .. AI is dM·in~. then JU " M ) dM is w~ the i,,~ra1 of / ~r M and ill writ\.en J~ J. Now , WI! " - ' that the li~ integnol J~ rameteriz.al:ion.

/

ill In,...n..nt under

"'''10'' of PI"

PropOaition 13.3.1 Linternoll Q/ R line! '" " o:>n.h....• ..... ~i"g from J <>Ill.. I . Thl~ (",. S) is d(M .. "') •• IIiI..t line! dM .. ",(d(M 0 ",l). ~. IIIMI = ..,(ldiM 0 ",)1) m....:..... AI it rig/lt-o:m.tm ........ "" la ,hj &t' '" is ~trictJ~ mc..n.;ng.

""'eli""

P ROO'" Clearly. !of .. '" i& .. funet;.,n ollocaUy boundndoo. that [r , :J"[ ill di M .. ..,). oe&!igible. Now ~ a and P he t"", points in 1 rud:t that a < p. C .... rly, (M 0 ",).. (0") .. M..(a ). Yo'hen If' hts i n J, then ",-I(1a ,till _ [o~ ,6j and

(M "",) .. (ft"') .. M. (.B) , 10 d( M o",)(",-I (]a. 8\))

..

M .. !p)- M. (a )

..

dM(]a •.6II·

On the other Iw>d, when If' doeo DOt hel"" g to J. "'~ ,fI')uJtr ,tr JIIt dIM "",).lntegr&ble and .t(M 0

",) ( ",- ' (

[a ,.8[J) .. d(M .. ",)( )a~ , 11')) .. =

(M o",) .. (.8')- M ..(o ) M (:1) - M . (a )

ill eqU&l to M.(P) - AI. (a ) ~ dM(]o .tIl), ~ I. n..er.fonl, in bath """"". d(M

0

",- 'Oa. PI) ..

",J(", _

I

fJ

iii the righwndpOint of

( )a ,til )) .. dM( 10 ,til).

Ne.rt. I Uj>pOI
d( M " ",m,,'))

_

( M 0 ", ).(a') - (M o ",)(~') M (a) - M (o) .. O.

Finally, dIM 0 ",)(,..,- ' (" )) .. d{M " ",)({o"H ill equal to M .(a) - M (.. ) .. <W({ It }).

'I'

ate

Ib short , ..., Iu.,.., 1"00(11 thaL .,,-' (8) i6 d(M o ",,}- Int~ !'or eech S...,.. E and 111M dM (£ ) ~ d(M 0<,»(",-'( £), II....,. ( .... S) .. 4( M OIO}-«uited and

dM ,. ... (4( M 0.,,). CIoarIJ', !dMI :S 'Pi ldiM 0 0;»1). Weshalhay th&~. € (e ,d[". su.tionary poiDI ... beoevf:. tbm! exillll! € J ~ thaL I > . and 'Pil) _ .,,(.). So. i£ I € (e .d! ie not &.Itat;ooa,y poilll. then (M 0<,».(1) - M .. (I • stAliocwy polint_ Lelting f tend til If in ,"" oct ofDOOll.adonary poi ..111 of J O]- oo ,.8'] . ...., _ ,hat (M 010)(.8'- ) ,. M {{J- ). [n p&rticular. if fJ e Ja, b) iI ouc:b thai fJ" belo'Wi 10 J and .8' < fJ". t!lllo

IdiM 0 .,,)1 (.,,- '({J»

-

Id(M 0 .,,)({ 8'))1 + Id(M 01O)({fJ"1l1

IM({J)- M(fr)I+IM.({J)- M {fJ)I.

IdM{ff.l))I - IM. (p) - M (frl l·

Now,

""~,."",

that At i6

.~ht.-c""dnooue

or." is itricUy il>tt
[ t .....

mal .... to be proven ,bat .,,(Id/M 0.,,)1) S IdMl . ...... (t, fJ be t10V poinlllin I SIItisfyi", Q < {JA.QIIte IIrst I,," !1" lleo in J. aoo let (... .. _" . ) he a oubdi,ision of [0 '" .8']. For NCb 0 S ; S n, let I; be ,be Iargeet or .,,-' (,,0(1,)). The I~ 11'_ 1 .• ,1 ( I S j S n) are dlIjoint, And

elt"""ut

iliao than IdMIOa ,81). We (t)"dude that

141M ., ."JI(",-I (]a .P))) S IdMjOo , ,1.1]). AMume ---..Ily that If" doO\OI "0' lie in J . toO Ih&t

IdM10a ,fJ() -

.-

(j _

b.

T~I

lim jdMI([a.:zj) ••<

-

lim Id(M ., ",)1 (.,, - 'c)(t , :zl»

-

Id(M"<,:>JI(IO-'{]a ,p[)).

••

13.( Tbe

Let

... " - - _ ' I ; " "

,~

or a Flmctio<>

m

On tbe other hand,

IdMI({PIl .. IM(P) - Mun l ,. Id(M o ",)HP'))1 ~

Id(M 0,>,)1(1"-' (11)).

! 'Pllh' - '{ I" ,PI))· F inally, when" lkolln / ,

)dM )Ha})

=

I M~{a ) - M (a)1

..

Id{MO I")I({"~lI Idl M 0 1")1(",,- ' (<1)) .

.. The proof ill complete.

0

Conooequet.I]y, under the .. bole hyp«besill of P roposit ion 13.3.1, and gi....., / : X _ F (X :> M el ), F Banach If*le) , f <> AI
10M

III 1- Ill..,"

PropOaiUon 13.3.:l Lel c /10: a poi..: D/ I"./I(. Lkno/.o ~ M, (~, I1t M. ) IN ruI""W'" of M t<> (a,eI (~'P<' [e ,h)). Lei a mapping '""" X :> M (I) inf<> a EJa"od, rpaa. Thm 1 0 M ;.,dM·inlqrabk i/.....t O!Olr if f 0 M, ;., dMi'~ J",. oJ! 1 S ; S 2. In tAu ......, 1M I ..

,100

1M, I + lot, , . P ROOF: C\ovly.

dM, .. (dM )/ " '» - (M(c) - ",+ (,,- * • •

w!>ere I I (retlI"""'i>'eiy, h ) "' tbe u 'p ',, ~ OpooIitloo foIlowa.

e. ,;

13.4

a

The Lebesgue Decomposition of a Function

Let. 1 _ (a , h) be .... ln~ in R (.. <: b in J[) , and~ by ~ Lebesgue mo:uure on tbe lIIItUra/ """,iring S of /,

Theornm 13.". 1 lid I' . . . wmp/e% ~ (1ft S .ne! M an ~ .." s'~ GI l' · II,. _/1. + 1'. ;, to\<: r;-..g... 01 .*",...ililm QI", 1IIi/h "''''"''' III it (_ IAoooI I if I«allr it·;,.t.,....bk .104 1'. ;, ~"t from it), th ..... I",. it.IIIm"" ~I if, M w oftri..zit« I(~) ill if. PROOP: ~6ne aD In~ J In R all koIlow1: J .. 1 if 1 .. open. J .. Ja , +co( if l -1o,1oj. J .. J - oa.i>l lf I .. [.. ,I{ , I.Ild J .. R If 1 .. [.. ,11\. Writ.:". fur t~ean<)<>1 -Ilnj«tlo<> fr<)m I inlo J. T 101" tile natural ..,."In ", In J , ADd It.J kor ~ "" 5m ... 011 T . For e""')' ""..empty T «I )c . "1, ".- I( [e , dj) .. I.Il S·~ , bon_ It ill ~ual to 1$ M (a) - l'(1al) on 1-"" ... \ ...ben .. li
1. 
   :r E J 0i0>d> Ih.t D( r(p)j(:r) ",,1m. :if ...,. deri ..u!oe D( ... (p»)(:r).
   
   poi"'"
   
   'l'btor8n 12.1. 1 """' ........ lhat . at It.J.III",,* all % E J, M haoi deri_ tloe j (:r). H..,."" at A.al",.. aU:r E I , M haoi deriVllti'"l/(:r). a
   
   n..llnltlon 13.".1 M: 1_ C is oaid to boo: aboolutely conlinU<M.III wben.eoer. lor tao;:b f > O. tber& eo:lIU , ;> 0 iIUOh that E ,:s;.:s;. IM (A) - M ("';)I :SO t lOr ~ fiDit
   E. :s;I:!O.(A. - 0,1 < 6.
   
   «",..
   
   - -'
   
   s....,'"
   
   In thil c_. If 1 io (Oml)lKt, lhen M is of bowt.d«I VlIriatioq.
   
   Definition 13.4.2 M : 1 _ C i0oi oaid '" be io>o:IIJb' &boo!u""b' CODI;"'- il IU reotrictlon to ~ (Ompod. ",billl..-vai 01 I ill abooIute1y _iIl\W)O. In \h\o cue. /of i0oi d Ioa.Ily ho"nded variation.. T heorem 13.4 .2 Ld I' ..... e»mpkt: """"~,.., <>II 5 one! M 411 ;~finiu. ;"h"wl 011' . .4........, ~ I' • ., ............, o. wn a Iiu in I . n .... M i. /.oaoll, uNlliftS, .....ti.. IiOIII if ane!1nIIt il I' ;, a m.eouun: ";/h lwm it. P It(IOr: Lee. f bot .. real-...Jued !t>lute1y CiOnUn ....... II and only II M It. By n..".,tm 10.:1.2, \Ill <). il and only If ~ 10 loo:aJ~ aheoIutcly OOftIiWOlllo, whldt ClOGlplelollbe proof. a
   
   'VI
   
   ate
   
   Deflnltlon 13.4..3 M ,1 - C is SBXl to be 'ingulac whenever it ill oIloeaIly boo.u>o:IM variatioa and hM deriV8.ll"" zero ),.~.~. Theorem 13.4. .3 Ld I' 100 4 """,pia """",...., on S aM At an i ruh/iniU inl'fNI oj 1'. "111 ... I' it fingulor (i.e. , /fuj<>inl fram ).) if and onlJ iJ M it
   
   riIIgtJ
   o
   
   PNXW: Thill foIlow-t Imm.dlatdy from T hoorem 134.1.
   
   M be G Jurw:n.m
   
   Itcton
   P~tlot! 13.4.1 Ld
   
   o/loo>ll~
   
   P ROOr: We may 8UppOoIe \ha~ I Ia romp.d. OJ,,,,,, • real /lUmber ~ by E,. the set of t"",," % E 1 - D such lha,
   
   .
   
   .,.
   
   hm3Up
   
   ,_.
   
   IMeV) - M{%)j
   
   Iv- %I
   
   T
   
   > 0,
   
   i
   
   > -.
   
   •
   
   F\>r e.ery V E D , let t, be the inte.--,ection 011 and the <:run~1 subinterval of R with center V and lm«'h 2TIM(v) l· Each % E E. beIooip 10 Infinitely many I" beca.- the", an. infinitely matIJ' /I E I ouch t bal IMM - M(% )l/lv - %12: I/T. T he function g - EoEV I,. ill lDIuIite OIl E.. But ).O(g) 'S EpEv ),(1,) 'S 2r . EpEv IMUI)I lot finil e, h«auee At lot 01 boo'nded variation, and "" """ cnncloo. lhal. E,. Ia ),.nts~Kibie. The JriOjkkilion follows. []
   
   tm.
   
   Theorem 13."'.4. ( Lebeque Oecompoooition of. Function) LeI At : 1 _ C 100 o/locoll, ""'Mod .",riIIlion, aM lei %0 E I . Tht:n At it u.. "'... M, + M. 01 " Iocollp ..booloJdv o:mIin ....... ~tum AI, and a ~T !tmcli"", AI. FI.o~ AI, aM M. are lII1ifadl1 dde",,"''' bov the """'.tiIioo< M ,(:ro) _ M (,..). At u d;g=n-tiaw., at ).• .JI pc>tm, 011 , it. del L :Ci,c M' (defin.et1 )....tm...1 e~) i. /«»lly ). . i..u,n.W., aM M it an N efinil.e inlesoml 01 At' ). if and onIr if M it kw:t&Ily ..b...tv1elr """,lin.""••.
   
   ..m..w
   
   PROOF : F\>r all :r E (a, "'I , ""rite M +(z) !or .he ~hand limit of M at z . If b 110:0 in I , ""'" M +(b) _ M (b). "Then M+ is I.ll indefinite intes:r&1 01 the S.;"itja ..-un: .1M. Now dM can be written I~ + 1'. , when: I is locally >.--~ and /'. ill d~nt from ).. Denote by AI, the indefuUte inlegr&l of I). dw . takes the .-oJ"", M("<:I) &t ;<0. Denote by M. lhe JlUlil of M _ AI + and the illdefinite inte:gn.l o f /'. that hM the ,-oJuc M+ (zo) - 111(;<0 ) &t zoo 'Then M , +IM. - (M - M .lI. 1/1 An inddinilrem 13.4.1, M+ hall derival;"" I (z ) at ).-&1_ all poin18 :r E 1. Proposition 13.4.1 . hm . " ... that M "'" deriYaI!ve f(:r ), at >.--&1mr::osl all pOinljl % E 1.
   
   :HIO
   
   13. St ..It;. I~
   
   -
   
   Finally, 10:\ M, boo a locally abeolutcly o:onU.DlIO\II function from 1 ill!.(> C "bid> W:eI thol value M!:to) at "0, and IoH ;\1" 1 _ C be rlincuW. Su_ M .. AI, + M,. At ~moM ..n :t E I , M', ..... detiva.tivtl f{~) , Th", M, .. M, and M,-M - M, _ Af _ A1, Finally. if At II k>c:alIJ abet>luldy oonlinoous. tben dAt .. f), . 0
   
   ,ba,
   
   -
   
   -
   
   _At,.
   
   We lIOte t b&t 11M: Le~ OOt:ocnpoIilion 01 _\I~ l enerall1el the Lebeifcue detompoait ioo <J fuoction<
   
   Theorem 13.4 .5 wI III be A.. ~ jotndiqrt on /. M ~, G fi'lil t 4riwtioe A.... oI- , !he. /toru:tiim AI' (~),·4 ...) io ),. ,,,/qftbk, <1M
   
   """"II,
   
   1#
   
   M '(I) dt S M{P) - ,1.1(0)
   
   ffl<' ""I O.
   
   11 e /
   
   MlIU/J'i>If 0 < 11-
   
   pftOOpo, By TI.eotun I1.U. 1M fUDCC;"" M' II dd'it>ed ),.~_ e:-o-eoy,,~ and IlIoo;a1ty ),.lntecrable. s;".", dM II ~II~. 11-' be writlf:n M')' + 1'.. whM! 1'. is rlinsular and pooil;ve. Hmot
   
   r
   
   M'(I) ~ " (At'),XIo ,M) ~ dM( )o ,8\ 1-
   
   ,II fF ) - M +(o )
   
   S M {.8) - M {o)
   
   o
   
   k>r alia . fJ e f at.,lofyll\11 <> < fJ O~
   
   ,ht.,. it AI : 10.11 _
   
   R is theC..,,,,. rlil\llUW fu nction, theeq....tlty At'!I}," .. .11( 1) - M (O) fo.il!l. Now we 00CMdder real mpm_ OIl S. If " is one 0<>C:b _ _ ure. then, "" e-.;h :t e I. 'fI'e defi"" Qp(:t) and Il,,(:t) at. foI:o::.....
   
   1;
   
   Qp(:t) _ oup inf " He .d) ) ~l:' (~-"I 4 - c Il;o{:tl -
   
   In!
   
   &up
   
   0,/:' «-"1
   
   I'tJe.dll, - t
   
   where (t,,,! runt Ib""",,, lbe d-. Dooe S«u
   o.,finitkm 13.4.4 Let M : 1 _ R. AI each point % E Cd .bI. M hu uwr and. ""- riehl-band derI... t! ..... ~!\.ned at. fol~
   
   .•••
   
   .
   
   ~- (l ,,,:t ., " m,up M (:t+h)• M (%) '
   
   'VI
   
   ate
   
   .-
   
   . . fM(~+hl• -M(:r) . D liZ () _ ,,mIn
   
   "
   
   ••
   
   LOkrwir<:. &leech point % £)<1 ,b). M h.., u~c and Iooo-er Lt~hand dcriva\i_. de6-' . . ~
   
   .-
   
   MD,:r)_, ',moup /If(:r+h}• - M (:z ) " '<.
   
   .."-
   
   ._.M(:r +h)_M(r) " D{ , z) _ lm,ru • . " "
   
   Clee.rly. if of 1-', tbtoa
   
   1"". """
   
   ~ on S and M .. """·vaIued iDde!inite imqral
   
   .wD(r):!:Lb« z ) .teechzE ]a .h)
   
   D.. (:r)~Jbi( :r) . teadt:re (a , fI( luch thai. ,,(Ir )) _ 0. N_ lei. I' be • politi.., linguw " I ' "'''' on S and At • """-valued lD .t ,...1lI>0II1 all poi"'" of '" • b), and it hao ri«hl-hand d/:ri..uive +0> .t 1""0'"", all r E (.. ,Ioj.uo::b that jJ({:r )) '" O. In particular, _""'" 1'".018 pofIItiw, ~ and 1IiJ>«uIar, 111 bat derivalh.., +0> &t ,,~t all poinu of J , heot<: at u""",""n\ahly """'1 poinl& For """",pie, tbe Can_ .!ncular fww:t.1oa bu derhuhoe +co &I UDOOUlltabIy m&r\Y poinu of , II. F'in.lIly, ~ ha.., tIM: foIIo..-I..,. im~ .... ult.
   
   to
   
   Theorem 13 •• •6 (Lebeo.gue ) Ld 160: .. !«GIlt )' .i..u,n.bk m""""'f from f into .. mol s.. ....cA.po«. Fur ,I.· at"",>! all:r E I ,
   
   --'-- . [
   
   II'-:t
   
   •
   
   !/ (I) - !{zlldl
   
   o
   
   PKOOt": Th. foIloon from Tbeot em 12.1.2.
   
   Coa!leq"..."ly, for A-a1rnoss. all :r E 1, (IICw - z »)·
   
   Ifr )"""' :r I ...... to :r.
   
   f: / (I)dl ron'bPS
   
   tQ
   
   282
   
   13..
   
   ~iohjeo
   
   InlqI'o!
   
   Upper and Lower Derivatives t
   
   13.5
   
   Lee 1 _ [a
   
   ,bI ( _itl> a < 6) be a (Ompact 'ubin~'II! in R. Md refer
   
   I.otbMcue ""'''Ire.,.. I .
   
   Lec 1 : 1 - R boo OOQUnuouo aad let
   
   \(I
   
   A ilr
   
   I _ R be regu!&led . SUJ>P lhal, a~ ~ % E 1 _ D , I has dcri\lllli"" r(%). TMn I (s ) - / (") " g(t ) dt fOr aU % E 1 (Section 3. 2). The following tMot en>. due I<)~, exteod8 Ih. T
   I:
   
   TI>..,. " m 13.~.1 LeI I : I _ R k ... n/in ........ 111M", ~ • .:ooo ntdit ...t...1 P GIla. 6{ ,..,. UI4l IJI (%) (rupocli ... t" O/ (S)} it jirriU /# all %it< [a •6{ - P, .."" i/ ~ /ofldWto % - 01(%) (,...,...,Ii.d'. :I: - DJ (%») from fa , ~ ...1<1 Jr U A.illcl",abll. ~n / ... "I ~ ...,;"1,,,.. Qrwl/{%) - 1(" ) < I. 1'(I) dt (rup«~, I {s ) - / (..) ~ f(t ) dt } /(1<'..u % E I , .muc /' kri9llGtu tile hri••ti"" 011 {4
   I:
   
   P IlOOP: SuPl)'IM I~ II • COUDU.bio: ""beet P of "'.6{ ....:to IIw IJI(s) .. flnlte for all s E (a,t{-P, Alto, a<>w<*' thal lbe function 1" % - Oi(%) from r.. .b{ Into Ji: . A·I~_ Lec (". )"l!' be ... t llWIlenl.llon oflbe POib~ In P . On- c > O. there ...... a ~ ,..,rni<X)ntinQOo.>l function, from / into Ii, Mrklty than /' al ~ pOIbt of [a ,b{ - Po ""'" tha. t. g{l) dt S t.1'(I)dt +c . For a bed 6 > 0,....."k\Ction FI:
   
   ,_tel'
   
   1 ~ %-[tI{t)
   •
   
   L ;. ..<0
   
   Tbo .... J _ h E / ; FIC"') C': 0 ilr oJl .. S z < ~ ) •• (Oml*\ interwJ [.. . cj. Suppoeo lhal c < 6. If c does IlOl ~ \(I P, ...., """find 0 < " S • - c anaU enoo&b to lbat 9(1) «0' II i°l'(e) o.od V (I ) - / (e))/fl -e) it NIlalJer than /,(c) + 6 .... aU I E 1e ,c +11); Ibw F. {s) - F. (e) • p08Il!ve few: ell % E Ie. c+ "!; bo a c+ It ~p 10 J , ... hleb contradicu tbe delInltlon of c.. On the at'hand. If c .. .. t lor """" Index 1. 1'-" u-. 0 < It S ~ - c ruch lhat
   
   " J. <
   
   •
   
   1(1)
   lor ~ Z E [e .c +
   
   "1,
   
   bouQf; e + " .ttiJ1 belo"" to J . whleb comrad.icu lbe definilior> of "We ooudude II>al c '" b aDd lbat F.(6) - Fd a) ~ poQtive. Si ..... ~ o.od € ..... arhllrMy-. It foI ...... that J(~) - 1(" ) s t. r CI) .n. . In 1M abo¥!! &'fIumenl , ....., "'"'Y ~ .. and b by arbitrary poln~ ~ , • (, <~) of t The ~ "''''Ion % _ I: I'(t )dt - (I{s) - I {") .. lbet. il>creMl,,«, and 10 l it of boo''''''''! vari&l1on.
   
   Noor IUppoII: tbert ii, COIInlable IU b8eI P of J.. , bI ruch thlt '0(,, ) It finite for rJl " E!o . bI - P. Also. SUppoll: that the Funcr.1on " .... 10(,,) from !o .bj into "R: ill ,\-Inttcnhk- Let ("')~~1 he III enun'1m\tloI\ 01 the poinu in P. Wlth ~. 9. and a ... hebe. doofjoe F, ... the function
   
   I H .... 1(6) - /{" j- [g(I) dl - 6{b - ,,)-
   
   •
   
   _.L>0 ; .
   
   It It easy 1<> _ thlt F, (.. ) "' ""IIati"", and the same azwumeo\ ... thlt used aboYe JIfO"1'I tlw the function" _ / (6) - / (" ) f (I)1lt iI iLKl 'i",. "IbeotMr_rtioOloltheR&temeDt IoIIow from I""equal.it;'" D f _ - D - /
   
   t.
   
   and/D ",,_-ID.
   
   0
   
   UI I .. (0 ,b) lao on inl<"n>Ol in R . oPltm. I' (tk/inetl"",rid. 0 x l) it 1oooIt. ~ _i~1 5' c~. Thtn l (fJ ) - 1(0.) .. 1'(1)
   0.
   
   U.~. l
   
   ,r.
   
   """"""*
   
   f:
   
   < fJ.
   
   P ROO'" It ",ffioeo 1<> prOYe tho.l II(1~) - I (ltl) .. ft(II " 1')(I )dt lor I':Vny ..,.,Ii...,.,... linear brm II on F . and lhe deoin!d .......It DOW foIlowI from Theoo(.u 13.~. ]. 0
   
   I
   
   Lot J .. ",.ij bo. """'-' iat.......J ill R (_ < b), 1; ...... ric ___ •• _ d........ . do: ..... Ilr p , _ M , I _ E • ~ 01 boumdod ..... ioa. RIo . - y IUbdivioiom 6 .. (...... ,<>.o J of I. pUI
   
   L" L "
   
   p(M (s. _,), M (s. ).
   
   '5'5·
   
   s.._ M • risht_i_ <xl ",. ij, P ....... bat, "".u 0 :s; V, < V _ V(M,lo , ijj, ,here 0." 6 ;!: O...a. I .... Eo 2: Vi "" <:Wr'J oubdiyioioc< 6 011 .boeo n-.h . I.-.ba.o 6 (lot V. bo ia lV, • VI """ 10< 6 ' .. (........ 4 ) be • oubdi..... lor which E". :!: V.; ,I. . . I < ! c fOSl~~_ ' (01,., - 01,), ouch .....
   
   (
   
   P M(d,j. M(~j
   
   2
   
   ~,
   
   V, - V,
   
   I :s; 2(]>_ 1)
   
   :s; z :s; 01, + I b -" I
   
   Lot A be I..ebet&uo,. , "ore <xl I _ 10 . I]. """"ioa &00.> I inI-o R ...... ' .... ' (0) .. 0 _
   
   Imp'"
   
   :s; j :s; , - 1).
   
   , be • / (1 ) " I
   
   ~
   
   ~_
   
   I.... ,-;"&
   
   lJ. Sti
   • no<m. ~ that ,be hlnetion
   
   I . Eqaip R· ... ith it.o E_lid
   
   M : r-(r , f(r )) ;, of """'tided 2.
   
   from I iD1<> R'
   
   ...na,,,,, tond.hat V(M . I ) S. 2.
   
   .u...- _
   
   ,hat I 11M dori....;v." 0 A' ),..01_ 011 poInu of I . Fi • • > 0 and. lor __ in• •, .. E N . dec .... by A. tIM Od of t _ r in 10, 1) ...... that tho _"""E I. u E I ... < :r S u, tond ~ - .. S 1/ "1,,,* (1M -/("1)1(0 - "J Cb.ootoo n 110 tha, ~ . (A.. ) ~ I - <. Write J lor ,boo Od of tJ.:. A E (1 ,2•.... n) .uch .11&1 I(A- 11/.. . t / nl _ A • • and ..!ite)' lOt tho complo "' c' of J;n (1 . 2..... n) S_1.hat
   
   s.·
   
   0-_.11&1 V(M .I) _ 1. ~
   
   """'p",:\ ...
   
   I _ R ho ()I)tIIjn_ who", I _ I.. . I>( 10 • b;mervoI of R. Dt£Do by fl • • ho. ... of'ilM 'I ""'ion of g Ie I . and by 0 _ 'ho Od <>fief\.. ""panobo of 9 ie I : r E ,," , l0/liot In n. ( ","p r ("'p r(z ).
   
   ld"
   
   ,Iou.
   
   p....., It..! O~ (tsp
   i,,,,..
   
   .,r
   
   e
   
   e.
   
   I .. . be
   
   lc>Ucooinc -.cloo . ..... in "";1<0 £'(g. ' i "') ond E""u.. / :.., .... ) """
   
   C' 4
   
   ld I.
   
   I :I _
   
   1.
   
   o.nd
   
   R be i..., [77inc &nd """,Ie"",," (1 _ ,," . I>().
   
   p.,. oll < e )O . +<»/. _ by E, .I>o .. t 01 in I. p""" t!>at (z e Io./>(: D' (r) _
   
   tw
   
   e
   
   it ..
   
   ),.."'cilClble.
   
   ,w.<-~
   
   +00, ..
   
   Included in
   
   01 f er } -,."
   
   n,.. t:..
   
   &ad
   
   ODd ..., I>< ....., ratior!.Ul .oeb t ha. 0 < " < ...,. om ... <>I'"" .,,) .. E;: 01,," . 10) _ 5oI1ow..: E; . )a . /o) &Dd, lor owry " e Z' . ~, .. tl>< oeion d 1- .8! "'.. 'h"""h tho eluo 01 011 "",,_ttd """'pane .... 01 E;:. SMw that .l{E: ) «, I...,," . (~ - ..). Oed""" that t ho ... (" e la . />( : ,D(z) < ,., < ..., < D" (:<)) io ),..~ &»d """"Iud< that ( r E )a ,101 : , D(z ) < D' (z)) II k'1llCibie. ld
   
   T'!
   
   :s
   
   ••
   
   3. 4.
   
   ~,
   
   z_
   
   -z)
   
   RepIaoo I by 1.he ' """,ion -II ftoao 1-6.- . ) 1_ R. &lid dod_ fro... pOrt 2 thot I.. E '" .10{: 0,1,,) < 0( .. )) io ~"'cl~bIo. UoInr; p&rU I . 2. and 3. P"'M' thal I t.oo. &Dito .... i_i"" .. ~.I_ aU psK"U of I . Deli ...
   
   i :R
   
   .. EI -
   
   00 • •
   
   - R by j{z ) • I f.. ) b aU .. E I . I I.. ) • I f.. ) lor 011 1. and I (b) Jor aU .. E ~.-t<><>( . ro. -=II n E N •
   
   1(.. ) _
   
   .......... tho Iw>o:tion ftoao T 1I'l10 R. II f
   
   aU ..
   
   ~
   
   :I
   
   _ a + io • ...:b that f fz) lI .ho dtri_i"" 011 ..... Jor ). •• 1 _
   
   e /. '-,hot
   
   rr
   
   II)".
   
   !'
   
   J
   
   ~!'!. In 4)., . -
   
   Let. I _ Ia.1I! ho a<Xl!*'l oubiD~ of R. " I.ebtIopo II ,,"' .... I I &lid i« / : I _ R + ho ... intqrabIo. Deli ... by F tho lw>o:tioG I :;u I . 1Io.oj tI,o.
   
   J
   
   I.
   
   FiJot,u· ....... lhat I 11-' .. liromiDIIiOUI.Let.(I' ) ' ~l bo1 ... iDa i", ooq_ 01 """,,i~ h",e,iono from I into R + .. _ "PP"I" on. I ....., II / . Fbr -rr n E N . - . , by ,ho funcl.ioG .. ' !Io .", tIp. FiDaU)r, i« 1'" : I _ R + ho....:ll.hat 1'"(.. ) II ,be deri..,i.. 01 F .. ....1 _ 011 psK_ z 1_ F .... 4). 5 _
   F.
   
   r
   
   e
   
   / (6) - / (. ).
   
   f l.
   
   ,ha, r _ _e.
   
   ".o.e.
   
   Let. I &lid " boo _ ill E 1<. . r.. Aloo. i« (J. ) .~. be ..... , .01 iDa i"C fwlct.ioao ftoao r iDto R + • ...:b ,hal I _ L..Z,J• .. 6";\0. Let. N be .. ,.. ~..,boo:< 01 I _ lhal I &lid lho I • .... di& ..",io;b!e .. aU psKnt. 01 1 _ N.
   
   E ,sos.lo. and
   
   _.haI
   
   L
   
   n 2: 1. PU' On . .. E /-N .
   
   t.
   
   Let. (",). l ' be. out-:,_ 01 I'. )n~' _ lbat I/ - ... I!' 2- ' b aU ,, 2: I. Oed ..... from pOrt 1 , hat 0'•• (:.:) """...... 10 f (:.:) .. aU rET - N .
   
   fo< _
   
   o'.( ..)!, f(:r) .. 011
   
   ,..&1_
   
   3. Coed""" \bat fi r ) _ tboooftm).
   
   E. z• r. (z ) b
   
   ,,·aImoot aU .. E I - N (f\tbini'.
   
   'VI
   
   ale
   
   14 The Fourier Transform in R k
   
   Thio of ,. po ' "mly P, <:aIlod .be _ _ . l< fImctIoa '" P, It .. >eO')' III R' Imo ",.~) .. on ru"",,,- cIe/i.!>(Id in R ', Thio wiD boo portlcwady ..." .. u ...., ... ben .. """", rm of ,. _""'utio:l ........ pn><_ ' boo duct "'. Il00
   -,,,I """- [. ,,-.Lo_ .....10'''... '''' _"_ t
   
   Mq'''''''' '"
   
   it ."'"
   
   Summlly In \Il10 - ,""" .... lioe _ "'" "y ODd ouffie\en. <:Ondi'"- "',. <1_>.., 01 ormIrin:cofa """"mpty """" on*," of R' , ... "" .. _ ..... (Propooru.:- u ,u to 14.\.3) . 1~.2 In R' , tho dio.ributl<Mo fwo:tioG of . proboobili.y defi_ on <.bo Borel" " s ' bn .. F : (,,', . .. . " . j _ p ( n «.~.l %,1). A oeq......,. of probabilltiM p~ with dlouibut.iooG hiDoI.... F. '-"''6 -.kIy to .. probabiU'y P .nth <J;ou;batiooo 1111 ..' ..... F if and oDI)' If F. (,, ) __ ..... to F (" j at oovery pOi'" o f _i..w'lI of F . 101.3 In thio __ ..... _defi_ t ... --.n...o.matri:d... • .5 im p<>rl ~ ;" tbo ,~ of proboobili'1. 1~~ TIlIo -.-lion io dnQt4d 1O •• bort 1Ii>dy QI.,ht nc>nnaIla.... ;" R · . ".1
   
   II'
   
   doIi_ "" .............
   
   - "".
   
   f
   
   en-
   
   -.,,,,,,-1>0-.
   
   o n .....
   
   m
   
   ..
   
   •
   
   14.1 Measures in R .I: LeI. fI be .. nooempty
   
   open rut....t of
   
   R· and S tbe natural !lemirins In fl .
   
   Now let " be .. quasi·measure on S. ",,"ume u.at ,, ( A~ ) teDdo to ,,( A), ro.. nwy (DOOeDIpty) S-rectang~ A _ n ,S':;:. )O; ,&, ) and nwy dec. oj", MICI' ......... (A~ I A~ ' of S·~ of tbe form A,. '"' n,s.s.j that A '"' n.~ , AA ' W. AhaI.I pro'" that " ie .. _~ To bqln. we PlO'" tho followlns ~1Ilma. Lemma 14.1.1 LeI A '"' 60: 0" ~ 8~
   
   .t..e/I tlt<>J. A '"'
   
   u. ~ ,
   
   n,<.:;:. )0; , &.I 10: a .. S -m:I4I!gk, aM" Id (AA )A~ I "/:S·m:lang'" 01 ~ I.,..,. A. -
   
   A... TIKn ,,(A.. ) _
   
   ,,(A ) ... n - +00.
   
   n':;:'s.la: .6;)
   
   can find ruJ numbel1l 0" . . . , 0 • • ucb that 0, < a" . .. ,1I. < a • • "d n ':;:;s .. (lI;. &.) II mnt&lned io n . By induction on 0 ~ I :s: .1:, we _ that, P ROOF; W.
   
   far.U
   
   Q .. ,
   
   E {IlI .. "br .. ,} .•.. ,.,., E {a• •bo.}.
   
   ,,(]a, , ~I ]( ... ]( I... ,brl]( ]0,. , , Q + d " ... " )lI .. ·.,.,D-
   
   L
   
   .....
   
   (_ I)I.Il . ,, (1o, . C/.I) " . .. " ).0 • •C/.• ]),
   
   I C I' ,·· J )
   
   a; ror all 1 ~ i ~ I belonslns to J , clI _ b; for aU I :s: i :s: I !lOt beIort&ins to J , and CI" _ C, for I + I :S: i :S: t.
   
   C/.I ..
   
   In pattiocular,
   
   ,,(la, ,bd " . . . " la. ,6.]) a
   
   E
   
   ( _1 )1/ 1. ,, (1o , ,col ]( ... "]0. ,C/.• I),
   
   I CI I.. .... ) .. hen! CJ.; _
   
   4,
   
   or C/ .< _ b; as i belonp to J or DOC. Similarly, for all n 2: I.
   
   ,, (]a~ .6,) " ... "
   
   )a;' ,b.D'"'
   
   L
   
   ( _1 )1/1 . ,,( jo,
   
   ''',,1" ... " ]0. ,,,,,]) ,
   
   I C P .·· ·" )
   
   when!",; '"' a: or ",.; _ b; as i ~ to J or 1lOC. Now, ror tva)' J C { l , .... t j, tbe MICI""""" ("'.• )~2: ' dec... to c/." kif all 1 ~ i ~ .1:, eo
   
   1I(]o, ' '''.11 ]( ... " ]0 • •",.. J) _
   
   .
   
   teDdo to
   
   1I(]o, , cJ. ' ) " ... " Ill • . ",..]) .. n -
   
   C
   
   Lemma 14. 1.2 LeI A '"' n,s;s.]a.. ./.1.;1 lie "" S_m:I4ng/
   •
   
   2M
   
   I.. The Foutior n....fonIo .. R'
   
   P IlOOI': For € > O. let (B,),s.s'" be • .Iini.~ putitioa 01 A, u...o S· ....".. nps wch ow. V,.(A,) ~ r; '$:O"!i '" W8,,)I+,13. We m.o,y "'1'1"* tIw An 8. ~. for.u I <" ~ q and thu A n B,;" for all 9 + I ~ " S M . for . .uilable I) S t :S At - I. For....:h 1 :S ,, :s t. B, baA the form n, s .s. Jc, .01,). when! Q.; :S c; < d. :s: ~! for.u I :S j ~ t. By LcmDIII 14.1.1. there ill all S· 11lCt....1e n:, of tt.. Wm n ,s 'S. Jc: .04). witb ~ " c, for all i. l uch thu II'{ Bp)1< \II(n:,)1 + €Ilq. Since An Bp _ O. then: exilill I :S i :S It oud> that (0 ~ &" and, lor R ~ enno"b . ~ ha.-e b\':S: hence A~ n 8'. _ •. n....b... if _ ....... R """"enno'''' 10 lhaL A~n 8'. _ . bt.u 1 :S: " ~ q, aDd lee {~ hs.-s: "" (.Me N 2: 0) be • putilioD 01 A, n A: !nib S-,u"n&I"". _ _ Ihu
   
   <,
   
   L
   
   V,.{A. ) ...
   
   V,.( A~)_ V,.(AI I
   
   'S·S···
   
   :S
   
   L
   
   L
   
   1,.(8,)1...
   
   'SO"!i.
   
   :S
   
   L
   
   :s: r;'l':rS:N ~B', n
   
   i
   
   , " ' S,S M
   
   11'{n:.)I
   
   L
   
   +
   
   .t:. )l 1or .u I
   
   11'(8,)1
   
   ,,'s.sM
   
   '~O"!i,
   
   Hilt Wn:,)1
   
   11'(8 ,)1 ...
   
   +~.
   
   :s: , :s: q. bflcaUll! tr. don "'"
   
   . . - A.. . IlD
   1Jo(~)l< \II(8"n A.)I +
   
   L
   
   1I'{8.nA~)I·
   
   IS' S"
   
   V,.( A.. )+
   
   L
   
   VI'{A;. l:S
   
   L (L IS'S"
   
   'SIoS.
   
   L
   
   1,.( n:. n A~)I +
   
   111(8. n A;. J1j
   
   ..'s,s M
   
   +
   
   Mil, n A.. )I + ~ •
   
   L • "'SO"!i '"
   
   ."" V,,(A.. ) :S
   
   L
   
   ,"s.rs"
   
   jp.(8" n ..01.11+
   
   ¥.
   
   Now, lor f!WrJ q+ I :S:,,:S: M . tbe ~ B, n A. - n 's,s. le. .inf{d.:.II:'I) deew ' 1!I1oO s,. n A,. n , ~ ,~, )t, , lnf(~ , ~)] M R _ +()(), 10 ,,( B, n A~ ) ~
   
   It.l McMu"", in R'
   
   t.o 1'(8. n A). Thm, b
   
   f\
   
   289
   
   large enough,
   
   L
   
   V,,(A.):S
   
   !p(BpnAlI+!,
   
   u '5~'"
   
   o Lo:IIUna '''. 1.3 Ut A .. en S._I<mgk "rid S·.....,unglu. n..... VP(A ) .. E.2;I VP(A~).
   
   (A~ )~2; I
   
   "pGNiti.m 01 A ",to
   
   P lIOOf': Si!lC'e,,",. quas>-me ...u"' ..... kJ.... tl1O.t E.~I V,.(A.):S VP(A ). Let f > 0 be ",..ea. Applyi~ LemmA 14-1. 1 t(I Bee tbol there "' aD S'reclanPo A' I<X:fI t hat :if C A and VII( A) < VII( A') + t/2. FOC' """"
   
   V". _
   
   •
   
   n 2:. I, t~ ",,!ttl an S-n'<:t3ng\e A~ such t hat A~ c A'. aDd V,,{A~ j :S
   
   •
   
   V II(A.. ) + fi/'r" '. Since :;P c U.~ I A' w, lor n latg.t ell(l"lh, .... ha"" :if c
   
   •
   
   U'~_5. A' .C U'S_~. A:".
   
   VI'(A j:5 V,.{A') +
   
   ~:5
   
   ""birb """" IIbon tho.t
   
   L
   
   V,,(A:') +
   
   IS"S.
   
   V,.(A .. ) H,
   
   IS" ,. _
   
   E
   
   V,.(A')<
   
   E
   
   ~:5
   
   VI'(A:, n A' j.
   
   IS",,· V,,( A):5 L: V,.(A .. )H .
   
   ."
   
   o
   
   Lo:ttlma 14.1." ,,(A ) .. E.. ~ I ,.(04.) ..nd"" 1M AW<>IM.u "I um ..... 14.1. 3.
   
   P IW:)Or: If 1'(A) ill • finite p&rtitioo A " .. " A •• tbet>, '" all n;<: l.
   
   II'{A) -
   
   L
   
   105_,."
   
   I'(A.. )I:5
   
   ()i
   
   E
   
   A into
   
   ........... -,,,,,
   
   S·~.
   
   1,,(8 11:5
   
   OO
   L" V,.(B )
   
   ill ' - than
   
   o
   
   14.
   
   no. I'ourie.- T.-.Jom. iD R '
   
   Prapoolt!on Ho.1.1 ~l I' he /I .,.....,.._~ "" ~ OWlllmliltmtring Sol \1 . " if" if ..... onJr il p.(.4~ ) k>l
   ......,g,.,
   
   n
   
   l£
   
   n.
   
   II _ n ':5,.ls. l" .. b.l Ie. ~~ ~ R' , lor
   
   &11)' !l1b6e!.
   
   <:1- (<1,) , S; ~""~ <1, -
   
   J of {~ .. :., kl
   
   .... If . belong! to J. ~ <1. .. b. ., . "'DOt bozl..." to J. 'Then. fa,. ~ .... rwx % of II. thf,nl is • unique, J c {l. . .. , k I for "bleb % .. <1. (- 111/1", <:aI 1o!d the fignum of :r, &nd iI wrilten scn,,:r. l.d 11 and 5 be as btl'OIe. and Ie\ F be" function from \1 into C , For evoery put
   
   S·roctanglo A . pm
   
   ,.(A) ..
   
   L
   
   (-I I"I. F(<1) ~ 2 )ogn ,,:rI' F(:r), • Jel ·.··..'
   
   Propooition 14. 1,2 '"'" fo&Wn 1' : .4 - ,,(A)
   
   from S int.> C ;, 4Jdit>«..
   
   PIIOOP; Suppaee lilat eod> .ode ).... b.) .. I, ol an S· reet;&llj!be A is panltlon.OO inti) n, tubinterYUi J;.j - 11.... - •. 1' J] ( \ 5 j 5 "')' ...·ho:nl ... - I ,A < 1;•• < . ,. < I ; .... .. h, . Then tbe n, ... ··· ". ~ 8 ;, .... J. - J. J. ,,·· · ><J.J.
   
   (ISj, S ........ IS}. Sn. )
   
   (\)
   
   parti\.ioo A_ Such , pm.it.ion II called rqular , We fine .,...,...., that " is additi""
   
   10. "",,13, ~"il~
   
   p.(A)"
   
   L
   
   ,,(8, ......... )
   
   (2).
   
   UO .... J.I
   
   The ri«!tt-hand !lid<. ol (2) II L:B L:k sgns "'-F (:!'). ",l>enl the (>Ute< tum extell
   Ls L• !I&'Is:r· F(:r) - L• P(J: )' L ~ns z
   
   (3).
   
   U
   
   ..-Jo.en,.
   
   OD
   
   ~
   
   of
   
   ex\aldo
   
   .ode, ,be ...,In onun e>:tends ~ eod> " thlu is & 01 the ~"'" B. and. fo, fi1«!d :r. tho: in...". BUm
   
   tbe right-hand ODe
   
   or _
   
   ~
   
   all 8 ol whlch " ir a ","ex. l'ow BU_ that" io a • ..-to; ol ODe or onore of 1I>e """"fit"" B, but Ie 1"10' • ~"e>o of A. Tben there mUlt be an I (I SIS t ) IIUCb that '" is neit""" ... "'" 10, . Fix one I (there IDA)' be ~",a1 ouch I). Tben '" .. I,,, wilh 0 < j < .... 1l>e """"'n&Wo (I) of whlch :r is a """'" therefonl OC<:W' in ~I ... B' .. BJ,., "'_ •.1"' . ......... and B" .. B;. _.Jo_I ..... . Jo • •• _.... , and ogDB"" .. -"&"/1"'" ThIJI! tho: inuer ""'" .,., t be rilJ*hand tiOO in (3) is 0 if % 18 _ a """'" of A_
   
   'VI
   
   ate
   
   lU M........ in R '
   
   291
   
   On lhe oI;her hand , If'' is & V<)tIex of A, lhen. f<>' e&d> ;, eilher ", ....... ' ;.(1 c. '" = b; ~ 4 ..... In thisc-, " is a of only one B oft"" Iorm (l )-Ihe ODe for _hid> j; .. 1 or j. ,., n; "'l 2; ,., ... or.l; .. !o, - &nd "1"8" .. 8&"A'" ThUll ,"" ",ht.-h&nd eIde of (3) Tedu<:es to ;«A), whid> ,,"0''''' (2). Now " " _ \hat A .. U, ~.<~ A • . where .4 .. 1, " . . , " I, .. an 5~Ie • .4... I'Or 1 u < n, &nd the A. "'" disjoint. fbr (1 '5 i .1:), the inlen'K1l h" •.. . , J,.~ ha"" h "'l tbeir union. o.It~ tboy ooed _ be disjoinl . In ""Y Cll8e. lheir eruipoinUl oplil I" Into disjoinl sublntervalri J •."." ,J,... $UCh that eew:h I;,• .. ,he union of . o",e J•.J ' T he _~ of lbe form ( I ) "'" • re:guIaz- pArtition of A • .... befonI; furl!':' " .... e. the ~ B COCIwncd in .single A. form & ~r po.rtition of.4". Si_ lhe A. "'" d ioVoint . It folloo." by (2) that
   
   "",tel<
   
   -=;
   
   s:
   
   I". " .., " I" .
   
   s:
   
   ;« ,01 ) .. L ;« B ) .. L
   
   ,,(B) .. L
   
   L
   
   •
   
   ,,(A. ).
   
   o
   
   De:8nitlo" 14. 1.1 F Is said to be right--<:ontinuo .... al. WMa F(, ) ~ to F(,,) "'l , t.,>tlo to " ..
   
   !I n
   
   (Iz, ,+001" ... " I"•. +oo[). Wtl 1M.
   
   P ropooldon 14. 1.3 ,01 _
   
   - , -, .....
   
   Th~ I' '" 0 " ' - _
   
   if F '"
   
   &
   
   point 2 of fl (:0 " ...• :<0) in
   
   .v_ntitma.ri.
   
   rig/>t-t>&,
   
   p ~~: Lee. A .. n ,~;s.la;, b;l he "" S· rect~le And let ( A. ). ~, be. deer '<\1'; 1Oeqn<:n/:1': of S-~ of the f<>rm .4 . .. n ,s;so!<>' ,v,o j, oLW;h that A .. n.~, A •. Then I'(A. ) - E JC 11 , ,.' 1( _ I)IJI • F(c1" ) """>CO I!"" to I'(A) .. n - +00. By P r<>p(ICitioon 14 . L 1. I' io • meI'\&U",. a
   
   In }M\tticular. taking for F the fun.ctioo :t .. (:t" . , , ' ''0 ) .... '" ... " 0. _ obWn (Ill S .. me&iI~ A. SIlli$fying
   
   >.(
   
   n ),,>'))~ n ,, -.)
   
   'S ;S'
   
   ' S' 5 0
   
   meu~ OJ> O. Bf """""""'" on S.
   
   for all S-""",*,,«I6!. that is, .\0 "' Lobftig"" Now..., otudy ""3""
   
   """""rwe"""
   
   FOr 011 1 $ ; $ k. ",rite Po foc the function (:t", . . . ", j .... ". on O.
   
   Propo<ion 14. 1.4 ld ("" j~;;:, l>e " ~ in M "(S) ~nd 1''' """"'" ........,.. on S. If (" .).;;:, ",n .~~• ...,...1)- to 1', 1Mn1'.. (Aju.w to ,,(A) .., n _ +00, lor 011 " .~ S -t'I'<'141\gk1 A. c.m_I~, if (",, ( ,01 )).,1:' am· """'" 10 ,,(A) for every S.rutGngle A "" n , s. s~ I... ,b;11aCA Wtl 0 n pi' (G. ) "n.tI fl n pi '(b;)
   PROQr: The Ii", TO: .I'ueb .h.!.t ..... ~ lie in E. for a111 :S i :5 k. N_ let A " ... , A. ( A. - n , ~. s. IG~.b~ b'1 S 'I S P) bte-..c.lnduded in asameC.1C\ It , ~"" , />' I. For 6-' I S / S t, write C(J (0 :s: j S ..... ) for ,he ... , />,.o'f.&~ UT.~ In IlICrtuin, oroc.. n- ta S....... ·n~1eo n,s.s. ICiJ.- , .4J,J (1 '5 j , :s: m,l k>rm. putitioo> ptA) of A. """ tood1 A., (I S II S p) is .. un ion of rntmbert of P{A ). Coo £.luently, fur 0.11 ..1 " .... .(. """.4 ,
   
   n
   
   O.
   
   c-
   
   II eil ber ""'PCY or. diojooinl W>ioD ofc-t.. We "'''y condu6e. dw Cut.) is
   
   • wm\ri".. Next. let '" : n _ C be cordi......... "'ith OCMnpert fllppor•. ~ e>::0WI • IInilO dlljoint suboollecdon 'P of C $U<:h t"-, ,,,PP(,,,,) c U" v 8. Picl B _ 's..s . "" ,l>,) In 1>. By b¥po. DI':/IMOCl R' the DOI'tn Izi - OUPI£.s,. I-<-,I. Gi ..... , > 0, t~eriIt>I ~ > 0 web that 1<;,(",) - 'o'UH s t lor all z , • e lJ a ti$fyinc ~ - ~ S 6. Fbr all 1 SiS k , 'here 10 .... bdiy ..... (c,.;. ~s. .... of la, . I>, I • ...-t. tht.t , Il0:0 Co.;,
   
   n
   
   s: ~ ro.. I s. j, 5 ..... For tad> j - U,,, .. j . ) In n' slli.ll . ' . , ,m,l, let 8 J be the n'<:Iangle n,S's'. Ic...;.-, ,Go",] • .00 let 'i boI_ 01 11>0 ,'lIJ_ t&b:n .". ." on BJ• that II'" I" - I:, 'I . I", I 5 ( . beIorI.I'o Eo """ leo", -.:..,.-.1
   
   10)
   
   I'b< e>-ery j,.be *"'lur:r>oe v..(B,») ~;;" com"""", Intqer N ::: I .....:h d .....
   
   \01> ~( Rj)'
   
   I!L'i' , I", do<. - fL, .}· 1",d~1 s.~ s...
   
   Heoce t heft", an
   
   lor aU .. 2: N.
   
   If (•.., -E, o,. '·,)"'·1" ·
   
   for all " ::: N .
   
   14.2
   
   n.e...f_. f '1"
   
   f 'P~ -
   
   tlw
   
   14.2
   
   OiI,ri"","" F\u>«iono
   
   tI,,_ oon~ to f ",·1" 11# III n E"u, f '" . I" 4,,~ <0<1,""" to f 'f' tI,,_ I"
   
   m
   
   +00. wl>en
   0
   
   Distribution FUnctions
   
   l..et B be the Bore! ,,·algcbra of R ' (k 2: I) and S tbe natural ..miri", of R', A pOtiti ... " .'''"'''' P on B Juch that P(R' )- 1 ;. called. probo.bility 011 B. If" it. pOliti"" _ r e on S ouch that ,.· (R ' ) _ I , ~ if. uro;que prot.,bility P 011 B...m that P(A)- ,.(A ) b-.oll A E S (s..:tioo 6.4). Defi nition 14.2.1 l..et P be. probability on 8 .
   
   F : (... " .... r. I_P{
   
   II
   
   n.c fur><:tion
   
   I-IXI, ....I)
   
   'li'li'
   
   !O . II it CAl ..... d>e M lribution fullCtion 0( P. A- n 'li'S. I<1,.b,l ;. • naoompty S·~ and
   
   fr(lrll R ' In'"
   
   If J • ruhect 01 (1 .2. .. ,k). "'" call 4 the _ of A ",boec ilb «>otdiDllte;' Ot belong to J.
   
   from R ' illl<> 1£1.11. Then F i.I <>1 G prob4bililll P if """ ""'~ if i.I/w 1M 1<>I1tnrint
   
   P ro_ilion 14.:U Let F be .. jvndi<m 1M oI;'IM~i<>n /tmCli<>n p'ope.liu:
   
   (.) 6."F .. EJ~ I , ..... , ( _1 )IJI . F(C:) ;" ~ I<>r ,,}.I S·rtet.ngJu. f') F ..
   
   "'t-c;oflliJo~ ......
   
   (e) F.,.. Ctoef'1j I :!>;.5 k OM I.,.. e.....-y (:1, .,,'.:1•. ...• :1. ) ;" R .... '. F (:I, .... ,l'i, ... , :1. 1 ",,,...,yu I.. 0 !l.> jI; - -"".
   
   (oil F (:I, . ,., ,z. ) I~ '" 1 ... "" ' __ .. :1. 1111 knd '" +00 . In t.W ....... , P(AI " 6."F /er GIl
   
   S·_~~
   
   A.
   
   PItOOI' : First, t"PJIOO!'" that F it tbe distribution function of. probability P . and It!; A _ llit!' J.... /ljl be Nt S·r"c:"v~1r By induction on 0 :5 J :5 t , _
   
   n
   
   ..
   
   ~
   
   POo-, .b,l)( ... " "" ,11,1"]- oo.co.,I " .",, ]- 00 • .,..1) ..
   
   L
   
   ( _l )t)!. F(d)J
   
   J<: (I •••• ))
   
   fur aI( t!J~
   
   _
   
   co.,
   ... ,I .... ,e. E I". ,b.), ...here~) if Mfino:d III fol\o:N~, _ II, foo- 1 .5 i .5 I, i ~ J , and t!,.• .. CO for 1+1 :s
   
   E ( .. ,~"I1, for j E J , ~J~
   
   :S t.
   
   ""ben / .. t, _ obtain P(A ) _ iJ. .. F. By the domil\llloOO oonVffV""" t l>eorem. "'" _ thal condit;""" (b), (e), and (d) hold. Co.,,_ly, &Illume !bat condiliooli (a), (b). (e), and (d) an! _islied. and doll... a measure II oa S b,- II{ A) _ 6 .. F for all nooempty S·~ A (P fOPOlIition 14. 1.3). Fix ~ '" (~ •.... ,~) E R ·. For e-.uy in~r 0 < / :S t. ~ (a" ... , Il/) e R' such d,al. a, < ~ kIr a111 :S j:S: I. and '"'""'Y IlUboIet J of (I •... , II. Ie!; e~~' _· .... 1 ... lhe pnin! In R ' """,- 1111 WULdi!W~ ill G; when ; belong loG J and Is ~ when i does not boo:long to J . By loduction on /- t. i
   
   In~.
   
   _ _ tll&l.
   
   II"(/<>' .~]" ... "
   
   I... ,11,1") - 00,11,.. ,1" ... ,,] -
   
   L
   
   ,", , ~I)
   
   '"
   
   (_ I )I.I!. F(e~"",,""J).
   
   Je/'. ··n T ........"..,. ~". ,
   
   ,."0 - '"'.~ 1 " ... " I -
   
   . . ~. all awr-;b
   
   +00, '" _
   
   ,",.~. I) '" F {b, •... ,~). N"". le!;ting tbal 1111 '" 1. D
   
   If F "' t~ dlstribuUort fuDCtioll of B probo.b!lity P, B ... " " I)' and .uffi,*"t ooodi'loo tll&l. F ... """tin<>WI at (:t" ... , Z. ) In R "' tbat
   
   )- ,", .z,l" ""1-
   
   ,", ,:;0:,) ba> I . F ~ be diaoontlo"""" at unD<>\lI1tably """'" point&. Propollll km U..2.2 ul (p.)~2< ' joe a ..... ~ "I pt.' . Win... on 8 , GOId~, P joe " ,. • to~lilr on 8. Call F. ~ Ji#lril>od .... furu:ti(m 01 Pu , for all n e N , GmI F 1M dwrib1Llion fMnd ...... of p, Thm (P.J.:>. .b'~ to P if "u an.Ir if ( F. (:cl). ~ l to F(,,) fM' ."";; of ..",linllitv,:z, of F
   
   """""''lOU ....
   
   """lit
   
   """""rgou
   
   P ROO'" By Tboc:nm 7.1.2. the condit;"" io ' .... e " y. eonv".... ly. IU_ it ill sati$Md. If A .. n's,p]a;.M "' an S· ...... · "81o> onxh that ,,;'(G, ) and ",' (~) an! lI-atgl!&lhIrL for aU I :S i :S t , """ can .... pI P~I A) in term!I of F~,
   
   and !llol.loor1 that ( P.{Al)~i! ' ()(I<1VftW"! I<>
   
   PIAl.
   
   P,<>yoo>ilion 14.1.4 tbert
   
   .now.. that {P.). ~I """ ••,"" --.kIy to P,
   
   1:1
   
   14.3 Covariance Matrix Equip R ' (l E N ) witll IU usual .... 1.. ptOdL>Ct. (:t.~) and
   
   r
   
   r~ '"
   
   E,S'S' ".~
   
   iu norm:z -I:zl- (""'l'n .
   
   U:c " boo • p "" of <>nIer " ,.Ite'''''~f
   
   r
   
   II'I ~ Jp(:c)
   
   :s :s
   
   :s
   
   La finite. I~ this~, " ill of ~nIer m. fGr esch I m n, bect. ..... Iz l'" ~UP( I ,lzl·) b" 0.11 Z E R ·. n,, ", of orOer l. S{!') _ I id ll.' eI" "' ~aIJed tbe mean of 1'. In what foIlooo1I, we ww · t b,u " ill of 0T
   ("
   
   J
   
   ~l - [,(z -
   
   m )J ,1:(", - m») dl' (Z)
   
   R"" R·
   
   II symmetric and pOSit i....,. and il.ll matrix (wil h MpI!Ct 1.0 tbe _K;aI boosis of R" ) ;. ealled the (lD\1lri.".,., matrix of I' and "' WYit~ D{P). D(p) hal! enl";"" Co,; - f(z; - m;)(",; - m;) ~.•)... "
   
   on
   
   c;....
   
   .;,.. - L a,.; .
   
   CoJ' a t';
   
   ,~. ,,~ .
   
   - L., a~.; . _
   
   0 ....
   
   J
   
   (z, - "I;)(:rJ - m, ) <4«z)
   
   / [". - .. (m).I_ Jy. - .. (m).ld(III'l{, )·
   
   N_, ..!>eo k _ I, 1('" - £(P ))'do«z) _ Iz'dji.(¥) - £(P)~" calI«I the ....n..no. of 1', and II written V~)_ V~)'fJ", lhe Jtand.ald dcvi.ation of
   
   •
   
   Oo:'hution 1...:1.1 For every m E R and rOO" ~" > 0,
   
   _ I'
   
   ~ ( _ (z - m)') ," "J2:r I' 20-'
   
   '
   
   r
   
   II • I'08itl'" musure on 5, l uch that 1 do< _ '- It ill called the normal Mrz "'re ..lib pa:ame\ero m and" and iI writll!D N, (m,"" )' In puticular, N, (O. I) ill called the st.nd.rd normal measun: on R.
   
   For ~ mE R, ...., (\mot" by N,(m.O) the _ r e QIl S, defined by lhe Wlit..- at m . The a"U"~ N, (m ,.." ) (m E R, .. 2: 0) are <;ailed w..:;>rmal _ u ..... OCt R. If n : :r _ "" + ~ is alii"", nIN, (m.,,"lI _ N, ("", + ~, ,,' ..'). In l'Nticular, N,(m.,," ) Is lbe image of N,(O, I) under z _ m N, (m , ..') is of order TO, lor all n 2: 1, with mean m and ~ ",'.
   
   + "",_
   
   10&. TbooFowiorn ........ . bin R ·
   
   LeI. me R. For ~ bounded C'O
   J/
   
   .IN,(m, ,,') ..
   
   (om (tglll
   
   J
   
   I (m
   
   + "V),
   
   I ;R
   
   _ C,
   
   ,J~ . exp ( - ~) d~
   
   I (m) .. " :> 0 .. ~ O. ~f(l~. N,(m.,,') H, (m,O) .. " :> 0 .. ~ O.
   
   I<)
   
   narrowly to
   
   OOfl.'
   PI
   
   N~ "" 'P; R 3 .z ....
   
   Jt···d{N,(O.I)I(~)" ~ J
   
   e'"' .exp { -
   
   It. differentiation under the inlqNJ si,pt
   
   V;) .lV.
   
   aoo..-s thAt
   
   .p(.z) _ ""pC - .z' 121· Fina!l)r. It"! me R .. 00 " e 10 •+wI, ..,1<1 ...rite '" fur t he func&io<>
   
   ~
   
   .. J
   
   e.... "IN,(m."' )Jbt).
   
   Si".,. N, (m.,,') is Ih<> ;mqe of H, (0, 1) IlDd
   -.
   
   ("''')
   
   ot> () .z .. ~exp - T
   
   fur all .z e R..
   
   14.4 The Fourier Transform Let.
   
   boonded tnoefIjjUre on II e natural ""iri", S of R · . ;Fp : .z .... f t - )l..• dp{)oJ aod 7,.. ,.z ..... f t"u, .l,..(,) ""l (.'IM tl>o: FOIl';"" tr ....1orm aDd 11>0: ;n ___ FOIlIier Y"anIiJorm of P. ' ........ Iimy. Deflnltion 1..... . 1
   
   p ba ..
   
   l.et. e' (R '. C ) he the norlMd""""" of bounded conl inOOO8 func&iono from R· into C .
   
   Pro_itio" 14.... 1 Far 0>11 p e M '(S, C ), ;Fp if ""iJarmlwamtin...".. ,,"01 """'.......,. n. Ii_T ....' p .... ;Fp from M ' (S. C) .nlO CO (R · .C) ;, .,.",n".
   
   -
   
   1
   
   PROOF: By I~ dominated COIl~ Ilteon!m, inf(2,61' I) dVl' (:) tcDdI to 0 M 6 :> 0 ..pprooclIeo O. Ci""" ~ :> 0, lhere lhuo! ""w" 6 :> 0 wch lhat lu.f( 2.61%() dVI'(zl:s f . Now, if l:, \I € R' S&lis!y jJ: - , I:S 6['h, the!> IF14") - T"MI f, 1>«:&_ It- - I I - I 10· k " dtl 1" 1 for all 11 € R.
   
   :s :s 'fl>oertfore, 1"1' _ unlfor",lJ' corui .."""". Nut, IT ,,(zll < 1 1dV I' for all z E R., "'hiob jXo_ \be oecond aq.tion. o
   
   1.01;~. or dz, ~
   
   Leboor;ue measure on R · . For all g E Lb{dz ), t~ fW>Cf,;"'" 1", _ 1"(14) and 1'g _1'lgdl:j o.re called 11>0 Fourier transform and I~ i l l _ Four;.r lransform of g.
   
   For o:xamp\e", :, -
   
   T ,, (z)
   
   ap(_,,1, 12 ) is dy-in~, 1"Md, '"' I, &nd
   
   0
   
   0
   
   II 1~-""'.' .ap(-..~)d\li IS,S· II ap(- "",,') 'SJ~ '
   
   0
   
   ,,(,,)
   
   for all '" ;" R ' (br klion 14.3). !II> T" = g,. For each" :> 0,
   P ROO<': We may 111_ lhat I' is po&iti"". lr F , R ' " R ' -10,+<>:>1_ oontln ....... with compo.el llUpport , lhen
   
   j j FC" "
   
   1 1 -1 1 '"' 11
   
   - l:) d;l (r }d\l -
   
   d;l(z)
   
   F (l:. \I - l: ) dV
   
   d;l(z )
   
   F(l:,\I)dV
   
   F (z ,V) d"i")dv·
   
   o Now , let I' E M '(S,C). For all,,:> O. deAne
   
   By fUbinl,. lheorem,
   
   ,
   
   ••
   
   .. ! dz(:t;~.· ., ! eo">
   
   . F9« :) _
   
   j ~_ .'u. dp(z)
   
   -F9,, (' ) . Fp(z)dz
   
   for WE R ' . Th ... p . g. '" :J'(F9<.- •F p ), and p. • g. ;. amtiouous and boImded . .....0, p. • 9<.- is dV-in1qn.bie and
   
   / 1p ' /h )Ul dJI .. jdJI!g.. /, - z l dp.(z )
   
   .. ! ..
   
   doI(z) / g.(v - ' ldJI
   
   / doI(.:) /
   
   ~(,) dV
   
   .. / ldp.. Now. Ioc ...:h I € C"{R ' , C),
   
   j I {p. · g.. )q. .. / dy! J{~19.(,- .,)dp.(z)
   
   .. f .. f ..
   
   
   rlJI(z )
   
   f IM~Cp - "l d~ f l(z + V)g.(W)d~
   
   !d*) JI(z +vl ' ~'9,(~)dv
   
   .. J J '" J +"11)9, (~)d dp.(.:) I (z
   
   I (z +<7V)91C,I) dy ..(z )dv·
   
   Hence, by the domiDatl!d COD~ liIoo<>uID , J IV. • hId, ImdII I<> J l (z )g'(V)d)I(z )c/V .. J I dol ..... ;>- n .p~ n. W~ """"lutk lhal '" • g.. )~ COD>('gt6 ",....-1), \.0 p.. Since p. • 9. = :J'{F J• • F p.). _ M~ lhe 1OI1owi"3 II.oot cm.
   
   Theorem 14.... 1 The ""'wi"f F : I' _
   
   u illjecn..o..
   
   F p. /n>m M '(S. C) irttoC'( R ' ,C )
   
   Up \.0 Ptopooilion 14."l, ~ I E J:hI~l "" .9~. For uoh:t E R ' , ~ by -,(.:)1 the !un<;tion , - I\JJ - z ) on R · .
   
   P ROof>: f'il'!ll., ~ tll&l I io amlin"""" wilh rompoocl.uppon K. Lee V. be. compod ~borhood orO. Sina: I io W1ifonnly CODlinl>Ollll on R~ , glvn> ~ > 0 tben: o::UsII . almpooct neighborhood V C \00 of 0 such lbat.
   
   1/ (,- :.)- / MIS A(K: V ) o Nth{:.)! - !) '"
   
   for all:. € V and o.Il,E R·.
   
   f I/ {, - :.) - / M Id!!:S;
   
   br &II:. E V.
   
   t
   
   Tbaebc, tbe mappilll( :' _ "}(:.)! io amtinltOUl at O. NO">" ~ tbe ~e"",&l case,;n ...hldt I E Ci, P,). IfU. ).~ , io .!tCquoeno:e in H { R t , C) con-stllll" to I in tbe mean, tbe ...wIQII
   
   Nth(:.)/. - "}(:. )I ) - N,{f. - I ) ahooos 111&1 tbo: 1Iequoeno:e '" futtctlons ~ .... '1(~ )I. amvervo unifomti,y QII R t to:. _ TCz )f. 111... ~ _ -,(:.JI ;. CODI;II"".. at O. The propooilion roIlo::oon
   
   euWy.
   
   0
   
   Remark d w, for """')' COffipoocl neighborbood V '" o.
   
   /.v·
   
   M.d!!-/.
   
   v'
   
   """.etp5 to 0 .. " We know thai I oi l .., thai
   
   "-'g, (!~}i'-l g,(~) d!! " (v/ . 1'
   
   > O.pproodteo O.
   
   . g. •
   
   (f.b: ) . fH 10
   
   A. in~
   
   for ted!
   
   (7
   
   > 0, and we
   
   f fH(,- ~)/{~) .u - f g..('+ ~)/(-z).u -f
   
   (f * M.-IM •
   
   /(, - z )g.(z).u.
   
   Propoaitiou 1f.•. 3 I ' 9. "'''''''~ to I in Li:{A) "''' > 0 .~ O. P _ For &II
   
   (7
   
   > 0,
   
   N,{f ' 9. - Il -s.
   
   r
   
   d!! f l/(II- "Z) -/{~)Jg.(Z)
   G;\letI t > 0, let V bo:. almpooct neighborhood of 0 oucb that N ,(-,{z )! -Il -s. t for &11 :. E V (LoIWII& 14.4.1 ). "0 > 0 110 thai. g. tlz :s; f for all " E ~. ".J. Theel. for all " e "0).
   
   to ,
   
   crv--
   
   f., M.(:')· Nth(" )1 - I) u S
   
   Iv.
   
   €
   
   ....
   
   'VI
   
   ate
   
   Next, the funn.ion Iz. ~l "" I III- z l ill (",)')@,I,..int~ablc, Ivoa,"'. Iz . v) f{~) ill (",).) ~ ,l,..intogable IP'vpo;eition 14.4.2), I'Urthe,wore , / fill /
   
   1/1, -
   
   z l- /Ml d(g.).)(z ) -
   
   -
   
   / " (,g.).)I:r) /
   
   /
   
   [/ (w- :r)-/(w) ldy
   
   ~(") ' N, b(:r)/-I)
   ... benet _.,. that N,(! . g.. - 1) :5 (I + 2",, (1)) _ lor <1 EjO , <10], Theorem 14.4.2 (Rlemann-Lebeegue Lemma) FM All _",;,;.a at
   
   f
   
   0
   
   E th().), F /
   
   iroJinity.
   
   PROOf', for all (1 > 0, tbe fuoer.ion (:r, , ) _ 1(:r )g.(W- ,,) ill by P,<>p
   )' @ )'.in~.able,
   
   F(f _ g.)(l) '"' / dv e-...••• / 9.(, - " )1(,,,) 11+
   
   - JJ~_ "'.n I(,,)e- ...·(·-d.g,,(~ - ,,)d:.4),r - JJ~- ....z·/("}o-...···g,,(II) d:.dV -
   
   F/(I)Fg.. (z )
   
   lot all. E R ', Now F(f ' /h ) COO',",&<' uniformly 10 F/ lIS (1 > 0 approoochal 0 (Propoei. tiot>o IU. I &I>d IU.3), &I>d F/h ~ at infinity, We may <x><><:luck that F I also ,....ishet! at infinity_ 0 Theorem 14.4.3 (Fourier Inversion Theorem) Ld" E M '(S. C ). 5 ..... poH /Jw FJ'. u d:..in' ." "k , n..,,,1' _ " IF,,)
   PROOf" Let COIR' , C ) be the norm«! Spo.oo:' of COIltlo\OOU$ fw>o:t;.;.... from R' Int.n C which vanish at in6nily_ By tbe domiMU!d COO'"""8ellOl! ti>eon!m , F g.. . FI' ron""", .. 10 F I' in Q,(d>;), lIS (1 > 0 a~ 0 (he<:auoe F g" Iz) _ g, {"I,,)))_ Therefore. " • /h - J!(Fg..· F,,) coave.geo to J"(F,, ) in r (R· .C), and W . g. )d:. coon.ugeo ~ to 'IIF,,}<£<. On the other hand . ..", Ir;no..' that V. ' g. )d>: "",,>utM .... ' ....... Iy t.n 1'. """"'" I' _ J'{F p )d.z, a
   
   For all (1 > 0 &I>d lOr all ~ E R ', deoote by g..\w-l the fuoer.iooo" .... "I, - "I from R" imo R. Fo, aU a E C Rnd ~ E R., ag.. III -) ill of the
   
   Iorm ~ _ !lnp( - " lsP + Ioz). wilh fJ E C , " > O. and b E R O, Co""e,.. I), lor all fJ € C, " > 0, and b E R O, tbe fuoc:tlon "' _ fJeo:P( - " ls l' + Ioz) can be .m.ten Qg,.b-), ';th If .. ,filii, II' .. 6/24. and 0 .... Itably c:bo:! : 'Tia;'elooe, tbe _ _ !I\Ibspeoce 0( C"{R " , C) K""", alt!d t,o tbe "'(11'-) iI &II
   
   ~ A (wltbout unit). M . hat !Cs, ) " 1(.,,). Hence we ~ coooide< tbe COn>PKt splICe R ' U foci , apply tbe S _WeiersI.,... I'-m, and .:....dude lhal A Is dense In C" (R ". C)'Ik foIlowi"l! d ..... em II fundAzn.o:ntalln probo.bilily lheory.
   
   T"",,",m 14.. <1.<1 Lot (I'~I~> . be" _""" of ""';tiw mMtd ..........., "" S, .11<1 Id p 6.. " /Ioo"n4td .....;.,.,., "" S. Thl:n v._). ~ , Qm"'"'9t. n'lllh'lJ u> I' if.-i ~ if (.1'",, )" ;: I WII""»" ,.,;nlriH t<>.T1'_ In til;, ~, {f"").~ I Qmocrpef t<> f .. uni/r>rmIp "" wmJlOd
   
   .d.,
   
   J
   
   PII:I)OI' : Fi~. iUppGI!Ie that (.1'.... ).;: , C(Itl• ."g... pOinlwieo. 10 f,.. I 111'. .. f I'. {O) (1)D~ 10 .1'I'{O) .. I III' lIS " - +00, an sup,. 11'0 I II Iloilo!. Now. "" tbe dominated C(Itl'" ,"gu_ t""""~m, 10< ~ tr > 0 and lor every- ~ E R ", (p., • , . ) (, ) .. ~~ e>;p{ _ .. tr2 1~1 ·1 . .TI'~(~) d.:i: aKlU'VS 10
   
   J
   
   J
   
   w ' ,.)M "
   
   ••
   
   fe ..... ""p(- "tr21r1') . .T..(~) th
   
   "" - +00,
   
   Equi_olly, 1'0 ~ (,- )] \ends 10 I'[g..( r - )]- II foIJowa tllM (p., ).~ 1 C(Itl'-"' ga& wee k? 10 I' (P "'!'OO11oIl 1 ,1.3). ThUllI' ;s poo'Clad \0 1'( 1) . we IDIIY mndude IM.I (p.,). 2:'
   Coc""roeI)I,
   
   IIui.l v..).» C(Itl""'lIEl1 natro'III'ly \0 1'. Lel. K be .. Q)m1M'"t ",t.t t:I R ' . Pul 11 '"' fh E W+ (R "), 10:::: 1) _ GI""" £ > 0, then! exi'll" 110 E 11 !Or which (1- 110) III' < (/3. Now s.here II aD inlecer m 2: I ""'" lhat { fl - 110)11"" :::: c/31O< all" ~ m. Finally, I",,", e:s:iosU II E 11, 112: 110, aucb Uw {(I - 11)111'. :5 ~/3 for all! :::: " :5 m, and _ { ( I - Io)d;i.. 'S ~f3 for all" 2: 1. AI % I'IUIIlhl'Otlgb K, lbe fwrlion!i, _ Itt,) - C,..·· bm • COI>I~ .... ~ t:I W(R" . supp(Io); C ). By lbe &n..::h-8leinha.. lheorem. I",,", aisII an lnttser .v 2: ! sud> IIw "'pp
   J
   
   :~ I
   
   f . .(JI'!f:-"'.... III'.{,) - f
   
   1o(,1)<e - 2i"". III'M! :5
   
   j
   
   for all " ~ N_
   
   Ot> lbe OIher hAnd.
   
   for All"
   o
   
   'VI
   
   ate
   
   JO'l
   
   14.
   
   n..~ nonobmin R·
   
   Theorem 14.4.5 ( Levy) Ld (jI. )~~, be: ~ ~ in M ~ (S) Juch Ih G juneti<m .., ok·..mw.t """'Y" 4h.o t . 1"'II<"fo ~ .n.to # E M ~(S) nodi tMt .., .. F # d:I:·.urn"'! ~uery>o/l.e .... ~, if I' ;, omtin...,.., al OGnd(F .... ) . ~ 1 """l>n)'" to I'~,
   
   ""eft u.~ ).~ , """....-,eo 114m >tdr I<> #.
   
   PROOP' 'Ibcre is a .n) ql>eDOl! v.,..).~ 1 ~ (I'~).~ ' 01 v.,.).~ , thlLt ~ w" kly to I' E M ~(S) ( P fOp(IISibon 7.7.5). Si""",., t..,1onp to ~(d:I: ), ,he function ~ _ ..,(:.-). ap( - ..I:.-I' ) is d:I:-integrabl.. Now, by the dominated ron~""" t~,
   
   I'~(!h (~ - »
   
   ~
   
   v.~ · ,')M
   
   ..
   
   !~~" e~p ( - "IZ"I') -.FI'~{:'-)dz:
   
   to f .';"" exp ( - "1:.-12 )
   1'(, ,(, -)) .. ..
   
   v. ' ,, )M !~"'·"exp ( - "I%I·) ·FI'(%)
   By Th,...,,,,,,, 14.4.1 , "'"" ""'Y <:Onclude .hlLt.., .. F I' d:I:-o.I n>06t ~~ " ext , SUppOlge , hM .., ill OXlIinUOU'l at 0 aDd (1"" .).>, (:O'''(L gee to '" elU r" ~, T~ E .. (% E R· : tinuouo at 0, """ only
   "'ti, *"""'.
   
   Prop
   r
   
   1.,1" dVP(.,) <:
   
   ... V·
   ! .....
   
   +00 .
   
   (u,\I) ' . . ( .... \1) dp(v)
   
   for <&II ~ E R " and <&II U, •.. . , U. E R' .
   
   'VI
   
   ate
   
   PIIOO!' : For eed> point (" , ·· ·,II~j in {R· j~. deno:M br T( .. ... ... l \I", fuDdioo (11, •...• 1I~j .... (11 " , )··· (,,~~j rrom (R ' j" IIIW C . n.a ruappillfl (", .. _,h) .... 1(., ...... 1 from (R 0)" into L "( R '. C ) iii mwtilino:ar. Ii. ",,",. for """')' l: € R ·. ~ .... i" . ~ .... T( ....... ) iI Now, by n..o.em 3.2.3,

   
   It-integrable.
   
   In ))BCticular, .. hen i: _ ! and
   
   r
   
   ~Ift 4V I'(z)
   
   < +00,
   
   Definl t ion ...... 2 LH I' be • po6ili>'I! """",,un: 00 S ouch IIw
   
   """ • aoIled
   
   :r .... ~
   
   r I do< '" L
   
   ,.It(;~) - / ~, •• dt«,)
   
   d>ata
   It
   
   It
   
   Si..::oe thedw-acterist;" function of Oetcrmi_ (by ~m IU. I), "'" _ that the title ~;". is &nlply J",,-tlfi<,d.
   
   l4.5 LH I' be
   
   Normal Laws in Rn &
   
   paoili"" measun: 011 tbe n&1unoJ IItmirillfl
   
   s.. of R " (n ~
   
   I) such
   
   t4 t r l dp - 1. Definition 1",5.1 " illlI&ld 10 be G ........" w bf,""'/ft, for "" R ", .. (p.) II • normal 5,.
   
   mMII""' .,.,
   
   ~
   
   U",,*, form ..
   
   Propo.ldon 14.5.1 I' ;. Ga....u.n if and <>nt, if iI iI 0/ ...-.kr 2 . M ito ~ic f-di- iI:r - ~ ....... ""p(C- 1/2):r'D..,). lOll""" m _ E(p.l, D ... D(Jo), aM:r iI ...,.nf«l ... .. m.a.I~
   
   """.mn
   
   :s
   
   PIIOO;»': Firot, ... ~ tlw " ill CaUOllW>. For all 1 i:S n, the imag'
   /
   
   ~Il .t,.{%) ...
   
   L
   
   ';s's_
   
   / :r~ do«" ) -
   
   L / zl 4",(:<.)
   
   ';s'S_
   
   :r
   
   It
   
   ill fillite, and ill of order 2. Put m '" E(p.) and D ,., !>fl')' Given E R " • ..,;w. .. for the Hnear kom1 ,"":r~ oa R " . .. (p.) ill normal .. itb n-.. mz and vari&n<:f: :r'Dz. be"""
   
   / i'·· dpM _
   
   /
   
   ~"( 4{.(p.))({I- ~imtz) · exp (
   
   _
   
   ~12" :r'Dz)
   
   for':ll te a (Section 14.3). Taldng I ~ I , "" <:OI>Clude that
   
   «::ba(P)](,) - ~' ...... exp ( - ~z' 0, ). Co.nCi8dy, OIIppOIIe lhall' is of order 2 and
   
   «::ba(P)I(' ) '"' ~.~ . exp ( lor all z: e a ", wben! m _ £(P ) and 0 = a li,-, form on R~. Then, lor all • e It,
   
   (o::ha("(1'))1(' ) -
   
   f c'''''')
   
   d,,1II) =
   
   ~~ or.) VVo). w .. : 1/ ..... r.1I (z: e a - ) be
   
   f c...•
   
   (d>a{..u.IlI{' ) - P (o(,o)}o . ap (
   
   -
   
   ,,",(wI
   
   ~ Jcba(p)I(.u)·
   
   ~ Var{b(P)) . & '), o
   
   and..w);. C·'.i' " .
   
   'on and ,, : a- _ a' io affine, thea ~(P) 10 G., .. joo. la (c, • . .. , e~) be tbe car>OlLk:al boosioI of a ". Ao n" n_malriI D. with ".,.J enl"'" C<J' it IUd 10 be Iymmetric and pooili ..... if the biUnear form '" on a " " a ". such lhat V{c"c; ) = "'J for alii S ',j S. ", is 'ymJIloltri<. and positive.
   
   CIoe.rb", if I' is (;0"
   
   T'-'nom 1".5.1 kl me a ", aM id D .. ("'oJ};.; 6< an n" n·"",/ri:r ..w. msl entn,.., .,........,Iric on4l'Q1ili.... Tlt.tn Ihere " a (unitruC) Ga"";"n meutI.., I' .... lIoat £(1') _ m .....t VVo) _ O.
   
   s.. .......
   
   P IIOOP: lAlL • be ,be usual ...·10· product 0lIl R " , and lee t/J (_I"'""i ...ly, 7) be the bili".. ""'" on a" " R " (uspeo:ti""lly, 11M: end<>m<>rpbl$m of R " ) _bO:NIe m.oI.riI witb r~ to (t, ..... e..) is V . Then
   
   for alii S. i. j S n.
   
   "" 4>(71,) ,,) .. ~"') for all z:, ~ e R". Clewly... is lIermilian (witb ,upect 10 40). There", &II Orlbooormal b.o$is (f,.- __ , I") in a " (0( ']sting of ~""""""'_ HeI>oe (flo- ... I.) is orthooorrnal for 40 and onbopuJ for t/J.
   
   h .... ,I.
   
   if )'ll''' P,),• ..., may IUpp<)e/: t hat then! "f;{/ .. h) .. ,,~ ,. 0 for alii S. i ::; P and V{ kh) _ 0 for all p+ I S. i S. n. DeDotc by C the di.tgoo&l n)( ,..mat';" Per,,,,niDjl
   
   o
   
   .'•
   
   c. o
   
   o o
   
   IU Nor....!
   
   a.-. In R "
   
   LH\ins" bt Iht ~"m"f R · ruoh that u(~l- I. for- t.IIl SiS n, i ... "'""" A wit h 1t..""",1 It> (fl . ' . .. e.l is orthogonaL Th", A' . A .. A· A' • I•. Since C ill the malrU:: " r ¢" (u )( u) with re;1'fjC1 to (e, .... ,f.)• ....." _ 111&1 A'·D·A.C. N....
   
   Wi
   
   . "'"
   
   the meMll. e
   
   .... S. , where to II the measure on 5,
   
   defined by the untl maa.t O. 'l'hen
   
   E(,,) .. (0 •.... 0), 0(,,) - C, and
   
   jcha{,,)){z) .. exp ( - ~,,~z:) )( , , , )( cxp ( - ~,,:z:)
   
   .. n:p( - ~r'o("l") br all z € R O. 1'bercfon:. " is C.U8IIiM . The irno.g<: I' of" under tbt ""W III z .... '" + ulz) from n " Int<> R" io GaU!l!rian. and
   
   .mnt
   
   E(#, ) " rn + .. (EI,,») _ "'. D(j. ) .. A · D(,,) · A',. D.
   
   o DefInition 14.5 .2 In lhe DOtalion of n...:...em \4.[•. \ . I' ill called the IIOTmaI meM\lU:ll
   Rder t<> t bt notation of Theorem 14.5.1. O t.erv<: that IIIfl _ ",b8,....,., of R· g1! .......ted by It . ,. .. /~ is the orthogonal a>mp!ancnt ol ""'i7), and lbat tupp(,..) .. m + " (flUPP(,,» .. m + (l<et(,..lj.L , FUrthe.mon:, p "' the rank
   
   "D PropoIIltioa 14.5.2 LeI '" and D '" GI in
   
   
   ~
   
   n.....:...
   
   14.5.1. 5.~ II1II:
   
   O. 1lII:n
   
   N.(m. DJ .. (b ) ....fa . det(Dj -, n . ""'P ( -
   
   _,hu
   
   PROOI' : From the equality
   
   C _ A'DA. in the
   
   i iz: -
   
   m)' D- 'iz: - m») Ih ,
   
   _ation"
   
   Theorem 14.5.\ , ;\
   
   A' D- 1A .. 04-' D-'(A' )- ' .. (A.' DAr '
   
   )r
   
   ale
   
   o
   
   c-' _
   
   ,
   
   o
   
   "
   
   r · A'· V -I. A · % .. ", 2:r~ " ... ""; ~.r! I'or aU r E R -, "'''''' Jet 11 : R~ _ 10 . +oc>( M .. Borel function. ThO'"
   
   lienee
   
   rh"" ~ j"II(m + U(%)) du(S) Ie "'Iu.oI to
   
   (2Ir6n. del(D)- '"
   
   r
   
   hem +
   
   tI{"»·exp ( - ~"'A'D-l Ar )d;r,
   
   Thor: k.nnullt. for W,...e 01 "". iahlc8 olKIWs I bM
   
   r
   
   II dj. "' equ.ol to
   
   (h )-an .doc (D)- 'flI
   
   x
   
   "II(~). ex!> [ - • J
   
   2(" - ' (11-
   
   m»' · ,4' V - I A · ( U - I (~ - mIl] dv_
   
   Slnoe .. - '(~ - m ) "" A -I (~ _ m), , .... ptOpOOlloo I..,UOOO$,
   
   (r , .... , z:,J -
   
   a
   
   (a,,, ,. _.. _<,,...., 0, ' .. , 0)
   
   LH M ("""Jl"CChdy, B ) be tbe m.uri. of ~ ( ...p
   s- II' _,4._ At · At' · A' _
   
   A· C - A' .. O.
   
   FuRI>crmore. the raak of B ill the rank p of D eo.."""""ly, let B be all arbitrary n " ".matrix of rank p, oueh that B · 8' .. D. If II> is the Jintat mapping from R' inl(l R " wi,II matrix B and if ,. io ,be im>«" <'II N,(O. I , ) uneW '" _ ... + w(",). thm ,. 10 G ........ n and D{p) .. B · B' .. D, 110
   
   ,. .. N.(m, OJ. At thlo poi"' , .... inlroduoe!lOme tolrminology IhIo! will be USOld tQ the !iC
   J
   
   or
   
   or
   
   ••
   
   1 Let I' M. boO!"! 1 ,6!! ' ... "" ........ ~'01 e'" ~~ ). Fill. " R.
   
   J
   
   I.
   
   ~
   
   _in ... S <>I R. ond
   
   let V' : , _
   
   0111 > 0."'-" lho.
   
   ~j'_. "
   
   :II
   
   -- "'~(I ' ~ 'fol~(lI~ - ")) 1I!,~. (I II " - G)
   
   1. DId _ _ put 1 .hat
   
   2 I. the -.tloa. of E"tEdoe •• let jl bot.be I"."ure""'llqal
   ~.hat
   
   2. P ......
   
   J c".w( . ) .. k<'(.lr lor 011. ~ R.
   
   
   ...L.
   
   (P{ (z )))' .. ..((0)) .. .
   
   3 Let 'I't be u.. fuD
   ~.. !
   
   j-.'
   
   1V'(.)t' d._
   
   Ii _)11). I I-> .' I(f) "" R . ODd lot /. : :r _ I ~"",::r .
   
   S_.hat F ... (~) .. 2_/.(s ) lor 011 z E R. ODd dod_.hot wc_) lor 011 ~ Ii: R.. ,
   
   ~ b&
   
   E,O\",
   
   J . - /. (:rl b
   
   ..
   
   that., ..
   
   Supp
   I.)
   
   ;o(1. )a l - E 'SlS. I,od,
   
   ""-ate Z',
   
   (b) .,.olII"un !c,... I.) " " _ t E N , Ie) ",(-I )-VII ' ) 1ot-0Il! ~ 0 K..."" tbo .. 0jlIt of .,...,. .. ... 10 .be .,.,.._ polyp whooo ... , :,. tIcIoo ....... _ _ - I" - ." .... """ _11,," d, ...... __ wbtu "",)ocled. "" the borboooot.oI axio.
   
   I. 8_.1t&t ... _, I ~ _ 0 .. " _-k>o.
   
   2. RIo 011 ; Ii: N , let p, .. '" - ".'}!" P""",.1t&t E,~ , 1 Let ,... boo .. '" ~ ... 3. PfQ'o'O ,It&t
   C') i
   
   "' .. I 1_
   
   lor-"tEN
   
   part
   
   I).
   
   14.
   
   J08
   
   n..Fo..no.- ......•....... in R· 1'11) -
   
   I>J.... G)'
   
   boll I E R..
   
   ,~.
   
   L,,,
   
   .. Lc< / be lboo fun(. iooo T p,I, / .(I, z) ("",.oh"n .. m ~iot 3) _ SM. 111&0 I io ftDite "" R - (Of and """", ......... on It.
   
   r.. ~
   
   i"roft 111&0 J io h-I~. ,!>at ~o.k: funct.ioII 0( J 4 .
   
   g.,_ .hai,/,: R _ [O.+co( ....... ,,,_ ..cO) .. !. a.d ..c-I) " ...el ) for 011 t > O. I.
   
   I J d.z ..
   
   I. _
   
   
   Io.ht dwonoo-
   
   ..'11 .....1 • ......." wtle..
   
   in«.
   
   " - _ , ..... b.... _ . .. ..,(I) .. 0. AM _ • E Z·. let "" bo tho __ tI.. ........... with.., .. - " t /'J" (~E Z· ) and 10 oIfI.. on.-h Ik' 'l' .(t -;. 1)/1"1. SJ.o,. lhat ( ..... (I).~ ........ F' 1L ' " ..,(1) for 011 , E R.. Dod...,. frem r.....,. ............. t ..... ,~ ... iot •• poo;.;..., _ u ... " ..... he ......ural ..... 0( R ,lid! that r i d" .. I """ ct...fI') .. ",.
   
   hidrI.-
   
   lrl""
   
   1. Uoo6er tho l\)'poI'" 0( pan 1. ...... that '/'. -.., io u-i"",rabIe. C-:I,ode 11Ii• .. to.ot-a I , R _ 10 .+001 .. ..... lr. ....... ....:I 10 finlt~ .... R - (0). S"- t loooo1 1(1) _ / (_1) tot 011' € R..
   
   ,hal." ..
   
   3_
   
   ....
   
   I.. tboo _0.1 ).
   
   ~
   
   _
   
   t ll&o ",10. _uriMic fu"",,,ioo> (PoIyr.., en ....
   
   'IF
   
   ate
   
   Part III
   
   Convergence of Random Variables; Conditional Expectation
   
   •
   
   Copyrighted material
   
   15 The Strong Law of Large Numbers
   
   (fl. F . P) ...b.,., fl io .. _ooa,pty oft, T .. O'• • bra ill. n, and P .. proI>oobilt. y, . ba• ., .. pooi.i>e _ ~ .. on F ""'" tha!. P{O) • 1. A ...... .10>0<1 .. rvIom varial>Io It .. _~.oblo fu"","",, from fl into R- C _ .. __ ; "' 01 iado .... Kk ...... ndtwn variabIL So. _ X, + ... + ~h-,,,,I pcoI>lom i~ probo.bibility.booory io to otUodJ" S. / n aJ>d dil!erent 171* '" """"OOK .. n _ +<>a. TIw " ""'I! Ia. «!up DB ...........,.. thai. U'tbt X. ba... lbo...,.. dlotributioot._ $. / n <00 .... , , ,_ lW'Oiy to E ( X ,) _ f X , n> .. A probo.biij,y ..,..,. io • ,,;pie
   
   ""'*
   
   x., ..
   
   ,,_ +00.
   
   Ji- -... fu...:l&mom.al deliDiliox.
   
   15.1
   
   Thio oet\lo
   iD.M.boory of. p< . .... ...1>0 dodinitiono 01 .. prot,.biJity ..,..,. aDVia.bIo. 1.&.2
   
   In UUt oet\ .... tho ["ndor
   
   We p.e bIIitJ' .I>oo
   . 1(0
   
   . ...
   
   t
   
   biJity,
   
   of i"d ."",rl< • io .,,,''1><'"
   
   ... "".... pIe '" "'!!fUlar IuDC'l;oo , ..D;h ...... ....uraUy from prot..
   
   15.4 Give- fl, ...
   
   n. • fl b" all n "
   
   N and 0" ,.
   
   n. 0.. n... ..."
   
   : t..:I milt
   
   !.raaolormallon " on 0" 10 dofinod t.,. (",).~ , ... (:Ire • • )'~l ' Tbe pn>bobi~", <Ji .. ... 1. . ...... ..-.. io oitMr 0 or I (P ii>!_ i< .... 1$.4.2) oM ... _ p i<. (P"'p< ,' c.ioI!. 1&.4.3)- _ Sldion 11 A. Then, U$i"., Bir~ e ,100 b apMOioo 01 z with -""ptOlic ",Lap
   t
   
   ,
   
   •
   
   15.1
   
   Convergence in Probability
   
   WI: m:aU that .. !DeII/Iurabie 'pIIO!! is t be pair CO. F) of .. ~"'pty ad 0 fU>d .. "-aJ&ebn F in 0. A prob.bi.Iity on:F is .. po8iti~ lMasu~. P, 00:F sud> that Pin) _ I. Then (O,F, P ) is <:&I~ .. prObe.bility space. We t8, &rid , fot 0.11 E e F , PIE ) eotim .... tho, hlolli~ tlw. E be re.o.liz«! wben the tx(>eoimcnt is porlormed . For illSt&nt"e, ~ tbe ""pe""""'t ooP'je\lng of ..... inli.tIite ~'_ of " solan ""bined coin . Imagine thM the IIidcs ha~ bet:a Labcled I and 0 lroaI.e&d of tbe .......1 b ....h and tails. For eod:t" e N , Let F" be the po .... . .... oliO, I) , and let "'" be the probe.bilit y on F. >kid> 111M ,,~ ( (o)) _ "",WI) - I{Z. Flno.IJy, Let F .,. ® d) F~ be tho, Borel .. ·~ 01 n _ CO, I)"' , .00 Let P be tbe probability On F r$rietion to fl.» F" is
   
   !Ie"""'.
   
   n.:I: '
   
   w""*
   
   (Le., P( A. ) - n" ~ 1 "",( A"), if A. c (0.1) for .o.lf1\ e N .00 A" _ (O,I) b-.o.ll but ~nlt<:ly many indias n ). E vidently, (O.F.P) .~ ? gEnu tbit experimenl.
   
   ®.;t. "'"
   
   Deftnltlon 1$,1 .1 ~ (O. F) .00 (n', F ) be t ..... me......"bl.. 'p&OIlO. A m&pping X , n _ n' ie c&Ued ....ndom ve.r\able (from (O, F) im<> (n' , F )) if X -I (A') ~ '" F "'" all A' E F . F<)3" 1_"_, <:<>DIIi0 mapping (%. ). :1::) .... ~;a;~. %" from 0 illto n' 01 • t"&odom ..nat.Ie. It rq>.-ntol tt..: number of obUi""" 0.11<:1" .t 10 .. In an Infinlte!leqUltllC1! 01 roin-.
   
   n.,finltlon IlI.1.l ~ 10 ,F . p) be a probo.bllity s"""" and X a random ...,.;."bio: from (0. F) Into . measurable 8pea! (n', F ). Then tbe 1map n-.ure of p under X (dtfi""" 0Cl F ) Ie .,,11ed tt..: Lo... (or t hot; distribution) 01 X. writt.;a Px . If (0 . .1', p) ill. poobo.bility spece. a property .. bleb holds p-.aJ."....; every_ ... hen: is said '" be lrue a1."....; surely (or "';th probr.bility I).
   
   Definition 15.1.3 I.,.et rr be .. """,,""loll:, me\riuhio: uniform space, and let. F he t!toe Borel "-a!«ebra of rr. ~ (0 , .1', p) he .. probabilily l<JIA« and ( X. )":I::' • geq........e (0/ .o.ndom variab1 ... from (0,.1') im.o (n', F ). Then {X")"~I io said to mnlOlrlle in probability to • random variable X from (O.F) In'" (n'.F') if it CO' ''dV"' '" X In P_IIWIII\II"(!. Equiv.)emly, { X")"~ I mn~rg'" to X in p.
   
   ,
   
   ••
   
   P ........ ltlon 15.1.1 Let (ft". r ) be CI ...... (0',.1") in lkfinilitm I5.U. Willi tM Ml4tl'<m 0/ tM ~. Id 9 b.. 0 8.-1 .....wing fr=o fl' into rr wIu.:II ;. amlin ...... , ton""'lK' ta g o X in probabi/itJ. PROOf': The _ 0 of pOilM of discontinuity of g li
   e>:1,_.
   
   Let. (0 , 1") be .. IIIN5IlI'able s"""". Givm an intqoer t 2: I. Ie\ F' be tbe Borel "-aIcebra of R" '* 0'. A random variable X from {O. n into (O' ,F' ) "cWled elmply a random variable (ot .. r-andom _ ) frotn {O, n Into R". Mot""....;r , _hen .. probabilily P on :F is IP""", lbe distribution fullC\ion of Px (0 tl~I""'y, tbe cbaract.erio8tic function of Px ) is also colIN! the distribulioo
   
   r'!diW'ly. tbe dwv:toeristic fun(,tion) of X . X is Nld W be of order n E N If Px ill of order n (Le., if IXI" dP < +co). If X _ 01 order I , ,be """'" f 0:1", dPx .. f X dP of Px _ called ,be ",pect«I value (ot mean value) of X , writ"''' E{X ). If X _ 01 order 1. tbe ~ nw.rilc of Px _ called lbe ~ ma,rUe 01 X. wri'\en D{X ). Hence Df, X ) ... (c,J ). ... hen: "''' ... f I X, - E(X, » ( X, - E{X;)) dP (.be X, .,., ,be """'~"tI of X ). Slmllwly . ..., Mline the variaIx:e of X , ...bon t .. I. Let >., be Le'-«uo..-un! on the natural .. miring S of R ' . In tbe aoo.... ootation , X tot Px ) ill aald to ha"" density g if , : R ' ~ 10 , +col ill a Bono;I funcUon, rgd)" .. l. and P{X - ' (A. )) .. f .. gd>', for all A. E F'. Equl~ly, p x /. _ ab80Iutely contin\lOUll wi,h relP«' to >." and any densily of Px,. iI equal >""Le. to.he Bono;I function, : R t -10. +cot. In , h" calle, for sirnplici\y • ..., 81m ... rite Px .. g>." .
   
   funttion (.
   
   15.2
   
   r
   
   Independence of Random Variables
   
   Let ({fl,. T; , P,)) .. , be. family of probability opaaw The ,,-rJpbra 0 0E , F, " tbe ,,_ring ,;entrated by tbe .. ~ OE , F, . Let. p be , be meNu"" 0 .. , P, on F, doofined in ChaptH 9 if I illini"'. and in ChOp'''' II if I _infinite. Finally. let P be the """" .. OM of" to ®'E' F ,. Ot.erve lhal " t.tId P ha"" lbe arne main probl,pIloo>.
   
   n..,
   
   n
   
   o.,ftn ll\on 15. 2. 1 In , be probabilistic <:O
   By tbe filial ""mark of Section 11.2, if (I; ); EJ _ • pan.ition of I in", I>Oft> emp\J' .... t.wt., then 0 .., Po .. ®,u {0 "" , Pd· Iie..... b,h. let (O. F . PJ be. jAut.biUty "1*"'. We now it1l.rod""" Ihe ~c"'''eptof~
   
   ,.
   
   .
   
   31~
   
   I S. Tho
   
   S."""
   
   La ... of Larp Numbon
   
   Oeflnltion 15.2.2 Let (0,)"", be .. !a.mily of ... a1gebnw; contalned In F. Tho: V , are MId 1(1 be i~ (with ....p"'" to P) ... ~ P(n. EJ .4,) _ n oEl P(A,) tor "''''l' ftnlt<: rut:.et J <:>f I and eYer)' family (A;).u <:>f ~ (indtn<\ by J) auch th-.l A, lieilin V. tor all i E J . Until furtber notice, I will be an index !let and, for all i E I , X, ..ill be & v&rl&bio: from (0 •.1) into & _nWIe II*'" (0".1'.), For ..... J><>
   ,.n<\om
   
   n,vn.
   
   ,...,br.
   
   110 th-.l
   1(1
   
   be i!ldependt;nl .."""",..". I"" ..( X, )
   
   Thoonom 15.2 . 1 TU Xi "'" ind."... rlml if .. nd emI, if PIC, -lor GIl """"....1'" {iroil£ nb!tI J 0/ 1.
   
   ®.o
   
   P",
   
   PI\OOO": Let J c 1 be /lulu and nono:mpty, PIC, and ® oEI P", ..,., IdeotlcaI if they ~ on lEI if
   
   n
   
   .1',. ,ha, ...
   
   p( nX,-'(St»),., P(X ,'IBI)
   
   "'
   
   '"
   
   f Is, dPx,
   
   - .II" I
   
   I ... JPx .
   
   - II P(X,- '(B.))
   
   ."
   
   ~1II1)',
   
   o
   
   In cnIer thu ,be X , be indepetl(\ffil;, it is _
   
   ' 1)'
   
   aDd
   
   oulficielll that p" • .. ® 14I1 p".,
   
   e,
   
   T heorem 111.2 .2 F.". -.II j E I, 1<1 10< .. F, . Th£ X, ""' i"up.,...unt i/ .nd MIl, if
   
   "'"l'I'~
   
   in !l; """" IIoaJ. o(c.) ..
   
   p( n X;-' (Ih )) .. IT P(X.- '(B,J) o€l
   
   Of,1
   
   /M .,.....,. {inU.£ '~I J 0/ I ..,..{ OW')' ( AlloEl 8IOd\ tIIOnetnpty. Denote by CI !I~c~ 'b'i,,« <:>fll>e..ell! BI _ B, . .. be", 8; E C. for all j E J. Then el is .. " ..)'tiI(:m and
   =
   
   n'EI
   
   J
   
   (Propooitioa 6.4_ 2)_
   
   0
   
   'VI
   
   ate
   
   Theorem 15.2.3 S~P1'f'H IJx X, en ; ndep14 if. for tGdI j € J. i} ;, .......tam ...new. '""'" (rLtl fI.; ,0 1o;1 Til into a ........."'w..op&<>e (rr, ..1j), U- 1M . ; 0 [X ,\.v, *"- ~., .. n """,:
   
   o Theorem 15.3.4 S _ tluIt, lor......,; E I. X,(O) ;, . , ...... ' """,,14!o'e • ....: (" ') IIdcmg$ IQ T; lor
   n... ,
   
   P llOOr: Obnoo.Mr..
   
   0
   
   Theorem 1Ii.2.5 G;~n kEN , -"'JII>.... D(E... , X; ) .. ~I O(X ,).
   
   PROOr; Otwloos
   
   C
   
   DefInition 15.1.4 A f..... ily (A' )'EI of T _ .. Jald 10 boo i!ld<po •.deu .... """" e....-I'(~ J AI) '"' n lf.J P(AIl for.......,. IInlte out-.t J 01.1. I.a Ihil . - . forNdl; E J, letT, _ (I, O,.4; ,AO bo I~"-~I''''''.''''' by A.;. By i..duc!ll)n" n, we s'..d that I'«(\.; , Eo ) .. lE , P(~) for.....ay famil)' (E,)If.J 01 evtnta, Index.ed by • lin;'" OUMtt J 01. I, ~tx:b Ih.t.t T, f.,.. aU; E J and Eo '" At b at. """'. n iDdiceo i E J. n .... bc, I""'" " W br• .1', &rII Inde .... odem..
   
   n
   
   e. "
   
   Tbeonin 15.2.6 (Seeond Borel-Cantelli lAm ...... ) II (A~)..!l ;, a ....... ~ _"'" oJ . .....to and E~! II'(A~) " +co. tAt:n J'(1imoupA,, ) .. I. "",."li",,,,,p A EO : 1..,(" )" 11M' /JVinitd, ........,n). R
   
   _
   
   ( "
   
   P R()()f': Slnoe I - .-' _
   
   p(
   
   n
   
   1: .-.
   _ ~ ~ ~ _ +.
   
   _ ! R ~ _+.
   
   .. ~ ~ ! ...
   
   +.
   
   R )
   
   lor all in\e&"flm (1im~pA~) - I.
   
   ..
   
   o
   
   'VI
   
   ate
   
   15, TIM: Stro
   31$
   
   15.3 An Example of Independent Random Variable:s Let b 2: 2 bot an lnteser,:r the~. !It!; of rr .. (0. I., ... b - I) . and let P he L<:b<:r;cuI: tDt&lU'" the Bon=l ,,-~, B . _ 10 , I]. For ~ E N , denoI.e by x~ the function from fI intO rr sud> th&t X~ (t) .. ~ for ~ IEO, ...here {Z.).~ I is t ile _'..,.minulrlg baiie-bapoD'ion of /.. We ... iIl Ibm< t hat tbe X. are i~ ,..""om ..... riable\l from (O.S) into (ft,F ). This examplo: iUun~ the _ioc of Indec>oe: if & r.,.J Dwnbo.>r t iI choootn u random in 10, II. knoo<JedK" of tbe ~rst (" - I) \.OI"ms of i\.o noru.ermlrwiq: bueo.b apAruion gi_ flO iDdic&tioa of tbe term of order n. ~ ";l~ in fact. prooe ~ Fi>: ... prolwobility " on .1" ....::Ie that "H" II .;. I for all 0 ::; .. ::; b - I. Wriu ,.. .... Mil). i_HOI of ~((1I1l. Put Q . .. ft . ~... P , and ,.... .. ". for ..u in~" 2: I. Th"" ®.l!: I~. is tbt El<>n=1 ,,-.bnt of the topQIoci<:aI tq)&Oe rrN .. n •. Call" the prob&bllity 181.", 1'. on ®;f' ~•. (z.).~, _ I:,.", (z./6"1 iI cootin"",," from 0' inlO I _ [0 .1]. as tbe uniform Umit of the fu"cUoiefuI!ta Integral of G(p ) takillJl 1M 'Ill"" 0 at O. Then F (II .. G(p)( 10,11) foe all tEl . FID&lly, denote by P the probability indooed by Gu.1 on tbe Bord 6 of 0 = JO. I].
   
   on
   
   c,
   
   or o
   
   n
   
   n.>1
   
   ".,.bra
   
   P ropOsition 111.3.1 F ;.,,, """"im",,,, ""'"""'''9/rme""", .Irid/r i"""",,,o ..., if tiM .,.." if p.. '" 0 I".. 811 0 II < b - 1. It'hen,.. .. l I b I"..
   
   :s
   
   ..u
   
   0 ::; .. ::;.- I, IJien F(II .. II"" 0111 E I , '0 CU.I ;, ~ mancn: on I. WIu:n!h~ ~ 0 S .. S b - I IUd! !h.u "" '7- l I b, Ikn F ;, tingu!4r, Mo' ...... ' , F(I ) _ PO + ... + Poo_ I + Po · F(k - tol
   
   I"" _", 0 ::; .. S b-
   
   I
   
   "nd......., I E l"l b,( .. + IlIb].
   
   PROOF: For all tE l , G-' (I) h.o6 at moot <"'0 ~lemenll>. 00 Gu.)(( I}) ,. O. It ilIlooi'f Ihal F ill oootinUOUll. EI>dow 0''' with tbe ).,.;'",aphic """". Now G is i""""""'n«. For z . ~ E O'N satis(yinc :< < " tbe pa and the proper bueo.b ex~ of ... """" t E)O,l ln B , w,,"",
   
   B_{; ;n> I .O::;kSb·,
   
   ki nu:ger} .
   
   Let n. k be lwo inl«enl of fY" o;:alisfyi"l! kl 6" S G(r) S (k + I IIit', lben
   
   (a)
   
   (to;);~,
   
   S (z, )Q' ... eI8e k/'" > 0 and
   
   of kl "',
   
   (z.),~,
   
   ",Ibe imp~ exp&n$X:>D
   
   15.3 A. Eu.mplo of iDdtpeidtul R'
   
   '., V.n.bIoo
   
   117
   
   (b) (%;)'"2. ' :5. ("" ... , ". , ~ - I. I> - I , , , .J, or . . (I< + 1)/"" < 1 .00 (%o);:! '
   
   II thiI: 1110»<1 ex!>BllSion of (I<
   
   (el '" - "I , '" (d) I<W'
   
   ~
   
   ,z. -
   
   0 .00
   
   + 1)/"',
   
   u".
   
   (",k> , ill lbe Improper e:<panoion 01 1
   (e) (I< + 1)/"" < 1 and (",);2. ' • lbe proper apIlnIOioo of (I< + Ill&" ,
   
   Co> q<>ently,
   
   -
   
   p" ,"
   
   . ""- .
   
   We may conclude lhat l' o;Io(,a 00( iDe.? ' Oe Rrialy if 111m! edil& .. " luch lbal,." _ 0. On 1M other 1oaDd, if .... ouch " and if I, I E I M'il;fy • < I, 1eI. .. d 0 :5. j < ". be I.... out:h Lb&t 1 - .
   "". 1&
   
   inti",.
   
   z. ,
   
   ".. ".
   
   P( X''''''' ''' ''X' ''' ''.)
   
   -
   
   p{Z, . " , •... , Z. _ ".) P(X; ", z.)
   
   '" II
   
   's·s _
   
   for all .. 2: 1 and all " Ir" . , Zo in 0'. Wa _ rel:W'I); to tbe fUllC:tion F, Wbea p" .. 1/ 6 b- all 0 :5. " < l> - I. lor each .. e N...., """"
   
   Fe;l) -F(;)_ ~ for all 0 :5. t < ".. ll-.e F(t /"" '" k/6" kit all 0 :5. I< :5. ".. Si~ F II coolin"",,", .... concl..o. ,hal. F (I) '" I for all t E l , and Gu.) • Le~ " .,, "' .1. on tbo: Bono:I",a!«ebra 01 I,
   
   in wbat f~ th.at P. '" l i b for Bt le&st one 0 ::s u ::s b - !. W""" P N DS thl'Oll&h rr, the ~venr. (X • .. P) >= ~I!O ,w iD
   c,,_ bo . [ F
   
   /I'
   
   ( . ... )
   
   'O'hU~ to
   
   -F
   
   ('0)1 II'
   
   _b" _P(:r, ) .. p(". )
   
   1'"'(1) ... n _ +<:00. Now F'(I) ~not be ~riclly positi~. for. In this case, c.. /(bc._. ) .. p(:r. ) to l i b .. " _ +<:00, which .-.01 ..... t all j € l....m lhu P (I) .".;.u and is finile, lbat is. PII) .. 0 for ~ " lDOIIt alii £ 1, &lid .., F 1.1 .inguLv. F inally, liven u Err , let I be "" de"",nl o f [ul b, lu + 1)/1>1· We ba""
   
   """verg'"
   
   F {I ) - F (u/ b) .. G(jJ ) O.. / b.1 t)
   
   .. 1'((" :G(r ) £ [!JI b, t [} ).
   
   BUI. if:r € 11'''' .. ouch tW G (:r) lieain )ul b , t), t!tenel:r) ~ "/b+ ( 1 /6}GI~:r), w~ vt,,) .. (:rH ' )Q: " Tbtrefon:, I'{:r : Glr) € 1" /6 , f J}
   
   j.< {" : G(v:r)/ 6€ JO , I - .. /6 J}
   
   -
   
   Po'
   
   ..
   
   p. . v!l') (h' G( ~ )£[O , k - u ll)
   
   is equal to p. . F(k - .. ), and
   
   F ( I)
   
   ..
   
   F (u/b) + p. . F (bI _ u)
   
   .. Po+···+p. _.+p.· F(k - u)
   
   a
   
   ... dEsired.
   
   We ba"", in (.ct, ~ IOmcthina: mor-e: if t E jO . II .. """" that 1'"'(1) exioN, it! finite, and P(X. Ct)) ". 1/6 for lnfinll~ly many n, Ilten P II) .. O. Note, 1ncldectaIly, IW tbe argument """"... contains tho: proof oIlhe cWm we made at the bqinni"l 01 this lO!!Ction . When 6 .. 2, we can, In fact, repn:I F ( I) ... the pn>b&hllity of 1Pf« &\. "bold pl.y' for- • pmble-- ",1>000. inlt i$! eapi~ "' t 1_ PatTidr. BUlinpley, P" bMlil,.1<4 Mid ,Oft, 5edb, 7, Wiley, 1986).
   
   15.4 The One-Sided Shift Transformation Let (O,F , P) be .. probability 'p""" and, fo r ~lIoCh n E N , loll X . be .. random nri.t.ble from (fl, T) in1o. measw-a.hIe 51*<' (fl., T. ).
   
   'I'
   
   ate
   
   n. ~, "'( X~ ,
   
   .+, .... )
   
   X ill called tIM: tail ",-field __ da1ed with tlM:!'tq~ (.X;; ).~,; its demenu aN: called lail """'u. DefInition 1$.".1 T _
   
   P rvpgs5tlon 15." .1 If ( X. )~~, iJ 'ndqlomd<m arut A u a !at A is a tail ~ Oi....., .t E N , if B I:..Ioop to "'(X , .... ,X . ), tben A and B at<: lbde~', bto""" A E ..(X . ... , .. .J. Hence A io iOOependo:m of "(v) by TlIo!on!m IU.2 (i.... , " (v ) and tbe PIIOOF ;
   
   "'-al&ebn. ( '
   
   . 0 , A. A<, ~ i.t>«~"""o1 ). But tben A ill indq;wnienL of ItAelf'. So P(A n A ) ... P(A ) - P(A ), which Imp/i.. thaI P IA ) ill either 0 or 1. 0
   
   It ill ~ OI'S)' to _ Proposition 15.4. 1 to j1i O"" t!>at a &I""" _ muIIt ha.-e pmt,.biUty eil~ 0 or I. 'The dil!ieuIty ...... hereafter in dtttrmi"in,g ..hich of these val ..... (0 or I) .... are de6lillj! "-;th. prot lp m, ill t.c\, may be 6L.lemely dif6cu1t to ooI.-e . N"", Itt (rt' ,F .,,') be. probability spece_ Put 0 . ... rr. :F. ... T' , and "- ... Ii, for all n E N , and t'O
   ~
   
   TIl.
   
   0 .?,
   
   Propoolitlon 1$.".2 p IA ) II dlh.r O or 1 l or t=Il " · in"" ....m _nt A (1."-, fo>r t=Il A E C ""'" ~ ,,~'(A) ... AI_ P IIOO' : Foe I'a<:h n E N . de6ne by z" tbe rr.ndom >ViAble ("" )'~' _ "'. from (rr" .C) Into (fl •. :T.l. Lo:t A be .... invariam """'I. Then A beIongo to .. (Z" . .. • Z. , ... ), to ,,- '( A) I;... in
   
   ",(Z,
   
   0
   
   tI, _ , _.
   
   Z.
   
   0
   
   ~. . . .
   
   J- ..(Z, . . ..• Z.... ... .J.
   
   Ind""tiwly, .... _ that A belongs to ..(Z ... ,. _... Z Ir+ • . .. -J lOr -=II iIlteg« k ~ II. 'The.do, ,,. A ill. tail eYeJIt. and ,,(A) io eilher 0 or I. 0
   
   Propaoltlon 15 .... 3
   
   It
   
   if p-.,-,ocIit.
   
   P IlOOl': Ld A be lUI _ audt IMt 1" 0 ," '" I" outside. ~~I* art N. n...., lOr .......ry ILI1eg", t 2: 0, I" <> .,. ... I" outAlde E ... ,,-( N ). Put A' -U~"'-'(A ) and AM... nJ ~",-J(A'I. Clcerly. I" , _ I" -Ollwide E . and I" . <> u" I" . outside E. On l be other hand , I,," ... I", outsldo E. Si...,., ,,- '(A') C A' • .... ha.-e A~ _ ",-' ( A~), to that j>{ A ) ... p(AM) i& cit!,... 0 oe I . Ncn , let B be a p·"",...urable 8e\ IIUdL tW 1" <> '" _ I" II-" mon everywhere. ,"",,,,, ex;.u ..., ~ A C B lor .. tuc:b B _ A i& ~~lPblIl.. n,.".
   
   u."",
   
   I ...... .. I.o~ _ I" _ I .. ".aI_~~, whicbtbow:a lhat ,,(8 ) .. "IA) ioI e;t bco- I) or I. P'OpO&ition I I .4.2 tben poo'1!$ that ~ " !' !,godie. 0
   
   Theorem 1$.4.1 ld (O.F, P) "''' JI"'I6obilill tro«. aM l R. A _ t f1Iou th£ X • ..,.. i4ID1li E (X, ) .. J X,J.P "' .. -0 +00. PIIOOI': Write" lor 'M pnK.,bilil)' 0 , 2:' PIC. 00 ,he Porel "..~. C of R N. Lott ~ be 1M llil" tTWlliIocI1I'1on (z')j~, _ (:1:,+,). » from RN into ItaIf. F!naIly, let Z. boa the fu""tloro (Z')'2:I:' Z . fTOOl R"1< into R. for all PI ;:: I, and pS' Z.
   J
   
   CilMhUsee almost
   
   ""eIJr 10 S (X , ),
   
   o
   
   (1 / ")5,, ~ to E( X ,) in the II'nll--.lllpoQe Cl..{P ). boa. . ( II " ) E,s.So NOd (1/ ..) L' St S_ X. 03
   sex : ) and EI X'")
   
   x: 'It ( p )
   
   In by- Pr<>pofIkion H5. n,., d · .;....' enmple iI 1M ottQtlflla.. oll&r# numt.er. lor Se, ....wlllri&ls. Here p(X • .. 1) .. p , p( X • .. 0) .. 1 - Pi ,~ lhe numbco- ol . .. ' . , ,I In n trialt. and (1/ n)S. _ p almoet tu.ely. n,., idea of prtJ.bility .. freq""""" de ......... 00 the _bllity of the .. ·, '. , at", (I / .. )S.,
   
   s..
   
   lea,."",,,,
   
   15,5 G!WD
   
   Borel's NonnaJ Number Theorem &D
   
   irI~
   
   &> 2, let P be tbe JIO"I'eT ""' of fI' ..
   
   (I) ..... ~ -
   
   I).
   
   'P"«,
   
   Pr-opoAlt ion I S.S.1 Ld (0 , T. P ) l>o .. pn>6o>bilil, 1ln/e ~ N.(.. " ", . "tJ lite ~"q "/1Ae . , IIOpIe '" IAe firll 1 lriolo. tMI u, ~ lnimkr ,,/m.1Ido IJo4J 1 S m:5 n .... X .. .. u""" X... N_ ( I /") N~ (.".,., • • J 00>n"""'JU .w...,.,t ",.el, '" p( .. ,) .• . P(K, ) '" .. --too.
   
   •
   
   n
   
   "+. -
   
   -n_. ,. ""
   
   P ROOF: For eacb ~ n ~ 1, put 11_ '" 0', :F~ '" p , and I'.. '" Px,. Wriw ,. for the probabiIily ®. ~ , I'.. on C '" ®. l!:' :F• . Finally, let Z. be the ,.ndorn ....n.bIe (:rlke ' - :r. from (O'N.C) inlO (fl'. F ). Omote by '" lhe funcliort ] 1(. ...... . )1 oIZ. , ... , Z.+I_,1 (,. By G.D. Birkholl"', ~rgodie theorem. (I / n ) L,S1S:. 9; UID""'WflI ,. " ~ ~here 10 f g, dj; '" P(", ) ... p(u. ) M n - +00. Nexl , for ~ry n ~ I, .. rite Y. for 1/(. " ....• • )) <> IX ••... ,X. +l-.1. So y• .. ". <> X. where X '" IX, I' J!:" Since P.r .. 1', ~ _ that (I/ n) L. s:.ts: . Y, ((In>(i l¢'l a1",,* ",rely 10 P(", ) ... P("l ) " n _ +00. a
   
   Now , lei. 0 .. )0 , ]1and lei. P be Lebetl&uc "",""un! 011 the Hom " . IlIpbra, T . of lO . IJ. For..m n E N . lei. X. he lhe fulictiort fIIom...-iabico from (n , .1") into (O',P). Set 8 - lk/&" : n E N , 0:5 k ":5 ~ , k in~) . OeftnlUOli 16.6. ] A number I E f - B is &aid 10 he rompletely IllImlOJ (with I"1!SpecI 10 the ' - b) if, lOt e.ft)I In~ k > I and ~ry k·tuple (u" .. . , Ut ) of ~b dizj!.l, lhe k-tuple .p~ in the bout 6 e.:pe.psIon of I with uymptotic rtLatlve frtq uency II II'.
   
   ........
   
   PropaolUon lli .li.:Z (80"'1 ) p ·almo.t aliI E f - B Grt
   
   COIIIpld~l~
   
   """"'"
   
   PROOF: Fh.l: ~ I and (u, . ___ .... ). For...,b integ<'. n 2: I. ~ by N.lll ,, · ··, u.) the frtquel>CJ of (" t. ... , ". ) in the lint n +.1: - 1 trialt. By Propooit iort IS.~.I, (1/ .. l N.( .. ,. ___ . ... ) UlDve.,.... p ·a1moot ."",11 c.o IIII- .
   
   c &er-cUufor Chapter 15 I
   
   Fi1I n E N aDd wnW G, b lhe poup III po«mutalo- 01 rr .. {I, ... ,n}. II " E G, ...o:t ~ .. I'"~ : r € Z ) io 11>0 .ulJsn>up '" G. by ... "' ,-..... _ rr tl>o . . .I.-M r.Iat"",: ... _ r .. 1>0....., ,bel-< ""Iau " . E .. ouch ,bat r '" ,,·(a). ~ "'lui" l" .• <' - b 1m. noIatioa """ callod .ho orbiu WIdot~ . Loot 0" ... . 0. be'be dl&rtn. orbtu ulldor~. ""u......u.l.., ,w 0, _"j .. • ho om·l"" e\omed. 01 rr - U's.s..-, 0.. b- 011 I s: q :s ". FOr -=:II I :s q :s " put r. _ 10.1 . ...o:t ......... bjI Y., • . +r._,+l (" ) .he 01 0,.
   
   _ .W
   ._11 ,.k_uI.
   
   FiDalIy. put
   
   Y., •.._•••• , " We .-y .hat
   
   (,,j .. ,,' _ 1(Y.,.
   
   +.,.,.,(,,)
   
   "" all I :5 r :5 r, .
   
   (y, (,,).... , Y.(,,» io lhe " ..... .,f c,clic '
   ,
   
   . . t"",
   
   01 ...
   
   1M F be ,110 pO & oet "'0'. F tbo pO' " oet '" n .. C • . ODd P t loo u.1Iorm ......'ily "" 7' f~ fa II • I/ n! lor all " " fl). l'bn:. lor .......,. ! ~ k S n. Y. iI • r&ndom .v\abI:o &om (fl. F) i,,10 (0' .F ).
   
   Ft:< <;Wrf I ~ k ~ n ODd lor all " " fl. 606 ... x. (~) '"' I 0< 0 .. Y. (a ) <:<>Iet<:o ... c:
   E,s.s.
   
   Ot"~ _~.
   
   ~ IS t , * tho X. (I to t , * all tbt
   
   f o<
   
   I.
   
   Ft:<
   
   r&Iidom .........,.. frrot (n.F) inIO R- w. ,1:000 . an: io:M,= '"", ....... toM p( x . ... I) ... Ill n - . + I). I:a.-. 11::0 Jamt diolribu,iou.
   
   . S
   
   S
   
   x.
   
   alii.;"
   
   i, ..... i. _
   
   X. 10 •
   
   ft.
   
   s ..)
   
   ,'*.ho.......... i 10;. 11
   
   (I ... . ... ), .. ' L., ."" ........ A.U) j" . ... j .... diM;""',"'"
   
   d;otl"", ~
   
   n A..., U. l)"
   
   p(
   
   (n n" )!'
   
   ' ~_S '
   
   , . n. k "
   
   N . GMt. 2, ..... be ."" ....... that
   
   :<._, in (0.1) ...... r> ..... "
   
   x,.",... " X.... ,_z._,
   
   -..:I
   
   In (2, . . .. n ), Itt A
   
   Y, .. I. Y. .. r:: , .... Y. ... V• •
   
   ,' n ... that A iI _ _ !JOy. Sd z,, '" I. Ft:< o.IL I ~ ~:::: k - I oud:.1::or. z . .. 0, ""I j. ...... , . 0.: ,1M: ot ...... IL&Dd. if L S . S k - I .. - " li>u io ,1>0 INs- h.i... < • fat .. h""" 7: . ... I. "'" ,I::or. A ... (), ~' S' - ' A.. u.1 """ p( A ) ... (n - , ... l)1fn'.
   
   " .... I -..:I If 0 S •
   
   ;. ... _, . s_ 3.
   
   <•
   
   UoiDc:' _ .,....., '"1M< -'2.2."lotI . 0-..:I:s p.:... <J. k_ ,".100 Iar-pst b:.... ill (0.1 •. . ••• - I ) _ "-,. Show tI... A n ( X • .. I) ..
   
   n
   
   04 .. (;,1
   
   'SoS'
   
   p(An(x. - II) " ..
   
   I'<><
   
   ~
   
   I
   
   . - ... ,
   
   :s . :s n ..........;"lI' """ .100,"
   
   I
   
   iN- _
   
   ,
   
   1>( 04) .
   
   04. ' - tb ••
   
   p(X. _1)_ n _ k +l C'oDdoo. .....
   
   p t A n i X• • I) .. P(AI P(X • • II kIL' - " A.
   
   6. Oed""" from Tbe::o.m 16.2.4 ,!:at IX, ..... X. _" Y, ..... Y. l 0tId x. &rcIopu ..... C.,..d ....., i>!' Ood .............. X, .... . x • .,., !-.kpe if"", lor 011 I S k :S n. l
   
   n .n E N.&ndIot(o'7'. p)booM;"£ " " I. fo< _I:S ' :S n. if
   
   ,
   
   .. (2)
   
   v
   
   ,
   
   En.. ' " for Chapter
   
   I~
   
   ... defi.... X. (~ ) .. thor u""""'"' 01 <1..... '110 beh •• n 1 Md k _ 1 lyios I
   (i " .... ... ) be • _ of "' ~b .. ,\>do ,hal i. belonp I<:t ,, -'(n), .... ,, -' 0 ) ,.;th tbe /oI..",.iOS PfOIII!rt"': (a) ,,-I (n) ..... _ i • . (b ) For -=10 I S k S n, ,, -' (n), ... , ~ ' 'in - " + I) .... diltil>Ct elemoeouo (.g
   
   .«.
   
   (c) For-=1o I S k 5' nand alii 5'! < k , tMr.! ...... _ .. -'(n_l} - ... _. _ m E (ft _ ! + ), .. . ,n/ audo ,hal a- '(m );> a -'(n _I).
   
   (A""" by iDd .... ioo:o OIl • . ) Coodl>
   " E
   
   n to.
   
   wlPoIo
   
   2.
   
   Deduct from part I ' ....1 ,be ""'Wios ~ _ (X. (a »),S' S' from n iIIto (OJ ~ {0.1} ~ ... )( (0, . . . , " - I I io bijeoti ....
   
   3.
   
   r.". -.y 1 5' k 5''' Md ""
   •.
   
   Prowo ,hal £ (5.) ... (A{ .. -I ))/~ Md , ...., \'~s.. ) ... O tn) .. (2 .. ' + loI -6).
   
   3
   
   Let (O, F , P ) be. pr<0. p..,... ,loa< P(lim ... p ..... ) ,.. 0 (S... lloftI-CameUi Irm.... ).
   
   ~
   
   Let (O.F, P) be • probabiIi'y~, and let ( ..... ).~ I be a _ ........ of - " ...,., ,loa< L'~ I P( ..... ) io infinite. Foo -=10 ;"""'" n 2: I tudt I ..... L, ~ . s. P( ..... ) ;> 0, PIli
   
   I.
   
   ,.,+.. ' +Po.
   
   for..-y n E N , put N.,.. I A. + . ·· ... IA. and ..... ... E( N.) _ (.. her< "' ... P( ..... ) to. oJl L S k S 'I). W"",,, m.;> O. " - ' t .... ,
   
   Var(N.) ... and 2.
   
   ~
   
   t . 2:
   
   i (N. -.....).~P ..
   
   (t . - I)m!
   
   I . Noor ... ... liminf._ •• f • .
   
   Le\" ;> 0 be Ii... a. for -=10 ;"""'" n . \>do ,hal .....
   
   ;> " , """'" ,hat
   
   ""
   
   ptN. S ,,) S P (l N. - .... 12: .... _ :or) S (f. - 1)( .... _ :or)t .
   
   Coodude lhallim iaf._ ... P(N. 5' :or) 5' e- I and pt. uP. N. S :or) S ._1 .
   
   3.
   
   Ded_ from pAIl 2 ...... pt ..... In!.iwly oft ...) ::!: 1 _ (e - I) .. 2 -" (~-L&mpertlloouna)
   
   ~
   
   CamdU ........... !>
   
   'oct.•t.. """,,.d
   
   WhftJ ,be eftIIU A.. .... iD
   ~""
   
   lei (n.T. p) be. ~111 ~ Wtl~ F lor .be <.... ofo.ll ... ~(ll(l' .. {O.II. Let ( X. )..~, be • oeq_ of ibdoe ... ' ...., U>d idontio:a1ty dlotribtltod
   
   •. . - ~ t.- (n.T) _(rY.F ). Ndt.!'.at 0<, _ PI X, - 0) < I. For oimplici.y. _ _ ..... (X. (... )).. !. c:ontai ... In fin~ Jna"Jf _ ..... ~ .... , •..., lor - r '" E f'l For - " n E N , Wi&bIo ond ptl" .. t ) .. ,' (1 p) ........ k 2: 0, So
   
   _oj""' ..... c:
   
   .
   
   Pil.
   I.
   
   Jr L..~. p" <': ....... dod_ from
   
   r. ....... 3 . ....
   
   PH", : I.!", )::!:-. 1.0.)) - 0 ( ..,,-,,;.0. io ... ~"" lor "inftnl"">' 01'1 .... ). 2.
   
   Aoo,"..... ;" .. hot
   
   101..,..........
   
   ,\ . _ (I.
   
   :L .~ , p.... _ +00.
   
   I...
   
   ::!: _.) _ (X _ _ .r. _ X'~"
   
   _ '
   
   _ 0).
   
   t'bo-.-,. .. E N . ___ u.ot
   
   L ptA,) + (L: L.P(A. ). ...Ls. ptA, n Aj) S .s. .s. P("" ))' + 21 ~ .S p
   
   l. If L..~, p" _ +00, drd_ from EJoer<:ioo 4 IU. P(I. ::!: '. ~.
   
   PTo¥o.Ut (I.
   
   1'.:..
   
   1.0.) ioo. toil ..._
   
   ;.0.)
   
   ~ I.
   
   for (X.).~"
   
   II ".,..Ita n ..... 10. I'. T ..... he I'I<weI .. ,""«-b<. In 1'1• ...,. P for I..obeoc"" " ...."'"
   
   _n
   
   ~F.
   
   FbI &II I"· Vi ~ ::!: 2. r ... E N.do!fl... X. '10.11- (O.I) ooloIIowo: X.(I) _0 .... 1."•• _0 .... - . ;r (z.).~, ioo U.. ..,il\ehnlnot;"" .....~c~~ of I . U.. 5.., oIo
   En,,_
   
   1
   
   We .... '" tt.. ...,....Ion of S«t .... I~-l. ...,. _ume that b .. 2. Shoow that F(I ) S I lor 011/ E 1 .... F II )::!: 1 lor 0.11/ E I , . PO S 1/2 or PO 1'.: 1/2.
   
   8
   
   In.boo -... .... ofS,,''''' IU, I .... ~ - l, ... S_ IIt&t F iii 1M ol~..- fuOl
   c.m....
   
   "tt-_
   
   B
   
   '"
   
   _
   
   1/2. .....! '" .. o .
   
   Ea.· · g
   
   10<
   
   ca.- 15
   
   1>. the _MIon 01 Setlio<> IU, 10< Q : I _ R he bounded. , 1.Idt tbao Q(I) '"' "" + .. . + + p" . Q(W _ ..) 10< _ry Int_ 0 $ .. $ • - I ond """'7 I .. 1_/ 6, (.. + I)/ ij.
   
   p,,_,
   
   \.
   
   p....., l bu Q(f) '"' Q(t/ . ") + p(.,, ) . .. p( ... ) . Q(b" . - iI) 10< all iN , , n :i!: 1, 0 5; k -< .". and aliI in Ik/.· .( k + I)/."I. .. here " , / h . .. +•• /." io .ho ptQI)ft . . po" . .. 01
   
   1,.·.
   
   2. S_tbuQ _ F .
   
   16 T he Central Limit Theorem
   
   iii'
   
   Lo\ tK.. )*'!:L boo. oeq......,. cI ;,,10, " ' ODd idmtically diltributed random ..n. abIoo, I~ their .... _ l~ R ' (wit h I ~ I ). - . . that - " X .. .. ___ lmev_ o...ote.". D " " " _............. Ix 0( X. 'Then (1/v'ii)(s.. _ nEIX. )), .ben X L + '" + X., ,, ....... In Jp, to Ibt -.n.aI law N.C o,D). T1Uo (_ ..... Hooit) .I>eoo'oa> o:od i\o ....,.....u.alion, .boo LII>dobtoS It ' '' .........
   s.. ..
   
   impoo." ·
   
   7
   
   "
   
   .
   
   In poobobjtil:1
   
   ' '-Y. ,.~
   
   '"""'"i< •
   
   _w...
   
   111. 1 We Mud,. _ po~; " .100 of t be ill law '" "'" 10" 14.2 Lot ( X ... ) _}, be ...........lu ....,.cl j,odepe..do."t, _lind, ond "'IU_
   
   'i' '''.
   
   In.....abk ' "" ' .n variabIot. R>r """'Y "
   
   uod
   
   s.. - !:' S' S"
   
   _,."..,uoJ
   
   ti,.
   
   I , ""';te ... ..
   
   [L' S' S'. Var( X• •• )1 If>
   
   5.,. .
   
   _it. .
   
   X..... n..W..... t>eo.<:I>OIditlon io.~lIId""t lor t OM., .... rn- . 16.2.1) , Nooo, i&cide,,' oIIy, l b • • hio .-.J. c-. (. . . '" 5 <__ ..... 01.. FO!Ior', tbroo<w,
   
   ill 10.. 1<1 tbo .... maI .... io..... "'1 ID t bo _ .~;n. • _ po""",, _I.
   
   l'.l W. -"to of
   
   <:
   
   tbeoot"'-
   
   ~
   
   .....,'..., rn.eo..,., '6.l.\ ). .. well .. - . . relI_
   
   16.1 Convergence in Law Oeflllit ion 18.1. 1 l.ef; 0' be • Locally tJOmpo.cl Ha.wlQrll ~I*'" ,..boot .opoI. OCY I1AII a <:<>un· · bIe - - . aDd J.,t I' I>e • prOboobility on the Bord "·...bra F of 0'. ~nal!,., \eI. ((n.. .F. , p.)).;!:! be • ""'11_ ot piwhjlity I~
   
   16, 1 Coo .... "'..... iJL La...
   
   UId, for all n E N .1et X. be" random ,,.riab\e from (o",F. )into (IY.F). n.... (X.).,<:. _ oaid to con~ in law (or in di!!tributlon) to JO if P.( X.) COD\Ctgos fW'f'O'IItIy to ". When JO is the Ia.. of a p;iYCLI random..m~ X , lbe ~( X. ).,<: • .. aloooaid to
   IIcnc
   r
   
   r ........ __
   
   P~F: For &II "
   
   E N , pul p... ,., p.{X.l, ~ by D lbe lid of dilcont .. nultl<S of , . If F ill. ~k: t subooot of 11". ,-I(F) io included in g-'( F} U D, ~
   
   Iim,upg(p..)(F):s lim . upp...
   
   (g-L(F)) :s I' (g-I{F») -- I'(g- '(F») ,., g{p.)(FJ.
   
   o
   
   and lhe oonduAoa foIlowi fro,,, Proposition 7. 7,6.
   
   'Theorom 16.1.1. tl>ough ~l/,tllO!nt&ry, io usefuL
   
   Propollitioa 111.1.1 Ld (IY . F) oM' ~ X . lie 41 in Dqildritm 1'. /.1. C ElY . kl ~ 1I>'l! "" F J~neJ ~r ~ ... il ...... at Co no.. ill lI'I'"IIer tluU (X,,).~ I om....,. in lei. t.. le. it if " ' 7 " • .,. ...... ~'" lit'" P. (X" j V) """..",. to 0 ... n _ +00. ler «leA """,pod ~ V <>/
   
   '< .. ......
   
   e.....
   
   PIIOOr. SUPI"'* the ooncIil""" boids, and let I E H (tl'. C ). GiY(ll f > O. the ... '"''''''" comJ*1. nel&hhoLl-.:\ V of e sudl that 1/ 1:r) - I (ell:s e/2 for
   
   .u"E V. Tben
   
   j l o X"dP,, _ I (C) _ /,
   
   (f 0 X. - / (e))dP"
   
   ( Jr.{Y)
   
   +/,
   
   (JoX,, - / (c» dP••
   
   ( Jr.f Y)
   
   IJI and
   
   0 X_ riP.
   
   - l (e)1 S
   
   ~ + 'lI/IP.(X;'(~))
   
   for
   
   ~l " E N .
   
   II I o X_ dP. - I(ell ill to. 1haD., if n io Iarp ~b.
   
   ••
   
   eon'eudy. -...... that (X. ),,2:' .............. iD wml*'t
   
   ~borbood
   
   Is equal to !
   
   on Y'.
   
   P(X.,. V ) <
   
   f
   
   law to !<. and let
   
   Y be ..
   
   of <:. If I: fr' - IO. IJ iuomillUOOllll..... ilM. .t c, aDd
   
   \.hen
   
   ,0X.dP. ..
   
   ... P(X.,: V) _ _ ..... toO
   
   ..
   
   f
   
   10 X.dP. - / (c)
   
   for eodI" E N .
   
   ,,_ +co.
   
   a
   
   ""i{<>nrI
   
   Theorem 16. 1.:1 Ld rr o.e .I«..U, COfIIPQd. ~We. ~ fJ/O«, .I\0Il1.01 i/6 &>rd .... ~~ Gi_ .. ,,,Hilil, Ipo>OI: (f1. F . P), ~I ( X. ),,;!:. k a I#JIf"M:'t« 01 rrutkm ..tiM/U fn>m (fl. F) illl<> (rr.Fl, aU lei X k a ",n4&rn wria.'/k fn>m (f1, F) in!. (rr. r ). II {X. ).2:' """,«i j " in f " hboilil, I<> X. IN... it """w:.,... t.. X in Ie.... ,~. if X iI oIm<>l1 ",,...1, .".... 1<1 ..
   r "'"
   
   c.m......
   
   P IlOOF': A.""", IMI (X.)_;!:' con.",,,,,, In pr"Obobilily to X . and let I e X(fr',e). / II: unl.bmIy _U~ on 0' ..... Ii""'" :> O. I","", ill .. ck lEd ~ IV of fl' rueb 1M' 1/(%) - f(~») :S 1/2 !or all (:t.~) e IV. Chooee ..., E N 00 that
   
   P ( {"': (X.(", ). X (",»,. Tbo,n. lor all .. ~ ........
   
   IV}) :S .I~M
   
   £0. all ,, ;!: "0.
   
   loa,,,,
   
   I!UDXo - / o XldPI,1
   
   f.\" • .JfI-ilW}
   
   +1
   
   I/ o X. - / o X fdP
   
   /X•. ''I''(_I
   
   l!o X. - / o XfdP !i I. '
   
   whid> J>IOWIIIh&l (X_>-i!:. -.~ in law 10 X . "The - . d '§] aion of lhe ""-1Qn: from Pr<>pOCilion ['-i.1.
   
   "Ie""""
   
   Noo< lake lOr fl'
   
   \~
   
   ,Z·l l-I ,', .u-). For
   
   a
   
   ..-Iii. aDd !or X .. . U><\om v&riabIe from (fl. F) into
   
   R uwlonn!y dil!;ribut.I. ~ to 0 in law. In ...hat ""-Ion..... Woe roT fr' .. Euclldc&n ' '''''''" R ' (t E N) .nd !or F the Bn 01 R ". In tho: """,hoo 01 Oo:finilloa 16.1.1, let F. be the distributioa fW>e\ioa 01 X ., tor all " E N , and let P btl lho: dlltribul"'" funt:tion 01 1'. n.en eX .),,2:' Q)'''''p:e to I' lII!o.w it aDd only if (P.(:z )) .:;,. """""rg<s to F {:z) for """'"Y pOint :z .t .'hlo:;b F is eontinuoull ( Propot!!tion 14.2.'1) . On 1M other lw>d,
   
   16.1 Coe .........
   
   {x~~, COIl'"' .... 10 /I. In 1&.... if ""d ODIy if
   
   ft!-· .t;.("' ) for every r E R'
   
   r-.
   
   m
   
   (j ~x"' dP~ ).1:' COilVUJ1'&
   
   (Tbeo;or= 14.U ). N.,... _ """"i<\er lim,*" poOjiOOlilioo!l, whleb &T It.. norm 1",1 _ ISlIp 11,1
   
   _be_.
   
   \'OT)'
   
   10
   
   .....tul.
   
   P~itloa 16.1.2 1.0<1 (P('))'l!' b.. II _ " " " oj ".Io~litiu "" F OOJI· d(J,;hf -.tIr I<> .. flU h~'t~ /I. . Fetr mc4 n E N . let (Il., F. , P~ ) k .. "..,... .Witt IJICI« • ...tld Y. k II ",n (1l•• F. ) into R· . FirWlr, Jetr eWo tt ~ I , It:! (x!.' )). (o.. . F~ ) into R' ..udI 00JI""'9U in 1e",1<> /1.( . ). A...""", tiwl
   
   Jetr nefll ~ > o. ~ (Y. )~;:, CIII"IUI<"JlIU in Ie.. . o /I..
   
   s:
   
   PROOF: Dt6ne on R ' lhe order:z :;: r if x, :;: " fQr all I :;: i k. kt pt.) be thoo diIltilMnioOl £uacUoo of /1.(. ). lor all tt € N . and let F be tbo! dlalribalion
   
   fuo>cUoo of 1'. Suppose F iI rontinOOUl! at :z € R ·. For ...:II 1ft! number h. put B. '"' (r E R ' : " S ..,. . h V lSi S k). Ex 0 and h- > 0 IUd> F and all t .... FI" at
   ,loa.
   
   p~(~'I"S.
   
   i) - p. (1x io) - Y.. I > h') S p.. (Y. S "'I S p.{Xi·) s 11") + p. (I Xi O '
   
   LeU;", n, and ,""" u. tmd to +00. _
   
   /I.(]-oo.")
   
   Finally, lettl", h'.
   
   Y. I > h") .
   
   obtain
   
   < liminfP.(y. S :z) S
   
   Um~up
   
   S
   
   /1.(1- oo.t/1 )·
   
   tend 10 O. ..., _
   
   h~
   
   _
   
   /I.(]-oo. zl) -
   
   p..{Y. S I )
   
   ,hat lim p. IY.S"' ).
   
   "--
   
   o
   
   Theorem 16.1.3 Fetr mc4 n E N , Id (n..F •. p.l k II pll h!ili" "",00: , ."" kt X •. Y. joe 1_ ....1iUm ""n"dlt. Jrr- (0..,1". ) into R~. Ij {X. )' 1:1 ..."",,u ill Ie.. .0 II proba~ I' 1I...t if IX. - Y. ).;!;. 00JI....,u in Ie.. 10 to, I.kn (Y.)'l:' ....."'"'l1". in Ie.. 10 1'.
   
   m
   
   1&. The o.llIroJ Llmil Tl>
   a P ropo,dd on 111.1.3 'JIO«I
   
   u l
   
   «n.. ..1"~. P~J) . ~1
   
   aM. Jor ail n E N . Id X. lao: a "'....."" ""rio.bIe
   
   >,
   
   u...t
   
   DJ proItability Jnmt (n.. . .1"~ )
   
   lao: a """"""""
   
   into R O. A, ... me (X~) .. 0!I10".."u in lei_ 10 G prokbiIitJ,... Then .... X~ ... ~o fOf" tIII.dI otqOI O.
   
   "'I>-
   
   P IlOOt': F;" 6:> O. Gi""" €:> O. cit<:><:.e r:> 0 oolhllt 8 _ !- r.r)· quadrablo! aDd ,.. (R ' - B ) S 1/2. For .. i-illtably cboltIen in~ ,.. ~ I. we
   
   ...~
   
   .. p.(I.... X"):> ~):S p.. (j X"J:> r):S (. in $bart. P~( la.X. I :> 6) to 0 ... n - +00. ~~ . .... X • ... to (Proposition 16.!.1).
   
   <:0 .... " .....
   
   a
   
   Tbeo .... '" 111. 1.4. kl (n.. . .1"•. P. » . lt,. (X. ).l!l. "nd,.. lao: '" in ;>' +001. Gn4 101 (&.. l. l!l lao: • oepenoo ""'~I\f to ~ v. R". Then (.... X~ + &"}.l!' ""'WCTJ'U in "'111 '" lite P" ,, ' Hi;''' .... :F' . the no 'j< 01", under:E _
   1\_.(. . _
   
   <>.X. +~ "'" .. '" the ""'~ t beo;wem. olX~ ... t. '" Propalition 16.1.3. HeDo!I ( .... X. +&.., - (aX. + ~) _ (a. - a)X. +(It... - ~) am· '''ga in law \.() to ... n _ +00 (P, Oj>Giilioe 16.1.1) . T hea, by n ......em 16.1.3• P I'IOOI' :
   
   .... x .. +&........
   
   16.2
   
   a
   
   The Lindeberg Thoorem
   
   Flxd E N . W";I-- 1%. P roposition I B.::U /..(1 «(n •. .1"~ . P~» "l! ' lie B ~ 01 pOl to~~ . pc .... AM, IOf" tIII.dI .. ?: I. 101 x . lao: " .... ...tom ....... We frr;m (n.. ,.1".. 1 m:.. R ". Fu...!Ir. 101 ,.. lie " p .. loloililr ... :F' . 17101 x • ... ,.. if ond onlr if IX. ... h,(ji) IOf"
   PItOOF: SUppo8e l bot modition boi
   tho: mappi", t beorem,.oo
   
   ba>cr;
   
   E(£,,,x' l _
   
   j .;,,~
   for oJl real •.
   
   v
   
   ,
   
   ••
   
   .now.
   
   1W~
   
   . .. 1 IWtbedl&rac:l.rillli<: function of X~ oon" tp pOint.. iIe to lhat of 1'. T1lt:n:fore, X~ ... IJ. Coa,ukly, if X ~ ... '" then IX~ ... h,tp) for t6Ch t IE R 4. t". lbe mappi~ th«ll"",. C By rn F, !II of P~lion 16.2. 1, ~n limit t""""'tIlI can be reduoed liDtIy to tbe ooo-
   Ie.. -
   
   z-
   
   ~nd
   
   f"(ltI.
   
   
   (izt!)'I- · ( (,,+I IZIH' 21~I·) _ )! ' III .
   
   '"' _ < mf
   
   '-'
   
   OS' S·
   
   (I),
   
   '"t""
   
   fof " 2: O. 1lepIaca" to,. ,, - I In (I), ooIve lOr tbe in1egnl! "" subotltute I.bil br the iolf:g7"&l iF> (2); this si-
   
   0"' .
   
   '"' l iz)' '-' kl "S' 5 _
   
   +
   
   j" /'· (r -.)--' (o" -l) d. (" - I)! 0
   
   n,hl , aod
   
   (3)
   
   i:or .. 2: t . f:M.iln..li"l tIM: inl~ in (2) aod (3) {ODrTIider ~"'Iy tIM: _ r 2: Oaod r < 0) ..,..1ads to
   
   Ie" _
   
   I-
   
   '"' Ciz)' < ...
   
   '-'
   
   OS ' 5 ~
   
   kl
   
   (1z1 . . . ' 2!!!:) . (,,+ 1)1' n! o
   
   li-"
   
   la """ Ia8I. iDoquali!y. tbe full! term 01> tbe "11Et sharp Nlimat.e lOr !r l -...ll. the ... ... 1 an ..... ima1e br Irl large. No- SUI'P'* that , lor _h II E N , X. ,' .. .. , X~ ... ..... random >VII.bII!I from .. probability apKe «(1", T", p. ) into R ' ; «(1~, T. , P~ ) mo.y chul" ..tth II. Such • coIlealoa iI called. IrJan&ular of .. ,·h" ¥ariablioo.
   arr.y
   
   ex..
   
   s.. ..
   
   'VI
   
   ate
   
   To _&bUsh the .-yrnptQti< ,oormaiiLy (jf S~ . It iIIlMJ< ' P ry l(j ccp&nd the cI!anocmillti< function at ~ X .... tolbe oeoond«der I,",,", &Dd t(j estimate 'ul ""nit: tbe remf,i<>
   S,,_
   
   I"" -'
   
   """''''<'<
   
   Theorem UI.2.1 (Lindebers) IIW, n. IN. X~." ..., X~ .... is i...uJOl'l'd'ellt "nd lIf.~fiu E{X~ .. l - 0 and O(S. ) .. I • . 11 I ipd!ft'j ...... tWoo Joo/u I"" 1111 e :> O. 1M.. S• ... N. (O. I. ). P JIO()'"
   
   Fint poor!.
   I" ... ... - 111,··· "'_I S
   
   L
   
   ~ I S k S ~~ .
   
   I', - "'.1
   
   put.,!.. ...
   
   (I)
   
   ' S' s" lOr.Jl O'lmpla DUmbtrs ., . ... ..... 1<', •. ••• "'_ of modullll 61 mo6t l. 100...:1. thit folio-.. by induction from
   
   ., . .. : .. - "" ... """ '"' (., - III.)Z,·· : .. + "" (""," z.. - ....... "'",1. ~mma
   
   N(>W, by
   
   16.2.1,
   
   le~' -
   
   (I
   
   + iu -
   
   ie"')1 S; min (lui' , lu i').
   
   Thereron-. the ctwa<W:riltk fuoction ....... of X~ .. "'I~
   
   1......,(1) -
   
   (I -
   
   Fer 0 < e S I, the
   
   { }1}to.• IS«
   
   ~!1"! •• )1 S; E [ min(lIX. ..
   
   rWrt-haDd $Ide oJ. (:!) it a1
   
   IIX... ~ .fP.+ (
   
   JIJf.~l""
   
   r. IIX... ~)l
   
   (2).
   
   n>06l
   
   IIX ....l" dPd:
   
   the.,:.. l<)ta\ to I and e It a:biITN)'. It tol.,...'1 by the Llndobt'l cond"
   
   Si ..... U"" ttw
   
   L 1..,.....(ll-(I - i,·cr!.•)I-O
   
   fOl"~~1
   
   (3).
   
   ' S_S r,
   
   'VI
   
   ate
   
   II (l - i1t..! .•) + o(nO)
   
   'S·S-.
   
   - II
   
   'S~ "
   
   e.: p
   
   (-i
   
   t2 ..: ... )+o(n·)
   
   ... erp( _ t;) +o(n")
   
   (4).
   
   For ~ > 0, ..! .• S ~· + fix • •1>< X! .• liP. , e.t>d 10 I~ foIlow1 by the LI»doebe<x <:ODdition that
   
   °
   
   n , 1 - (1/ 7 )1· ..! . _ aU between and I , and by (I ) 'Po .• (I) e.t>d .< 1 - (112)t1 ..: .. ) differ by at maR the 111m In (3). Thls mta.bllolll!lllhe first 01 the uympU>ti<: ~1aI.1ono In (4). Na~ , (1 ) aIeo Implies that
   
   For
   
   larse eJlOII&b
   
   n,:5.!S"
   
   n,,s.,s•
   
   J II exP( - ~f·":.•) - II (1 - ~12..: ... )JS '''''''' ',s_"•• For " 2: O. le- ' - I +"1- II: e-- (>: - II) dul :5 I: (>: - 11) dll ... >:"/2. Vol", tlLlo h. tbe rl&bt-baod member of tbe ",wedi", Inequality, ..., obtai. the bnund
   
   L,".,s .. .. S.
   
   ,IL"
   
   (5) and the foci that ! ... ... I, IIUJJI wlLid> the -.:.nd "'Iua!;ty in (4) follows. Now-, the ch&t&clt"ristic flUloCtion
   (1/8)t
   II"'" '" 0,
   
   n,,s.s••
   
   S , 00 pan of the proof: We LIOW ~ '" the cue where d 10 DOl ,+ ' nly "'Iua! to I. Let I E R~ be g\ve-o , JUCb that !t! ... I. For ~ n E N , put .. Var{tX .... ) for all I :5 t :5 r•• 10 that
   
   ! ... ...
   
   L ..:... ... D{IS.)", ( D(5..)I -
   
   ',s_". The
   
   !II' • L.
   
   LiD
   ~m •- Tbe' efo~,
   
   L ( 'S ' S"
   
   11,x. , 1><
   
   )tXo •• 12 dP. _ 0
   
   for ~> O .
   
   IS. ... N ,(O, I) .. h, (N~(O,l4» . By Proposition 16.2. 1, ...,
   1M)'
   
   a
   
   Now "'" Ulu$uaw lhe Li~ theon!m by two examples. Fir.ol. . ..., maill the OOI.al.ion or Chaplet I~, Ex=_ I, but "'" ..riw X~ .. ( for 15 k 5 n) i_nod of X •. and p. inst
   s.. ...
   
   LX.,.., l:S ' :S ~
   
   L
   
   L,. - £(5. ) =
   
   ,! - Vou{5~) -
   
   L IS'S.
   
   (lIt ),
   
   i- L
   
   (I{").
   
   l:S'~'
   
   ... .. [o!]'f' . ",.", "'" appI.y the I ,iro.t..." theorem to (X. ,.. - E(Xd~"' and""" t hat (5. - L. )/ ... (»D,~ io Ia.. to N,(O. I J. Sinc-e 0." ., ...{ n OOQVO!'lItS to 1 and b" - (L,. - los nJJ.,IIiii1i """h'"", to 0 "" n _ +00, "'" «",dude thai. (s. - Log ..".,IIiii1i ... 0.,,(5. - L.){ ... + b" _ N, (0, I). Thi 0 that is,
   
   p.(I!"n - II > e) _wroed>eo 0 .. n -
   
   +00,
   
   ""* ... """wionl of {I, ... ,n} Iut."" about kIg n cycles..
   
   Nert, ..... .- the _ &lion 01 Cha~ ili, E"",," 2, hul "'" "';1" p" in$.ead of P, and X ... (for 15 t :S nJ iusta N ,(0, I).
   
   _...
   
   ,
   
   ,
   
   ""-'f'(s.. _ ~ ) _ N,(O, I).
   
   16.3 The Central Limit Theorem Propooltkm 16.3. 1 Loti (0 , T, PJ !Ie G »r"l>IIoklil~..,...,., mnd I.., Y !Ie • ru .... .tom....,.;.,Wo! oj Md(!t" 2 jrrmI (O,T) in'" R " (d E N ) ndt !hat E( Y) _ O. Loti 8 .... d x p maIrir, ,.;tI! ... not p • .noch tMl 8· B' _ D(Y ), and "'I (i , •... , i ,) Q/ (I , ... ,
   ... ...b"'_
   
   1$..) Tbo
   
   C.~lraI
   
   Llm!1 Tbeooeh'
   
   11M pod X .. [X" . ... x.l. Then D{X) ... t. 11M Y .. BX
   
   l»
   
   ..m.o.t ftn'!Ir.
   
   P ROOf": f or all I ~ j ~ II, denote by I, tilt jtllltnl" 01 8 .. ("'u). Oi"", J € ( I .. . .. II) - Ii" ... . i.}, theft .....J ~umbon I" ... ,I. lIUdt thal hi., -+ .. . -+ I"J.. .. ', . Let 8, be.be (p -+ I) " I' mat';" ...' - ron ~ I,,, ... , I•• •I}. 'Ibm
   
   ""'*
   
   ....,
   
   ..
   
   (f" ... , I•. - l) ·D(fY,""', Y4. Yj j)·
   
   ..
   
   (f,. .. ,I,. - I). 8 j
   
   •
   
   O•
   
   ...... ,.. (I, .... . t•• - I )· B j
   
   .. O. ~.
   
   Fh:om tbe equ..oJ!ty I, .. 1,1" for all I :5 k S 1', and 10
   
   al.,X,
   
   .U,.
   
   Yi .. I , y' ,
   
   "
   
   -,'.
   
   "
   
   -.'. -+ ... + I,Y..
   
   01_
   
   + .. +1,.1•• foIlowI "'I.' .. I,,,,;, .• + ... +l,.a;, ..
   
   + ... -+ a , .. X , .. t, (a ". ,X, + ... + a ..... X. ) -+ ... + t,{a .,. lX , -+ ... + a ;,.,.x.).
   
   wl>eoce ...... thai. Y _ 8X almo:ooM. ... rely. Ckwly. X ill . rerwlom YariabIo of Older 2, and
   
   a1"'(111t Shrely,
   
   D(fl'j " .. . • Y.. J)" M · D{X )· M ' M · M' _ M ·D{X )· A/'
   
   'I'
   
   ate
   
   336
   
   16. The Omtrollim.i. TI"",,<m
   
   O(X ) = I •.
   
   a N..... let (O, ~, P ) M & probo.bili.y $p8l:1e and (YR)...~ ' an independent liequeM! of identically distrlb,ncd random variabLo:o (0, F) into R", Sup. »OIOe l hat IYd' dP < +00 and put 0 ... O(Y, ), Wri~ k>r Y, + ... + y~_
   
   from
   
   r
   
   s..
   
   Tho".., m 16.3. 1 (C e ntral Llnu, Theoren. )
   
   7,; (S~ - nE(Y, ») ... "'..
   
   (0, D J
   
   PROOP' We may IIIlpp ea<:h YR. AlI!ICl(;i~ x~ as in P")j>
   ,
   
   ,
   
   - 1, _ If ",, (X , + ... + X. ) ... N,(O, I,), ,hen J,i( Y,
   
   + ... + Y~)
   
   I..... to ... (N,(O, 1, 1) _ N,,(O, O) (... ben: "' ;, tbe linear mapping from R " illto R " with matrill B ). ~Iore, it is~ to~ tbet~ ... hl:n E(Y,) _ 0 and o C Y,) ~ 1,, _BUI, in ,h;, ca81!, put X~.1< = Y. /"" for Mdt II E N &lid eacll l S k S n, "" ,hal. O( X. " +.. ·+X•.• ) '"' I". By tbe Linde"", .. theotem 5.1"" .... N,,(O. I .. ),
   
   OOD~ in
   
   ... dmind.
   
   0
   
   In fact, tben: is • simple proof of 11>0 CE1I,raI limit theo",,,,, which ...,
   
   g;~
   
   ~.
   
   By PtOp(II'itioo 14.4.4 , tbrc dw~'M: fun(;. ion 'P of Y, - E(Y, ) ill f\lI"\t","~, 0 ",,(0) _ 0 and - D is ,1>0 ""'trill of D''P(O). Th....
   
   t(:r:) '" 1:r:1-'(.,,(:r) - I
   
   +
   
   e' _
   
   (1m'" D:r)
   
   :r approoocbN 0 in R " - (0). We put f rO) '"' O. N_ tho, cha:r~'M: fo ..... ion "'" of (I/ .;;t}(8. - "E(Y, » io :r _ (
   oon'dgte to 0 as
   
   11-
   OOD~
   <: I . Finally,
   
   to ( -I /1)r' Oz ... n _ +00; 1>0""" \lo. (:r:) -
   ("log 'P{:r:/ ./ii) )
   
   .~ ""p{ - ( 1/ 2)r' Dx) , ami (W.).a' am=gc9 plin''''"''' to t be clw". acterlttll: funetion of N.. (O, D). If..., ~oror.:...!.he bypot~ of the oentflli limi. t'-rt:m, 'M oor,dusiona are It~. Th _ thioJ,..., followi .., ,...u1t .
   
   '*' ,""
   
   ••
   
   IU The CeDI,.! Umit
   
   nx a..
   
   337
   
   Pro.,.,.ltlon 16 .:U Ld (n.F, p ) lie" ~ 'P'<'t. ,,1III1d X 10:" ""'........... ~ 01 <>t'Ikr 2 from (fI ,F) into R' ..... ilia: E( X ) .. 0 "lid IN. o\anao:tm.lic: ,..ncOOn 'P 01 X iI
   fa) D iI in....-tiblf: ,,1Id./or 0'!Gd 0 < .. < infj. j., (fDC), ~ aUt. p > 0 nell Uool lo,o(l)1 :s ap« - 1/ 2)0111' 1 lor aliI e R~ ..m./,mg III < p;
   
   (' ) lor mc.II r > 0, ""1'1.." . 10,0(1)1 < ). P ROOF: By I~ Fourier invemoo rormula, X hu., COIIlin...... denaily. lei. "J be tbe bilinear form on a " " a " .·haoo matrix ill D. AMumo thai. (\d (D) ,. O. Tben "J ill d",UNUIe, aDd Ibtte ""'''''- I Id R" _ {O) IUCb \hal ,(1.1) .. rDC .. D(tX ) _ O. II"""" IX ,. 0 allllOlt ...my, r.nd X taks Ita ...tUN in lbe h~ ortJ>osooalto I. al""""1 ,,,,",,ly, ld COdU1d'CUon with the that X hu., CO
   r.cc.
   
   .,1. -
   
   oJ;(t ) - I + ~t'(O/~ - Djt + Illt~(I), wbtte ~( I) _ 0 .. 1 ~ 0 in R". P ut m _ Infl. ,.,( rDC). Cb-ty.
   
   r {ol. - D)I
   
   :s -(m _ 0)111'
   
   for all te a '. It _ \.aR I .. that 1(1/ 2)1'(01. - D)II :s I, _ obtaln )01;(1)1
   
   :s II + ~I'("/~ -
   
   D )I I + 111' 1« / )1
   
   :s 1 _III,[ m ;
   
   " -1« 1)lj .
   
   Th.. lhe", aiR.o p > 0 l uch that loJ;{t)1 :s 1 ror III:S p, .. Meb PlO_ I. Nat, IU""""," that je 1 ". O. Tbcn then! t:Xw. e R IUCb that E (e"X) _ ~... .... "'IIIiVlllenl1y E (e'< 'x - " - I) .. O. A fortiori, E ( I - CO$(IX-.))_ O
   
   coo{lX -.) - I allllOlt ...my. Hence X t&kI':o Ita ...t_ ill tJ...:,z {" e a ' : t.z .. . + In_} , -In>(IOIt ",,",Iy. Since Ihie loon oo:!; 10 dz-nrglisibloe r.nd X hao ., """,in....... density. _....,.;"" '" ., ,uuradiction. In obon., )0,0(1)1 < 1 roo- t.II I e a " - (OJ Since V' , ...... tw. '" infinity. _ <:ODclude lhat ""Pj'I". lo,o(l)1 < I t.... eacb r > O. 0
   
   Theorem UI.3 .2 Ut (O,F , P) 1o:"~q<J«anll { X. )~" , unindtpc. :i.-l ~ "I ~Ir oIull'ihld "'...tom .... ric<.hIu '""" (O,F) inlo R'. S.ppOK Ihlol IX oIP < +00 _nil IN. """"",tm.tic fu.ndtort 'P 01 X t u h-Wqr' die ThDI (1/ rn>(S. - nE(X t)) MI. <»r>l1n""'" ""'"' ,.,. lor lilt n e N , _rid (f. ).
   J' ,t'
   
   N,(O, D ), IOIIm!D_D(X ,).
   
   P ROOF, We ""'Y JU~ thAl. £ (X ,)., O. Since the
   .. (t E R"' ltl
   B.
   
   ,.
   
   (t E R" : ltl ?:Nn) ,
   
   .00 p is p ...... by Pro~hlon 16.l.2. For,..""," f E R~, 01>.. . 1.. , (1) ~ up ( - (i{2)t'Dt)
   
   &$
   
   n tends 10
   
   +00, by lbe ceDlr.! 6mit lbin "tn, FUrthermore,
   
   100 {~•. I ... ).~ ,
   ~
   
   Now Ie\ c _
   
   U..... u".
   
   OIl" 11*
   
   Ia. ~(IJI
   
   to , ... exp( - ( 1/2)t' 01) in the met.n , by thol dom-
   
   )1 : III ?: pl· n..a
   
   til :!S t!'- '
   
   I J,.) dI 1... (
   
   1
   
   t!'- '
   
   n""
   
   _ 1\,' I a. coo-e.1I'» to 0 In Ll::(R", a ). I~ (~~, ODlhUga to I .... ap( - ( If.!lt' 01) in
   
   !Iirod.
   
   n.. fourier Invmlon fonnul&.1>ow5 that
   
   I 1'o'(1)I dI. LMR" ,a ), .. de-
   
   a From d~ Oemr.! limit tlw!on!m. _ Do$ Md l.pI_
   
   deduce 1Iw! foIlDwilljj .... Ull, due to
   
   Tbc!ori!m 16.3.3 Fiz l' E JO . II . p",....a. II E N , 1«1 Bt",p) .. u... ""omia/ 1a.. ..itA,.......~ n • ...1 p (""'pia~, c:Ue f). an4l«~ joe ~ abiJitr "" R, 1M irn4f< Q/8(n.p) tmdu z .... (z - np)/ . , f l - pl. ThnI (,.,.),.~, 0Im""","", L I~ I<> N,(O, I ) .
   
   s...
   
   u...
   
   PIIOOI': Ltt B( I, p) he tbc p<.~ ... hj6Iy on tlw! ~I " .'Igti>tll of R oud:> \Iw BU,p)({I )) _ l' and B(l,p)({O}) _ 1 - I'. Ltt ( X' )..o!te common law is
   
   u.. ..- 'E for Ch'p'Of 16 8(1 ,p). For every n € N , tbe la... of X , """,,raJ ~mit ~beoie""
   
   ,
   
   Jnp(l- p}
   
   + .. . + X_
   
   139
   
   is B{n,p). N_, by tbe
   
   (X, + ... + X. - np) =- N,(O. I),
   
   o I'IIr aampie, c...;':\e, an exchange which .. I ..... n ~ At • Ii""" Ii..., • e&l1tr bu tbe prob&bility p of ...", hill tdcpbone. TIle alii _ ... .. ,mOO 1.0 be i~ . ....'hf,t;. tbe number 01 e&Ils tbe erchonge must be &bIo w handle oimul"rw,,,·ly .. tl>&t .• t & P""" time, then! ill. probability ~ 1-<> th&t all e&l1er1_ Nt.tisfied? Let I _ {I .... , n I and n • 10, II'. TIle ootoDll>m 01 tbe ... ot experiment.,.., tbe rt!I:Ordiop of c.aIIs.t tbe exchange, that ill, tbe ('H ),S . S_ E n I if tbe hit caIItr For all I -;;: J: -;;: n. let be tbe !Aut»bilit,r on ....::h that + p, and define P ... (00 tbe tbe MAbie
   
   {:r. _
   
   "<
   
   ...!...1-- (-!·'l '" ,fir _
   
   2
   
   by tbe 0. Moiv...... LaplAoe tbeon!m. Tbe, eFo ... , If n is IMr;e mou&b and
   
   """dw
   
   It
   
   ill
   
   [ -' ~exp(- ~:r')d:r _ a.
   
   "'" ,-.I oaIy !.aIoo c > np + It Jni>( I - pl.
   
   £ureUu lor Chapter 16 I
   
   lAo. N be'M _
   
   of otri<»i.i ... int<s<"ro. T .1>0 _ _ of N • ...d, 50<
   
   - ' " e N . let p. be.he probobility 011 :F • ...m ,bAt p.({i)) .. l I n lor 011
   
   I S i !: n.
   
   I. Put M. _ , ... : k e N I for - . .. e N, '" ,bAt P. (M.I .. (1/" )1"/"1 b aU n ~ I . II "'. n ...., '"'" in' V ' ~ I, _ "'me ",I.... 1>0, ... ,.,
   
   n, _ _ tho let"", p for rwu-. No.- b .. e N , aud _ r.har. I _ P.(~. M.) • n ot. f! - I I p). If \PI .. ) io tho DIUDbor of otri
   ...
   
   d~
   
   .,(,,)/.. . n
   
   yr
   
   ale
   
   2.
   
   R>r _ h poi .... p .
   P~(6,...
   ·<'tE l ) co,..ug
   
   I<>
   
   D. [~+ ('- 4,) ('-~)1
   
   .. .. -+""l.
   
   L« /. bo lbe 1'Imet"""
   
   m-f.Mm)kc(I - ;) "" N. Io<-" ~E N. and Itt / bo Ib.e fwor:lioa
   
   _Ihal _~ ,
   
   /CI. - ndP• • !>pI'
   
   h.. 0 .. " - +00 .
   
   • _ L« (X.),. be a famil,o 011"<10,,,_ .• ' - ' 1 -w:.&eo !tom .",.,.,_ ~lily _ (0. ..4. PI Inc<> (0. 1\ . ...... U>&t P(X • • I ) . II ,. ro.- -" ~ ill N . lot ;{o' bo "'" d"ttibutioa 1\.0...,..,., 01. xto ) ~ II , ). that. foo- - , . " R. I .(m) "I) . . . _ .... <0 ~.) (.,) . . . . _ -too- 1ft 0<'- ......to. foo- Roo"" u " N • • be " 'U'" ,.!.. ) (on I.... EIoteI ,. .'c ...... 01 R ). "'" " - 01. p. u..... , ., "... ... , .. ~t,. .... _ +00
   I::.sox. kc(i-
   
   P_
   
   E
   
   P.I{ '" .
   
   :s
   
   xt o' "... . ax.. p .... _
   
   ... ~ .. u _ +<». &nd.,. ,. f.o<:t • ..... . " ,.. 1ft 10... (_ P. Bill'-ep·'oy. TI« . ... 72.0).
   
   Oed""".hat
   
   a For.-ll .. e
   
   N . Itt .... be 'be ...... 01. P. u.... .... 16.1.2. dod_ t.hao v...).~. """'0" -'<1,-.
   
   1
   
   2
   
   Loot
   
   ,.
   
   Uoi", p,opooi •
   
   (Y"I...l/ml .. l • 10M allmn. disUlb.o •. , in ,ho _ofCbap. _e,e.-iooa
   
   Co.od..de tI>&t
   
   (X... ) ' 5°1 ".'__
   
   bo . t ........"" an-oy 0I....u ....... ~ ....
   X ... , .•.• X •••• 0l0II Iodc;>c knt foo- ~ .. E N . Su_ tbal tI" IX ... I~·· "'" ~ foo- _ 6 :> 0, E(X• .• ) -0. &nd L_...,..·. oondi.ion
   
   ~'!'-
   
   L
   
   '~' ~"
   
   p E(( X •.• J'·· )- 0
   
   .
   
   F
   
   3
   
   Let" be • ~, "" R. .. ,.. _ P( I..)). Let , bo .be fwzcI.iozz
   
   It
   
   b Cbo.pter 16
   
   :WI
   
   n E Z· .
   
   
   Gr.... r E N , ... _ . h a l 4(.,) < +00. Prore Lhal , Lo 01. d-. \hal gr.- Ir ;. oD.,UnllOIe Dol 10. I], aDd .hal
   
   C--',
   
   w_
   
   lzattd 011 Z*. RII" _
   
   je·... · d,.IZ)- .L"".·
   
   ""10 . 11.
   
   ~
   
   r .,-'
   
   , 1<' (1 ) _
   
   ~'I(1) """1 bo ..
   
   Flea
   
   j- zi.. -
   
   I )·· · (z - r + 1) 4 (.. )
   
   +0<>1·
   
   · ..... d~rn-._'I .. ionol..m. ... '" ......... '1 and.nlo' ... !. "'_<4.
   
   01. dioliacl..1o: _ ",. .hal bo.... bo" ... mplod ;. r. , ........ I S r . S ... !At So be ,be <'" S u _ Lhal r . ....... with .. r./ n _ " (0 < " < I). What Lo t.hoo ~ diI!
   IUII.il . 100 ,,1Ui>bn
   
   "" .hat
   
   1)1-
   
   ( 1, ... , .., " b wlUd! Ilwl t ) : t ~ n.. fI:w _ t E N , Io:c ". be t.hoo WIibm pt' "'blI;'1 "" 11 . ... , .. J. n.cn n ..... __ .... t:.ot 01 .ho ,n" nl"c'" ,,' ...,. (I .. .. ..., " and rr io 181'2' ". -ots'iliblot. Let P be .ho probo.bi~'7 iDd~ .". 181' 2' " . "" .bo Bord ..•• .. ,bn l' 01 n. " - we dom ~ X ••• (I S k S r .) from (O. n ;<40 i\ _ "', 0: X • . ,(... ) .. I ; (X ••, + ... + X ••• )(...) io ,100 smOIlleoL iII~ j .""" 1hal llwill .... ,wUlIl .. t. I . Leo Y. be lho . 1> , 6,. _ II> • Bez.......u; _ wit.h pmbobili'1' (p EjO . l l) b ' W' p' PCy. _ t ) .. , . - ',. ....... q - I - , . Show ......... ho X .... (I S k S r. ) ...-eiDdt... m · · and , hal X • .• ;. d~ri""'1«1 .. Y,. _ •• 'I, • . 2. Put So .. X .. , -+- •.• + X. .... .... .. E(So J. ODd .: _ V..-cs.). SIz<",. .hal
   
   .~ ..
   
   L t : l(I _.t: T'· 'S· S' .
   
   . /.'
   
   '.... /.1.. -111. (l _,,)t-~....
   
   . . .. .. • ( I _ 'I._~ +n ,
   
   .....
   
   ,
   
   + ! '.
   
   .- I (I _ .- 1)-' r ..
   
   + Ro .
   
   I:
   
   Dod_ ,bI .~ _" (~/( I - ~)') d)I .. " - += Sim;w!y, 01..... ,1>&• (1/( 1 ~ :1"»
   ..... -.. I:
   
   3. For 011 I' "'10 , 11. ohow .bI E ( Y. ~ 1/1')' )
   
   K
   
   q,.- ' (1 + 7q
   
   +.,') (_
   
   ~ 3) .
   
   4.
   
   m.... put 3.hat Lyapou __ •• rondi. ioD hoklll fm ,be ruMIom van. obIoo X • .• ~ n/(n _ k + I), aDd £or ~ .. 2. Cooclt>de ,hat (s. ~ "'. JI.....
   
   Dod_
   
   />', (0, 1).
   
   I.
   
   Ld k '" N ... '" N , .nd Ie< i , •.... i . be pooi.i .. in....... ouch .ha\ ,,+ i. < k + n fa< 011 I S .. S k. D
   1'1(1. ~ i.
   
   ..... S . )1'1 BI ~ PlI... i.
   
   "" S k) · P(B ).
   
   For 01) ;01 V. '" ~ k + n ODd i H •• ... , i . ?: O. ohow , ..... ,be ..... ~. (I. .... .... + " S .. S m ) beIonp to c.. Dod_ , ha\ C. <01IUi". ,be .in!; U.»•• ,,(I • •• , .... 1.. ) . .....J ~n.oIJ1 . h... C. _ ,,(I. H . . . .) (_ .be ., ~ .lo .• ..., .",).
   
   2.
   
   be'_
   
   ,,(1, .....
   
   Ld . ?: I. n ?: 1 i
   ~
   
   1'(A )P( B )I S
   
   L
   
   IP(I... i. ..... S k) n B) - PlI ... i.
   
   ..""'" .". oum ."tendo ".... ,be ole" .."" (i" . .. • i. )
   3.
   
   L
   
   '~'S'
   
   0(
   
   ... .. S k) PlB )I • H . From put I.
   
   P(1.?: l: + ,, - ,,)S I"':. . J'
   
   Pro¥. ,ha\ \he law of ~ • • I' H' ...1 dooo .." ~ on n.
   
   S,- til .. E(I, ) .. 1'/ (1 - pl'· ~. Lot n Ii" N. 1"'-
   
   4.
   
   f,,-
   
   PlI, ~ 1) 41 .. 1'/ (1 - 1') ODd
   
   ,bo, Vor(I,) ..
   
   11,1...., dP .. L:.~ f. .•,/,I• • , dP. Oed_
   
   J
   
   1,I. HdP ..
   
   L
   
   'S'S·-'
   
   ;1".' +
   
   . bat
   
   L .(i - n)p'(I- 1') . , ~.
   
   ....... .-
   
   , p.-. J.-
   
   J..
   
   !
   
   zlz - "l dF{zl - l !
   
   .."
   
   zdF(zl-(,, - I )r=- '
   
   (I, _ E (I,) (I.... , _ E (I• •• l) tiP ..
   
   ,'"pI'
   
   (I _
   
   (2 - p).
   
   Coacludo,hat
   
   ,.
   
   -
   
   VN{I , ) + 7 L : j (I, - E(ItI)(I ... ,- £ (t.. .. ,I)JP
   
   ."
   
   - P(l - "l-'[ ~ -2(p-D'j.
   
   II
   
   I.
   
   Deli"" <> .. l"rIQ ~ 1 ( I - (.i~ II/I) . f\x ""J' p«>I..bilicy I' "" R, with _ ..:\.Orio4;' fundion " . _ "",.4 " 0 , _ that
   
   2.
   
   Ld (..... ). .. , be._ oi' c:b&r~fwIct""_tbat ( ..... (I) . l l
   
   '" I'" ali I;" _ - (p'--bwo.blUly"" R wb<.dIu...:terioC.ic fww:I;.". it ".., _ thai [,.,. )'l l io 'W>L) ooant,
   
   T Lot (0 , 1', P) be a poot..Nb,y
   
   ""*'"' add (X. )..... ' ... ; _...odo", _ _ 01 s.. ..
   
   nodorn _ _ _ (n,Fj iD
   abilit,.
   
   17 Order Statistics
   
   0.. "' .... _ .... ." ...d ..... io to 01><01.., c:t_ Ido..u.:.ll;r dlotributed nndocn ..n.bIro X" , ••• X" .. alldo ~\ioI'I .. P'*iblo 011 their com""", dlotnool"" '" n.. oo.-.~ (X, .... , X. I """'", .......,pIool.,. In , bill cb.o_ .... Ii... ..,.,.. . .;~ ' 1:2.... I&I'7 ...... ~ la tI...

2. I::b. .bat Xo(' ) '" ... '" X ", . ) _ Tbt 0::>0«11_ 01 ,,,- ,....tom _bIO
   
   caIIod.""
   
   defilld X .
   
   p_.I>o<w_
   
   17.' We iMrod....,. cht _ion ol
   
   Li
   
   'ia;n
   
   ala """plt lind
   
   p... _ _ 00<1 __
   
   17.1 Definition of the Order Statistics
   
   DefInition 17.1.1 Ir ((1. F . P ) Is • probe.bility "p""" and X , .. . " X" &n! i ... dcpcndenc random vambIeI; from (O,F) 1010 R ' wJw:;.e com""", Law "' 1'. then (X L," . . X " j III c;alled an n·eample of /'.
   
   H• .,tdxt h, ""PP*' that tl '" I and I' ;. dllfuse. Writ. F lor tbol di&!ribullon flldCtiQn 01 ,.. Let an ,,-sample (X ,. ... , X. ) ()/. I' be si"=- S;na! P(X , '* X,) _ ptx, ..1II({(I,I) : I E R }) '" 0 for o.U disUn('t i, j In (I. _'" nj .
   
   lbe IO!I. of lboile '" E fl IUCh t hat X , ("' ), .. _. X~ (",) an: all dial""l h.a8 P~bIe - "r 'l "(ilL For "'el ) permulation " of {I , ... , 'I) , we pul fl_ _ {'" , X<1 ' j("') < .. . < X.- Ia )(",)} . T be lIeU fI" an: mutually .mjoint. I"",, bdons 10 T , .00 lbeir un"" hal P -negUcibiu Ct'1lL Now let X be .. raadom variable from (fl. F) Inlo R & equal to (X <1 ' )" . . , X <>j_)l OIl ~ !l".. !ben X can be "';uell (X11l•...• X I. )), and 1M X (» . fur ! <: I: <: 'I , which ...... almoA tun:ty
   P ro»OI'itkm IT.I .I Fbr c....,. ! !O k !O
   
   •
   
   II IJ. ..... "
   
   • .,f - ( t
   
   .'.
   
   !he dillriht>cm fo<...a;.".
   'I,
   
   1.'(0) U' .'"
   
   _ l )!( .. _ t )J ~
   
   (Borel;'n) """';t~ .". : t - (I: _
   
   I
   
   - II
   
   )0-' " .
   
   tII11h rupo:l 10" , !he.. Px" , ~ ~ knIilJ
   
   1)7;'1 _t )l F (I )-t-- '( l -
   
   F(I)) - · ° l(l )
   
   X; ..
   
   X,
   
   "I1!Vf1tJd to " . PROO~:
   
   Fb 1 E R. For all ! ::: i ::: 'I , II.aU )" II .. BD
   Sinoa tbe Xi an: i~lIl . Y - I: 'SiS. I )_~\ O X ; hM 8 (n. F(I)) .. law (Chapter 6. Exe" ·11 2). For "'"Y '" E fl, Y(", II In., cardlnal llumber of {I ::: j::: 'I : X i {", )::: I} . 00 Ih.u ("' : Y (", ) ~ t} and ("" X I' I("') S I} dilJeo- only t".. .. p.DOcglf«jble "",, ~, if ~ . II l be distribU\\on function of X I' )' ~. (I)
   
   _
   
   8 (.. , F (I))( I: ... '
   
   ,'In
   
   n~ r )! F(t )' (1 -
   
   .. L
   
   rl ( ..
   
   F (I)r·'·
   
   OS'''_
   
   Now let • be In., func:tW:>o1 1_ '('"
   
   ,L.
   
   ' ( 'Ii
   
   o",,!;_
   
   T, n _ T
   
   ),1"( 1 -1)- .
   
   from [0 , I} Inll> R. Si""" 0(0) _ 0 .-'(9) .. (I: _
   
   ro. e-.-.ry 6 E (0 , I). we _ "')
   
   "\
   
   1)7;.. _ 1:)1"
   
   ·'(1 - 9)&- -
   
   that .,
   
   .. (I: - \ ).(n - .t)!
   
   [ .- " , 0 "
   
   -"
   
   )- - - /Iu
   
   .
   
   v
   
   ,
   
   ••
   
   "'
   
   1.1'(') ..l _'{1 -
   
   ~. (I) - fA- _ I)'(n _ t I l 0
   
   u)-- oIf...
   
   Nttt, -..-, th.o.t I' ho.s • deu6ity ! with roopect 10 4z. Then F is ina ' o;n:p: and at..Iul~1y cootiauous. 0" the other hAnd , • is abeoIutely oontinuo<>S. It fo!Joooo-a 111M . has derlval;""
   
   I)~n _ t )l I"{I)· - ' (I -
   
   .... (tl - (t _
   
   F (I)t - · 1(1),
   
   and lbe proof is romplece.
   
   0
   
   PTopooillo" 17.1.' Lt.1 B _ (("' ,. ... , r. ) e R" : TMn Px .. nl · 1" , 0 1').
   
   <.u * ...
   
   %,
   
   < r , < ... < :t. ).
   
   PROOF': For 06Ch Bon:I ftmr:tiooo It from R " jnt<) (0 . +00 1.
   
   j · lt o XdP _
   
   but , rot oJll? e G. , /\x, ... -..... ) _ PIC, 0 ... 0 PlI . is in""""", au_phw.. (:r,. , , , . "'.J .... (:r~(I)"'" :r~(.) of R " . .,
   
   r _n
   
   hdPx - nl
   
   j "(It · III )C,. ,.··· ,:<. ) d;.o(r d··
   
   uooer
   
   lhe
   
   dl' ("'. )'
   
   and lhe 1>f'Op»i1;"" 101""--
   
   Now ....
   
   2: 2.
   
   Propo<lon 17.1.3 N
   
   D ... H. ,I) e R ' : ,
   
   < I).
   
   n~
   
   1\x,,,.x,. ,1 - .. C n - 1)(£ (1) - F(.)r - · · 10(1,I) d,,(. ) <4>(I). P llOOr: f or all k e N ..00 lOr all a € (0 , IJ. the function I _ (F (I ) - <.I)' is an ioo.finile integral of t(F {I) - Q) ~- , dF(I ). For e-,.. .y (z.~) E D ,
   
   'j)1-( .....lwl-.. .o1 n(n - I )(F (I ) - F (.)r- 3 . l D(f , l) op (, ) dll(l)
   
   -f~
   
   ndll{') [
   
   (n - 1)[F (I)-
   
   F(')I~-' dF{f)
   
   17.2
   
   -L~ '"
   
   eon",,,,,...... 01. ... Empirical ModiN>
   
   :l4r
   
   n1F(,) - F(r ))" - ' dF(. )
   
   _ (F(~) -
   
   F {. )t [ ...
   
   _ F(~)" - ( F(, ) - F (r )f ·
   
   Oa the Olher 1wKI, lOr e""ry (I' . ~) ED',
   
   j'fII- ""~. I-"" .r! n(n _ I )( F(I) - F (' )f - " '" [.., ndll(l)
   
   l ..
   
   l o(. ,I ) dl'(.) dl'(t )
   
   (n - 1)[F (t )_ F(. )r-'dF{. )
   
   '"' [oo n F (I)" -' dF(I) '" F(W)" · NOfI' ~ F(, ... ) he the d illriw.ion func t ion of [X (I)' X I",)), Ibm
   
   Fi, ... )(r , w)
   
   '"
   
   p( (X l') :s 1')" ( X l,,):s ,»)
   
   '"
   
   P (XI... ):S ,j - p({ X (, ) > :t)r1(X (,,) S
   
   '"
   
   F(, )" - p( X (I) > z )r1{ X (,,)
   
   ' I)
   
   S' I)
   
   b- all (z. , ) E R· . H""""
   
   FI, ... )(z ,, ) '"
   
   F(, )" -
   
   p(
   
   n (I' < X, :5 ~») ,:SiS "
   
   '"
   
   F(, )" _ (F (, ) _ F (r »·
   
   if z < ,. and F(, .• )tz , W) '" F (, )" If I' ::: , . We a>IICIude thai. F I, ... ) io the diMribution func&ion 01
   
   c
   
   17.2 Convergence of the Empirical Median LH (0 , F . P) be • po-oW,bility r"""". DefInition 17.2.1 If X" ... , X" are random variablo!o from (0 ,.1) Into R , the fuJx:!.ion F. ,(1,<01) - (lIn) L:,s_s" 11_... ~I(X. (
   ,
   
   ••
   
   br (z
   
   the meMUn'I (nn the &~I ~-&!gd.""" of R ) defiu.ed br placiD« lhe ~ I at :r. N""I, oboerve thai . lor CYeT)' '" E n, F. L ,-,) io the distribution fUI\C\ion of I lI n) L' :5' S~ ( x . I",) " ow let (X. )>>, he.., ineO: of random , ........ blee from (n , F) inlO R hav1na lhe"""", dilfuso: law 1'. .....ri~ F for the distribution function of 1'_
   
   For all :r E R ,
   
   ,,"oow
   
   Definition 11.2.2 For all .. ~ I, lee X U ) • ... , X(?o+I) he tbe order OIatis\.icl of ( X. ).~ ,. 1'Iwo M . ;.
   
   alll>06l.lUrei y ~ .
   
   Definition 11.2.3 E--,. reo.! number t ouclt that F (I) _ I/ Z is calI«I a modLan of " . Pl"OpOOIltinn 11.2. 1 S~ tItt'n: 01 " '" ""..... M E R ,1IdI1IW F (M ) .. l i Z. 'Then M • ...... tlCfJ'f6 aI_lI...m, 10 M "" n _ +ce . P JU)()P':
   
   Denolu: br D the tot of t'- w E f1 !Iudt tloa.t X ,(w)
   
   ~
   
   X Awl for all
   
   d;'t.!nct;, j E N_ Recalilhat D haa. P.negl~ble oomplement . Now lee • ....:I I he t_ ,uiona! numhe .. ouclt that. <: M <: t _For "'..,..,. '" E D UK! for evuy n E N, the reJuion, <: M~(w) 'S t ill equivalf:nt to
   
   For ~ all w ED, F",.,( •• w) a>Il""l'I"6 to F(. ) <: I/ Z....:I F",.. ,(t, wl """'~ to F (I) > 1/ 2 .. .. _ +00 , br the "'rung 1& .. of I&:p numbcn. He ...... lor allDOlt oJl w E D. 11>.
   Iim lUp M"(wl:5 M :5 lim in! M. (... ).
   
   o P ropo!I1UOD 11.2.2 S"I'POH. IIW ,, 11M G tI"'qw ",<:diGn M OM G ~ I ";/11 fa,..,1 Io .u ...hidl .. a>ntinvo", <1/ M and ,trieU, po":lirlf tll M . '1'Mn
   
   z• .. "'" + i ( M. PRDO..: Tbe fu nction
   
   M) ...... tICfJ'f. in Ia., 10 N, (0, 1/{4/1(M)) ).
   
   io .. de"';ty of p ... .ntb .... peeI. .. density ,. ,,~ by
   
   ,., (11)
   
   -
   
   w.n,
   
   by Propoaitioa ]7.1.1. H...... P,. hili
   
   (2.. + 1)1 I (nl)'J$I + I . 4-
   
   "(4F(M+ $+ i) [1- F(M+ $+i)])·'( M+$+1) - a. ·I¢,.(M U- f (!It + ,,2.. + I II
   
   )
   
   .
   
   We ohow thai.. for all A> 0, {g. ).~ , alIlw.g... uniformly 011 I- A , A ] the """tin"",," denllity ol wmuV-
   
   ....
   
   N, (D, I /( 4r(M )) ) . T hill ..-IU
   
   prove tbat
   
   w
   
   (P, . ~ I
   
   ~y \(I N. (0.1/(4r(Mj)) .
   
   Si""", n! _ (n/ e)-.,Ibii .. " _ +co (Chapter ]3. Exti ciac II ). ....... lim
   
   0-0
   
   0"
   
   _
   
   from R into R. Tbcn,
   
   I I (M )
   
   :r
   
   ror n € N
   
   "(M+ 0 .,nn +
   
   1
   
   [ F ( M +:r1- F (M ) _ / (M )[ "
   
   aDd for U E: R.
   
   ) _ I + b /(M)[ • .,( ,n" + 1
   
   "(M+ v'1Ii•+ I )[' -F(M+ J2n•+ i )[
   
   ..'
   
   •
   
   .f2n + I
   
   )[
   
   [ (. J2f1 +!)['
   
   - 1 - 2n + ,/'(M ) l+r
   
   Recall that, for all :r E 10 , l i.
   
   .....
   
   !Qr;( I - :r) _ _
   
   ' J.
   
   :r _r
   
   I
   
   I - ,,,
   
   dt< -
   
   J.
   
   . •
   
   dI,
   
   G I - b: I
   
   ;;"c.:C.;;, 'VI
   
   ate
   
   If .... aloe n
   
   Iarce ....... ,~ 1101.1
   
   '"'
   
   [ ( J\!n+1 . l]' " ,..(a)<1
   
   2n+IP(M) 1+. Soc all ..
   
   E[_A , AI. .... obI.ain
   
   r
   
   lnlos (~.(a» + 2~:'] f' CM) {I +. ('12:+ I) !or all lIE I-A,AI. So
   
   con,..,,.. unlkonnty to 0 oa [- A . AI as .. _ +00, Then. evidend,.. .. Joe (o/I,,(a») + h' P CM ) con-.. Wlilorml,. to O. and [¢.(..)I" con'~l p uaifonnly to op ( -:hi' P (M ») _ ThcICIott. ",(a ) con"'''SCi unlfonnly to ";2/~ exp ( - lu'P (AI»)f (M) 011 1- A. Al u .. _ +00_ The proof It comp!o:l,,0 If I" " - • Wllq .... EDMW. M 0J>d if M ;. uno-n, P1 "P'-'ticJ1:16 17.2.1 and 17.2.2 pr(WIdo: .n """"",Ie of AI br II .. <>llhl: <mpll'i<:'&l median. ID tbe poo!1.lcu1ar CMe that " .. N,e m . ..' ) It normal • ..., obtain &II toli","" of ~
   
   Eureilef for Chapter 17
   
   00q"'_
   
   1 Lot (O, F . p) M. probot.i!ity _ _ ( X.).~, .. b ' ... _ of ....dom _iabIeO from (0.;') ....., R _itlt .,.,.",.,;;., diolributioo> fullCtioo> F. Fb< &II .. E N, ... F. M tI>e _pIricaI diotrihu.ioo> function ror X ..... , x •. 1. Fb<..-y .. E N . 01.- tbat D. : '" ,--.ndom...n.blo from (0 .;') inlO R. 2.
   
   _ _ ••• IF. Cz... ) -
   
   F(~)I it •
   
   a:>r! ••'.1 1 to FC:.) k>< &II .. € A!. wt.o.o A. io P -ftOdi(iblo. Sionil..-l.y. ( F.(~ _, ,,,»).~, a:>r!""'" to F(:.- ) ""
   
   RIo - r ~ € R, . - tloa< (F.!" ,,,,»).~ ,
   
   B:. ",t.o.o B. 'it p-.,.tisiblt. S. For __,. ~ EIll, II. P'" <;(~) a 1ft! I" : .. ~ FC,,)). SIInw &II ", E
   
   that
   
   ryl
   
   ate
   
   ~
   
   ~
   
   t S m _ I, put " .... _lI'lk/ m )_ Next. b oil Inc'ga ' " ~ 1 ond m ~ 2. d.no\oO "'" D...~(",,) .be maxinnam of lbe For 011 intqel1 m
   
   2 aDd I
   
   qua.o.ioioa
   
   b 4.
   
   I
   
   ~
   
   t S m _ I. $b(w...hat D. (... ):s: D...~("')
   
   s_ Ihat. 10< P _oImoot 011 ... e
   
   + 11m
   
   n. ,110 fuDCtioao
   
   10< oil ", e n.
   
   F. !., ... ) ""'_ WIi-
   
   bmI)I '" F (Cli~1li . boot_).
   
   ~
   
   2
   
   o..l""" from pan 4 lhat, £or p...:;....,.. 011 ""' (1/ ,,) E, s ' S. f Jr. M .... ... 3 weoJ<J,. '" " .. Pr , .. " - +00.
   
   t..ot In.F, p) 110 • ~ ! i\Y _ _ _ ( X ,. ___ . X. ) • ...,- - . ' 10 .... [1:11" . tho ",,~_io.I low with _ _ ... " Wrl ... XI')" . .• X I.) 10< ,ho order _ _ of (X , •.. . , X .). aDd _ Z ... X I') _ X (•• " rc. oJI 2 S i ~ n. I.
   
   p """, t hat
   
   Rm~
   
   of
   
   (<> > 0). Z, .. XI' )'
   
   flf, ..... Z.I 11M. de"';I)". and aompu... it.
   
   2_ Show that Z,._ . .. Z• .... lode ... DdedL"'" that
   
   Po • .. ,,(.. - k + l) u pj- ..(.. - k + I )f )' 1" .....I(f ) "
   
   rc.
   
   r«ry
   
   1 S k S n. I .. othor
   
   ~,
   
   Pf, ill tbo exVO",,,'ioI 10• • ith
   
   _ _ ,,(n _ k + I).
   
   3 C....., I > (), lot (n,F.p) be. prot..hilit)" I ,,"'" lot X " ... . X. (with ..
   
   ~
   
   2) bo _
   
   ... ' ,., '" ... . , _ f r o m (n.T) in", R w.... ..-""'"
   
   law 10" a (1/ 1) · 1" ~I dz. Writ. X ( I ), . • .• X , . ) 10< tho order _ ......" oltbo oo.mplo ( X , •.. . , X .). Put t . .. XII), 1.. .. XI
   """".hat. tbo poi"'" x. (... , (";th I S k S n) lie ill 10.11, ,,- X. (..) panitlon 10, II In", .. oo: ;-.e ,,.. , • 10
   
   L.., .. 1- X •• ,.
   
   ... dilltiDC;t aDd olln~,,"
   
   s... 10<
   
   0Vft}' ...
   
   t, (...1.... ,L... (....).
   
   1"00- oJl z E R. put
   
   "+.. o.op(z, O). t..ot z " .•.•
   
   %. . ,
   
   bo pooitlve. S - tbAt
   
   PCL, ;> %., ...• t •• , > z •• ,) .. ,~ I(' - ." - ... -
   
   %. . .
   
   ).1'_
   
   18 Condit ional Probability
   
   ~ .. de.""",, ... __ '" variabI<.o X and Y . an.! ...... _ A...:I H, ... w\oII ... ~ .... ~b'l U... ' Y E' A ~_"" u... X E B . TIl. ~" 00I""! ...bon ..... _ '" X 10 fin ite; .100 potOb. .wity ,1I&t Y E C ~ ,hat X .. z . <1<..,"« by p( Y E' C1X _ ..). it "''''JIb' p(C y E C) n eX .. z)1 P(X ,. zj.
   
   Ttr.io
   
   ""'*""'" ..
   
   ,--
   
   18. 1 Loot (n . Y. P) bu prOOabiUIJr"'*'"' &nd ... 1) be .... M ··VI,.. of T. For ...,. _ (n , T) ir>to R' , ....... uio ........;q... (up to a.e . ... ,...h.y) random ¥Viable Z &000 ttl, til bt<6 R' ouch ,lw. J~ ZdP _ fA Y o..-.. ....... ,1>0>.
   
   ...-..n.w. y
   
   ex,••
   
   -. 15.3
   
   "''''pM' '"
   
   7
   
   " " "
   
   Kent,.., s<_ alboe Jer-n·. Ineq.......Y to_ "' ~ upeotood .... ,_ 11 '" io
   
   _ut" ODd Y io P.i~abIo, "''' £ (Y ID) 'S E(Io>\ YJID) (P"-',..... 18.3.1).
   
   ...
   
   18..1. w. oJeA ....... _itloaal _1«1 ~ ol Y IP""" , be ,..n"' .~ iabIo X. Whoa t ho "'"_ 01 X 10 &booiuteiy OODtlm_ with ","pee' to I.eboos""",...
   
   ....... _".., -"p"'" ElYIX) (.... l ion lU
   
   ~ .,.;, )
   
   _ _ ..,....
   
   p" .~\si bio
   
   ...
   
   (~
   
   I ).
   
   1M nio ~ion io
   
   ~ to
   
   ,100 ""'" oJ. 1M COt>dit.oo..alla.. of Y ( i - X
   
   1&.8 We _ ' " _ ~i'''''''' Ia.......... a P" io <>ti"-'O " 'tb _ _ to • pnod ..:t " 1> ~. 18.7 W~ _ . ho nioltflOt of condit ............ when Y io ... R' nI....:I ~ tn_ .." 18.7_1) ,
   
   ,.
   
   ....... r'
   
   !om
   
   18.1
   
   Conditional Expectation
   
   or
   
   LeI. (11, F. P I be • probability SI*'", :D a ""1H:>4ebra F , and Ptv lbe pooo..bility E _ [ll$dPon l). LeI. Z : (I _ R {t 2: I} be ~ D. Pz Is both tbe imace ~ of P under Z and the imace mea/iUre of Ptv W>docr Z . SiDoe tile idelltity 01 R - is Pz- _u~ , \(I ....y it ill Pz·l~ _ u.u Z is P-!n\q:rahlc, or tluot Z is PfV!m~ In ohon., Z io PfVlnt~ If and only if it io P- inWcrable, and lbet> f Z d(Ptvl" f Z dP.
   
   T t->rem 18 . 1. 1 l.d Y lie .. p.~ ..n.dorn ~from (11, F) ;1tt<> R· (1 2: I). ~ erUu .. P-~ ",ndom ... riGbIe Z from (n , V) ;111<> R l ,..m Ih4I fA Z dP ... fA Y dP fM..I1 V · ..U A . Any","""'" ... ' ·ow., Z' fr- (11. V) into Rl h Z P''''''''''!...rel,.
   
   ,..,pmu,
   
   P NXlF: W~ may .... _ that Ii: '" I . 11>00 _U«! (Y P )tp Is ahoIoIlltely continUOUll with, pect \(I Ptp. By the Radon- Nikodym tbeon:rn, Ir-.: ""~ lbel tr.:...". mo:a8Ui"abie D and P~int.egrable fllr,edon Z from n into R sud> that Z (Ptp ) .. (Y PJ/ f>' Then ) Z· I .. d( PfV) .. f y . I .. dP lor all A E :D; t!qw\"&lently, Z· l .. dP .. f Y . I .. dP ..... ..u A e:D. if Z' 10M. tho:...",. properties than Z. tbet>
   
   N_.
   
   f
   
   he""" Z' .. Z p,.,...oJmost surely, o.od Z' .. Z p . &imooIt .... "'Iy.
   
   0
   
   Deflnitloll 18.1.1 III tile lIOtation of Theo<em 18.1. 1. the eta. of Z (for the odatioo of almost "'"' ""IuoJi1y bet ....... ..-urabIe D functloole) .. ealled tile <:OIIdition&! expected value of Y &I"'" V , and II ,.";t \ell E (Yl1'j. Z Is. , ...... " of E (Y ID).
   
   Improperly I~, _ oball ofI.en ~ tht.t Z ill tho: <>J<>
   ,.um
   
   P roposition 18.1.1 Lei C Joe .. ..... in 11 ,..m tIuoJ <,(Cl .. D. If Z .... P.inl."cble .....tom ....-"d"e from (11, V ) into R · ,iOG/I IIldt f .. Z dP ... fA Y dP f""..u A € C, tlw:n Z .. E(Y IV ) 4Im
   "'*"""'"
   
   P R(IOF; We ...., lluot ~ .. I. SO""" Z (PfV l and (Y P )tp ~ on C. lbe, an: identical br Proposit.b:o 6.~.2. 0
   
   P i"OpooIi t lon 18. I .Z Gi~ R · 1M Ew'HM" ......... , "nd "'I Z ill< .. verrion of E(YI1'I. "TMII IZ I E(lY1ID) .. /mO,1 ".,...1)0.
   
   s
   
   'VI
   
   ate
   
   PlIOOf': fOr all A E D,
   
   I.
   
   Z
   
   ~p",)
   
   -II.
   
   ZdP -
   
   I. y"l' I.
   
   IY "P ~
   
   By
   
   n.ro""" 6.U
   
   ,"
   
   IEIIYI IV)' (PfI»]{Al ·
   
   coru:lude tbat
   
   £
   
   IZId{P' DI :f [e (I YI tv)· IPn,lf{ A)
   
   lot all A E D, &lid IiI\fJIy , .... ~ IZl· ( PfI» :f e llYI ID)· ( PfI». Hoentt II I S 0 EUYI i1') P,val_ .rudy, &lId P-almolet surtly. CooikIer ,be ..e1&I.Ion of P--&I..-t 1<1... equali, y t...t~ P·;nwp.ble random >viM'! , from (n ,.F) into R - {o.l jNCl i.... y. from (0 . Dl into R ' ). It io .... equl", I!'Q reialloa, &lid ,be a1t~I""'" CJI "'lul~ cis ":1 an be idenllMd wl1b 1.' (0 . P; R ' ) (f 2JiA't.IwIy.... illl & lub6pooOt 01 t. '(O. P: R ' »
   
   SI".,.I IZ I elP .5 I IYl elP, lot ....y .""",", Z of E{YIOI, tbe linear rno.ppi", Y - E(YI:D) from L'(O, p, R ' ) lb1O iUd! ill continuo..... \"'1Ien Y .. ol orOO 2. '~iLI & elmple cII&noc~ ... lion 01 E(YI:D )• .. hich
   
   _p ....
   
   DOW.
   
   Tbo!ore m 18. 1.2 Tk do"" t. J.". 1M ............. ""rialokJ Z fram (0 , D) ,RU> R' (lJ <>nItr~. ",,~Jtitoou a <10M! _1M Me ... CIt H OJ L'{O. P: R ' ) 0"", JM ~ Y E O {O,P : R ' ), tAe ~ .. ~ <>J Y "" H ;,
   
   "'>ch "'"
   
   ,*:o...
   
   E{YIV). PROO" IU H \0 t.ome!.rit. t<> 1.'(0 , P, P; R ·l. il Is eomplete. &lid hence clooiotod In L'(n , P: R " ). Tha prO)eaion Z ofY on H iod1atscuri~ by lbe poopttly tt..~ f{Y - z .~) elP .. 0 lot aU <,?E H . Fio; A. E D. Fo<~ 1 .5 1:f ~. ukirog lot 'P IIIe !uncUon whoee jlk
   _ I..
   
   c
   
   If Y ... politi .... ~b'e random ,vitobie from (O, F) Imo It, IIIen E(Yi1') it clterl:y a I _ ....... y poo;:Ih.ce. If Y if & p.In~b'e n>D
   w"",
   
   P ropoAltlon 18. 1.3 LeI • lie .. /tit;.,_ .....PP"'9 from R. X R' ;nl<> R ' , ~ p , 4. k Gre IIridlJ JIGIiti ... inI.,...... lod Y lie ~ ",,"""on ... n..we from (O."!) iRU> R! ",,01 Z .. no"""m ~fram (n.n into R '. , , _ ...... , Z """ IY [ IZl _rt P_H S' tk E(. o[Y .ZlIP) - h [Y. £ (ZIVJ! ~ hln!l,.
   
   :nw.
   
   P ROO!': F"il'Sl. , SIIP1""1" I~ Y ;. Z4imple . lben Y - L'E/I/; . lA" when: I ill 6nite, I/; ~ c.o R ' lor ~ i E I. and the Ai ~ dilj(Hnt 1).!IeU. N",,'
   
   i
   
   . "fY. £ (ZI'DJldP -
   
   'f,hM.. "'(I/; , E( ZIV )) dP
   
   - r.l 'El
   
   ill ~u.oJ
   
   c.o fA . " IY • Z! dP. for all
   
   4>(I/; . Z)dP
   
   AM..
   
   'D~ II , .. hich J>I'O""S that
   
   . 0 IY . E(ZW))
   
   ~
   
   £ (. 0 IY . Z]I'D)
   
   "_~':" CIIIie ....... oboe"", that . since IYI .. the upper ocqoon", ( X~). ;?:' of poIlitive '/).simple fUIlCliono, upper e",...1ope ofthe X. '£( IZ II'D) (if .."".boo8e £ (lZ II'D)
   
   f f
   
   X. ' £ (I ZII'D) dP - / X. (Z l dP. X.' E( IZIIV ) dP., / (yIIZ l dP,
   
   and the moOOlO
   Next . then:aiwla ocquenoe ( Y. ). ;?: , o(1).";mple functiono from 0 inl<> R' ..hid!. ..,."elg Y . IkpIo.cing Y•. if ,... ' 'Y. by Y• . I U>'.1";11>' 11, ..,. may SUPl""l" that IY. I S 2)YI. l1w:n
   
   l. o fY.,£{ZIV lIl
   
   S S
   
   U. ' · (y.I·IE{Z(V ») 2n• • ·IY I· £ (lZ II'D)
   
   a1mo&!. Itm!ly. Because IY I · £ (IZII'D) is P_in~ble. t .... dominated conver-
   
   ......... t!.eooCIQ pl'tM:S that . o IY. E(Z('D)J ;. P-inl
   L
   
   h
   
   "
   
   ~u.oJ
   
   c.o fA •
   
   [Y . E(Z1'D)) dP
   
   -
   
   lim 141 0(y• • £ (Z''D)) dP .-+ "" A
   
   - .liTo.!. 0
   
   4I o(Y•. Zl dP
   
   IY • Z] dP fo r all 1)...,18 A. and lhe proof ;. mmplete.
   
   Propootit ion 18.1.4 Le( (Y.).~ , !o£ ~ ~ "1 ",ndom ~ '"""
   
   (n,n illUl R " ,
   I" £ (YI'D) ... n _ +<X>.
   
   Y'
   
   ate
   
   !.56
   
   18. CoiHli,iooal ?I"QbabiIi'y
   
   PROOf ' SUl>P<*- tMl
   
   sUP.~ I I Y~ I . ~ni'"
   
   everywhere and, for every n e N, pu~ X~ _ euP~"" ,1Y. - Yf. '!"ben (X')~.2: ' com~", 0 almost """,Ir, .... d
   
   I£ (Y. ID ) -
   
   £ (YID)I S; S{ X .ID)
   
   a1_ """,Ir. It suJ!i<::f&, tloerdo"l, 1(1 """'" I"'" £ ( X. IV ) convclV" 10 0 a1_.uoeIy. \I.e ~ IU(>p {he ialol><er envelope. We Iut.~
   
   f
   
   {riP S;
   
   f
   
   £(X.ID) rlPa
   
   f
   
   k>r all n 2: I ,
   
   X.rlP
   
   and the _comet",,,,,,,, 'MOI'(IIlI~"1 t hat Ik""" { riP _ 0, and { _ 0 a1_ """,Iy.
   
   J
   
   J X . riP -
   
   0 M n _ +CO.
   
   0
   
   N_ our ob)ecti".., "' "' define tbe conditional ""petted ""I... 01 a qlOMiIn~ random , ........ bIe. Let Y be G p- ~jnUgnlNt rnndtm! ~,..,.., (O, ~ in/{> It ~ aW. o.....tom ... rl4b/e Z In- (R V ) in! .. R. meA 1M!
   
   P ropoSition 18 .1.&
   
   J; Z riP - D. Y riP , ... ~ V·ootll A. (O,D ) mu. R ..-1It IItt _ Z '" p." ..j·in~.
   
   1/ z ' if ~ rnndom ....-Iow. fr<>m ,.""",Iou, Ihm Z' _ Z ..m.ou ....-.I). M""", .....,
   
   PROOr. Fi.... ""PI""'" tha. Y "' potitj~. Yor e.-ery inlece< n > O. put Y. _ inf(Y.n) and Z~ _ E( Y~ IP). Tbtn Z_ S Z. oJmosc. surely. for aliO S '" S ... TIle fUn
   "
   
   ZrlP _ OUP { Z.dP.,up { Y. rlP .
   
   .J..
   
   . J..
   
   (" Y dP
   
   J..
   
   10.- all A E D . Now lei. Z' he _"'" m -- 'url'ble V function from n i",o it ........ tbM J~ Z' riP _ J~ Y dP b- all t4eu A. Taking lor A the set f... ' < 0), ~ _ tlw Z' is alm(JO!t """lly pOeIti"!- For -=II intqer n <= 0, put A. _ Z- '([-oo, nf). TIle function Y . I ... is P_integntNe, b«:t._ Y riP - J~_ Z riP ill 6n1te. Ftom tM t'qualit y
   
   Z'e... )
   
   J;"
   
   f... Z'. 1,,_
   
   riP
   
   - l' J:
   
   ~.
   
   •
   
   M~
   
   - LL
   
   Z'rlP Y rlP
   
   Y . 1... rlP
   
   •
   
   Z . I ... rlP
   
   ...
   
   'VI
   
   ate
   
   lot' aU A E D , illOllo:Jw. that Z'· I ... if P .intfCl"'.bJe and Z' . I...... Z · I .. . aI_ rurdy. II....,. Z' :5 Z almost ...,..,Iy. LlkuAoe, Z :5 Z' almoot lumy, "" that Z' ... Z allOOIFt su,..,ly. We now pase to the ~ (:a8e , in ..hit:b Y ~ I P-qUlSO-inucr-ble random ,"IriIbJe from (n.T) into Jt To fix ideu, M1ppo111! that Y - .. P.ln~ l'hereaisl. ~ D function 2, : n-IO , +ooI-:b thal Z,dP _ Y " dP lor all :P.-.. A, ..wi I _urabIe D ..wi p.in~bIe fWlCtion Z, , £I - 10. +oo[ ouch thal f .. Z,dP ... f .. Y - dP for all 1'-oeu A. Then Z ... Z, - z, .. _urabIe D..wI
   
   f;
   
   f;
   
   L ZdP ... L Z,dP-
   
   f.. Z,dP
   
   ... f..·Y "' dP - f.. Y -IIP ... L YdP lor aU D-aeu A. Next. IH Z' be another """"",,"bIe D function from £I into Z'dP ... Y dP for aU D-aeu A. Thm
   
   f;
   
   f;
   
   R. such
   
   thllt
   
   L (Z'+Z, l dP ... LZ'dP+f..Z,dP ... f..' Yd P + f.. Y - dP -
   
   L Y ' dP
   
   for all:P.-.. A. II"""" Z ' + Z, ... Z , allIIOA &limy, and Z' ... Z almoot 1"",ly. C
   
   Deflnlti<>n 18. 1.2 In the J>OI.ation at Proposition 18. U. the claeI of Z (lor tbe ,..,wion 01 11,"* au,.., "'Iu.ality) is aill calltd the conditiooal expected val"" of Y si~ D, ..wI is written E(Y jD). Z is 1 venion 0( E (Y ID). 18. 1.6 l/Y , Y ' .'" tw9 fIUI-'I"'~' S,uWe ",Mom ~ from (£I, F) ,nID R IU14 if Y :5 Y' "'mo.1 ... ,.d~, tMn. E (Y ID ) :5 E(Y'jD) 0>/"",.1 Pl"OpOlll t~
   
   -.
   
   P ItOO" First.. M1ppo111! that Y , Y ' .,.., random varia""'" from (fl, F) into
   
   (0 , +ooJ such that Y :5 Y'. Si""" Y~ ... inf(Y. n ) :5 Y~ ... Inf(Y' . n ) for e\U) h"d.fIM n Ieo from (RF) lato R. oucb lhal Y :5 V' . "Then Y " :5 Y'''' ."d Y -
   "'Iha~
   
   E{Y+IV)
   
   <
   
   E( Y·+ fV) .J~
   
   Slln!ly."d E(Y - IV ) ?: E(r- IV )
   
   .u.
   
   _ . urt!ly. Sir>ee~ of E(Y +fV ) - El Y- IV) ."d E(Y'+I V ) - E(Y' -I V) ill dane
   o
   
   18.1.7 l d (Y~l~> l ". G ,~ Qf 'l""ri.• m~ rnndom --w.N<, jmm (O,T) in!.,. R , ~ich ..."..".,." ~ 'wr1~. II Y. dP ,. - lXl for" """ ~ th ... ~ ~ (E( Y.tv») ' ~ 1 in<:r$LI~' alm(I.! ProJ>05l~io"
   
   """'~ !.o E «wp~,
   
   Y. 11V)·
   
   By bypott-;s, Y.-
   
   PIWOf":
   
   Z_
   
   r
   
   ,uP~~ 1
   
   ill integrable for n E(Y. \V ) and Y _ .up Y. , .ben
   
   t..:K'J t ROUgb . If
   
   f' Y. dP _ f' YdP ~~, f.. f ..
   
   " ZdP. w p /'" E(Y. \V ) dP_sup /...
   
   1M all
   
   " i!:' ..
   
   v-u A. HffiOO Z _ E (Y \V I allllO!lt surel)·.
   
   o
   
   l8.2 The Converse of t he Mean-Value The()rem The Iollowing addillontJ ftOjulu 011 measuro!l will be ..-I to prove k"""Il" inequ&!ily for con<JihontJ expecled vaI~ (Section 18.3). SuppOtOne unions and int......,."ions:
   
   (a l Fllr any Increasing
   
   ~oonoe (A. ).~,
   
   in C,
   
   U.~ I A . lies
   
   in C.
   
   (bJ F,... any de.:.-Ing ""'ll>enoe (A." J.~1 in C, n..~ , A. I;"" in C.
   
   In p&rticnLu • • ,,-ring ill .. 1OOD
   °
   
   OM C
   
   Q
   
   monot""" el4u t;l>nlQin,,'f
   
   P III)()F: LH m boe It.. minim&! II>(IIlOlOt>e eM """. R , the inlt.-o\ion of &II ~ cia ., CO
   beta""".
   
   For any X C n. pm
   
   Zx = (Y c O , X n Y<€m , x
   e m, Xu Ye m I.
   
   ••
   
   Clearly, Y E Z]C ;" "'luivalentlO X E Zy . lei. ( Y; j". \ be. ~ ( iDo ? Fi~, for eu.mple) sequence in p ta ). For..u X c ii,
   
   xn(Uy.)' ;i!: ,
   
   x
   - ."
   
   n (X n y;<),
   
   -
   
   UW n Y, ),
   
   ' i!: 1
   
   xU(UY.) - UIXUY.). ; ~,
   
   &""""" that Z]C is a
   
   ;~ ,
   
   eta.., ... henev.:r it is oonempty. 1£ X lito? in"R. Z]C clearly oon'.;","R, and \bffl,fote m as _n. Now, lOr - r Y E m, Y liN in Z]C •.. X liN in Zy. Si""" Z,· is. rnonotonr! d ... .,...,. "R., it oon"I
   monotOl1O
   
   P ropo!Iit lon 18.2.2 Ld S" Aumoring ina . "R 1M ri", ~ bJ'S, ."" ,. • JOIitiw _ _ .,." S. At.o, Id I k "" UH"r1li4II~ Jr- illt<> a mil &ooocA..,...,., F. aM Itl D "" G _1.. . .4 nokd 01 F. II ( 1/ 111.10» f ... ldp Iiu if> D jo.-"!I A E "R.n<eIt tMl ,.(10 ) > 0, V- I '(F - DJ
   
   "·jnq,,We.....,.,...,
   
   a
   
   "1«aI1~ ,..~.
   
   PI\OOI' : Fi:z E E S . The eta.. 01 \I>oee A E belnr>p 10 ,.(A)D 10 • _ eta...,...,.
   
   iI. (1/11(10»
   
   5 suc:h that A c
   
   f ... I dp
   
   'R.,,,.So it is "'lu.al to 5,,,, that
   
   f... I dp I;.. iu D br ew:h A ~ 5 . ueh thlt.t A c
   
   n.ereen.~. ".~bIo
   
   E and
   
   E and ,.(A) > O.
   
   ...'-t N E Sol E.ueh thlt.t I , B- "
   
   iI measurable
   
   5,"_N and I (E - N ) is oepo.rable (Thc!orem 6.1.2). He""" 1- ' (8 ) n (E - N ) liIoI in 5 f","every BoreI_ B C F . Suppoooe that I (E - N ) n [J< iI """""'pty. and Itt 1»-.: m :!: I} be ..... n.. in I (E-N )n D" . Defintr .. , ball m :!: I, M tbe ~ of t,,","" real numben r > 0 such thai. B{"", r ) doo!o not mee\ D; put Z.. .. (E - N ) n r'(B(~. r ..
   
   ».
   
   n.... I (E -
   
   N) n C1' is included in U"i!: J 8 (11)-.... ) . ,,~, Z ....... tal .. (E - N )n /- '( D'l Fu. , we shaw that ,.(Z.. ) .. 0 for..u m:!: I; It wi!! tbon be obI-..... Ihlt.t E n r ' {IJ< ) iI ne&lisible. AIoume that ,,(z.. ) ,. 0 br """ m :!: l. n...n ~ .. ( I/ ,,(Z.. » f._ I d" Iitor in D. But
   
   bea u? : II - ""I < r .. on
   
   z... nus ..-e ha,-e • c:ontrao;lic:tiort,
   
   o
   
   300
   
   18. Condi ....... Proboobilil)'
   
   18.3
   
   Jensen's Inequality
   
   P roposit ion 18.3. 1
   
   u , H ..... .w...l """","" . 01 in R ' , op .. I~ um;"""'·
   
   hA"""', """_ ~tiOfI
   
   "" H, and Y .. P·intm (O . .FJ .011" R·, ..w. ...l.u ifllI. TIotn E{Y ID) 14A:u. oImoIt 'M~lr. u,,,.,j. ..... in H . "' '' y i4 qtoUO.i"~ lInd "' '' E(Y ID) ::; E(op " Y ID )
   1'1100>"": By Lerrun.o. 3.7. 1. ~ ui$U an affioe fllno:lloc 9 ; R· _ R ,todt .IIM .pfH ::; .... H....,. <'>" Y is bounded below by lhe intep"able fll_ion .:00 Y. Th., pI",M that ... " Y "' quaei-ID~ Let Z he. v., ....... of E(YIP)· By P ropoeilkon :16.4.
   
   P(IA )
   
   L
   
   ZdP - PI'.. )
   
   f.
   
   Y dP
   
   liel in Ii
   
   tMoeI A ~iofyI"3" PIA) :> O. N(JW ...., may cboooe Z wit h val_ in H (Propoeih"" 18.2.2). W.. """" IbM opo Z ::; E (", 0 YIP) &1_ ... ~Iy. FiI"Olt. ""1101_ I .... L "' '' Y iii Inttvabk. Sin«! [l' .... " Y ) 1&Ioeo! It>! """_ in 11000 ch .1 COIl""'" 9i1beel G _ ((r . l) E H )( R , I :> ",-{:I: )) of RH'. the mapping (Z. E( ... " )'IP») _ E()Y ."''' YlIP) l&kes. ~ oucely. it>! ,-.1_ ... G . Henoe 'i' 0 Z ::; F(op 0 YIII) a1moet ... ~Iy ... dcairnd. Nut. _ \.I'eO.' the '"'>tEal CMoe. in ""hid> ", ,, y., not nee ,unly IDt.egnlblo. Ci ..... aD inlqf:r n > O. put A. _ ('" € 0 : E(Io'0 YI1))(",j < n) end ........., tllM p(A,,) .. MOOly """i.i\le. 0..-., by 1" the probability A _ P{ A)l P{A.) on {A e ;F: A c ,01..1 , and by r:r the ... ring (A e: 1): A c A.). Finally.... rit6 Y ' for U.. """'riction of Y to A •• end Z' Iot- . he E"tIOtrioctioo of E(YIP) to A". Thm for
   
   ~
   
   ' Y'dP' - p(' l'YdP - p( ' l ' ZdP-1 ' Z' dP' l .. A. ) .. A.) ~ " for 611 A e 1)'. Thuol Z' _ E(Y' IV' ) almost surdy. 1.11.". i9t . the re!lrioction of E(", ,, YIPI to A. is eqlllll to E{,.o Y'W). almo8t surdy. Since
   
   " . <poYdP .... PIA.) j "<,>o Y ' dP' - l'(A,.) '1" '1.... EI"," YIP)dP • finlt6,

   0
   
   E(Y'III' ) ::; £ ('" 0
   
   r iP')
   
   1"..11I'I(IIIIl .. ~Iy . .o,., o £ (YIII) ::; E{IO" YI1» &lmOIIIt em-ely In A•. We dOl(lo,
   
   t"" ",., E(Y ID)::; E(", o YIP) almost ourely.
   
   CQn.
   
   0
   
   IIll .!<noea ', lneq..ality
   
   361
   
   Propoxltlol> 18.3.2 A_me /JI4l '" io Itric:tl~ ""'va .... H. Then '" .. E(Y IV) .. E(", .. YID ) ..s..-I ....a, i/ aowl on/)' i/ Y io, ..m..m ...... /" /IfIUIl1D a ",ndom wuwble from (n. V ) ;,w, R. PROOF: Let Z be. voen:ioo 01 £ (YII' ) witb vaI_ in H, and _ume tbat ", 0 E (", o Y IV ) &1_ ~Iy. Flm, ... WUOt ,bioI ", .. Y ;" intqrab!e, and a mal number a E)O , II. Tben "' .. (oY + (t - o )Z) '!i n . ", .. Y + (1- 0 ) . ", .. Z , ... hieb PlOItS lhal "' .. {o Y + (I - a)Z ) ;" P·intqrable. Since
   
   z ..
   
   r.x
   
   "' .. Z .. "' .. £(oY
   
   :5
   
   + (I -
   
   o )ZI1')
   
   E(.., .. loY + (I - o )ZIIV) :5 E(1a .", .. Y+ (I - 0 )"' '' ZItI»
   
   almOIrt
   
   .. "' .. Z
   
   .u...ly. H"""" f "' .. laY + ( I - a )Z)dP .. f la . "' .. Y + ( I - n )· "' .. Z)dP, .., .. (oY + ( I - o )ZI .. 0 '",,, Y
   
   + (I -
   
   0 )' "' .. Z
   
   01_ ouretr. ". "' ;" A riclly con"",. _ conclude Ih.at Y .. Z alD>08t ... ",Iy. NOOI' to pi""'" lho ~ <:Me, in .. hieb "' .. Y " nof, "U ' 7wily lntqrable, let A., .. ('" E fI : E(", .. YjV )(", ) '!i n} for 10\'<1)' in""er n 0, lhen y, ... ill &lmoet . "",]y equal in A., to. ~ V , ... . fUDCdo", and, ~ O - u.. ... A" " P.".,'ipble, _ conclude tbat Y ;" &1_ ...reIy equal to a _ ura6le V function. []
   
   WMIl /;: .. l. _
   
   bIo....,.
   
   ~htJy ~
   
   geotr&l rerult.
   
   ~Jtlon 18.:1.3 Ld t .. (o , b) (wW! a < b it! R ) ioo ." ",tenoa./ it! R, .., • mntJtZ~ .... I , aM Y • p."" ",Mom ......bIe from (O, F) ;,w, R, .,,;/11 toOhou '" I . Then E(YI1'J Iaolu, 1 ftr'I'Ir, it...uvu '" 1, "' .. Y if ~int;s' .We. "owl "' .. E(YI1' ) '!i E(", .. YIP)
   ,,,w.: .1,.,....
   
   R.
   
   PROOF: Sinc:e "' ;" """'""", lho li",l~ G; the ..
   
   It
   
   Wben G
   
   op(.) - r,o( .. ) < <,:>(.) - <,:>(1)
   
   .-a -
   
   ,- I
   
   ,.
   
   .
   
   J61
   
   IS. Condit"""" ProI:>&bi61y
   
   and Jinally IIw 'P("~) < I... &( .• ben
   
   .,.{lj - ",.) < ",' - <;:t, ,
   
   -,
   
   s.. 'P
   
   _. .-
   
   (.1Im! ~( M ) ill the Iclt-hand der'iVllth.., of ... M .. ) 10, all I in I n I - co , uf, .nd (<;(1) - ",")l/C' - ,,) o Y dP 2:1"(j Y dP). n- e""du&ionll ,,(l lIiIl "'h~ u ., .. ",,, _ b. E( Y ID) \aku. a l _ lureiy. he VIII"", in t ho. d C"I,,'" 1 oil ",lali~ Ii> R.. If .. btIonp to 1 - I and if _ pul A _ E(Y IIJ)-'(..l. lhen oP( A) = fA Y dP• .,. f (Y - ..)dP .. 0 -'- P(A ) .. O. Similarly. if b btlonp 10 7 - J and if_ put E(YID )-'(b), l hen P(8) .. O. Hen((! E( YIP ) takeoi , alnl(lOlt ...",1)',
   
   1i ..
   
   ill vaI_ in 1, ~ by ';l! tho.
   
   rontI" ....... function from 7 into It eqw.l to "" on la,&(,
   
   and doIine G by;
   
   G ..
   
   ,.
   
   {(:z,I)e1x R : I
   Clearly. C ie. ~ """vex 1lUt-t of R ' , First, !Ill......... that "" 0 Y ill Integn.bI/:. [Y •.;> 0 YJ taking il$ ",Jue; In C . 15(YID). E{"o y fPlI ......., .tmoot 6"",ly. 115 ¥lOl,,,,,, in G. T hU9 ~o
   
   £ (Y('D)
   
   ~
   
   S(IPO YIP) al_ OIUrdy.
   
   Whca a lies in I and 1'(11) > 0, whcrfl A ,. £( YIV)-' (a). lhen Y ,., a AI· ..... oureIy in A. If [" ill tbe proMhj!;,y E _ P( £)/ P(A ) on {£ e T : £ C AI, if 1J' .. (£ E P: E C A ). and i f Y ' is It-e ......ictlon 01 Y Ii> A. t!>cn E{Y ID1' A .. E(Y'IV' ) ~ a [" •• Imoat 9<\",ly and E{""o YIP),..! £ (11'0 Y'I1") .. .,.{a) p'·a1moo' . ,,>dy. Th,.. 1"" £(Y IP ) ~ £ (1" 0 Y IP ) a1_ oureIy in A. U loe...., 1"" E{YJP) ~ E(lI'o Y IV ) a1moM ",,,,Iy in B .. £ ( YJPl-'(b), ",hen h!~ in J. In short , "" o E (YlDl ~ E(.p o Y ID) al.- """,Iy. Now. up", Ali In P"),,,..iUQ" 18.3. 1. ...., __ I hal \I .. lUI inequalily ~ e\'OeD if 'P OY ill nolln~. f"inally • ...., note that ,·he last '''. llion loUo.... frtlm tbe _
   
   &rIumetlt OIl ,hal in Pr-opoosilioll 18.3.2.
   
   []
   
   18.4 Conditional Expected Value Given a Random Variable Defl nltLo .. 18 .•. 1 Ltl (0 ,.11 be .. measurable "I*'" &Dd let X, Z be t.-o random"4riableOl from (0,.11 inw measurable "1*ft (F. 8) &Dd (G,C) , ,..,. lpeeti."ly. Z it; sald to be .. function of X if li>ere exilllll" rancIom...uble 10. from (F, 8 ) into (G,C) sud! lhal Z .. 10. 0 X .
   
   n...o.... m
   
   18 .• . 1 1I-1I.en G .. R ' (fUPOCli""ly, .~ G .. R ) and C ;, 1M Bam ...aigdnI oJ R ' (fUPOCti"'I~, 01 R), tMtt Z ;, .. "'nc~ oJ X if and on4t i/ 1M (Z ) (covuti.., oJ 1M r '(C ) Jor C E C) ;, ind...u.t in ..-eX ).
   
   ,,··'st.,....
   
   e .. R. Awume lhal ..( Z) C ..( X ).
   
   P IIOOF: We will ..... s·,'" only tM cue FIrst. IUP1"*' IhaI. Z ill sim~: Z ..
   
   L ..,O; -1 .." "'Mre 1 io finite &Dd the A; are diajoint .. (X~eeu. For every ; e I . tMre " istl B; E B oucb lhat A; .. X - ' (8, ). Tben 10. .. L"" 0 ; . lB. ill .. random ,'Viable fl"ODl (F. 8 ) into (e ,C) , &Dd 10. 0 X _ Z . N<>w , for tM ",,,,...! CUO , ot.c ...... thal. I""'" "".. loa .. - ....1ICf! { Z. ). ~ , of limple raooom"4riableOl from (O... (X») inw R which ""'!>u p point ... ,"", to Z. Since Z. ill oimJ>le, ..., can find . for every n ~ I, .. random variable 10.. from ( F, B) into (G. C) IItLd:L that h" 0 X .. Z~. Let L be tbe oet of .....,, ","',"" or the lC<juenoe (Io.. ) .~,. Since L .. {Um,uph" .. lim iof ..... )n (l lim inf 10..1<
   
   +00),
   
   ;1 lieo in 8 . Moret:NtI". X (O) io inclL>ded In L . If ..., .w,fino: 10. by 10. .. lim h" on L &I>d 10. _ 0 III> L', ttw.! 10. 0 X .. Z , as deslred. 0 Now let (0. F, P) be .. probability 51"""'", X .. random .-..riabie fl"ODl (n. F) into ......... urabIe rpM>! (F, 8 ), atod ..{X l tbe ..·...,bn."'nerated by X . Ci....., k e N . the ltiatloa 01 Px ·" l _ ,..".., equalily between random "4riabIeOI from (F.8) inw R ' ill an "'lui......""" relation. If Y ia .. P.i~ random variable from (n , .11 into R · , ""'y Px · interp.bIe rancIom ,-.riabie 10. from (F, 8 ) Into R ' ruth that I B hdPx .. Jx - '(B) Y dP for all 8 e 8 ill called ..... oioo> 01 the tonditiooal ""peetOi!d val ... ol Y p."" X . Thus .. random .-..riabie h from (F,8 ) into R ' 10 .. ~ of the <:ODnaI ~POO:t«I val"" 01 Y Ii ...... X If h 0 X ill .. oasion of E{Y Io-{X )); the elM!! of 10. is abo called the coodllional of Y Ii"'" X , and ill wrinen E(Y IX ). We ofUm idcm ify h witb lu duo. Gi."n .. ".I)'Item T in F .ueb that " (Tl " B... px·intqrab\e ....do." variable 10. from (F, 8 ) into R ' ill .. veraion of E(Y !X ) ~he_ hdPx - JX - '< Bl Y d P for all B e T. If X ' ill .. rILOdom variable from (O. F) into (F. 8 ) which io a1mtl1il lu~y equal to X . tben E(Y IX ) .. E{Y IX '), a1l boo,p .. (X l ", ,,(X '). Si.mi.larIy, _ can deli".. EIY IX) lor ILilY P-Quui-iruqrable random variable Y from (O,F) into R. h : F _ R iII • ."roion 01 E(Y IX ) if it ill wMUrabie
   
   "".....,ted ,"",,,,, J.
   
   8.., [B,
   
   lIdP}I _
   
   r Y dP lx -'(9)
   
   !'or.n B E B.
   
   In .lIio cu., Jet Z be. ~ of E(Y"," CX)). 10 1h 1duI, ..~ lUppoet that tbe nqa'ive pari of Z is P.in\ql'abie. PUllin« A .. (z ~ F , h(%) S O) , _ oI:J,oo,,,,, that f~ II dPx .. f ; -'tAl Z dP II no( -00 ..... d tlw lho: neptl"" part of II ill Px-i~. Now
   
   r,. hdP" .. 1;- ' (8)(1> 0 X ) dP for all 8 E 8 ; thuo
   
   " " X is. ~ of E (Y lu(z)_ 'Ibere ex;n I'flOU1Ia anaJocouo 10 thole of &o:tions is.1 &Dd 18.3 for eo:pec.ed ...J_ pOOlrt X. \\'~ CIUI compute E(Y IX ) .. rolJoo,oos. Propooltlo" IS.4 .1 Ld (fl ,F , P) Ioe a ~p IpO« aM X a ",>WI"", .....;..w. from (O.n into IV {j > I) ..,.... 10", Px ;, Qb>I~w, omIin ....... lriU\ ruporcI 10 U,,"tgUe ........... Aj on IV , Fin<>llp, iel Y I>e " P .'ll.IquaIW. ",Mom ..-HI. from (n,n ;"1<> R· (lr ~ I ) ..... II a ...,..;on 0/ E(YIX). 1'IIDI _ 00" fowf • Px ,".,1igi&K &m!/ "" N ""til 1M JoJ./oftng a I._ /tw all % tN, an RJ ""'ldI IIIriItb 10 % ......1., (I f ),; { B. J) Px ( B. ) kndI u. a . trieUp fO,;m. n"mIN:r mod ( l{Px CB.))Ix _' llI.) YdP 1.<1141 10 11(%) ... n _ +00 .
   
   0'
   
   ,OJ,
   
   Let I boo • l r 1nt<&r"bk 80rd fuoction I""" R ' Into 10 , +00 [ ""'" that P., .. fl.,. Since I """ hI an: l,-im..grable. there io • ).rncsli&l ~ Borel IIe\ N' ,uch that, lor all % f. N' . Px (B.)/).,(8~) and ( 1/ ! j(8. ) fB., hI tlAj CI)rn~ to f(.,) and h(.,)f (%) !OS n _ +<0, for ~ (B,,).;t , of s.-t _ .. hid> $hriob to % 0""'1y. N<- tab: N =< N'u f - '(O). 0
   
   PIIOOt":
   
   ""'I""''''"
   
   18.5 Conditional Law of Y Given X Let (n . .F. P) boec • ~bill~y (0 •.1') Imo the ~
   
   os-e
   
   and X . Y 1_ random variabU from
   
   ''*'''" (F, 81 aOO (G,C), "",pal""l,..
   
   18.5.1 Lol (so. )... ,.. , ...... r E 8 ...wI F' ........:... Px , l>o .. lamil¥ of ,""IMbUlltn " " C. Fi% .. ,..•• ""11; ... .... ,..u. !hat ,,(ot) ,. C. 11 ~ ",Jtdi"" F' " " _ ~. (C). F' j" - 0 .... verrion of E{lc; 0 YIX ) lor e....., C E +, /Un ;1 U" ......... "I E (lc 0 Y IX ) /or ttICJ C E C. P~ltkm
   
   by , the c:oIlOletion of tht.ioe C E C for ",hid> ~he function r":o: .... P. (C), F' 11" _ 0 16 .. version of E(lcoY IX). Then ill. A..)'It"'" aOO il romaiDl ot, .. C. 0 PlIOOP:
   
   ~
   
   c.
   
   c. ..
   
   'I'
   
   ate
   
   o..l!nltion 11.5.1 In the notation of PropoRtion 11.~ I. if. for 1lI1 C E C, the function F" 3 " ~ II, {C) . F' ~ " _ 0 II • ....moo of E {l c " Y IX ), then (p, l.v' II called • conditional law of Y ,,~ X_ We .t.all <>fUn write II, '"' ptY IX '"' ,,).
   
   ".".km
   
   T ....... m 18.5. 1 Ld 4> (rap«ti...",. 'II) ". a ill F (rap«ti...",. ill G) n

   
   M
   
   +,
   
   Ute /l<Mtion F' 3" - II. (C) . F' ~,,- 0 it mell5 ... dl~
   
   "
   
   ~x .YI( B" C) _ jBII, (G) dPx (" )
   
   for all 8
   
   E 4> ."'" G E • .
   
   PlIOOf': Clearly. , he conditloru: ..,.., .>" " y. Coo""""'y, I UppIW: they ..,.., .. tis6ed. For every G E . , the fundion F' 3 z _ II, (C). F" ~,, _ O . urable B and m
   
   J. II, (C) dPx (z) .. P(X - ' {B ) n Y-'(C») • J. "
   
   Ie" Y dP
   
   X -'(B)
   
   lor IlIl BE • . Hence. ror every C E • . the function F' 3 " _ II, (G), F' ~,,_O • • ....moo of E( l c" Y IX ) ( P~ti<>n 18.).) . From!'ropoIi,ion IS.!>.! , we dIld\lCle that it • . in fact,. version of E (lc o YIX ) for 1lI1 GEC. 0 U",il (U"her _ice, we oball III""""" that . lor ci vm X and Y . tllne ex • • modltional Ia.. of Y civm X . We loet (s.<. )~,. he such a conditiooll Ia... Propooltlon IS .5.2 Ld / : F "C -10 . +0<>1 ". .......... rublc: B 0 C ( rup<'£' ~. ~x .I'1 ..........nobk) . ~ !h~ /rInction F' 3 " 1(" .l') dll, {'), F" 'I" _ 0 it .......... rublc: B ( rup«tiwl,. px ·~). and
   
   r
   
   r
   
   /d/lx .y\-
   
   r r dPx (:rJ
   
   / (" .l') dll.(l')·
   
   !'R:OOF: For IlIl A E B 0 C, we k_ that 1.,(:o: •. J II mea..urable C for every z E F . Denote by (. the collection of thoooo A E B 0 C Iud: that the r""", tion F' 3 :r - jIA(:r, II) dll. (, l, F' ~,,~ 0 II measura ble B. and 8UCb that j ] A dl\x .I'1 '"' j dPx (:o:) j ] A("" , J.111,(, ). T hen (. • a ). .•~m and It
   
   oontaI... B K C. Hence C. _ B 0 C. Now. if I: F" G _ 10. +0<>] • ImUU!"llble B 0 C. i\ is the upper . " .. Iope of ... l.ot" dlll!.eqr>enoe in St+ (B 0 C). The fin!. rtioo of the ...",,"""t
   
   --
   
   Finally. au""""" that / II ~x .y!,!DeUunbie. There ex;'tl a ........ unbIe B 0 C function. 9. from F" C Into 10 , +0<» which ~ with / ouuidc.
   
   1\" ,I'1' ''''Il!£ihlf, "" A E B 3C, SiDCe
   
   r r
   
   I,,(l:,/I) dll.(~) -
   
   dP", (l:)
   
   0,
   
   the ~ioo A(l:, ') is II. -r rgligible fo r all l: E P ... hlclt do not bo:l<>nl! to oome P,,' nos!i«IbIe lubeet N of P . Fo.-.....,.,. :r E P - N , ~ have
   
   r
   
   I (l:, ~)"".M -
   
   r
   
   Al:,~) d".{r).
   
   • . ( F" ,. :r - F/(l:,iI)d".(/I), \;:'. £',/% -0
   
   r
   
   o;-dPx
   
   r r '" r -r
   
   ..
   
   gtl:,')"".(iI)
   
   dPx (:r)
   
   9 d1\x >'1
   
   I dfl.x .1'1 '
   
   l'Tol><>"ltOon 18.li.3 LeI I toe • ra: H. Thn
   
   ._IJ
   
   ~fin04
   
   PItOOf';. U I
   
   "'" F', if
   
   f
   
   .. fl.x .>'1,;"1
   
   o
   
   SOobU: mOl ping fn>m
   
   u... .....p)'ing % -
   
   F ~ G inl<> J 1(l:.,) 0I;.. (1I) .. P" ,4/mo.t
   
   P" 'int
   f
   
   I df\..t ·>' 1 -
   
   if B @C";m~
   
   % -
   
   dP,, (l: )
   
   f
   
   I (l:, ,1 d;.!.(iI)·
   
   J I (%, /I)"".{II)
   
   if deli""" on l' and io
   
   P",lD~ Furt~ ,
   
   f
   
   Idf\x .I'1"
   
   f
   
   dP,, {l:)
   
   f
   
   f{>: . /I )dl'. M·
   
   Now , lor tho ~".., "'" fiDd. ~bIe BGC Dl&ppi~ 9 from F" G Into II wbk::h ~ witb I f\..t .I'1,a1m'ral!D06t ourely to 9 and IUd> thal !f~ 1 os: 2191. T~ io a Ilx .y!,neg!i«\bIe Aet A E B0C JUdo
   
   "'"
   
   (a) (J~(:r. r))A~ 1
   
   """''''gf)I to 1(>:, /1) !O< eaeb (>: .,) E A':
   
   (b) 1/.1s: 21/ 10<1 A'.
   
   18.5 Ccnditiooal Law 01 Y Ci_ X
   
   3167
   
   r dPx (z ) r IA(z, II) ~. (II) .. 0, lbe lIOCI.ioo A{z ,·) ill ,.. -nq:li«ibie fer all z E P whic:b do _ be""" 1<>..,.,.., Px-nqli,sibie suboet ,v' of F". On ,he od.... hand, r I/ I{% ,,) d", (,I) ill finite for all z E P ..hlcb do belon& to
   
   SiDo:e
   
   not
   
   """'" Px-r "C!j~bIe ... t.eI.,v" of F". pu,,v .. N' V N~. For ev-ery z E P - N , the I e q _ (J~(z,.»)~~, ron" .. "", to l (z"J p.-almoK sumy, aDd 1/~ I(z,·) ill domin· 'M by 21/ 1(z, .) ,..--&l.moIl. .urely. H.""" I (z, .) ill ,..-intqrab!e aod / I (z, ~J <1#. (11) ... Um,,_+<» / 1.(1: , II) ~.(,11, Nez\ , that the functiono z _ 1/1.(%,,) dp. (IIll from P - N imo R are dominated by I: _ 2r 1/ 1(1: ,II)d". (II)' So the % / I {I:, II)
   obIe,,.,,
   
   f
   
   dPx (%)
   
   f
   
   ""'PIli.,.
   
   I (z , II} dp. (-,)
   
   ..
   
   .E.'!'"" /
   
   .. f
   
   dPx (%)
   
   J
   
   1.(%,,) 41'0- (,}
   
   I dl1x .YI·
   
   o Now .... ...., f>t-opw.ilion 18.5.3 to compute conditional ""pecl.e1 .1Il.-.
   
   Tbeon" .. 18.5.3 Ld I : F " G - a · "" ~ 8 0 C ,owl I1x ,>'1,101 •. 11\e1I z _ / I (z , 1/) ~.(,) .. Px -"""'1 ftffd, de/i.... "" P , "owl iI .. px·a/moII .-1, 0f\'4l1<> E(f 0IX , Y )I X ).
   
   .... ,!
   
   PR:OOf': By P""""";tion 18.5.30 z _ / l (z,I/}dp.(I/ ) ill Px-a1...-...my de6ne1 on P , is Px-inl~bIe, and.
   
   Ie
   
   dPx (z ) / I (z , , } dp.(,1) ..
   
   / dPx (z )
   
   J
   
   I (z , , }'
   
   ..
   
   /1 ' lh Cdl1x ,y)
   
   •
   
   /,
   
   IB.c(Z'II) ~.(')
   
   l olX ,YJdP
   
   X - ' (8 )
   
   ferall B E 8 . n.... lhismapping is Px-almosl ru~ly<'Qllal t.o E(Jo[X , YJ IX ),
   
   o
   
   Simll .... ly, ... bave the foIlowina; ..."It.
   
   PN>pQIIltlon 18.5." U, I : F" G ~ Ii "" _noble B 0 C tuIII I1x ,1'1pari-"'".,!.~Ic Then 1M /IindiD1I z _ l(z,,) ~. (,1) /rom P inlo it .. epa! to E(f 0 IX ,YJ IX ), Px ·a/mool ....mll·
   
   r
   
   P IIOO': RISI., ruppoee l bat
   
   '""""'
   
   F' ~ z_
   
   l is
   
   positive. Then, by Propolitioo 18.t>.2, the
   
   j"1(z ,,)dP. (II)
   
   F"~z _ O
   
   ,.
   
   .. ~\11~
   
   I;
   
   r
   
   B. MOI_r,
   
   dP,, (:<1
   
   /(z.,)dl'z(W)
   
   ...
   
   r r
   
   .r
   
   
   f(I.~) , 18(" ) 4,,. 11,)
   
   j'!'hO d'1x ,y)
   
   - j,'
   
   f olX ,Y l dP
   
   X- '( 8 )
   
   lor aU 8 e 8, wllmc:ot the dai.....J _wi. N.... . If lie f\x .yt"'CIIIe. wll """ _ <>f Sene<.l!ily. that 1[" .yt-lntegral*. T hen r (z. -) is " •. in~bIe for oJl z e 1'" ""let> do IlOl. ho!Ions 10 """" 8Uilahic Px·negliVbhllU~ N of F' , For all z E 1'" - N , tho function I (z.- ) ... r (z , -) - , -(:r. .) ill " •. q"......ln~
   
   r ..
   
   "'"
   
   r
   
   l{z.,)<4<.(, l ... EU' " (X , Yll Xl(z ) - E(r .. IX . Y!lXI(z) ... E(J .. IX . YJIX)!z)
   
   p". almoou """"ly in
   
   r.
   
   c
   
   The foUowiJlfl: O>roIlaries to TIw:orem 18.M .00 P""!><8ition \8.H i _ify tbe terminology rqardln& tho <::ODdllioonal upected val .... .... bliohI"lIiu .... l&t~hlp ';I h tho ""poec:U>d val"". ..",hit C ....d 1\-.in'1"...", s.....J.
   Olron .... y 1 Ltl II , G -
   
   a " lot _
   
   ECho YIX J(z)",
   
   f'"
   
   PA"
   
   f
   
   hd".
   
   ."tm.n, IIlI ,...,.", :r_
   
   P IIOOr. Thia ioI .... I-.nl ~noe of Tl>oo«:m 18&.2. boca_ hI,) ill meuurable B ~C and '1... .1'].;I11-.;.... bl
   (l: . ~) _
   
   Corollary Z Ld" , C _ It lot m ....".....w., C 1..u1 I\·.qIOUi.;n~. E(h Y!X ){z) ... h tip, Px •.....",t GI/ ,..,."tI r E F'.
   
   r
   
   ,IJ'"
   
   0 Tht;n
   
   Dellnlt lon 18.1i.2 Let (0 . T , P ) be .. probability .~. 1) a subo .. ~of T , u>d Y a rllIJdom varilt.bIe fn)m (0 . F) Into .. ""'...... ""bIt ~~ (C,C). Writ<'. X lor tbe random variable,., _ w from (O,T) into (fl, 1). EYer}' conditional 10.... of Y gi>'tn X .. <:ailed a condition 10.,.. of Y liven 1).
   
   ,,
   
   18.& Com"" .... "'" of C<mdi .........
   
   18.6
   
   t..wo
   
   Computation of Conditional Laws
   
   Let (U,.F, P) be • probability space.
   
   <>nd Y be t"", .......w... 11fI.ri<>Mu frIY""'" X . Tbeo..,m 18.6. 1
   
   ul X
   
   c
   
   PROOf": Obvious.
   
   ...w... ...
   
   ut
   
   Theorem 18.6.2 X 4JId Y be tw.> .... ......wa fr< G .... (0, +oo{ Ioo .............we B0 C ond ,, 0,,·jlll~, IIIdl IAat f\x .YI .. 1 · (1'0 " ). N_ tkjiM. '1>'" IMf!mditm z _ I{z , ,) dv(~) fr-'(+00), .. I {J(z ,·)/
   r
   
   "1 ..
   
   PROOF: By P roposition 9.2.1. "';. _urabIo! B and finite I'-al..,..,.. - ,. ",here. Let '" be a .......urablo B function from F lnto (0, +oo! ",hlch ~ ';U1 '" on F - '1>-' ( +00). n..n Px .. ",,,. Clearly, '1>-'(0)" ",-'(0);' Px ' nestipble. SiDtt I ; " (o,. a 1 d(1' 0 v ) .. 1;"(0) ",d" .. 0, / ... n; . . . . I' <$ .... 0.1_ .... y .. ben! on '1>-'(0) >< G. For 0.11 B E 5 and 0.11 C E T.
   
   f\x .YI (B
   
   ><
   
   jriao r a 1(.... , ) d,, (z ) dv(,j _ jrr f{ .... ') diI(... j dv(W) i (""F') . a
   
   C ) ..
   
   .. flln" [Ic '~~i) dv(,)j ~z)dl'(z)
   
   ""','I ...(.,
   
   - 1M,[10 f\mbemtore. for every C E T . F' 3 z
   
   -I'~(C) .. (1/10( ... ))
   
   Is measurable 8, 1"" 'J"beorem 18.5.1 law of Y ~""n x .
   
   r
   
   1 · l p. a(z"j dv(W)
   
   """,,.00.,,
   
   . Iw
   
   (I'~)~E"
   
   ill a rooditionol
   
   a
   
   ,.
   
   .
   
   P ropooitioll 18.6.1 I.d X "",, Y !Ie I..... "'''''''''' ""riobkJ frr>m (O, F) in/.(> R J a"" R ' (j . k E N ), ",.ptct;""'~. S.."po .. 1M", auto a fam'l~ (I'. ).., ,, of "",ldiliUu "" R ' triIh Ute joiJqu,ing propertia;
   
   (a) B .... 8 ..... , .vboel <>/ RJ.
   
   <m
   
   ..nidl Px
   
   ;, <>:>nCt"IIlrnUJ.
   
   ~) F
   (e)
   
   f'tx .Y)( ]-00 , _]_ ]-co.1]) ~ ~_ .., " . ( ]-co , II) dPx (r ) /f)f" aU _ E RJ andIE R ' .
   
   ~
   
   (P. ).HI ;, " a>ndllumalla .. of Y ,
   
   Ii"""
   
   X.
   
   PROOf' TI,;" /oIlon from Theorm> 18.5.1.
   
   A, lUI ~nmpJe, let (0 , T ,P) be" pmbobmly 'PI'Il', and let X" .... X~ be l~ random ..... i&bIcs from (O. T) inw R with oornmon (Onl;""""" distribution fUllell"" F. Put X .. oup(X" . . . , X~ ) and leI I ~ be an io~. We ,..;"h '0 a>mpute tt.. conditionalla... of Y ~ X. &i'''''' x. rOO" ..n real nurnbtTs I , I, put G(., I) .. P(X S J , Y S I). If _ :S t, sinoe X, S X ,,,,, ha.... G(.,I) _ P(X S') _ F(.)~ . If , > have
   
   s: :s "
   
   I.""
   
   G{• . I) " P(X , S ., . .. , X . S I , ... . X~
   
   s: .) ~ F (I )r--'(. ).
   
   Now dF" .. PlF" - 1 dF &nd dF" -' .. (PI - I )F"-' dF, .,·twe dF io the Stielt;'" """"""n: nx"'~ with F . I!~, lor "'""ry I E R, l he fuDCtion , _ G (. , I) is lUI i~nit
   '1_ .rI(") dF(,,) + (" -
   
   I ) F(I) F"-~(r) . l ~ ._ 1(") dF{" j .
   
   Fo< 1\11 or ",..:h ,hoL F(r) > O. put PI_ I
   
   I
   
   1
   
   " . .. --;;- . F C:r) . ' 1-00 .. 1dF + ;;t•. ... lth !. the" " ' r
   ".{I -co ,I]) "
   
   n- I PI
   
   F {I} 'F(,,) . ' Io._ [(z) + 1/-"" ,'1(" )
   
   ro, &II I E R ; "" ". (I - co . III dP)( (:I;) .. nF"- ' (z ) - 11_... .'1(" ) dF(r ) + (" - I )F (t )f""- ' (r ) - II' ,,,,,,1(" ) dF(r ). We UlDClude that (1'. I",,,, ie a eOI"ldi.ionol is ..' of X. gi.... n sup(X ' , •• • . X ~) . We reman tbM , lor """" I E R , 1'. (J - 00. Ii) =
   
   E('1_oo ~1 0
   
   X. IX )(z )
   
   ••
   
   18.1 [.:\0.. 1 " clCoodi'lioDalr...... botz G_ R ·
   
   for PX·a1mool all
   
   %
   
   171
   
   E R '. H....,., for eocb I E R, the function
   
   , n - I F (!} R II %- - " - ' F(%} • I ]< ._ l(%}
   
   + 11_....11(%)'
   
   ilI."""';OO of £( I I_"' ~I 0 X. IX ).
   
   18.7
   
   Existence of Conditional Laws when G = R '"
   
   Theorem 18."1.1 Ld (O, F , P I .. G prtIkbitify.,..ce, X a .-....10m .,..,;"bk
   
   fr<mt (O,F)
   
   .,..ce (F .B ), and Y G ...ndom "" ·0w., (0 , F) into R· . Then ~ au,", m""""!lanai "'.. of Y X. into
   
   G .......
   
   ""'w.,
   
   ,.tom
   
   n
   
   fr<mt
   
   -
   
   P ROOf : For eocb I ... (I" .... I, ) E R · . I>Ut 1- 00 . I) .. , ~.~. ) 00 , I;J. Let A .. n , ~,s .)o. .ho] be I noz>empty ~c.angLe. By induct;"n on O :s 1 :S l , we ".~
   
   E(IIo, "'I ~ • '"' "'I ~ J-.. .." • .I • .. • J-oo .<. I" Y IX }
   
   - L
   
   (_I) I.lI. E( II_ ....,]" Y IX )
   
   .IC I '.·· ·)]
   
   for III q+, E (01 H . It, .. zl" .. ,e, E 10" ~, }. when! ':J iI doiinfd OIl follows: ':J.l - '" lor i E J , ':# .. b, for 1 :S i :S I, i,. J , Ind dJ ,; '" Ci fod+ l :s j :S l . h, piIrIicul.lf, E(IA " Y IX }_
   
   L
   
   (- IlIJl. E(lI_... .c1lo YIX ).
   
   .Icll •.. .•• ]
   
   when!
   
   <1,; - 0, for i E J Ind <1., ... ho for i j
   
   J.
   
   For every r e Q ' (with rllional coordinl""'), cboo8o I
   
   """"' :z _ F.(r )
   
   ol E{ I )_" .rI" Y IX ). By the preced;", ~ph and P'O\>'AIIloo 18.1.4, for III % in _ ruitabie 8' e B talT)'I", Px , the folJooori", are lrue:
   
   (I ) IJ.A F. " L JCI,._.oI(- I )I.lI . F. (<1) 2: 0 for every .......... mpty A _ n , s ,~. !o; , ~J wtzo.e vmica """'"e nuional coord ilLlts. (b) For every r E Q' , F. (r + {l / n " ...
   
   lin» ...." u p t<> F. (r ) .. n _
   
   (e) For every r e Q ' Ind for every 1 (OO~
   
   to
   
   0 ... n _
   
   fflCWI&Le
   
   :s
   
   +00.
   
   i :S 1:. F. (T" . . . , ., - n, ... , • • )
   
   +00.
   
   (d) F. (n, ... , n) ron.(lgee to 1 ... n _ +00.
   
   :s :s
   
   N.,.. Ii. % e 8'. I.et • e Q' . I I I: an Inl.,-, &nd T, I nulonrJ Dumber In ] - DO ,,,I. For III n" . .. , "'_ L, n, ,,,, .. . , n. in N , ooneider t he fflCWI&Le
   
   "I.... . . . . ..
   
   J - ]., - n , .• zI
   
   " ... " ]r, .,,] "
   
   ... "
   
   I., -
   
   n • .• ,1·
   
   31'2
   
   18 O:...,HtiolloJ Prnl»bility
   
   Noti", that lI. AI~ •._ ............ )F. !os pOeitiw, Md !Hli, . »I ..... . p-' ..ely n , ... _. n'_ I. n. knd to +co. "'" ..., !.hat F. (.) - F. (. " . . .. 't. _.. , h ) it pOSil iw.. Hence
   
   "I.,... _.
   
   F.f.) - F. fr) -
   
   L: IF~(', . ___ ."-t·"",.,. ..-.••) - F. (_ , .. .. . '1 _" .,.... ,....... ))
   
   'S'S' ;, po;!IIlllw 100' •.Il " • e Q ' such that r, :!O " fo •• n I :S i :S .t. For ~ I e R O, ~ F. (l ) _ infof:Q' ..~, F, (. )_Dy lhe ~i~~. _~ to R' tbt tuno:tlon F, initially ddlned on Q ' . fU -=It I e R ' .00 100' "''ft')' I > O. there e:><*- r e Q' , , ~ I, sud> IhaI. F.(r ) S F. (I) + InS m. lhere e:>
   + (1/10, -. _. 1/IIl) S
   
   F. (.) +~/2
   
   s F.(I) + ~.
   
   'Then, 100' all. E R ' NliPyi", I :S ' '5 • + (1/ 10 • ... . lI n ). . . !).o.w F. (l ) < F.{. ) :S F#(I ) + t. which poooett F. 10 ~t-ccmtinuouo at L Noooo it it _r .... ..., ,hal, 100' ~ _"""ply re<:t&nJle A. _ f1 ,,,;,s.lo•. 1>.1. lI.A F. it
   
   ,IIa,
   
   pOeiti ...,." , . B, P .OpQeition 14.'.1. F. illbe distribo."1on funr:tion of. prob&billty P•. For all r E Q I . the func\ion
   
   8'
   
   ,,:< _
   
   I .. (] - co,rl) _ F. (. )
   
   ... by _noction . • ;e."iotl Y IX ). Sll'OP; the eeu I - DO . ",. lor r E Q ' . oonstItU!4' c~ .00 toI_ ,he "_ri",, Jtt>t. at«t by 'his c_ a,......m !oslhll ~I ,,-·~bt. 01 R' . Propoeit;oo I.U. 1 ~'S lhal {st. )..,siI. conditional .... 01 Y Jiven X . C
   
   Theore m 111 .1 .'1 Lei X orld Y 6<
   "'""*'
   
   lor Px '4IbItoo/ all :< E 8'.
   
   E Q' . IIt.e function F' ;!%- jO. (J - DO.d). F';t: z-O aad lhe fulltllon F' "" ........ 0 - DO , rl), F' ;t: z _ 0 ..... two ,"IoiotW e&!igJble IU'*' N, e 8 01 B' tudt lhaI. ". (I - DO. "I) .. ,,#(1 - co . TI) for oJl % e 8' - N,_ ThIll. if % tiM In B' _ lJ..,Q' N.. _ bo.,.. ". (] - 00 •• J) _ ,,~U - oc ,r!) roo- all r e Q • . .....J to Paoo,-,
   
   (or
   
   all.
   
   ror~ry .
   
   e
   
   R~; It.
   
   'lIlo
   v..
   
   []
   
   Even ~ lhe reeultot of Theor-etntt 18.7.1 .....J 18. n ]Xiln' whoM R · It I'eplz oed bf a rompIete ~ IDO!tnc ~. "'" ohoJI neil.ber I'f'l"'e ...,. UIle t~_~ ..... _te-
   
   Euod
   
   , 10< et:apt". 18
   
   373
   
   finally. ~ pYe' cltri:,l eu.mplc in ... hich lhere .. no condilionalla... of Y IPYetl X . lei. .I. be I..ebeo:cue "",""u", on the Borel .,-.bra 8 of fI _ (0. I]. GiYen • lubeet E of 0 Iud> lhal
   
   .I.. (8 n E ) _ .I.{8}
   
   be all 8 E H {Stct.ion 403}, domoo! by:F II.. mlloo:tion of tbe _ (8 , n E) u (8" nE") .•'hen: 8, and B, Nn Ilrrou«h H. Hmoe:F .. lho omallrst "-al&o:bra conulnln& both H and E. Clearly, P : A .... .I." { A n E) ... pmbabi~ty on:F exteodi", .I.. Write X lor tho ,..;wIon;; vari.IM ... .... ... from (fl. F) illto (0 , H) and Y lOr lho random variable ... - "' from (0 . F) iru.o (0 . F). SUPSX*' there ... condilioDalla.. (.:.:..)••• of Y IPYetl X. who", D, e 8 .. .I.-nqligiblc. n.... ["",( E) .u(...) _ P(E) _ I . eo ~ _ {:.Ie 0_ D ,: "",(E } < I } .. >'-!IO&Iigihle. Let C be lhe ct- roosi:8ti", of t&..e..u \1>,'1] and [O,T] mcb that p, q. T ..., ,..lio".1;; in [0 , II and p 5 q. n.... C ill • oounl.&hle ".~ .... and il "" .. the Bon:l "-al&o:bra 8 of [0, 1]. For e¥trY 8 e C. lhe fuPlCtion 0, :1:.1 .... "", (8 ), D , ~ ... _ 0 ;" • ....-sIon of £ ( I s IX ). Si""" "",( 8 ) .. Is{...) for >'-almo::c. all ... e n - D" then: eri.u • .I.-ntlJi&ible ... ~ ~ e 8 of 0 - (D, u D, ) oucI: thai. "",(8 ) .. I s("') for all ... III 0 - (0, u D, U ~) and all 8 E C. Now, b e-vny:.l e 0 - (D, U 0. u D:.). let ~.. be lhe - EFSun! on T &.fined by the unit ~ II :.I; "'" and t .. '«ret on C. and eo on 8. In particular, "", (1:.1)) _ I. But, Ii""" "",(E) .. I. we a>nclude lhal ... I... in 8 . III short, n - ( 0 , u ~ u 0.) c E . ..hich ... <:Odtl*liction boa,,.. .l.(fI - 0 , u o.U 0.) .. I woo .• .... (E) .. O.
   
   ;;0-'"
   
   ,.1M
   
   Eurc:Uu [or Chapter 18 I
   
   !At (fl. T . P I be • ;>robabi~'Y opo.oo ..... X . ... n • _UrabIO 01*'11 (F. B).
   
   I.
   
   J
   
   S ....... h.at l ho ...... Ian. functioa z _ Y dP io .... ' 'H' of £ (Y IX ) if Y: (fl. F) - R ' io P-~.....t X.....t Y ~_pe,j , ( _ IbM Y io .he poitIt-.ioe limlt of. OIlq-.oe (Z. ). ~, of .,(Y )-tjmple moppillP
   
   """" IbM (Z.t 511 YI). 2. Show.bM.I>o" ro .. functioa " _ Y dP io ......... of £ (Y IX ) if Y : (fl,F) _ I{ io P·qud iDtqrabIo ..... X ..... Y ..... btdq;... ' (_",-.bM y : io p.- :waNe).
   
   r
   
   2 !At (R T .P) be. pmbebi,jl1",*""."'" ~ X ,. __ . X. (.. ~ 2) be iad&pr '""t I ' :lot, >Wi&bIoI!0 _ ot.atiotlco X U) •...• X,.) arioln& from (X ,., ..• X.). By Pros-ition 11.1.2,
   
   f\%",..".x,." _..I ' Ie(", .. ... ~.) dF(" , I"
   
   . ~F(". ).
   
   18. Condi.ion.o.I P ........bl.1itr
   
   374
   
   w'-
   ..
   
   < .., < ... < ... } . H....d.,"'I>. _wM II,,,,
   
   ~2.
   
   I.
   
   r.., R' _ (" E R, F(..) >0) Fot all z E R' • ...,.;~ B, .. (C:;r ... .. ,z __ ,) E R·-' : z, < '" < ... < zo _, <:;r}
   
   ,,,.'"
   
   "".~
   
   !)od,n _
   
   . 1•• (~ •... , r "_I ) dF(" ,) ··· dF(.... _,).
   
   n-' ,
   
   18.6.2 t hat (p.),~ ... io ...., ..(III _ Lo.... of [X(1) ..... Xi_,,1 Ii..... XI_,· I.. 001..,• ...,...., 1i''tII ""'..- Xio, - r • •1>0 random _ .. hie [XIII. '" • Xlo_"I. dio. rib,II.,.t • [Xi",,·,. Xi. -,,1if IX; . .. .. X~_ I) io "" In - l}.mp&o of.1>o Lo.w (II F (r ) . 11_ ....1" F . 2. Now. put X .. oup( X, ... . . X.. ) and Y _ inf(X... ... X.). Wril. D "" ((:;r.,) E R' : ~ < :;r). 8, P""",,",ho'" 17.1.3, 1\.<.>1 h. t ho "'"-it,
   
   Ir . ,) - n( .. - 1)( I"(z) - FI,)r-' . 1,,1:;r,,) "'ith _po>Od
   
   ".,
   
   "" .. I"(:;r).. -I ( F{r ) - F(~) )' " · 1,__ .oI(V)dF I, j.
   
   !ltd..... &<>rn TIoeortno 18-6.2
   
   ,ha, (p. )........ ........diliooood law of Y 11-
   
   x. s.... that ".(1- ..,.JI)" I _ (I _ F'(,)/F(r):-' "" ow.." Cc>odudt
   
   in R.
   
   ,bat.. ;r ( Xi. .... X'_ I) io an In-Ij-Nq>Io of (1/ 1"(%» .110.01 dF .
   
   u.... iDf(X : •. , .• X~_ , ) """ '" .. 10.....
   
   s_ """
   
   .. _ _ tbat. X, ..... X • .... ~ 5 (X, IXioJ) <1000 _ depond on t (I.:S t .:S ~). 01 po.rt I. and definl,.. by ~ t ho fwoctlooo (.01.... . z.- .J _ z , ... .. . .,. r._. f""" R O - ' R. £ (Xm + ... .. X<__OIIX
   1. I " what _
   
   !,.,.,
   
   io equal to
   
   u.. """ -.......
   
   _.lw.
   
   S IX; .. ". + X~_,). benao.., (n _ 1) I:'
   
   dudoo thai, "" Mdo I .:S t .:S n,
   
   f
   
   «I Flz )) dF I/)· eor.,
   
   I f'
   
   ,
   
   - I S (X, IX' ol) lr ) . ,, --;;. F (r ) _.. I dF(I ) + ;;Z
   
   4.
   
   b dF· · ' _ all "E R'. eo.. __ ..-i,/O "'" final ""mple of SoooI .... 18.6. _ taM x .. OOip( X , .. ", x.) u.d Y • inf( X , .. ". X. ) ... bebe Sh<,~
   
   3 Let 10 . T . P) boo .. prol>etJlitJ" _ . and let x,." . x. In ~ 2) boo i-=ltpeo ....... randoa>..n.biot fn>m (O,T) into R ... _ _ _ _ law io ( II I )' I\O .,, ~ (10< .. Ii- I ;>G). Fot..-,G<:t < I, ~< B• • ((Zl ..... :;r._,) E R' -' , 0 <:t, < ... < r __ 1 < :t )
   
   _,..., «.. - I)l/:t--' ) · I . . ..... _1 (wlotrfl .... _, io ~
   
   " " ' " , " , (111
   
   R
   
   O -
   
   ').
   
   En..
   
   tor Ch'poor 18
   
   I.
   
   DedIK1t ""'" EurtiM 2 that (" ' )'< 10 ,'1 ;, • """"itioMl low 01 IX~ I)" " , X' __ 1)J Ii_ XI' "
   
   :t
   
   a.o- poolt i.......mbon z " .. . . zo oI "" .bat ZI + _.. +"0.' < I, aDd put A. {I) " ( {rl , ...• ~) E R
   
   O :
   
   rl > ",.r> - ~,
   
   > z, .. . - . 1/0 -
   
   r.,-,
   
   > z o.I - ,. > Z... ,}.
   
   Fl...tly, put ' .(1) .. Flx", ..__ .>:,.,I (A.{I» .. (,.,/ IO) .I.,,(A.. (I» """ '. _ ,(,) .. ((II -Ijl/,"- ').I.,,_,(A.._,{,» . Frnm PtopaitiDlo 18.5..2""" pUt 1 dtd ..... that
   
   By iDdUdiDD, ronclud< tbat '0 (1) .. ( 1/ 1" )-(1 - '" - ... - h . t)- .
   
   S.
   
   From
   
   put
   
   2, dtd_ ......
   
   p(X(I) > z , . X II) - X (I) > "t, ... , XI_) _ X lo _ J ) > %., 1 _ X ," ) > " •• ,)
   
   " ~(I-Z'- "' - Z"' ,J~ tor 011 '"
   
   ~
   
   • _ fu. tad
   
   O•...• " • • , ;2: O.
   
   n,,_ I> . ud! .hAt 0 < 2h. < I, aDd...
   
   (II. < X,
   io.be
   
   II
   
   . lee .100 ...art A ..
   
   VI!O i 5 ,, ). ~.bat
   
   "uro
   
   c_
   
   •
   
   (n,T, P) 100. pobobiljjy opoco. UId let X aDd Y 100 two random...n.hleo ""'" (n,T) InCo tbe _ . _ (F,B) aDd (O,C), retp«\i~. 8<1_ llI&t ...... io. ooncIi,ioDIoIlo..- (p,,) ... ,. of Y Ii_ X. ~
   
   Leo 11. : a _IO. +oo) IwI"""unbIo C. s-.hAt £ (11. e Y IX )(z ) '"
   
   j"
   
   ,,. (1)
   
   -'e II +001JJ 4Ato._1(1) >
   
   'VI
   
   ate
   
   IS. C<>Ddi' ....... r ........ ti'y
   
   376
   
   5
   
   """*"'"
   
   ,h.o.
   
   Lft, i!: 1 &<>CI 9 ?: 1 boo '''''' ,...u ouch 11, + 1/9 .. I. Ch"" • p<"Oi>oJ>Ili'y """"'" (fl ,F . Pj, lot X boo • ~ .....w:.Io r.- (fl.1') _ • _ _ ( F • .$). &<>CI .. Y &lid Zbe,_ .. _ _ _ (n1')
   
   ImI4 R ""'" lh&t
   
   r Il'Y
   
   liP < +0:> aDd
   
   r
   
   IZrtlP
   
   < +00. P...... , ....
   
   £(IYZllXj :S. [£(iYl'IX)],,' . [E(lzrIXj
   
   j"'.
   
   P... · , _ .unIy (~. r>&llo.... of IY,Z(IP""" X ). II Lft (fl, F . P ) be • 1'" 01 . '"U\J" . , ...... A ............ ( A_). ~ , 01 F ...,.. 10 O&id to be m,., 6with_..... ~0;rli m. _ 1'(A. ., £) .... P(E)boJ1EET. I.
   
   2.
   
   3.
   
   'Iro&' ',..... __ J' . X tiP .. r.- (n,n im<> K
   
   If (A~).,~, io "'I~I ....."-"......uno. !I . _ _ X tiP lor .wry P-....... abIoo r&Z>dom ..,iabIr. X (I.., ........., X io pooi\i".,).
   
   .. f
   
   ,rw
   
   T - " ,bat fI _ ,he .... Jt" bdoo. to ,,(C). So_ _ ._I'(A• ., E) .. ..1'(6) "" oJl E E C. S'- ,hat Iiooo,._ . .. 1'(Jt" n £) .. .. I'(£} ..,.. olf £ E ,,(CI (_ 'boo .._l thoooo ...1 _ ,r... f.~. X liP. J~. .!,(X \o'{c)IIIP '-" ' ''' to" f X tiP .... _ +00. lor.....,. P .illlopabio " ~ . , • . - _ (fI.1') in!<> R. Coatl_ , bat (Jt,,).,~, ......... i>ti ....
   
   Loe<
   
   C
   
   be ....... 1 I
   
   ,ha, ....
   
   Let II;. be •
   
   ;.
   
   prob&bi~.y OCt
   
   F , .t.:>Iut.ly """'""""'" ..ith _ _ to P. • >q 1 1\.
   
   ,0.1..,.
   
   s- u...t ..w .. 10 ... : r" .'0<1 If P
   
   "'*".
   
   T Lft {fl. F . PI boo • p<"Oi>oJ>Ili\J" &<>CI lot (X .)..~, be ... i.ode>: ' ' rot _ ~ d ....1caIly diotr\lioll.Od A ..... nzlab'oo f ..... (fl. 1') 1_ R. Su_ d.... X , 10 01. 2 &<>< _ b ~ E N . PU' S . .. x. +. ·· + X ., Y... S.J{".;n,. aDd (1f(".,Iiij){5.,- Sp,.oI ) (wbor-o Il
   "'*'
   
   z. ..
   
   I . SIoow , bat {Yo _
   
   Z.~,
   
   2.
   
   o"d...,. &.- _ . 1 ,bat
   
   s.
   
   R>< &ll z e R.
   
   ..,...... ~
   
   iD p .l'f"')bebili
   .. N, (O, II (_
   
   n..or.m 1$.1.3).
   
   .mt.o". lor N,(o.l)(I-""'.:rJ). S - that, "" _
   
   P«(Z. :S. "l n E) '-'u,
   
   to ... p(£} .... _ +<x> ( _ .... I" ,
   
   E
   
   EC.
   
   • ZL • , ••
   
   cU "' X .). .. p...,..., ,hat., b ~ E E F. P«(Z. S zl n E) -.".".... to ",P( E ) .. n _ +00 ( _
   
   r..
   
   E>:tmo. 6),
   
   Lft 1\ be . . . . .bi!jh>Wy (oIIOI.i.II."'. wi.h rto', r , to P . b.u '" E R , 1\Iz. :S. %) - ... ( _ Ex d .. 6).
   
   sa.- ,ha•.
   
   S. p _ ,bat {Yo - Z.),.~, "Thootom 10.3.2) .
   
   """''''id;~ ""prOboJ:iili~
   
   1. Cooodode l~ i Polr.... N,(O. I).
   
   10 (I ( _,...... 1 and
   
   Part IV
   
   Operations on Radon Measures
   
   ,.
   
   Copyrighted material
   
   19 /.L-Adequate Family of Measures
   
   r.- _
   
   oa. ooor l r f t _ '" _ ..... . beory ~. ~'2 :1) tbat of N. _bald c.... '/v ' " ..... · p' zu ~, 7, aDd 8). [a t~io cbaj>ter, .... ;",1'0<1...,. , loot _iA of ' 1:;"''' IlomIl7 of Radon _ u .....bOth , lor uampio, oimplifioo .1Ie .. -..17 of
   
   , .
   
   7
   
   .......... _ _ I... ,·S •
   
   '--'
   
   IIlI Tho- Radon 771' .., "'" tim
   .. Radon ai '''un "" Y _ IoIIow-o. If I 10 _ ia"""'" _ Y with «)al.pKt _ I, doll,.. , t.,. j'(%J .. 1 (>: ) . _ " E Y _ g(>:l • 0 atll." w' T1ooo. f~ J,410 1.. " 71'ft iadaood. t.,.,. 011 Y .
   
   "
   
   1~_2
   
   ,....... X _
   
   ~
   
   ,omi'''' of """'.,... ...... i ... rodooced Ia Ihlo oottioo>, .iU be 7 22 .....
   
   laC« I. 1M ,..,.
   
   """ " and .. ,.~ <'- 1) of """'I*' ..... ,bote ;, ... ~ foJnity c,...) •• ~ of _~ ... ....::t. 11004 " - E " A",,, . loa! ... pp(,o..) bdocIp IO :D, and ,bat lbe ..... ouw(,..) ""'" _1oaIlr UIIIDlabIt eM lU
   
   G;w" .. """,,lWl Radon "
   
   (1_ .... 10.3.1 ), ItA
   
   Ia" -,ioo, _ ..1OCIJ1m
   wit h ~ ...
   
   10 .. ,. 5....,' _ fomi!, of "..MIl_ We r:onU U)i .a {Tbooo,m l i .t .2J.
   
   J ~dj.(I)...""'"
   
   ........It dido io oimllar 'Q
   
   _
   
   A.
   
   ""bini"
   
   •
   
   19.1
   
   Induced Radon Measure
   
   ».u&dorif
   
   Lee. X he a locally CIOOIpooa Spoooe. I' • Radon mee3Un on X. and Y a JocaIly compooa oubol!*,", 01 X. For ~ % E Y , t""n It • n.e~ W~ of % in X IJI><:h that W~ n Y It compo.ct. and hence cloeed in W• . CI........ ..... opeD Deighbo. bood Y.ol"% cnnlalned in W. ; thtn v. n Y _ V. n( w .n Y ) io a:-I in V•. Punllql U _ U..,y Y• • _ see thal: U .. open wei Y is c"-d in U. In short. Y is the i"ta_tion oI .. '*-Iwb8e!; of X and an open oubiH
   
   oX.
   
   FO<" all, E 'tty. C ), ~ t". 9' lilt functioo from X Imo C which au'O'itb, on Y and vanisba ou~ Y. F..,. all, E 1{+(Y ). 9' io upper ~ conti""",,""" .nth compo.ct fUppOO"t. It tollow&. to< , E 1{(Y. C). tha.t g' is p·LnV:ograbie. f'urt~, If 9' .....1 s f 1,.ldYp S bU Yp{OUPl>,) Dcllnitlon 19.1.1 fUn Ind\OCOld
   
   ~
   
   t". I' on Y
   
   R.adoo ~ .. : .,"" 9 _ .00 lr:;
   
   ",rit~
   
   py
   
   0<"
   
   f 9' dp on Y
   
   io co.ILed the ".,..,..
   
   I'l l"
   
   Wt thaIIl\I.ooy hi dec,lI . in ~""" 20.1. the inttsration with ~pooct to POll ind'-'O!!d ..-.oIra. fOr """'. _ thaII ""!.. in only the IoILowiPlPl ...... It.
   
   Theorem UI.l.l
   
   !!>yl_ !Ply.
   
   "k+ CY). n..",nloU., fO< t~ £ > 0 ... function g E "kCY. C ) ouch lbat 1, 1S I and Ipy lU) Ipy{g)J +t. Since 19'1 ha...... p~ Let / E
   
   s r. _
   
   s
   
   s !!>Iy(f)+, and , .. € io rubitrary. /P y l(J) s 1"ly(J)· On tho:! other Iwrd, let K he t"" ouppon of /. :wd lei. U he an open n6elrboI{U - K ) s t. For eod> E K . tben:
   
   ...~ ,...u~ IpylU)
   
   en.. • com!>"", nei&hborbood
   
   %
   
   V. 01
   
   %
   
   in U oucI: that V. n Y io CIOOI!>"",_
   
   •
   
   Wt let ("j)~1 be .. 6"itt f::.m U)" in K ouch that lilt V •• cover K. He""" V _ U,,;, V•. io .. nei&hborbood of K, Irrcluded in U. a.nd V n Y ill <:Om!>""'. By the not ... ~ t~~m. t""n irI .. ,,,,,,,In __ funetion h from V Into exl.eDdo / , Vny . If '" irI .. <:OnU""""" fun<:tioa fn>tn X into (0, IJ ...hid> vo.nioboo oul-Jide V and lr:; equal to I ot5 K . then do-li.... / , t". ,, (:r) _ h(%}P(:r) b..n % E V and t". M:r) _ 0 (0<" all" E X-V. CIe&riy.
   
   ""'' JlIoC\
   
   IO.I/ II-hid.
   
   I , bdonp to 1{+{X ), crteods I. and io.uch thM 1M - II I. Nooor, Ii....,., ~ > o. ~ ca.n find h, E 1{(X . C ) JUd;. that!": 1S /, a.nd !!>I(I,) S Ip(h:}1+C. Wrili", h for I"" """rlnlon of h, to Y, ~ ha,... h l;: 1{(Y. C}, Ihl S /. and ,,(15,)_ ~y(h) _ ,,(h , · l lJ_ I<). nu.. !P(h:) -py (h)} S 1I1·lpl{U - K ) S ~H/D. On I"" othe< hand . Ipl(f,) -Iply(f) .. 1P1(f, . Iv _I< ) is pOSIti ...... We CODCloo.:
   
   ....
   
   IV:
   
   ate
   
   :s:
   
   b'yl(1l H{l + 1/ 1)
   
   aDd ... ~. arbill"'),. lhat IlIly(f)
   
   Lemma
   
   o
   
   < IlI y l(f).
   
   Ld X .. ~ I
   ~
   
   pomir>o
   
   (_) For """Y......,...,I (rup«ti!IdJ, 0JIf!R) .'" H in K . I',, ( N ) _ ,.{ H ).
   
   (i) A n .bI", N 0/ K it I'" ....,u,;w.
   
   if _lid on/) if il it I'.~
   
   (e) TIIM...,.,."., 5 011''' it 1OoneJIIptr, ouPP(I's) _ 5 . PROOF: (. ) We m.y CONIidcr oaly the cue ill ... hid> H
   
   ."""1*=\. Dmou. "" /Ihe
   
   illdK:.tor furoetioa of HOlIK . / • upptf oemioon' inuotll, &Dd 10 the lowe< ~..,Iope of. downward-di~ ou~ 4I of 1(+(K). SiDo::e the IDd;c.w. fUlldioa IH of H 00 X • the ....... r ~.... bpe ohhedownwwddirecwd _ .. .. It: , e . J of upper ..."icontin...,.. functio
   ... 1,d,.K .. "'. It
   
   "IC (H ) .. inf
   
   int
   
   dl' '' I'(H ).
   
   (b) If N • ,.."""ilible. tbo= aiIu, for oech ~ > O. &II open nei&hborhood U of N in X ouch tMt I'{U) S ~; iii""" K - (U n K ) • deduce from pe4 I that I',,(Un K ) :s: ,,(U) :s: t; 1m. N II I',,-~ble. Coa--ty, it N ill I',,-nqlilibie, t ...... exiIKI an open nei&bborhood V of N III X ouch tMt " ,,;{V n K ) S t; by pe4 I, ,,(V n K ) .. ",,{V n K ); iii""" t Ie arbitruy, ,,(N) _ O.
   
   """'1*=\, ....
   
   (e) Fa< cd> open lie!. U In K
   
   Inle~iq
   
   5 , "K(U nS) " 0, bee>ce
   
   ,,(U n s ) .. 1I(S) - I'(S - U n 5 ) _ 1I,,(S)- II ,,(s - U n s ) .. II" (U n S)
   
   • o. Si""" every - " ' p 'y open _ III S can be "';t~ 0ptII.et U ill K ..... coodude IhIIl mPP\Jis)" 5 ,
   
   u n S for _ miu.bIe o
   
   382
   
   I'. I'"A ,Io.,
   
   0'. fimiIy of
   
   ~'- .....
   
   19.2 /V"Dense Families of Compact Sets Let X be . 1<)<.oI1y roml*'- H.lII!do
   to . tdlire
   
   ~x.
   
   Now lot A be •
   
   "'~UBbit
   
   Ia A wilb the k>llowIlIfl
   
   _ 1.
   
   EM:b
   
   2. EM:b
   
   o~ llUbilec finlt~
   
   .... becc of X . and let tJ be • claos of com~
   
   prOJl"It~
   
   of" -o...et is" "/).«L
   
   union 01 ~ III .lJ..RI.
   
   Propooollkm 111.2.1 Tho: /oiWwing ......ditto...
   
   a",
   
   
   (a; F"" ""'"'" ...... rt • ..,ibl, """'JICId HI K in A. !MIl. C K ,,11<1 pILl > o.
   
   (6) F"" a.!'I"J """',...:! ,...1 K in A ",.. ~ r th.o.l £ C I< all
   u....e crirll I. E 1> ndl
   
   > 0, 11>..,., crirl> £ E tJ nu:A
   
   (e) FM" ftoU)' com,...:! ,...1 K in A, ~ uiUI" ""'711""« (H.).2,' "f 1Mjoint tJ'loell _I· ..."" ... I< ..,dt. th.o.t I< - u..:;:1 H~ II I'·nes'i;ihlc,.
   
   (4) A ......1 B I. E tJ.
   
   "f A
   
   " ~I, "'"lS'igi"'" ..h~ /-I"{8
   
   n L ) .. 0 flX"..u
   
   P IIOOI" First, "".111" . t hat (. ) boIds. and let I< be .. compACt ... becc 01 A. P ut (l .. llUPLol>.I.<; K 1'(1.), and let (1.. ). 2" be...., It>t,t I '.inc 6«4"'"'''' of t).,ojeu imo:ll>ded in I< web that 0 : OIberwiol: , ';fL u t.. ) > Q for n ~~. T1>t1 ~fuoe, B io II· oocli«lblo:, Q .. 11(1<), and (b) ill ... tisflod. NCOI'. If ( b) hokII and I< ill a -_ such that H" .. , C K - l.I,.s. H, and 1'(1< -lJ",s. H~) '!O l I n; ki"" ~ition {ella abo satian..:!. Fbr """'l' mmJ*M. oot H In X , A n H 10 .. union of coun\aI>Iy many die;':';nt compACt IOU and .. "'"",li&iblo:..c, 10 (e) lmpl;" (d ).
   
   u..
   
   FInally, (d ) obviously lmpI>N (a).
   
   n.en u..»
   
   0
   
   Deftnltlo" 111.2. 1 When the wnditionl 01 Propoeili<m 19.2.1 hold , 1> 10 Mid tobe~inA.
   
   1£ A .. X and tbe oe>nditiono ol PIOpoIIitlon 19.2.1 ...,. N-llllr...d . D 10 Mid, limply, 10 be p-denw. Propoaltlob 19.2.2 S_P7<'Je V .. II ·kf>M i .. A. wt D' joe ~ dd .. "f """'pad ~II ... .4 Ad! IMt -.-. '/<",d ~ "I" D'.,...t .. ,,1>' ·...1 .. owl ,1Idl 1M!. ..0. finite "";.... QI1>'·K/.I .. " 1J' .,...1.. fl. fM" ~ch I< E 1>, II>~ 1J' .K/.I ooIIid> ""' m.c:A~"" in K lomo • 1'." ...... ""'"' in K . I/o.... 1Y .. ".",,_ in A .
   
   P ROOF: l.el L boo .. oxnfW1. IUh!le1 <>f A. For ~ € > O. lhere 6 .' 1 K €.1) ..,mIMI I< C L .:>d ,.{L - 1<) el'1. l"61 . I~~ e~is'" H E. 1)' lOCh that H e I< .:>d /,(1< - H) cl2. ~ow H e Land I'lL - H ) c. whld:l JAMei tM~tioOl. 0
   
   :s
   
   :s
   
   :s
   
   DefInition l i .2.2 A cl..M C 01lIII:\.0 in a ~ __ fl iI Aid toO) ~ locally COWlt.ohle if, lOr t8dJ 2" E n. there cxillu a Ilf4I>borhood V 0I :r which " . p'1 on1r .. wpn'"bIe nl1lllber 0I1~ C-.." . If C is locally OI)Unt.!)le. """'Y compooCt ""bIet of (} imm .:tI only .. ooum .. b~ number of tbe C·oetI. CINlly. I~ unlon of a local!)' COUIllabie do... of 1'"......un.bIe ..,u .. 1'"" "IIrabit.
   
   Theorem li .2. 1 S _ th4I P .. $<'dt;nu in A . Then: QUU Q Iocal/)l OI>IIWWC dIo# C c P. """"'h"f oj ...........p/y ""Ii mllt....!l, dUjoint ..lor, nodi """I A - UKa: I< it Iocal/, I'-Mgtigibk "lid IhGt tIIPpUlK) - K J", .1/ K EC. P II()OP: ConIIidet tbecl ' C C P , ~I .. of JIlUtuoJly d~ ..t _mpl.y 8?!ta.,..,m Uw IUppV.K )" K lOr 0.11 K E C. Tbe8o: clum form .. ... beeI. . of lei of o.U .,,!.>d0' ol V . .:>d fo .. nonempl)" sll:lOt II Q)II1ai". lbe ""'pcr clMl. Or&!r • br Inclusion. Then. by Zorn'. kmm.., • hao • maximal clemclll.
   
   I""
   
   C. Foor t:W<)" :r E X . lot V be .... open noei.shbomood of :r . with oxnpooct cIoeo, .....
   
   c-
   
   If (K ,).,., .. a li.ftile !&mil)" of distlrlct at>d ~1IoCI 0001· t&i ... a 1>«1: K that III DOl ,,·~i&ible. [f L It lbe "IIpport of I~" Illpp{jid - L, .:>d C U {L} belongs Ul . , ... bleb contradicu tbe maximality
   
   :s
   
   E/f;'
   
   "K,
   
   ole.
   
   0
   
   Propoooltion 111.2. 3 Let J It( Q " " ' " . " " , /rWI It inUI • "'pOlo j ' xI 'PO« F • • 004 !eIV """"" ./~ _p,'d " ,/t.M;tI K 01 It -.'I1A4J f , K .......s..-.
   
   Tk joI~
   
   raJ V
   
   ~.,..,
   
   ""';...!cnL'
   
   .. wu- in It.
   
   (t) Eller)" ............... oj I to
   
   A
   
   .... toNe A .. ,. . .............I!
   
   ""')!ping g
   
   Jrvm X
   
   into F IhGt ;, ........nt
   
   (e) Thm: ;, Q>IUt",,/
   
   I>
   
   p · ... ee ."
   
   ""tMc A.
   
   '* n\Gp!NAf
   
   9 : X _ F thol aleruU I OM ;,
   
   "*""'>t
   
   P JIOO, : IbM CUldiUoa {al ~, and let 9 00 an ('k~n 0( I \(I X , <:1 on AO. Fo...,..,h <X\DlpA('< ruho!et K 01 X . 1\ n A and K fl{ X - A) p.in~le. Ci""" ( > 0, ~ist tompac\ _ P e l( fl A and Q c Kfl{ X - A) IUCh .ha~ p(KnA -P) <: e/2 and ,,(I((1A<-Q):5 (/2. On the (ltberband .lberef2I!Ua<:<>mp"cUubeet H of P l ucb .hal.p (K n A- H) <: t /2 and l i N 10 conti ......... lbra I>f. K - H U QI :5 l BII(! g, NuQ ito ronti_. which IIhow:IIIMl g ill p·n 7'ur&ble. Coa~noeIy. {el obrioulIr lmplieg (al. []
   
   U8
   
   1"'=""
   
   ,..,I'J('
   
   Ci""" thf: condil~ d. P'<>p<*lioo 192.3. I '"' SBiod 10 be i"""bie OQ A. When F lo melrin,*" ,"" definition ~ "';11, DefiolUon 0.3.1 . Propooolt Lon 19.2.... WI C lie .. Ioo:all) """,,14ob1o! thIN 01 p· ....... ""'No .~bI ..' 01 A , nell thaI A - U~ B .. t.mIl~ ".~ lO9'igo~ ThCII " "'. , ~ '"" / from A imo I> 1"""'~ ·001.,...,;, " ....... ~ (>fI A if (1) ru1 ON' iI) 1, 8 "" . ........ ~ om B , /11<' till B E C.
   
   "H
   
   pROOf": Foo- all 8 E C, ,boec alWl*ropoeitlon 19. 2.3). SQW let K boa. oomf*Ct ru'-'- or A. 8 f ~t-, the"" .".10... ie, Bn ) Io l.-rqligible For all n > I. pu' K n B" n ( U, o So N° and the c~ l(>trn a poIIftl, ion of K Into ,,-I~ feU. Since V". lor ,,-deuoe in 8 ., 1i>tTe ... ie. "the.> NU{ u..<:. N.) ;; ,..nqI~bIe. Tb. 1*0""' tlM.t /It "."-u",,bIe "" A . []
   
   (l)
   ...m
   
   c.. ..
   
   n.t.
   
   u.."
   
   <"010.-.,
   
   " maWi"f! I from X IIIto • topoIogIetJ ..... ~ 10 aid to be unht, " If -.n.bIe ... bChfr"'!r lt 18 ... mtfllUfabie lor ~ Redoa ,IOt,IlI",,, 011 X . Proposit ion 19 .2 . ~ Ld I Ie .. ,,· _ _Iok ""'","i' from X in/oJ I> ~ kt · w".... F . 71It:Jo tII(TI'l X _F. Mnitot:rMll, moo •• Wt, #1IdI t1IU mill> ,,-4Im<>at ~w,~.
   
   <:nato, :
   
   1-,
   
   !.a tI be ,be daM ol.hoie oorn~t oet& K ia X for ",hio;h h" iI t"QlIlInlloOUf- Si_"D iI ~ there txis~ B 10000ly coun, .bIf /lUbel ... C of P ~r:
   
   :D. <:Oi
   
   ,. '''11: of muint _
   
   . such that N .. X - UK I! K Is locally
   
   ,.."""li&ible (1lw!oortrn 11).2.1 ). Let 1> be All .Ieonent. of F . Put ;(,z) _ btlo"f'
   
   \(I
   
   U««
   
   /(2) If z
   
   K .:Id , (zl_ 1> Olhtnv;"e. Then g halo ,he dellrtd propeny.
   
   o
   
   19.3 Sums of Radon Measures Let X be a locally oompact H............ fI .poooe. For aU pooill ... Radon m pz "","",
   
   X And •.1.11: X - (0 ,+00), I"1!eaIllw /" 1diJ.'" .up,,/" 1 · I"dl' • ... ben! K ""'" \.h~ U~ ciMo 01 axnl*=' .w-u 01 X . Loet 1'1 , 1" be two P'l'iti.., R odQn lDI\aIIu,"", 01> X , And put I' .. 1'1 + 1',. Then/" 1tJ;. ,. /" /df'1 + /" /d1'2 &nd /" Idl' '' /" ldiJ.l + /" /diJ." So< /:
   
   I'
   
   on
   
   X _ 10. +001. A mappi", I from X 1n\0 a ~ tpoIC"e io I'"measurabk if and only IfI~ II hoI.b 1'1" ill m urable and 1':t""!MMurabIt. Now I.t 1'1. 1'2 be two <:Omplex Radon " VI,"", on X . and put I' .. 1'. + p,. A I from X IDIo a Banodt "PH'" io (11'd + 1I'.1l-lntegrabie ( idpt&
   
   -wms
   
   d\1!lJ'. --.tially Integrable) II and only If It iII lDtognblo: ( '""'~tI..,ly, .-n­ t!all)r Intecrabk) with ..,.peel I<> eacb 011'1 &nd 1'" In which """" I dl' .. I~I + 1diJ.,.
   
   J
   
   J
   
   J
   
   Pro.,.,.it!<m 11I.:J.1 Ld H k"" 1IP_nl·
   PItOOF: \\''-1 bo:Ionp to "' +( X l, tbe rel&tioa I'Ul " WP-"'''W and vani:I.beI outside a cnmpo.ct _, .tid ,",n_ , E ". . . . Ihu I :s g. Oi""" e > O. there «is", ~ E H that ..(I) :i!: ,,(I) - (, &nd ...,.. V. - ") "U ) :s V. - .. l{g) OS ,. So .... (Il <:: I" U) - t. Thill yiotlds OiUPwEH ,,' (J) ;: ,,' (Il, and the f"eW:"'" ;ntquality It c ...... Nel
   $"""
   
   ,,' (I) .. sup ,,' (J~ ) ... IUp sup ~. (/~l ... IUp sup,,' (/~) .. lUp ,,"(I). ..
   
   ~
   
   .., N
   
   ..,N ..
   
   Finally, "ze••~oo.;ng aU f"l'SlriclIobo 00
   
   I'. (J)
   
   We Qbee . ... that dee I ).
   
   -•
   
   --
   
   ,,"VI It tIOI.
   
   _
   
   -
   I,
   
   ~PI"{/ ' lK I
   
   ~""p""{/ ' I") ~.
   
   -",pI.. p,,' U · I,,)
   
   •
   
   •
   
   'up ""Cf).
   
   ~.
   
   o ' P"rily equ.aJ tn SUP"';N ~' Ul (e/. Em--
   
   P roposltloll II1. l .2 W, HaM" lie.., ill Pt~tiOfIl!Jj.I . o,."j let f lie a m.pping /room X mu. G ~ "....,.. 7711: .. / ;, I'-"'~ if (an;/. ...u,. if) it ;, ".~ 10.- alJ ~ Iii H.
   
   P IlOOFc Let: K be • """"""'" _ In X . Gi,'efl ~ > 0, Ihtfe oW ~ E H such INl v( K ) ~ 14K) - ~/2 . we ean find B 00
   .oo
   
   ,,(K _ Ll -
   
   U. _ ~)(K -
   
   ~)+v( K
   
   -
   
   L)::;~.
   
   c
   
   which pio ".INI/ IJ l'"'..-..rBbIo.
   
   Deflnilion 19.3. 1 A family (p.,,)~ .. of !lUit''''' Radon ........,mo on X ill IIBid to be .... mtnableo .. beo>e.", v...(f»~ A\m",,.hle for all / E H - (X ). The mould", Radon " Uf' l ~ / ... ~( Il (f E 1'({ X , ell ill tBIled tbe AUlD of lhe p..". and \0 _,;IWI ~ .. p..".
   
   !:...s ..
   
   . ..
   
   SUPl_ that (P .. }... A Io! ... rummable r....lly 01 j)Oiltl~ Radon '0(, Wf$I and ~ TMn " ol/) .. L..t: .. p.; (f) for &II x(0.+0:01; ... II frotll X; nlO'" IOjXilogi<:al p.n~ if &tid only if II II ",,· _,calk lor .0 (t E A: " mli.ppl~ I from X into • r'e&I B.nn tpIICii ill , i ,tlt.lly ", i~ If 0.Dd only If il is _ ttollr ""i~&bk lor all I> E II - ' the..,.. E.... .. (lid,... IJ Mlu. in whid! .....
   
   ... X.
   
   deft""" ..
   
   .. ",".
   
   _PP;:,..
   
   I'
   
   'paoli"
   
   r
   
   / I "" ,.. E..., .. f /oIp..
   
   Theon!m 19.3.1 La " .... - , _ R.>d.m..........re ""' X . ould 1) lot: .. ,,·4_ d4H II! CIlI'mJIiId _/I in X . ThcI ~ mmabk ,....., (p."l.. .. o/ ,oHI!toooe R.kn _tuU on X , ~ /MIl''' aM fDCil t/o.oo.Ilhe ....~ <>/tIw:·-- .. u "" iorlcmg'<> 1) onJ 1_ & I«toJly ..... n~ famiIr 01 "'''1I
   """II" ....
   
   E... .. ""
   
   P!lOOt' :
   
   Consider ... Ioulty t'OIIntablo:: family.
   
   (K~)<>E '" U,E" K~
   
   oI'>O<>toap',. and m".
   
   "wi)' disjoint v-u ou.ch lhat. N .. X illlouJly wnqlitihlc (Tboooe,u 19.2.1). Do:fino ... II F ' ure PG (XI X by I'o.l/l .. f J . I .... "" for I E HeX , e l. The .... ppon. 01 "" IJ i!lJCh.lded in K... and the",lOre ill, v-t.. II n:maIDI to be ""-n tM' L'"'! .. "" f f) .. j.l/) kor tad> , E 1f+fX l· Wrl'e S for the 1Upp<>r1 01 J and ,oJ' b- ,he (""'U1IAb1eo) ..,beet of A """'itAI,.. or " ' - Q E A JUdt IhM K" illltl_U S. Since !V n s ill ,,·Mg!!gIbioe.
   
   I'fn .. / , , 19<4<
   
   ..
   
   - L: j,.lsn . .o"" - L: j"!K. dl' ~
   
   ~
   
   ..
   
   ...
   
   '" E /I.I ... and..,
   
   "en .. ~A ","(f ). u dtsi......
   
   o " '"
   
   c
   
   •
   
   19.4 It-Adequate Families Lc\ X bt .. locally oornpllct H,usdorlf space and j..I +(X ) the lie!. of positi"" Radon __ u~ on X . In t his !lectio". j..I +(X ) ",ill AI .....)'" bt equipped with the I<'IpoIosy Ind!lCJl!d by the ~ topology of j..I eX. C). 10 My thal. a _ ... pine " : I .... .I., from tbt locally OOnlpllct lI,uodotff"1'I"'" T into M +eX ) ill """t;noouo ........ tbe...r""". that. for evny I € 11"' eX ), the ....... .,.J.....J function I .... .1.,(1) ill COIltinuous. We , hall ...y. in thill.,....., thal. II io ~y """"inoo.... To Illy thal. " ill ~ .....&Sur.bJe (...1Im! I' io. poIIiti... Radon measun: on T ) _ lhal the collection of all compact IUi>eeUI K ofT lII>dIthal. tM fftIlrietion of " 10 K is va,g~ly COIllin"""" "IJ~IIR (P ropooilion 19.2.3); in IhiI~, ..., ohaU My that " is ''agIJCI)' lJ·meuu",bJe. Lc\ " : I ~ ~, bt • _ppiog from T into M+ (X ). We shall My t hat II II ...I· rly .-:ntit.Uy intqra ble for p if. for.....-.y I € H(X. C), !be fUI>Ction / .... .1.,(1) .. . nti.ally p.integrable. Tben, cJ ..... ly. " , I - J ~, (I) dlJ(/ ) io a pmiti... li.-r form on H( X .C ). Thus il is .. Radon """'"ure. We ahaIl wnW " _ J~, dp (/ ). For / E H(X . C) . the OMeIll i.a1 in~ J ~,(I) w.. /",. eveYlI " E H , OM if u.. Kt If.l.,
   j
   
   .l.,
   IJUPjA, d•.'(t ). ~'E H
   
   P ROOF: Verifyiog lhal " is .... 1..1)' etooenli&1]y ;ntqrabJe for a poeiliv.libdon _un! pon T amouat.110 "";(y;og \bal t .... .I., {g) is p-m""'lUr.ble and that .I.,(g)dp(t) ill finlle, lOr ew:ry 9 E 1(+( X ). lie""", t he pooposillon foIlow:o from P ropoIIi\ions ]9.3.1 and 19.32. 0
   
   r
   
   Propoooltion 19.4.2 A ........e fhall' " 1M 'um "/ " ... mrrl4b'" I"mu, (IJ~ j""A
   
   ,."u,,,,, .......
   
   0/ ~ru on T . Thm " "oaotorl~ uunl""lw 1J-,nCbk if and onl, if il io-lOIIl.orl) ~""" p., . ,~ /"" IllI '" E A "nd if u.. lam. IIJ (J .l.,dp., (I))~~ A of m ............. "'''' .....IIk. / .. u. .. .-., J .l.,dp (l ) ~AJ .I.,
   fo/'-- from P rop<>oition 19.4.1.
   
   o
   
   DeI\"alo" 19.4.1 LeI X he a locally compact lIaUlldorff .pou, " : i_A, a ... 1"ly e!IIIentially l'.in~"'bJe mappiog from T inlO j..I·(X j , and " "' J .l.,dp(l ) tbt in~ <>f II. II .. oaid to be lJ-prHdequ&u ..he"" ...... for eYefY
   
   IowI:r 1Itn\lcoat1D........ IimcticJIII
   _,,·_umbIe!ln
   
   J'
   
   J'
   
   I (z)da«z) -
   
   d,,(I)
   
   funnion I ....
   
   r
   
   A .. Mid 10 N 1'·6deq""t~ ... ~ il io
   
   'f II· 11>11 1oI1ooaina:
   
   X , t be
   
   I(:r )d.\, (:r)
   
   r
   
   I d~,
   
   ( I)
   
   I"·proodoeq, ... te
   
   lor
   
   C'\'Cry
   
   peelU""
   
   meNu", "
   
   ~11on
   
   ofit'D makeo it \>'Mible 10 d>ecl< tbM .. ,h_
   
   mappilql io jl-adequate.
   
   .\, k a """,,ri, _tWig ....."....., '""" T i...., M +{X ), .. Ad plOt .... f .\, IIp(l). Pro~;doll 1 ~. 4.3
   
   C.d A ; 1 _
   
   r..) fl A u ~I, amlin"""', ~n I _ ettety "' .. ~, """""'1;"..,.., ~Iion
   
   J' (6) Ij A
   
   I (z)da«z)
   
   ~
   
   ~; (n
   
   u ~ ",mi(onIin......., /...-
   
   1 ;:: 0
   
   """
   
   r r
   
   u ~ p.~,
   
   IIp!l)
   
   ".'n~
   
   X.
   
   and
   
   I (z ) d.\, (z)
   
   (2).
   
   it it " .• deqoo.ok.
   
   P IIOOF: La I ; X _ 10. +001 N ~ ~DtinU(\\I&. &nd let P be tbe _ro:kIi....,.od oct of fun(tiono g E 11" +( X ) ,och thu II S I . r .... /1 E F • .!e.Jcn.te by II. tba function I .... .\,(g) on T. Similarly. put 11 / (1) - A; (I) = ""PoE.' 11.(1). W~ inln>o!. lbe IoIklwl"t; h,rpOtt-io., " ' 4< thaD lhal iD (a); GOlly thu tbe ....criaion of A 1.0 S _ ~ly oontinuout, 5 bel..., & ~k7-;1 Ahocc 01 T a>nUJnm, the ""wort of p . ror g E F, define by Ii, lbe function ... bid> Ifl'- .ilh II. 01> Sand ,"""" tbe val"" +00 00 5<. Put
   
   au_ X, _
   
   Ii"
   
   thai. I, .. II, on s. Ii. is ......" oemJoooliltooUII. for noer)' 9 e F. Tb.. Ii, ill lower """ ......... tillU<"lU!, and tbe family (Ji,)oE l'" , being up..-ardly dil"1!<.'ted. tat W'>loe ""PoE."
   
   tc>
   
   /O"Oi, ) - oup ,," (ii. ) - sup II ' (4.) ...up ..(g ) - ..'(1 ) .~ ,
   
   "'"
   
   ,.."
   
   (Tbeorem 7.2.2). SiDO< 1 and Jif are lower ..:micominuoua. the poeooding relation give6 the equality ,,' (Ji,) .. .. (P""pooitOoo 3.6.2). SiDO< Ji,,. 4, jl-a1moM e"o'ef)'",here, _ conclude ! ha' ..' (n _ ,," (h,), .. desired. ThUll A " PO P" ·""qWlte. In tlIiI &rJWll'I'fIt. _ could have ~ /0 by " ~ II . and .. by II' _ J A.dp'(t), bee" . . A .. &1.0 ",&I'''y ~ly ".inlqn.bio> and S """WM the "'ppOft of 1". II followl l h.u A " ,,·6deqWlte. .u..."", that " 10 vag""ly contin....... We can tab: S ~ T: then 4, - 7i, .. .--... leI"IIk:oge ODlIIpect sut.ecu K at T such that the nnr\elion of A 10 K " (9Dlinoous, " ,.. deI>ooe. t~ex"'" lUm"",bIe famlly {/Oo )o€A ofpeelliWl_..... 0<1 Tt-ucb
   
   ·In
   
   IhaI. P _ E..UPo and lhe .u~ S,. of Po ~ to 1) (1beo=n 19.3.1). F"" ~ry 0 E ,t, h II ... I_rly .-mia/ly Po.in1ovable. &nd .... pul ...,. _ I .l., dp., (t); lhe family (..... ),.EA is summable, wilh sum ... ( Pl"OpIMiliou 19.4.2). If I : X - Ill. +oo! .. Iowa" """ironIin........ lbe lim part of lbe proof .. wiled to tbe"......,... Po and lbe eloeed!IeU S,. oIJaq lhal
   
   (e) h, is Po ·meMu.... ble I"" aU 0 E .4 . &nd lhemo ... ,..."""",urabie; and (d) r I(r) ......,. (r ) - r dp.,(t )r I (r)d.l., (%). Formula (1) IolJo,wo by summalioo ........ o . Applrilljl: lbe p<MOdi"" ~ to &Q,)" ............ p' p, .... find lhal ,.. ill ,....deqUloU!. ThillllfO"'S (b). 0
   
   s
   
   In lbe ""'.... 00.-. of tbis-cioo, .... define by h: I _ .l., • ,...v< 1eqUloU! ....ppi"ll: from T Into M+ (X ). and by ... Ibe intqral I .l.,d,,(I) of ,...
   
   p ...,.,.,.lllon 19.4 .4 Ldl Iot_,,"'l:titm/romX mc<>rO.+ooj. 'nK>o
   
   (-)r
   
   I (%) J.,(:I)
   r
   
   dp (t ) J' I (:r}d.l.,(:r);
   
   (.) ",.. " ........ /r <»nl'n ....... ,
   
   r
   
   I (:r ) .wez) ~
   
   r r "-'(I)
   
   I (z ) ,u,{:r).
   
   PROOF: LeI. g be .. ~ -rumntinUOUl functioo 0<> X ouch IbM f S g. Sina! l(z)dJ.,(z) S J
   r
   
   J'
   
   r r dp(l)
   
   I(z) d~,{:r) S
   
   by formula (I) . ... Itio:h
   
   r r d,,(I)
   
   J{z ) d.l.,(r) ..
   
   r
   
   J{z)J.,(:r)
   
   (o.). i""'luality (b) is j UII. .. easy to ""I .. blish, "';th fonnulot. (2) Olllltead ol IonnuIa (1». 0
   
   n..
   
   pnl'o"aI
   
   P...,.,.,.itlon 19 .•. 5 ull lot Gil· ........' dl.1II4pIIing frrnn X intD _ top:oIo,icalqaa. 1dridI;, "",,141Ot....ui.k a II·......uro.u ML DenoI<".,!oJ IM .., <>1 u.o.. leT f..,. f ;, ""' .l., . ............w.e n... M " 1oatU, P-~. _... it" p.n.,tigtbk if ,.. .. wogooel, connn ........
   
   """eh
   
   P ROOF: S;na! eodI ....;ntqrable _ .. Included in _ .... in~.bIe ~n Bet . I .. constant 0<> tbe ...... pk",."t B ol. countable o.mion ol ... lntovabie opm _ . There b:iou. partltioo of X _ B ct"'"iwillll ol • ... ·negli3ibie _ N and K, Ie mnt.in......... t:leoo\e by
   
   5 lbe _ of Ib<. leT for ... hicb ,v iA Il()\ ~,·~it;lbIe. By Proposilioo 19.4.4, 5 Is Io<::.Jly ,...~bIe. and it .. p · neglit;lble if ,.. .. vacuely
   
   n.. _
   
   cootlnOO\l& K,. N. B &re universally m ",,"able, and the rMrictioo of I to ucII oftbe", Is .l.,-"""",,,, .. bIe IorI!YCfY I ~ S. Thus lis .l.,-_urable for~I"S,
   
   0
   
   ,.
   
   TheoAm 19.• • 1 utI: X _ (0,+001 k "."'_~ ~no! ".~_ ~I~ ~ AI Ik lOtI QI /JIooe lET 1M oJIidI I if nql .I., .",...,..".w. ~n.:I .I., - tnDoImlt~.
   
   (~)
   
   AI to I400Ilr I'.~, 1_
   
   r
   
   1(:r) 4-{:r) ..
   
   r
   
   n,,)d.l.,(~) if l' ·mtGI.....we. "no!
   
   r r dl'{l)
   
   I {:r) d.l.,, (:r)
   
   (3).
   
   (6) 11 A ;, ""9V1J """'U.......... u.... AI .. 1' ...qIi,ak, Ik J(:r)
   r
   
   r
   
   1{~}dv("I-
   
   ........
   
   PIlQOP: Replacin«
   
   rr
   
   I br Ioflj, n ) In
   
   d,.(l )
   
   1 (:r) dA.{:r)
   
   ID~
   
   ~titnI
   
   I _
   
   (4).
   
   
   I.el N and (K ,I.{1 be .. to t"- argument Df " ...,position 19.4.$. I (%)d).,(",) - E,*"f"U' IK.I(,,) d)., {~) for ~~ry 1 'I. s, boo- ""," I - E ... ,I · I N, ).,-...1_ evt<)'Wben:. He""" _ . - i ooIy _ t"tbetet ... wl>tn I~ ulzl! • coml*'t lei K such \""1 I ONIis1>t& 00 J(C aM
   
   r
   
   such lbal tbe Iftilrktloo 01/1<> K iI finite and """'tin....,... I.el G be ...... inl"C'.t>ie.open lei ronu.lnl"fl K , ~ .. mp" ' ''' which domin.alell I . h. lhoe k>woe.- .elIUolo.t iDUOIIII fun<'er. I. ~, h. "" .... ~ Apply lormull. (I) (I'e'IPflClimy, furmula (2)) I<> tbe Iiooot< ~inuooo fUDl:tlo<>t h. aoo 11. Then, .,..;demty, I I (:r )d)., (%) 10 ". -tn'urabie and we h&~ Ionnul& {ll (miPfICI;my, fonnul& (4))_ Finally, If A it ~y mpt.inOOUl, I _ I (:r )d.\,(%) 11M .. IInlte uppn In~; th"" it 10 ,,·modrr~ 0
   
   r
   
   r
   
   P ",_itlon 19.4.6 UI I ..... ....",m, lrom X ."", .. rwI &nadI .pa.... .. ·""",.~rdk "no! ..· "'001.....te. ~ I ;, ". 'nl~ if ....01 ..u~ if dp{t ) 1/ (,,)l d)., {:r} ;, fooik-
   
   r
   
   r
   
   o
   
   PROO' Thio foIlowt immodiatO!ly from TIooo,etll 111.4_1
   
   Theorem 19.4.3 ld I k .. v·inl..-W. ..... ppmg /rvm X mtq" rcoI B..n<do "...., P , .no! Id H k t1uo.d QlllIoK t E T 1M oJIldo I .. Mt).,·i..u,noWe.
   
   r.) H ;, /otDl1r 1'."";,.'W<e, I - f I {%ld.l.,,! %) (dtfi~ I''' ' t IUlIJ l'.inI~. and
   
   / I {%)du(%)- / .t,.(I) /1{Z)d.l.,,(:r)
   
   (5).
   
   H) if ........
   
   19,~
   
   M
   
   II " ;, ...,...,1...... tin..,.." (tkfi1lftJ 1M' I
   
   ~
   
   ,...Ada!"'" Pain
   
   then H "1' -ntg!iglUe olUll -
   
   f
   
   391
   
   l (r ) oU.(r )
   
   H ) ;, I' -,~,
   
   PROOr: The ",",ult ill"'" ..hen Iii" ~ili'~ fUl>Ction (T heon:m 111.4.1 ). and the ... lore .. hen I iI rea)·vaIu.d. Let H(X . R ) 8 F be the ... ~ of CJ,fI') a>DAiati"l of lhe Ii....... c0mbinations, witll cotlf">cienl.l In F, of fuT>Clions in X IX , R ). By li.-rity, 1M _~t io valid when I beIonp to X IX . R )0 F. N_. for e>"U)' I E C~(v). lhere exists " """""""" (J~ )~ ~, in H(X. R ) 8 F with the followin.s propert~, (e) (J~ )~
   < +«>
   
   (Tbeootm 3.1.2).
   
   Da>ote by N, the"'" oft~1 E Tsucb lhal ),; (,) ., +«>. By Iormula (3) (respect;""ly, IomwJa (4». N, io locally l'- nesl~bIe (""'peeli",,),., 1'""' Fgli~bIe). For t ~ N" the tnapplnp I .. heIong to CJ,(>., ) and the 1ItQ""""" (J~ )AU",," in CJ,(>., ). Let AI be the _ or t"'-e z E X l uch lhat (J. (r)). ~ , "'"'" con"""", to I (r ). Aa AI is .... neslijpble, the let N, . of I"",," l E T for ... hkh !of io "'" >"-'''-'&i~bIe, ill locaIl,. I'""neslijpble (""'poeti....!,., 1'""""lIli&ible), by P ropOoition II .U . S~ that t dooM not helon.s 10 N, U N • . Then the ",",,'IHQ' (J. )A2: ' con.u",," in eJ, (>.,) and It con.tl",," >.,·almo:wt evffyWhen: 10 I. Hence I beIoop to eJ,( >., ) and f Id)., - 11..... _ _ f lAd>., ( Pf'IlI'06ition 3.111). So the _ H Ii!I oonW., (r ) brina; , . . <4ially I"~ (_pccti""ly> l'.intqn.bLe) by ~ 19.4.1 • .., can appI,. tbe dominat'" OOIJ'"tT'nce theorem. Then t _ f I (r ) d>., (r ) (deli ..... for I ~ N, U N. ) io _otwly I'""integrable (""'pccti... p-intqn.bIe) and
   
   r
   
   r
   
   I,..
   
   j dl'{I ) j I (r ) d>., (z ) Since I IA (r ) dv(r) oon'E'",," 10 """"'-
   
   -
   
   A}i.~OOj cIp(t) /IA{z ) d>., (r )
   
   .,
   
   A}i.'!'".,j Mz) dv(r).
   
   f I{... ) dv(.: ) all n
   
   tends
   
   w +<x>, furn>ul& (5) 0
   
   19.5 I.l-Adapted Pairs Let X and T be locally oompact H"uadorff If- ... I " ",."pins from T im<> X , and g" fUlx:tion from T low 10 , +«>1. Fo. ad> t E T . write t . ( I ) lor the Radon """,,"ure on X delined by tbe IJl.UII 1 at ,, (t). and >., for " t}t' I' )' Finally, let I' he. posit;"" Radon rneM= on T .
   
   n..:
   
   Definition UI.lI.l pair ( ... I) .....1"« conditions bold;
   
   ijI ... id
   
   to "" /'"wpted ... he ..ner t be !o!•
   
   (a) .. 1lIId ,.,., ,,·_u••bk (b) For all 1 € H +(X ), tIM! function t _
   
   1 ( .. (1))111) it _mmiaIIJ ~
   
   inlegr&bie.
   
   Prop S .. (I E T; g(1) > 0) if " ·"'tGI.....bIe. PIIOOr. Su......- that (r.,) .. " .adapWd. CltwtY. " ill· !an, _iaIly ".~~ The <»mpOI(:t JUIJoi&I K 01 T ~U<:I> that tIM! rQltrictiono of" and , to k are concinUOUl kwm .. ,,-denee clo8s In T. If K it ,udo .. CO.,(1 ) .. g{t) "' Ilk. and.., S • ~" . " J,able. Now 1M GII""_'.eta K C S luch Ihat "" ill Ulntl""""" and /\1" 10 ~ ..,..t l _ 'om> .. ,,-den..: d . . 1) In S ; If K E "D. the _trkUon 011_ ( If ri!»'\" to K io ~Iy coRdn"",.. which 1mI'll.. !hal. " / 1r it oonl iOOO\l!.. T'herdon:. 1rI!l is )...neMur&~ 0
   
   ,,·_a.•
   
   <.,.) ..
   
   We will _ LhaI ....ben (".,) io $I'od&ptfId ..... Section 19.4 'or" ; I - ~ .. g(1 ~.(,) . FI,.,. . .... r>OlIld the ~lowl"4" ....suIt.
   
   caD im~
   
   1M .-.lta 01
   
   ""i"...,....
   
   Lemma 19.1I. 1 AU~"'$ tIuI4 r II .. ,,11<1 P"*'" (i.t. , ";' d OH I .nd ,, ~ I(~) .. Qo>mP/'CI/... oJl .. E X ). F in..II~, lei h : T _ Ii lie ~ _konliJo"" .... 1/101 u: 1r"" IDf",~_l(.) h(l ) .. /q=r oemi<;Qnlin~
   For todI real nurnt.o.r ". put A.. _ {I E T: " (I) :!O /1) &DC!
   
   PROOP:
   
   B. _
   
   {o: EX: ~("):!O /1).
   
   Sinoo 1I(1r{I)) 5 /0(1) b all ' E T , .... """ ~(A.. ) C B,. On Ih< other hand, 10< ew::b,. E B., ,,~ I(:r) ill tiOll(iDply lind 00"'1*1. and ~ then: erio9. I E ~- ' (:r:) lor which tr{:z ) - hll); Ita In A.. and ,,(I) = ~(A,, ) .. Bo · B• .. tJ"",blt oioted, which \*0_ that" i j l l _ *,trU<notinuQU8. 0
   
   1r.'"
   
   Theore m 19.5. \ f g(1)f.«j 0/01(1). F",
   
   r
   
   SlippOle tIwot (". 9) .. 1 ' X - (0 , +00],
   
   e......,
   
   f(,.) do-(,.) .,
   
   r
   
   ",.,..t.
   
   1{,,' )g(I )d,,{ t )
   
   aM
   
   leI"
   
   _
   
   ( I ).
   
   •
   
   P ROOP: A. firsc.. I hu" has eoliliou 19 . ~.4,...,......t only obow lilal.
   
   "''''''*''
   
   r r
   
   r
   
   I (..-) ""'("- ) :OS
   
   r r·
   
   r
   
   l (d )g(I) dIJ{t).
   
   t he symbol on the ",ht-Mod lido! of the equation !MY be , in lurn , f"eJIl-l by Br definition of tbe u~ ;ntecnl, it • ..mo:. 10 verify tbe w",,",
   
   r
   
   1(.)dll{:l: ):OS
   
   r
   
   h(l)d,,{I)
   
   (2)
   
   IOc- ~~ fuo<:tion h, ~ ..",icontinUOUl on T. whlcb domiDal." 1- l (rt)g(I). Now, &1""" t >- 0, let U be lbe function (h + ~)/" which iI k>Joer oomicxnUnuous on K. FGr -..y I € T , u (l ) 2: 1(..1); iD- O. Write or(..-). fO< a11..- € X , lOr lbe Infimum 01 {utI) : I € K n ..-' ({..-I) }. The lW>Ction .. dominotee I by tbe pr"'-"'CIio«, 1\ illoom" ~u.ous on X by kmma. 19~. l (appt;ed 10 " I K ), and ..., h.t.~ .,(.-' )g( I) :os ~I) + t lOr "'""'l" I E K . Applyio« 10 v lOnnuL. (I ) 01 Section I!U. we obtain
   
   r
   
   1 (. )dv(. ) :OS
   
   r
   
   v{. ) dv(. ) -
   
   r
   
   v{ .. t )g(I) d;t(I)
   
   :os
   
   I;
   
   (h (l ) + t) dIJ(l) -
   
   r
   
   hdIJ + t ,,(J).
   
   Sinoe" iA bounded and r ill arbitrary, iDlqu..!>lity (2) """,It.&. R N(fOf """' pM! to tbe~....,..,u~. Tho>ee compooet _ K in T lOr which tbe _,"",ions of .. and , to K ""' COflIinOOYll! ror", a /,-d.onoo! .t-, "P, ia T . By "Theorem 19.3.1, " iltbe rum of a 81,mmab&e familJ- (p,. )~u of m mre8 carried by- P-IICI& As (.-,, ) is ",,+adapted. .... can oomidef ". _ f g(1 )e-. (.) dj.,, (I). By part A, I(z) "'v~ (z) ... 1(.-t)g(I) ..." . {I). FinallJ-, """"" tbe _"""'" v.. are .. ,mmobloo, with $um .. ( Pn>poIil;o., 19.~. 2), n
   
   r
   
   r
   
   Jfz) dv(z) -
   
   r
   
   r ... L
   
   ... L
   
   -r •
   
   I (:z)"'v,,!:z)
   
   r
   
   1(.-I}g(t )"'"" II)
   
   1(,,1), (1)11',,(1).
   
   •
   
   o P ropooillOQ 111.5.2 S.~ tJu.l". io <>ml;......... alld' )Ii ",",. , 9 it amtin .. • ...... , ond,- '(o) .. " .j,"~. Theft C . ,,1 ill'. adapUJ and I( ~) dv(;r) ­ f (..t )g(t) dp(t) Nr eutr)l /' X - 10,+=1(1lIht:rt: .. - g(t )t . (. )d,,(t )}.
   
   r
   
   J
   
   r
   
   1(+(X ), "" ... 11&/1
   PItOOP; 1'00: e>"1!t)' '" E
   
   D.:linitioo
   
   23_~1 ).
   
   ..
   
   r
   
   r
   
   f (:r) dw(:r) 5
   
   "(1)01,,(1)
   
   (3).
   
   1';'; • ,,·!nlqrab!e open """ U rool&initlg ,-'(0). Given ~ ;,. 0, put h' _ " +€. Itt_ As h'/ g io ~r ~nuoo.tt, tho:: runt:'lioll 7 dtAned on X t". 7 (", ) ., Int ("'(tl/gll) , t E " -'(z}} is ~ ...",icontin""... (L...""8 19.!:..I). Moo."""",, 11., fullowl'lll p'OpCHiee hold: (a) 7(",) ~ I e",) for all:r E X .
   
   ( bl 7 (1rI)g(1) ::; " '(I) lor all f E T.
   
   I'f'OI)
   r
   
   1'.4.3~ , ~~.
   
   f (r )dv(r ) S
   
   r
   
   7 (z) .w(z) S
   
   r
   
   that
   
   r
   
   7 (a-I)g(f) dp(t)
   
   r
   
   1I'(I) d,,(I) =
   
   r r
   
   r
   
   1o(I)dp(l) H " "CU).
   
   s:
   
   Since f II arbitrary, "" obu.in / Cz ) 4v(z ) 11(1) <1;«1 ). .. dl.'sired. Next, fl"!)U' (3) _ I .. I CI: ) dv(z) 5 1{.' Ig( eld/.CI), and.be ..,'UII6 i~uaJity IoIIowoI from PlOp
   r
   
   f
   
   Henco:t'orth , " " _ I""~ ( .. ,,) io ,,--.dapUd and ... rite .. for c 0(U 9(1) 4I« t).
   
   t.e a ""'wing from X into .. "-""";-0-1.,...,.... Th~ /., .. · ... eo7~ .... W. if aM onIr i/W:
   ThlllO.... m 19.&.2 l.d /
   
   ., wrn"""",,1oIL
   
   PJI()()f': F'I.. , ' """me that / It .... _urable. Since ... :... ,,·!I>elblu .... bIe, tbe COOIpKt ...b8eu K of S ruch that ~/I< io coni;" ....... forma l'·den:led_ D in S. Let K E V boI Ii""" By bypothErlis. """ ucl> (T>rnpooa ~boc\ H 4f K . tbere exlstl .. pUlition..t ..( H ) COIlIiisting of .. ~-II, of a:)mptTL:t 9tU Iud> thoot the restriction of / Ioeoclt C. io continuous.
   
   Under-,'- C01>dit ionoo, H n ",- '( N) .",01 the oeta H n ", - ' (C. ) porti!;"" II.
   
   . -1
   
   Now H n ", - 1(1'0') iI p·ntd~bIe (Tt..on:m !9.S. I ). tile !let. H n (Co ) an: COOI..,:t, and tile ratriction of 1 <> ., to eao:h of lhem • OOIIIillUO\lll. Thus I'" mmpM:t!let. H in K for whid> / 0 "'I H ill conlin"""" !arm a p-dmoe cLu. in K (Propo.;lion !9.U ), .,hid> pnM'lII lhal / 0 " 11 • /O-_ .."rahIe (P",p"ailion 19.2.2). Con~!y, JUppoee that 1071, s is l-'· meMurable. Mne C. .. I'" oo!Iection of mm..,:t """ L in X for which / I L • conlin"",-,,- II • efIOUII:b 10 obow lhat C. • in X . Let N ... a ~ut.e!. of X luch that N n Lis ... not&l~bIe for all L E We ha,,, to I>fOV" that N il locally ... not&l~bIe or, equivalently, lhat ,,- I(H ) n S ill locally p.""SI~bIe ("Thoe<:nm 19.5.1). N..... llie compact . ubleu H of S ouch lhat lhe n:s\rictions of 71 and / 0", 10 H are <:OIIlin....... form a p~ claM 1) in S. [t su/6",,", tM,d"" " , to I"""" that ,.- I(N ) n H ill ~neFl&ible for """lY H E 1) (P~tion 19.2.1). If T .. an ultra6ltn on ,, (1/) <:OII""'3in& to z E ,.( H ) and if U ill an ultrafillH on H a>nI.&inin& {H n ,,-- '( o4 ) : A E T} , llien ,,(U) _ T and U """HlgtS to a point t E 1/; tlnlo ,.(1) _ " . and, oi""" (f <> " ) (U) OO'''(I,*" to / (d ) _ I{"), we a>Ddude lhat I {F) 1(" ), He""" the ....Iriction "'I to ,,( H ) ill OOIIIin.......... [ootbet _de, ..( H ) belonp to andsoN n . (H ) . ... r>eg\l&ible. By Tbeoran 19.U, .. -I (N n ,.(H » n S ill locally 1-'.""SI~bIe. C"Y Flunltly, H n .. - ' (N ). which is inctudlld in .. - I (N n .,(H » n 5. Iro p·negI~bIe, and proof ill """,pIoN. 0
   
   u..o.e
   
   ....x,_
   
   c..
   
   "",,,co,aw
   
   c.,
   
   t'"
   
   If I w...;tII val .... in a Banach "I*'" (or in R ), I"'atatem.-ntll "/ 0"11 ill p-_urab!e" and '11 " .. ), is p'me&6\U"able" _ equlvaitnl , ~u&e 9 iI p' _rablt, doeIc not .-anilb in 5 , and \.."isbmoo T - 5 (Propoe.ilion 19.2.3).
   
   Tbeo",m 111.5.3 Ltt. / bo! a I if UHnlioll¥ ,,·irl/qnlbl-o in' Si±~'; in 110.. C/lU,
   
   f
   
   m4pptng from X if
   
   0"" ml, if
   
   inlo a rwI Ban<Jdllp<1a. 7700> t _ / (,.1)911) .. eumtioI/r p .
   
   1(" ) d,,(z) - /1 ("I ), (I ) dp(l )
   
   (4).
   
   IIo4t .. " rontin ....... and p'''''''' , , " ootntin ....... , ,,"" , - ' (0) .. p.~. Then I .. ,,· ;ntqrabk if and."." 1/ 1 _ / (..1)911) ..
   
   04_, m4,"""" p ' in«
   
   s' oabl<.
   
   PROOf":
   
   A. Fu. , _umo that I-' .. carried by • """'~t _ K on which 9 .. M,nder!. [nihil.,...., " ill MU"W1 (~m 19.5.1 ) and ..... can ",pIIooo , in 1M ""~ ,' ",dally l~· by ·in~·. If I is ... llUecrable , lben, by n .... ew 19.U , 1 _ / ( ..1)911) io p.ln~ and mation ( t ) boIdoo Con.eoseb-, If I ... / (..1)9(1) iI ~~abIe, \.hen I iI ... rnoeaaunbIe (Thocon:m 19.!> 2) and f"j/ (a-HoW(z ) _ f"j/("I )I p(I)dp(l) < +00 (Theo""" 19.!>.1): tbue/" ......tlaUy ... I~ and , In r..tt .... I~ble.
   
   B. NUl, we <X>n8i
   C. The II'lOI)Ild pao1. of lbe _ _ _I folk>a-a from tlot 11m pan and lilian 19.5.2.
   
   P~
   
   0
   
   Eurciu! fur Chapter 19 1
   
   [,g 0 ~ .bo.loca1ly _ptoet"""", o:cnoidenld in &.ere ... 3 of Cb&pter 7. Fo< .,.. inleCfl" " ~ I. let p,. boo tboo RodotI _ uro 0<1 0 doli ..... by , "'" ,.'3 m
   
   11m' I.
   
   ot the pointo """'" l hal
   
   (If ..... ~I'"')
   
   lor 1
   
   :S m :5 n.....:l t E Z.
   
   u.. "'pnmu:II(If lp~ • n ~ I ) in .M ~ (n) ;. lbe RaD _u",
   
   QONi<W«d itt ~ .... 4 of Ch.pt ... 1 2. St- that p: (D ) ., 0 lor .0 n
   Po-,.....
   
   2
   
   Let. n &O>d" be .. in Exe c... I . I.
   
   Let A boo .... claoo of ~nl'" _ In a Fl>o-..-...y " E A. dot,,~ e .". f~ 1M iod_ fua<;tion of {OJ " " .... 0, SIoooo- tbat tbe "~I phe '" _ (f. lwlt . " !f<)aI n intO a " it .... _uroblo.
   
   ,_ Show ....... 3
   
   r f. 4 - 0 100-
   
   all
   
   0-
   
   E A. bu,
   
   ,ha. J' l....... " 1.) 4
   
   _ +0-0>.
   
   -,.i,..
   
   l.et X boo • Ioo.Ib ....pert _ • ...,.,. " • Radot> rr • • ,,,,. (II> X , ...d (I. ).. .... ... u~-diro<:ted!amily of~i,.. functiono "'" X . S w •• be ooawi"ll ' _ (1.(. » •• ,. frum X Inl<> it A ;" 1'-_",abIe. I.
   
   Show tbat
   
   1 . ........ ,. I. :. 1'- " " '1uro.blo .ud ,bat
   
   'I'
   
   ate
   
   20 Radon Measures Defined by Densities
   
   W.
   to,_ "
   
   _.u4y
   
   l ~kMo wlt~
   
   " P ....
   
   $"mm",],
   
   Let" bo .. R..don _""' .... a IoaJly compo.« o.p.oo X . Let Y be .. locally oom ~ M,N, • 01. X , 1M with _ _ "" . be iDd~ _ " " ' " .. 10 doo-. ill u.. _ ...,,,,,01 __ (TI-- em 10.1 .1). P*:iac t<>ptbe< Radooo _Uf'Ol. _ de.. '" ,"""" ew :IIU.2, io _,~..tuI m aaal,roio and di&o:....1o.l ",",.try. 'lO.1
   
   ,,,,.tiaa
   
   Lot " be .. Ibdoro • ' ''Ufe "" .. locally -..poo« -"" T . A ,..""""'" J " - T C io oaid to bot IoeolIy I ~~ .. be.o ,II. io 1'.;1ItOfJ*bIo lot or-y -am1wI<:tioe " . kIt """'f'K' ... ppon:. I~ \hlo ~ , ,, ' ~ ,10 4 io .. Ibdoro _~~ W. ..1Od;r _ ""I'" $ ' ' . wi.~~ ('l'booo._ $U ond lIU . ~) _
   
   n2
   
   _
   
   to,..
   
   2IU Tho IlIdooo- Nikodym ,booo r hold b Radooo ' ... _ :lOA !'\un Cbapter 10 i>II<>w _ 10 .. RadoA _ - .
   
   20.1
   
   J
   
   and ~'. ,:
   
   .. "I'OI'it;.;..
   
   reoulu,,", tho d..ali\y 01
   
   "« em
   
   oUlI
   
   Lcv.).. _, ...t..n"
   
   Integration with Respect to lnduced Measures
   
   U:t T hi: .. locally Mm~ H. ,....y..« $pII<:1:. X .. locally CO0 !I-'I'1! lJ>doood by " 00 X . tt.aII thai. \l
   20. Radon M 1"_ Defino
   398
   
   ~
   
   I'
   
   x-
   
   [O,+co>[, .be. t",""
   
   f' 1d11'... I- f' /,~I
   
   (wbn'e
   
   r
   
   is the
   
   function which I&"..eI wilh I on X ..00 VIUlIshtA out4ide X ). A mawina p from X into &lOpQlog;caI &po.ce is I' .... """",mable if..oo only if il is ,,,,"e..,urable on X (T~m 19.$.2).
   
   &.....:/1 , po«. I ...... H..n..llto I'x. om!!.' We iJ ..,..I onIw iJ ill {CtJ.nonierJj =tewn!' .. U«IItWly " . "" j • • Irk; in tlti' - . f Id" x. - / !' 04<. TOOurem 20. 1. 1 Ld
   
   I .... ,"owing from X inlD ..
   
   PIt()()F: ThIo ....1.. from ~'" 19.$.3, ... ben I' is p<*.i"". Tbc fin.-... .Ion is \hen otwious 'or Cll tbt fact thai. I' is .. 1'-< rombi .......... of fou r p
   11'1. O~
   
   0
   
   tbat I'Y .. Vox. )y , wbt_ Y is a locally Cll,;"': t bat. 1",,,... 1 dil' l fur aU functioo,. I froru X i"to 10. +001, and that .bt "'Iuallty boIda wben X is open. II dot>! not _I" l ..ilJ" bold wben X is (QM (Chapt~ 7, F' Id .. 4). Tbc f<>llowing thtoooetu los "'"" In IDIIr\y br ... ",""" 01 ..... t.l>ematica.
   
   f'
   
   sf'/'
   
   T~m
   
   20. 1.2 Ld T "' .. /.ocaIlr ...... po.ct Ho...un-JJ IJIG« And (U~ )..u. ... .".... toWI "111 '" T. F~ ado <> , kt I'<> .. 0 RJ.Un rrw»IUr\' "" U~ .1Och tIt-', for <JAdI,.ur e/i"# ,.. o. (J IM""Nt;J. U", n UI' "", 1M _~ ~ to,. 1'", ..M 1" "" U.nUI' _ Tl> E A.
   
   tq..w.
   
   P ROOP: F im. _ ohaIl show tbat o:acb fw>ctioo 1 E H (T.CJ can be ..Tit..,., in the bm I _ Eli'S . I, wbere, for e.d> lDdu i, tbtno uiou; (I. E A..ucb thal. I, E H(T.CJ and rupp(h) C U"". For this purpOlle . ....., ot.~ .bat, 01 T CON.ain!na tbt . uppOrt 01 I , l ben \~ txln if K is .. c:om~ finitely maQJ" lo>U(K>J mappi~ It. : T I ) aud> .bat OUPP(Ito) is almjlfid &lid is <W!Iained in U", far 1 < j S n , N'ld ~ud> tbat EI S 'S~ lI,{t) - 1 For aU I E K . n.en the (UDCtionl! f • .. fll, oatOsfy tho> roquimd condil ione. Thloo! .oI~ pV ..... lhe un;q.+1 '" '" b , by dMinilion. 1
   til'*'
   
   to.
   
   JOIn -
   
   E' 95.1'(/1I.;)- E, <. n ?tIT. C ) wltQoge ...... riction to ?t (U""C ) is 1'.. for _h 0 E A. (H(V""C ) is idcnli&d wi, b ?t(T.U..; C )), il JUfIiceI to te1ablisb the following ....,11.\0<>, Ii''"'' ''''v fini\( ""'l1>('llU$ !9;-)' S'5 " and (h,), 5.>5_ 01 funct ....... In H(T. C ) eud> tbat 1lUpp{g.) c U., for I < i f '" and suw(h;) C U~, ki. 1 S j S n, ....:l ....,h thai. :C, S' S_ !I;{t ) ... 2 '5.>5_ h, (I ) .. 1 for all r E JUpp(f), the-n
   
   L
   
   1'... (19;)-
   
   L
   
   I'~, (lh.).
   
   •
   
   Similarly
   
   Since 1lUw(/ f;IIJ) ill OO<1lal!ll!d In UG • n U~" it tOi~ from ,be hypothooooto liw. """ (J "II; ) ,. iiI, (J I),II,). and 00. , T ,bon ~I& 1\ 1'UI\&l!>l to be u--n Il1.at the lill<:&l" form Ii !IOdefin..:l 1!l • R8doa IDOIII\lf'e. i.e( K be. (Cmpooot ou'-t of T , and define lhe UG • and I"" '" N at the bq:InnlPIII ot the prOd, It H, .. auw(II;). ' ''''no br bypolhetit, Lite.., .... iII l &; ;!: 0 such tlw !.u...u)l :s .. ,IgA for allg e 1'({T . ' I'i C ). tIeD«, for ~ I E 1'((T. J( ; C) , _ Iur......
   
   1......(fh.)1 .., lbo.,
   
   20.2
   
   s ... a/"-. S ",II I•
   
   11i(J)1 :S (E ,s,s~ Q ,)III.
   
   o
   
   Radon Measures with Base
   
   jj
   
   LeI T be " locally COUIpecI. HaU8dorff """""" and I' • complex Radon wcuu..,
   
   ooT Propoorition 20.2 .1 Ld , 1I< .. mapping fn:>'m T ;nl .. " BaMdl/j ~ foJloci.ng <»nno .... ..",iUlLlcn.l; (. ; ,11 ;, I' -;nl
   
   M , . I",
   
   j1
   
   ,« Then
   
   dl, lor IJU II € 1'( + (T ).
   
   ;, Ii-inl.,..w.: lor all .........." IIINtIl K
   
   "I T .
   
   PIIOOF: AIBume tlw condition (a) II _iofiod . and let K be • """'pooot oubroH otT. Tberettilu II E "'+(1') ....... that II .. I on 1( . and.., f ' II( .. (~). LK .. ji-iDtec:r&bLe. C"',"'Ciluy, tlw "'-'ition (b) holds. 'J'btn I iII Wmeuu ••ble, '" gil .. ~~ for 1111 II € n+{T). Si"""
   
   _wnc
   
   r
   
   r
   
   !9l1ld11'l .s: M ila
   
   ,11 iII .... ~abO..
   
   191' 1_.)dII'l < +00. 0
   
   Dellnit lon 20.2.1 Undtt lbe a>ndilions of Prop<)liilion 20.2. 1, 9 ill I&ld 10 1M: locally ".lnl.e«' &b!e.
   
   III tm. __ , U>!! tinew form I _ f I,dI' OIl H(T.C) it clarty,. RAdon 011 T . ca.tled 11M: JUdon n_ure ... ilb density 9 relll.ti~ 10 1'. and is written 91' (or II '1').
   
   --=
   
   P fflposltlon 20.2.2 A_ 191'1- 191,1,,1· PJtQOf": Wriu
   
   that 9: T _ C .. 1Gml1" 1""'1'11"0""'.
   
   B lor {h E H(T. C ) : Ih l ~ I}. Tbon. for e>'try I E H'"(T).
   
   J l/lll dll'l" !.~1J I'hd;i\ - :~ (T1>$)Jtlb
   
   Th~n
   
   7.U).
   
   Now
   
   f 1191dlpi - f I 01(""
   
   f Ihd(g,, ) ~ J IdllIl'l '"'), aDd 10
   
   19,,1 - 1, 1. 11'1. a
   
   T hM>rem 20.2. 1 A H..me w.t 9 l' _ C it IGmlly l"rnt"1fl'bk. ~q 1 "191'/ - IlfIdlpl /or 0011 J : T - jO .+=1. M~. if 9 it .....U"""' ... lM , - '(OJ ;, I'.ml,¥,,~. 1
   r
   
   r
   
   r
   
   r
   
   P ROOF": Wril.e f lor 100 identit y of T. Th<:n ,t.. pili. (1. 1, t) "' r" l-adopltd &ad 191'1 .. f ~, I,' (I)dII'I(I) . TIoeott'U :2Q.U f<>Uooo"', d.e.efOit. fmm 11-. ....., I g.:i.1 and PropoosiUon 19 5.2. a
   
   Theorem 20.2.2 04_ tJUJI. 9 illoc..U~ wmu,mbk. Then
   
   (a) 2 'A ,illg J ""'" T inlII 2 ~ 'j:WJQI! iI 91'· ......... robie if anl~ if I i. I' .........""'w.: on S -il E T : 'It ) ~ 01; (~) " .....p"ng
   
   1 ""'" T inlo 2 &naGh 'po« .. UH>111411w 9J, - in~ JII" ......"tioIl, " .inlqildk;;n lIlil .... ~. f Id(g,,) "
   
   .mI'"
   
   if .... f Jlld;i· P ROaP':
   
   .u..rtioo Ca'
   
   n-
   
   f>I>d tt.. flr"8t part of _
   
   rtioo Ib) Mult from rf:IDfl UI.~.2 &nd IU.3. Now f 1<1(,1') .. f Igdl' iI true for pcsitl ..... , and pOIitj~ I' (..."., rtf.. ena). wheoCE it I"U",.,. lhat il >flua"" valid 10.- <:Ompit" 9 and (l)Olpie>: ,. t.e<:o'l$I! I' .. (Re 1')+ - (Re 1')- + i(lm 1')+ - i(lm 1')- .
   
   o As .. ~ of lbeorem 20.2 2. in n rd<-r that • mapping I from T 11110 a Banach jpo.ce 1M: II"" ? 5 ",.abl.>. ;1 is ~ec ' ry and sufficienL that 19
   
   be
   
   "._urt>bk..
   
   'I'
   
   ate
   
   T ........ m 20.2.3 ul" , T ~ C lie 1Do>l1~ ,,·int'l/l"llWe. 1'hDI v. ' T ~ C ;, 1oooJ1, " " .int.,. W. if Gnd <mI, if I,V. ;, Ioa>/I, I' .• nt~ In !II;' ....., v.ClJI') - (g,v. )" . P ~f'
   
   o
   
   Obvious,
   
   'The_ ~!I') oIloxally J"lntegr .. bIe funct;"'" from T into C it. complex _ lpooee, and II>< wappina: 9 - 9" from ~u. ) loto M (T ,C ) it li.-r. gIl' - 9>1' if and only if I" - v.1V" - O. Ihat is, If II' - 9>1dV" _ I d( tg, - " IV,,) - O. Equivaltnlly, ,," _ "1' if and only if 9l Iooolly ".,b,,""I .....-rywt..re.
   
   r 'I _
   
   r
   
   Oeflnltion 20.2.2 A Radon I_ute" OIl T it Mid to I>< .. n>eMun: wilh ' - I ', Oft abilolulf'ly continuous with mopoet to " , wht:neYo!r tt..re it a Iooolly J"lrrtegT&bIe function 9 from T into C .""" lhat .. _ ,,,. ",.,., , .... hicb 10 dt:fined Ul> to a locally J"rwgligible ""'-. it called .. denl!i/y of" r-elati,,, to 1'.
   
   Hencefooth. IU _ that" it positi.",. If g: T _ C it locally J"j~ .. bIe and,,, it posili."" tbm ,,, -Igil'. and Oi) 9 - 1, l loeally J"alrno:>Aeoccywt..re. In lit!>.' words, a '6 '., &nd .ufficioent condillon lhat,,, be positive it lhat ,be positive locally J"almoot .....-rywt..re. F in&IIy, Dotice that
   
   for aU locally J"inlqrable functions 9 .. V. from T into R.. A family (g.. ).~A of "."""""arable functinol from T into [O,+ceo[ it Mid to be locally a:ouDtabie .. henever tbe family (,;'()O. +«O[)). u it loc:aIly OOUMabie ( o.6 nition 19.2.2).
   
   Propao;ition 20.2.3 u! (g. )GEA lie a 1oooJ1~ """,,/4bk Jamu, G/'-II, ,,_ in! J " 61, ",,,,,ticru from T into [0. +«0[ . Th< /oIlea.., """"i""", ""' "",oi>-
   
   ""'"
   
   (oj 1'11= eN,., , , T - [0 . +oxo[, Ioa>/Ir " .~, n'litJo ~ "';!II E.~ A 110 '-Itr "."'mGot ~.
   
   l'J Th< "'mar (gG " )..-EA 0/ .........""
   
   ;'...".m
   I.. /.'I;, ...... ,,, '"' L:..e AIG'" PJIIX)' : By Ecor<w'. lheox,,,,. It '"' L:..£A'~ ilWmMllurabie. Th.. condi· tion (a) is .. t ..1Ied if and only if ,,' (jlt) is finite for every / E H+ (T). Since (0) E A : 1110 " 0) it countable, ,," (jlt) - l:..EA ,,' (j ,~ ) (P rop<8ition 3.6. 1). N"", I>\It "a '"' 110"; tbe condilion ,,' (jlt) < +oxo io equivait-nt to the condition (j) < +ceo; lo other ....".js, (a) holdo If and only if. be flUIiily (". ). EA it lununable. Then, .. riling" for E'EA ".. the ..,ooedI", COInput.l.tioo IIv.. v(J) - I"(jlt), and lhe proof it complete. a
   
   E....A ...
   
   ,.
   
   20.3 The Radon- Nikodym Theorem
   
   -
   
   Let T be .. IoMolly romp&("!. Hau
   Lemma 20.3.1 IAI (t . . a )1061"1;"" I>otmdt"d m.... ~,.., "" T , and leI (I /I( a """,pkz _ on T .o.ch u...I IP! '" Mo . ..... _ AI .. 0 po.II"... o:>OUt4nl. TIle .. tMrt nilil an o ·~bk """,~.....s~t"d ",nen"" " 8",,11 u...I P _ UO . P ROO~, LeI. ,
   
   e £bIT. oJ: eiooe 9 if P.'neaw....bIc anti
   
   r
   
   Igl~ dlPI '" .II
   
   "' finit
   f ''''!'
   
   !to
   
   r
   
   1I I'do
   
   ill C,b( T.P): ""'....... r,
   
   < (/ "'.,,)'
   
   < (/ ' • • ) (/ ",' ,. ,)
   
   , "'(/,,,,)(/" ,'''') N.,..· the mtlppill& I - J ,~", a oontinOO\llt 11'-' Ionn on 4 {T ,o ). Sin<:e LMT.o) "' B H!I~ sp&o:e, then: aist& u in 4(T.Q). and therefore in 4{T.o ). ",oeh ,kat J gd/J - J gudo for every 9 e 4 (T .o ). Applyin& thle rd&I;ion !'or 9 e "H( T. C ). " _ tl"'t {J _ 1'0. 0 Len,m . 20.3.2 l.d,. Qnd ..... 1_ ""';tiue R,.,Jqn _ _ ft.I on T . Au..tII
   Q ......
   
   tul AI 2: 0 ....,.., IJIcoI. I", ' v '" AI . I", . " .
   
   Thtn V "I"~ ito T . P ROOf":
   
   I[
   
   K is B D«t at>d A
   
   ~
   
   .. Bor-el subolct of K. tben
   
   I .. . ... .. I .. · ( IK .v) ", !of · 1.. ' I';
   
   union of t_ a:hiu&ry ~ K .."d K' io; a D«t. becA_ IJ 0 rontai ...... compact""' K e V such that ,,{Kj > 0 (PropoSilion 19.2.1). Ch<>oooe M > vlL)/I'I L) and apl'lY Lem"'" 203.\ '" lbe po;aoili"'l bounded noe i"", <> _ 1 ~ · (v + U p ) and to lhe rneamre {I .. \ L'(1' - At I' )' Modifying, if _ : 'Y. tlte fUI>(tion ~ a,oeh 1.... ( {J .. 00 , we "",y _ II(
   
   I~
   
   20. 3 TIw; Itodom-Nillodym n.-.m ou"",,*, tll&t .. ill reaI-.....JUI!d, uoivn'OalJy meM",able (Pros-ition 19.2.S) , and that it .......... ""wide L. Si""", If .. (! € T : .. (I ) < oj ill ... -...able and """"ined In L, v(H) ill the supremum of the numbers v(E ), _ben E " o r ~ the 01_ 01 all O. Since
   
   IK · v - /If · IK · " " IK .(1'- /If,,) ~ 1K ·iJ- IK .(.... ) .. ( IK . .. )0
   
   ..
   
   nqati~,
   
   o
   
   K ill . V-ae\, and the proof is oomplt\e.
   
   Theo..,m 20.3. 1 (Radon - Nlkodym) 1..:1" " M" Ioe "'" T . '"'" JoIlotoing amdilimu a,., """""I"m!: (a) v ....
   
   IMIUIIn!
   
   M
   
   ~I~
   
   E--,
   
   ho(J
   
   poliliw: "'.......""
   
   _til ... " .
   
   ".nqliglb/.e.d II
   
   (c) E--, ".negiigibIe
   
   Ioctdl~ v. ~.
   
   OlI'mpGCt , t ! ..
   
   ,,-nt9"o'gi/IoIe.
   
   P ROOF: Clearly, (a ) implleo (b) (T heorml 20.2. 1) and (b) impiifs (e). Nat, _lOme 111&\ condition (e) holdA, and define D .. In u..run.. 20.3.2. Let (K .. ).... " be a locally oounlable family of mutually diljoint 1)."""" aucb 1hl.1 N .. T _ U..... K" .. locally ~negli«ibl.. Si""", the family (K .).. .. locally couotab'e. N if unh,,- ' Ily mh ",'a blc, and 110 locally ... neglicible. As t be fw>ctiorw I K . Iorm • locally count&bI. lamily, ",boeo 110m .. "'lual 10 I locally a1.....t ( 'CI,.1Ine roo- " and roo- ". Pros-ition 20.2.3 lmplleo that and II " L ... " v.. ( whe ... "" .. IK • . " and II.. .. IK • . v ). On the ctheT hand, hy de6nition of D, the", ""itIU, for MC:h 0 , •
   " .. E... " ""
   
   - ,
   
   v .. .. 90"" '" 90 ( 1 K• . ,,) '"
   
   19.. . 1K. )"
   
   .. 90"·
   
   By Pros-ition 20.2.3, then: ""i$t8 a locally ~;nt"",bIe function 11 from. T inlO (0 , +ooj luch tll&l v .. 9'" .. WIll to he ahown. 0 P~ltlon 20.3. 1 LeI I
   
   T . TIIen tM:n-. ...... " ... t'ef'KlJ, ......."nObIe "'''''"''''' " ItIdl tII4t 1,,1 .. l, I .. "III, aM 1'1 _ \III .
   
   Ioe .. 0lI'mp/.ez RAdon "' .. ","'
   
   (1ft
   
   P ROOF: Moo B,. for I ~; ~ 4, hy I , .. (Re9)+, ,, _ (Re9)- , ,, _ (1"" )+, &Dd I. _ (lmlr. By 'I'Iw:>r= 20.3.1, the 9. ate all .....""" with buo 11\. Hence there ",jog a locally 1I1-i~ function U l uch that. B .. "III. Now II I - fo!l lIl. and I!O !E>! _ I locally 191-a1E1>1)$\ """YW1Ine. Mo"""",
   
   li6 _ (l:1li)191- 1' 1. Finally. 11'0) nII\Y IN ppOflt l hal " is univoe.Wly ","""ul'IIbk: and Ih.ot 1"1 - I e.t1) whe", (1Ia/!'" "f3mrIffi1 l1li l hat of Pro~1ion 19.2.3).
   
   o Now, It is easily lee<> lhat the Radoa-l'ikodYID li>ooum .....wn. valid for complex ~." ' . • .,.... I" En:r cl. I; 2. _ si,-e ... OI;~ proof oflh .. romlt. Theorem 20.3.2 ~I,. De a pOIiti~ &U/"." m.... ~'" (>II T and /d" De A mil RadIJ1I "'...._"." T. TIle J"'~ «>n
   (a) For we>"lI UK1IlNIlI, ".;"Irs; !.I.. I~~n J ""'" T into (0 . +col and }cor t1>tr7 ! > O. th~ aistt 6 > 0 .nidI 1I\.oU 1M """lit>... 0 ~ II ~ J a nd
   
   1" IIdp S 6 imp/r r 1141...1 ~ !.
   
   (b) " id.>ngI to /Joe '-d P<"'tnrttool hJ,. ill /Joe Riut IPG"'" M (T, R ). (e) Esdi " ' ''~ _pco..ct H I ;, v·nqoIigiWo.
   
   (d) v ;, .. " ' _ ath kM II· P ROOr: We CM ......" " ouf'l"'l_ 10 t he CAlle .. he",,, io ~ti'-e. By PropooUk>n 1..3.3. " toeI O. ,1>0: ... n \8t.s 6 > 0 8IIdr tlw the rdaliord II E H --(T) . 0 5 II 5 f. and f li d,. 5 6 imply f lid" 5 t . Th.. CODdI...... (a) imp/iN ..,...jit Mlr:r (h). Next. ~mo tlw ..,...jitlon (b) holds. and ~ K be a "'~i&ibJecomp6Ct ooel. Cb<> -- J E H " (T)...dr thai J '" 1 on K . 0 1....." > 0 ..... 6 > 0 he ... • 1. .... We QJ1 lind II E Ji" (T) such t hal II 5 J. ,. _ 1 on and S 6. Then v(K ) S /lib; 5 f. This ........ t hal K if "'~i&ible.
   
   ,..,td
   
   K.
   
   f
   
   f "d,.
   
   N"", (e) imp/iN (d), b, I"" HtwIon-N ibodym l l>eorem . F'lnally. (d) ;mp!k8 (a). by the .am.. ~n. .. t .. t hat of Tbeoreln 10. 3.2.
   
   o T _ Ra.dd .. Bre I!qul~ if and only If there aists a k:>caIly ,.. integrable fu!IC1ion 9 from T into C ...... thM .. _ , ,. 00Dd 9 1' 0 b:.Uy """'Y"'boo",.
   
   ,..&1_
   
   1'."
   
   Dt:llnltion 20.3. 1 Two Radon """""'.... on T are ... id to he d isjoi nt (or mutually oin,suW) if in! (11'1. 1<'1) ., O. a"d, in ' hill CMt. _ ... rite I' .l v .
   
   Cb.rly. P ,"""';tiord 10. 3.t . 10.3.2, aDd Tbetum 10.3.3 ",mal .. .-..lid lor
   
   Hodoo _""",. n.,flnil;On 20.3.2 A Radon measUn! l' 00 T is oaid to be w ...... ,lIt1Oled on E C T (01" (&/Tied ~ E) ",he"" ,or T - E Is Icoc&!Iy ,.·necliiJt)le.
   
   If two Radon
   
   ~ .... 1' .
   
   "011 T are a>'O!f1Ie propouy II t .... br lbe followi,,& P mpotitloD 20.3.2.
   
   A eec. £ c T .. Melt [otc, secu ~ COOI(*:t eec. K c Tin .. IknI seI. It oaJd to be loc&I.Iy BoreIian. Lei A be tbe "-aI&eb
   P ropoooltlon 20.3.2 wI (p.,.J.~, he ~ ~ oj RJuUm m.,.".- "" T tudo ami Po .. ~ muull• ....,..s..r for all Ntinct p E N ..... , E N. n.... ~ e:n.tf .. ~ ( E,, ).~l 01 cIiIjorint!.>cAU, B.....Iia" ..t. nodi /JI.4l MdI p." ; , """"""",Ied "" E".
   
   tIWII,.,.
   
   PROOF: AM""", tbat
   
   posit;",Puttins: IJ " II, + II'. Ibm: uiat locally #"i~bIe functio .. f " h from T 11110 [O, +«>( IIiOb thal '" .. /,p.. We may IUppooe that II and h ""' """",urabIe .A (Propooltioo 19_2.S). Then p . and ", are ""'1Oe"1"~ 00 II E T : /t (I) > O,/.{I) .. o} and f1 .. {I E T : /t (t) .. 0. 10(1) > O} , p." ..
   
   Ef ..
   
   [ .: """ Ii >1'1)'. For eadlln~ j
   
   and Ft which any 1', and P;. j . . "",ti~ly. Then PI io ~ralcd 011 E, .. nJ~' E{ and . be
   
   > L, there elCist disjoint A_
   
   Ef
   
   P, areCOOCJtDtrued on F, .. U -~, F'f. In the...,.. way, _ can Ii.;;{ t;, C F, and F. C F, ouch tlLat 1'. ill carried
   
   Eo and 1'1 carried
   
   by F. lor all j > 2 . P' ","ELJi"3 wilb tbe COOItmaio<> oc.ep-br...up, _ (6" ).",,. tbat hM tbe deoired pooptrtieo.
   
   by
   
   Now. Ior...,;t, t E T, let t, be tbe Radon
   
   obc.lU.D tbe
   
   _uen<::e 0
   
   _ute
   
   on T dt6ned by tbe unit . . - at I. If"" .. an ~ Radon measure on T , then 1",,1if the fUnt of the IUIIlIll&bIe family (1",,({t)) I£')'tT' Each pooiti"" diff_"....ure on T ill dilljoint from all the ~, . Therem. it is disjoint from 1",,1('fl>eon:nt l.2.1). Hence &II MOmIc Radon ....... ure and .. dilf_ Radon ............ an! difjoilll. Nexl , let " be .. n.don " ' Mure OIl T , and let "" be tbe n.don ..-ure defined b,< l be ..-,,({t) .t~ t E T. T hen (P - ",,)({t}) _ 0 for every t E T. 110" - "" ill dilf""". We amclude thal" C&II be written "" +I'.I,"bcTe "" if atomic and "" dilf_; nlOlWLtr, tbif decompa.ilion if unique. Finally, POlice thal resul~ e1milat to P~tionll 10.4.1, 10.4.2. 10.4.3, and n....uu 10.4.1 exiH for Radon.-ure, and ....... In " $ ; ' T to ....... 1ICe.
   
   r...a.
   
   20.4
   
   Duality of lJ' Spaces
   
   n.
   
   LeI. n be .. Ioc&lly comPKI H. UIIdorff _ and " .. Radon meuure on Gl-en p E {I , +001, let q be iu mnjupte exptAknt. For all, E L~(P), ... rite 9. "" tbe ronlinllOlll linear form I - fill dp on L~ (I')' Thon, 9 : g _ '. ill an *"""try from L~(P) onto ...ubip!OOt of ( L~(I'»)'.
   
   Le. iA be 1M maio proIoclption of 1'. When 1 < P < +«>, &ppIyI", T'heo<em 10.5.1,
   
   LeW) ""I(>
   
   mo."" L1::{ji)( L~u.»' ,., (L~[j<»'. SiDll1Mly, wben p _ I , 6 mo."" LeV.) onl(> ~_
   
   that 6
   
   (Lhv.l)'. Next , .he ..,.ulU of!i«:tion 10.6 remain \1lJ;d f(lr 1'. M<><e(I\~r, (L~v.»)' .. Lb{ji) If "",d only if lbe suppor1. (If I' is ~nile. All tbale ~ulU """ be pr(l';'e<\ di<e<;tly, "i'''''ut ratOtting to Chapter 10.
   
   Ezerciu:.s jor CMpler eo I
   
   I' be _ """'~ Radon """""'"" on a loeaIly cornpoo<:< Haosdorll" opaoo. and lot , , 0 _ C boo IocalJJ' I'""mtogr~ . ""... p .... _ bOW proof of .be 6:;>!looring
   
   l.g
   
   ......ionoc
   
   ,.)
   
   bo,,1- III '1,,1·
   
   I' 'ellrl'l .. I' ' "IeII,.1for all f
   
   (0) ,<)
   
   f
   
   ") 1.
   
   : 0 - [0 , +<>cl·
   
   _IaJ.I,
   
   A ...awing 1 from. n 1__ eomp .... Hoo...h _ _ 10 U'" in~ if aDd ""ly i f Jg 10 .-...ially ,..illl __ blt; in .blo caM, J _ ~" ' I _10 If knDIy if f II ,.·~OIO " "'IrE n, ,(r) ~O} .
   
   f
   
   ,,,·,,,.arorabIo
   
   Let 5 be ,ho _rios in n «oDdo'II111 of ,"" K n L', .. here X ODd L r...,.., o....r tho..- of """,_, .... in O. Ddl"" _ _ ,"" " "" 5 by E _ j 1. 4. IkDOO,. ODd " !la.... lho ...... _ i a l probop'ion """" 8.l .1). and , 10 IooaIIy o-;"I<'pabk. For. R-I ......pKI ... X in lhere "". . a ooq.,...,. (U. ) .~ , o f _ ~_ of K _ that "'I' CU.) t.4(K ) + l/ n and II,.I"CU.) III'1(K) + l i n'" all .. ;?: 1. Cotoot."", _ [0, II, _ th&l ~ I. if «Iuai ILl I on K ao>d h.. eom.,..,. conu.l""'" kt U._ Oed"", .!>al. l,,: 4(g,,) ~ II . 1« d,.. C"'..d ..... I!>al. ,,. ODd 9" ho ... . be _ _""I ~io ... _"""'" iollow ..... lint . h_
   
   rn.e.
   
   n.
   
   ~.
   
   2
   
   .s:
   
   -
   
   p~,
   
   .s:
   
   _
   
   '''1'POrl
   
   f
   
   f
   
   ••!la• .-enion (d) 10 ' """
   
   Let,. ODd ~ be ....., complox Radom _""'" 011 " IooaIIy Odiliblo W~ Ii... " ""'" j>I"al. ~ .. a .,.....,.. with ...... 1'. 1.
   
   Let E be " ,.. . . .islbIo .-.I&ti .... ly """'J*:' _ , 000 lot (U. ).~, ......... ~boodo of E. wi,b """,pact dot" .... ud> ,hot VI'W.) """usa to 0 _ " - +00. Show .bel V• • 000 "" E it_ , II .. m:~l~bIo. Conel..... ,hat a.ch IooaIIy J>-oqIi5iblo ... II Ioc&ll:r
   
   q""""'" of _
   
   ...
   
   2.
   
   n..,
   
   ~.
   
   Let 5 be ,1>0 _ring ;., n co",....i", "'f .... K n L' ( X, L compKI .... boed 8 for .100 _~
   
   of {l). Wri~ <> for .be " " " ... E _
   
   f
   
   E..... fl."
   
   OIl;
   
   Dod..". If.- Th
   
   s. w.. ~ ,ba!.
   
   11 _ Ii and jJ,." (Thoot_ 4.1.1).
   
   .'" lD.2.I and "l"boo<= IO':U , .... ,bon orloto aloco.lly ... i~~ fwK:tioa , lor ..\Ud:J. {J _ 1/1>. N<)W u- ....... Ji1 ., iii - , and lbat . " _ " .
   
   21 Images of Radon Measures and Product Measures
   
   boo._
   
   2 1_1 Ld T , X k>,-" , be i ............""'.1 _ [(Ju) oiJ>. of ".D,abIo
   
   "
   
   r
   
   fJ1w--
   *. '_. . ._ .
   
   21.2 Woo .. ""'" ,be dw:>topoo;t"'" of. _ure in,./;coo. 1'1110 be . - i. Chapter n. 21 .3 w. doII ....Iot ....,.t. . 01 ,_ RAdo
   m'.,"..
   
   _~t""
   
   .ilI
   
   21. 1 Images of Radon Measures
   
   'wo
   
   1I,,,,,..,Lnb, .., loot T and X be loeaIIy <»mpoooct " "'>edorlt' ..,....., ,.. .. OOtDpIex Radon It : 5pure 0<1 T , and ... B I'-mtfIIIunbie ...... Wi'" from T in«)
   
   X n.:.flnltlon 'l1 . 1.1 " io *ient ODnditlona bold:
   
   (al R>r all / E 1f.· (X ), 1 0" ....... ntiaJly ".inlegrabit-.
   
   (b) fbf all~!leUK c X , ,,-I(K).
   
   :nt ialIYl>"in\~
   
   In th .. calle, the R.adon 1IlHOI_ H(X, C ) ;I jm.", 01" WIder" aDd it wrill'" rVo). U~tjl
   
   I _
   
   J I " "d"
   
   ill call<:d the
   
   fUI1b.. _ice. _UIM thai." iI p(altl"'l aDd:t Is "'prop<':\". 11>en l he
   
   pair (_. I ) it ,...odlpeod (in lhe
   
   IIeTlge
   
   01 Chapkr 19)• • "d
   
   ""'}-f
   
   t:01. [ cIp(I).
   
   Cot! luo:n\iy, _ ob(.ajn lhol foIlowitl(ll retul\.. Tbeo....m 21.1.1 Writ.:" Ir>r "v.).
   
   :nw:..
   
   {.} r !(r ) dv(r } .. r 1(,,1) dl'lt) Ir>r Gill: X -
   
   [0. +oc);
   
   (,J (I ...."pillf / /n>m X j'lL«'" "."'........-..bIo: if 0114 0fIIJ ill"" ... " ........rdoIe; (~) 0 ",appi"f / from X inl<> a,.-,,
   ""ink,,
   
   J(foo)d,.. Moow,,", , if" .. """'in........ ond 1" I : X -(O.+<x>I·
   
   opt,. /h.m r /<1" ... r / 0
   
   We !>OW return to the calC In which " iI .. aDd ot.en-e tht.t 1"v.)1 :S: "(I" Il.
   
   ooml*'_ RIdon
   
   0:
   
   djJ Ir>r Gil
   
   n_u~
   
   OIl X
   
   Theorem 2 1.1.2 W I /Ie c ....pping /n>m X if\/.o 0 Banac4 ,pace F . II I " " .. 0UCtlti4l/J ,,·in~. til ... I ... eumlWIt o:v.).intqrUl< .nd
   
   J I &,("v.» - J I " o:dj..
   
   PROOF: T he Ii...,.,. ""'wi"W' g J gd{o:v.)) aDd g - J g" o:d" from q:(-("'Il) InlO P are oomlnuouo and they ...... on "h:{X. R ) I1j>P. 1O they are ident\cal. 0
   
   Notice tbat , II/ iI intqral*, 10:0 ",utoa!.
   
   ~.m.i.ollJ
   
   ",v.rintqo::rable.
   
   I " '" to """
   
   _tWly ,...
   
   Rllfulu limila. to P ropoolitloo18 H. I. l 1<0 1I .1.~ ""iot for Radon """ .. '_. We loa,.,. tIM: d.:taito to tIM: n:edco-. fOr example. _ obWn the lo::>IIowi'C .....wt.
   
   r.
   
   P ....,..itkm 21 .1.1 Ld r. 'r lie I/o...., IGmIlr"""'pod Honde,e 'po>OU. " (I pouit; ... .........", on r , '" G ,. . ......... nsI:ok m4pJ1ing from r inlo 1", ,.. .. fflGwV 1"'.
   
   0"" ,... .. ,.. ""'.
   
   pnI"",", iI .. CGK.
   
   ,,'p'ope' , 011<1 u/int I" _
   
   o:(Pl. Fr>r 11 10 lie 1". , ., " and ,..j}icitftt tlwil 11' toe 1' -1"01'... . .nd, ;" III"
   
   (0) S~_ t./uit '" "
   
   11' V<) - 11 (1r{I')) .
   
   ,.
   
   (6) Auo.IM IMI r
   
   .......,in...... AM ","
   
   ioo 1'.,...,..." u.~
   
   u ",v.l·,..,,..,., .. ~ .... v.) .. r (",v.l).
   
   '"
   
   U 1' -,10"':'.
   
   r
   
   P lIOOr A8iert1on (al Ito obvioos. For (b), Itt K ' bea~liUb9tLofr. Thoen K~ .. r( K' ) "com~l. Henes tt"-I( Kff ) is ' t1i1i&1ly 1'-I0lqrU>le .. _II .. Illi ""bote! ",-I ( K 'l. Thm '" il l"propel , wei _ ..... "'" Il llion (a) to oonclude. 0
   
   Decomposition of a Measure in Slicest
   
   21.2
   
   Let X bon io<:&Ily a>m~ 1I."".1....if ~, '" a mlt.ppi~ from X IRio a 1ouIlJ' _I*l lla~ $pooiI T , I' a po6it i.., Rtdon .....,.."re 0>11 T , " : i .... ~ a from T ioto M~ (X ), ..-!any _otwly I'-inlep-ab/e and vasueiy Jt-~ Put .. ,. . -1(1), -1Ia)' fhal
   
   mlt.PPil'll
   
   J....
   
   tbe bo .. I"" .. J '" <1,,(1)11_ " dtwcllposit ion of ~ in sliooo, ... llh ""l>"1l'i • <. Tbor: feilowlnl reorulllto IMui.
   
   _i..-
   
   Pl"Opooolt lon 21.2. ' S..".,~ 1M! .I., .. ~",I
   If N
   
   ~}
   
   1/ I
   
   e T;' 1DooI/» I"~ cl(N)
   
   u . 1'.m..,....... 61. .. · m.. Ie o6fe.
   
   u IooJIv ... ~ligiblt.
   
   -nn"f '""" T into .. ~cnI """"'. 1 o ", u
   
   I.et II coosisl of Ibo8e functions II E H -( X ) oud> IIw II S I: footrvuy II E H , .... ~f ,..{i) .. ~ (II) foo- aU 1 e T , d liv*'" by " . &pp/og I .... ~~ , and drootc by ... the!ilU6U.-e J11.1., dl'(l) OIl X . P lIOOf';
   
   N~
   
   r'. . . . r r <11'(1)
   
   l d(U.) "
   
   I ~.(II)
   <
   
   +<'>0.
   
   to(.lU.,. It boondcd wd ., Ito .,..P«'Il<'t. On the Of;her bud. i10 .. _wly p-lllU;f;l'&bIe. ew:ry / E 1-/.{T),
   
   rot
   
   IU o",)
   .r r
   
   .r
   
   
   °
   
   (f ,,,)<1(11)., )
   
   /(1)(11 .... )· ( 1)<11'(1)
   
   .. 1 /(1)",(1)<11'(1)
   
   21.3Prod""'olRadoaM........
   
   411
   
   by 'Tbeorem 19.4.1 and lhe facl. IMI loA, is """""Dlr&ted 00 ,, - ' (I). T berel"ore,
   
   ,,(.... )- "1'.
   
   'l'hoe" nrs ..... b 10 E H. constitute &1\ upward-directed oubooec. of M"' (X). Let I E H"' (X }. Since l he colLection "P, of Ihcee CODlP'ct.u K In
   
   T""""
   
   Ib.at "'" 10 ~r ODDIinUOLl!L. is
   
   v(1) -
   
   1'""""""".
   
   j ".1., (1 ) a'p (r) ... "u> sup !. .1.,(1) a'p(t). "
   
   AIoo,; lhe 61\e1 of _;ons of }{ , tbe _ppi", t .... (1I.l., )(1) _ A.(1h) ,,01'1>('ld \0 t _ A.(f) unlfumly on every K E 'D (by Oini'. lheorem). Hence, b aU K E'D,
   
   !.
   
   A.(1) dp (t ) - ou p
   
   "
   
   .... H
   
   !.
   
   (hA. }( 1) dp{, ).
   
   "
   
   We conclude IMI
   
   n.adore, " _ IUP.... H "" (Lemma 1.3.1 ). If N e T 10 locally I'"nt&lwble, ,hen. £or -::ry 10 E }{, 1\ is locally 1M'" nedicilM, and ,, - ' (N ) is IoeaJIJ" ...... nqli«ible ...1>...... ~ dedoo: thool. ,,- ' (N ) Ioloeally .... """Ii«ible (Propooit>on 111.3.1). SimiLarIJ", if / is "'P>eIIOUfablo, then / 0 " is "" •..,...,urabI<e. at / 0 " ill
   
   .... meYu~.
   
   21.3
   
   C
   
   Product of Radon Measures
   
   In Ihlo _>on, X and Y "'" 110'0 IoeaJly comP'ct Ib.uodorff"fW"l FOr at\Y 110'0 oomP'ct oubeeu K, L of X , Y . define ali ....... ioIotDdry 101 from "H (X" Y, K" L;C ) onto "H( X.K ;"H(Y, L;C)) by (1oIf)(Z){W ) - I (z,w).
   
   ,,,,JIGOI!
   
   Lemma 21 .:U ~ doted VI!d.o1" o/"H{ X" Y. K" L; C) ~Icd ., {, e 10 : g E "/-I( X , K ;C ), h E "H (Y, Li C )} ill "/-I(X " Y, K " L; C ) IIMI/.
   
   PROOf"' Ler. I< E "/-I{ X, K ; FJ. where F _ "/-I(Y. L ;C ). P~lioa 1, l.le ....... I","" &i>"tO t > 0, lbe", nisi. fl .... , ... in H (X, K ; C ) and h" •.. , h" in F oud! IMI III«z) - L, :s,:s~g, (:r;)h; II:::: t for all z E X . The......:lt 1'oI1owo. c
   
   Lemma 21.3.2 wI p "'" 4 """"""" R4dMi ..........., .... X . For ......., / in "/-I (X " y , K " L; C), the w_ JI (z , , ) a'p (z) /.0 H(Y, L; C ).
   
   "'rIC"""
   
   "lang.
   
   PROOf': Put II'" wtll- Then j ~ t.:)
   o 21.3.1 l.d I' <1M v be 1_ ttmlplt;< RGd..... ............,...... X <1M Y , ~""'Ip. ~ eriJt.o G ""ique /U&don ....... A .... X " Y mcII tho.l AU ~ 1I1 '" p.(g)o{lI) I~.u g E Ht X, C) <1M Gil II E li(Y, C ) Mo'w",", . A{J) ... j dv{~1 j 1(.: , 1/)41'1'(':) I~ all I E H( X X Y. C). TI>eo~m
   
   g'"
   
   PROOF; De6ne by >. tho funct"'" I jd..(,)j/{':, I/)dp.(': ) from H(X X Y,C) inliJ C If K (~iYdy. L) Is compe.ct in X (....spedivcly. Yl. tbeQ .... "ILl nUlDbet lpe<> tlvely. 1v(1I)1 S bd lll) b all g toq ... tively. 1I);n N{X . K ;C ) (.... pteli,..,I)', In H(Y ,L;C») . let I E N(X X Y, K" Lie ). Then I j If.:. , 1<1,.(.:)1 S "",1, 1 ror all ~ E Y . .., 1A(!II:so m,..hD/ U, "h;cb p1"ll>"e!l that >'le .. Radon me-4U~ AA .. ~ 01. Lemma 21.3. 1, ...., _ that>. iI the uniq"" Radon _ n oox:b that >.(g <8 II) .. ,,(flv(1I11or' aU 9 and ..u II. 0
   
   s.
   
   Deflnition 2 1.3.1 'rbe ","""un >. , written
   
   01. ,. and v. Thea."", 21.3.2 I"
   
   uu. ow:>ttJl;"'" I" ® v i ... 1,,1® ivl.
   
   PROOF: let I EO N+ (X " Y ), W~
   
   "®", ii IOIld to be the product
   
   and let ~ E N(X " Y, C l be !IUd> tnat IPI S I .
   
   ha,..,
   
   IAMI"
   
   /41',,(:<) /
   
   ,(",p).t.>(~)
   
   Sf<11,,1(,,) / 19{"',~)l dlvl{~) .. (1,,1® 1"1)(191) S (1,,1® Ivlj{1J.
   
   1'l>enl1'ore, I>'Uf) S (IPI@lvl)(f)and,finally, 1>'I:so 11<181>-1· N_1ct. II E 1'('" (X ) and v E li" tY) - Given! >- 0, there exist and VI EO N{y,C).udt lhat Iu,1 S II, Itotl S D, and
   
   2: 1,,1(11) -~, 1.-<",)1 2: Ivl(v) - ! .
   
   Ip{II,)1
   
   ~,
   
   E N(X , C )
   
   21 .3 Prod"", 01 Radom M.......
   
   1)'1(11 0 ,,) ~ 1),(11, 0 ",J1 -1,,(11, )v{u, II Sir.:e Ih. II; lrue for all t
   
   1.1.1(11 0
   
   ;>
   
   ~
   
   (J"I{II) -
   
   t)(I"' ( u) -
   
   413
   
   t) .
   
   0,
   
   u) <:: 1,,1(11)1"'1") "
   
   1.1.1( .. 0 v)
   
   ..
   
   (1,,101"1)(11 ® u)
   
   (1,,101"1)(" ® ,,).
   
   Thill eq..-lity, In f~, holds for .. In H(X ,C) and " in H(Y, C), and L=m.t. 21.3.1 now 1M".... thal lAI_ 11'1® I"'. 0
   
   ctw,. our DOt&lion.
   
   Let 0' , fl" be two locally axnpooct H...... dorff '!*IN, let ,,' o.nd ,... be two oompltor: IUdoa measu... on rr and fl" , • w"""'ti...ely, and denote by " the RIdon . . - u n ! ,,' ®,,~ on n .. rr )( fl". If f : n - jO . +<>oJ • ..,...., oemit:ontllll..OOUl, then r ... f(r ,. )dY,," II ~
   
   We now
   
   r
   
   iIeInIcontlnUOUll and
   
   r
   
   fdY,, -
   
   r
   
   dY,.'(:i)
   
   r
   
   f (:i,·) dV,,·
   
   (Lomuno. 7.2.1 and n-m 7.2.2). 'l'bc resullll 01 Section 9.2 (with the ""oep(ion of P IOpOSilion 9.2.7), .. ~U .. n.--m. 1O.~ .2 and P.OpOSilion 11 .1.5. ..... true.,....., for IUdoa ."FI F'.... We t>«d only return to the pmo& of the Iollowl", ptopooiliona. PropoIltkm 2 1.3. 1 wI A' (rupu:tiwl,. II·) 10: .. ,.'·modcn!Ie 1IIh",' 01 rr (rap!Cti"'IJ, .. " • . .....ten./.e ...."'t 010"). 171m II' )( A· ;, " .~/.e. PIIIOOF: We m.o,y f"fS\rict 00. attent.J.on to integrabLe IUboeIlI A' and A~. Let V' and V~ be!Jltesn,bIe Open leU com.aini", A' I.nd A". respectiveLy. Since lv' ~v- dYp illinite. V' )( V" iI $I-;n~. 0
   
   r
   
   P ro_IUon 2 1.3.2 Let /' 10: 0 ,.' . ........ "nobIo: ""'pptng from 0' inlo .. 10fI0' 1ogioaI.,.,.,. F . ~ f : (r'.%"" ) .... I' (r);, ".~. PIIIOOF: Let K be axnpooct In 0 and denote by K ' tbe projection 01 K on 0' . 'Thtre ensts &eequcnce (K~)..~. 01 d~ compooct . ubIN of K ' iIUCh that li ~. iI continuouo for all n 2: I aod iIUCh lhal N' .. K' - ~ . K~ II ,.'·nqli&ibie. N..... K n(II' >
   P ro.,..ltlon 2 1.3.3 wt II«:. ,, · m"" '" &k ""'""" from 0 inlo .. ~ iaI.".,.ce. and GUIImI!: !loot f ;, on 1M. """,PkmDtl 0/ .. ".modercI/.e ",L Writ.! N' lor 1M. "" ollNue :r' E 0' It
   ..,.,.,,,,nt
   
   ...........w.. ~N'
   
   ;,,.'.~.
   
   yr
   
   ate
   
   P ROOF: Let (K. )~ >l be .. 8OWl £or all " ~ I. Domne
   
   N_
   
   {:z En - U K.: I (r ) ;i z}.
   
   There is .. II . negli&iblc .... boIe. II' or fY ouch that N (r' ,') is aU r' f 11'. Then, for all ~ f II', I(~ , ' ) is ""'fIlt6S\Inblt.
   
   "H . n,,&li&ible lOr a
   
   Wb", .... hA,... done r.... tbe prodl>Cl. of t"", II"ItaIIUre!I atmdo easily to fini te produet of Radon ~u"""
   
   n b. .. """'_, H 'l'e<\orlI " _, .-,,"'" n Olldt Ilia< r I tip - I .
   
   1 I.e<
   
   aDd ... " b. • dilf_
   
   ~tivt
   
   R.adoa
   
   I.
   
   If K " • """'pOd "" 'GO on 0. of .. I'""'l.u.od
   2.
   
   Let K b. • """"PK' "" in n. W aD ope<> "" """'a1ninr; K .udt ,hat ,,(K) < ,.(W) , aDd ~ .. real fIUIIIber o-*;,.(yiq ,.(K) < ~ " ,,(WI . "'" .....,. ~ > 0, _ lho. ."1tO;""",, d ~"""rablo "flO'" ~ U, V Olldt , hal
   
   .""b
   
   K c U e U e V e V c W and 6 - . ::; /'IU ) " 6 " ,,(V) ,.;: (+. ("""'".-uet V § ...... and nort Ul· l. Let K . w . 6 boo .. in pv\ l. S - l hat I""'" ."..... lMIu.od,obIo open V b ..-Md. K cUe 17c W on
   ""*
   
   iDductively • family jU(.I).." or " .quadrablo opeD (a) V (O) _ . &<>
   ""'.
   
   Olldt that
   
   (0) 17(. ) c U(" ) b all • • " IS' D Mliofy; .... " . ' ,"-, !or """'l" I E r - 10,1), put U(I ) "- U... " .•s,U(r). and",-, that lho. bouDdal)" ol U(I) ;. ,,"~bIe aDd Ilia< ,,(U(I)) ..- I . p,.g,." ,ba\ 17(1) C U(f ) b 0.1.1.' •• ' IS' I Mlio(yinr ' < (. 5. Ret.aininr;'b. -..ioa 01 part t . . - Ih.t t ho. fitoction " : 2 _ im { 1 E 1 : r E UrI))
   
   _ 6,
   
   2
   
   n - . . (0 , I ( aDd that it io ro,,,i .. ,,,,,,""
   
   ~ lha. "~~ '«" .!oi) _ ~ - .. £or all Co ./Ij in lho. natu,al oeminn« of I. Condudo I ~ _ " ' " on I ;. lho. i _ 01 " u- . . ...
   
   Let [) and" be .. in ",rr ' P'Utobie.
   
   ~
   
   1. S - ,"'" ,"""'."... ... bo
   Ex"" , , lor Ch.oplft 2 1
   
   :s
   
   Le< (} ...:I f( be 1_
   
   ( 15
   
   IocaIlr """'1JOct
   
   H.uodorff _ , ...:I lot jI, j/ be 1_ =' ... on (} ."d rr, _pectivelr· Awu .... I"-t _b 0>mPflO\
   
   ~;w, Radoa,
   
   wt.<:< of n 10 _Irinble Le< I be. m,w", r..... n .. rr into.
   
   metrizoblo _
   
   P ooodI.haI
   
   (_) l(z,,)ioj/,_~,.w.,Ior~zEn , (h) Ih z')O:<:OoIII_b~r'€rr,
   
   ,,* ,,'
   
   1,1,." I'f'OVe that / is ·_rable, 1. Fi>: .. <XIDIj>IOCI lilt.<:< J( of n ODd .. distaDol! d on J( I<>Inc:I of K. fbo- edI 2: I. \eo (zl"'), • o. be .. ""'I"""" ID
   
   ict_"
   
   zr·~
   
   K ..... lbal K 0: .ho WI;'" of .be <>P<'" be.lb B( I ',,) (1:5" i :5" .1:.,1. Foo- _ r € rr , roa::t.ruct .. _ i " , /. ( .. ~' ) r""" 0 InIo F 00 tW I. 10 (I., . j/. .......uabIo> ODd (I' M ' """ ..... to I. Cond1>de 1M< 1 10 (IK · "I."'·_~'obk Le< IK. I.u be .. io<:aDJ' -.nloblo family of diojoi ... «llIIi~ . . . . in 0 ...... IW n-U.'AK. 0: locally ""~. Sbow.Io.a!." .. u I.,•.". • haI. E"A(l 2<. · o.nd IIw 1 10 ".,,',~
   
   "I.
   
   2.
   
   ,,*,,' ..
   
   4
   
   1.0< n be tboo inI.......!
   
   E..
   
   ,,)* ,,'.
   
   to. I), with'be ..ual t.o]>OIop. and let fl' boo , boo iLl.........
   
   to, 11 eqwiwed wnh l be d"""",,, lopOiosY.
   
   ...:I let ,/ be lbe
   
   ~~~
   
   LeI" be Lvi , ... u U ",," on n, that ,,'((r)) .. 1 Ior..u z' E fY .
   
   on rr _
   
   "e,,',oq1i;Ible. 2. Let 6 ~ ((r . .. ) : :0: E n I be l be ~ of n, ODd wrl", /Ior Iu i""itaao< ru<2(t1o
   1. fu
   
   ~
   
   r EO. oIiowlhal 1,,1" fY io _
   
   6 0: locally
   
   r
   
   r
   
   "e ,,'.~"iblo. Con>puo .1,;,. -..It _itb 1'I:ooo..em ' .2.5,
   
   22 Operations on Regular Measures
   
   u.t.,. _ _ on oemil'iDp, we P"'"""'" o.ppI;< •• iono or _"'" ~ ... "'. .,...q_;.... ' be d;.&;i."'iation or funct ...... _ .he SormuJa fo< +'DI!"
   
   '1 .... I I
   
   or _A'" ... ...........10k, _
   
   . '"
   
   ,..or". 1~ tho .,G...., ,. of ftadooI
   
   For .klo, ,ho - - , _ "'" oboc.roo:'c" .... _ "" be com>o
   5~
   
   _
   
   _
   
   _
   
   ....
   
   ,bo - - ' __ on Radon __
   
   Summ.'Y
   
   mChaptot 8 ,hoo, _ OM _ W l .. """lao- _ .... ,. with a _ _\It'll .... u..... iIl, .. comin..,... ~_ "'..,.~ "" .. _ "'-.tiD"",," fu"", U" .. with .... pod ""I'PO't- w.. deoo:ribo llerein ,be fu DCUooaJo ~ willi u.. ia:I .. ' _ ..... "y, t ile" .., .... gil 111&. b.r,,,,, .. _'f ..ith "0; . ... ,..
   
   22.1
   
   \\oe haft - "
   
   Mod 1M ..... m T' u" .(00). n .2 w.. doIiAo ,100 ".riDe: of. ~ .... ;n a Iooally .....,.,... . ~ 22.3 Woo _ obo .... u.I.. olSooc< .... n _l ... . - ,hat . IIo: P">
   ctw\fo <>I ...n.bIe bmw...
   
   22.1
   
   Some Operations on Regular Measures
   
   T" ~n. Itt X be .. locally C« 10 X , ,. .. rquLu mc .. UM 011 S, Y .. locally rom""", ""boI~ of X , and Ie<. T be .. ""mlrin,c In Y such ,hat .....:h rl\a.ined. in .. """,nu.bie union T..-. Ir I'fT i. is ~lat, l<>deed, .. rit~ ~ r..... be
   
   or
   
   """'18,
   
   x arisiI>IJ from ,,; by the ruleoI of _nlW Intqmioa with I ph t to " IT and .I." , / : y - C ;' FDtWly "/T"ln~bIe If and only If It ;' MllmtiaUy .I.".i~, and then J / d{p1T ) _ J / <1.1." .
   
   Radon
   
   ~
   
   OQ
   
   On the other "-"d, the foIlo.-i", propoe:Itioa Iw:>IdoI. 22 . 1.1 Ld fllol! ~ Ioooll~ compo•.:! H~.."u,rg tJMI«, $ 0 oem>n,., in fl , I' ~ 'nI, if) l" IOIK) < +00 lor all compact x II K . In /hit CGK, ~ eriJlI a Iotallr I" ;~ /unctiMt. 9 ,..,... fl mu. c #t
   ".
   
   S
   
   PROOf": Let E C fl be I"~ For 0.11 A EO 5 , thmo ex ..... B EO aucb th.at B e E n A and E n A _ B 1$ I",...'jp;lble. Since E n A _ B • IlIto "'M&li,p;lbie, we cond..de t~t E n A ;. "'i~ and E 1$ ",_u...bie. By n...:....m 21.2. 1. tben! nist • • loco.IIy ""',,,'hIe c!.oM C 01. diojoint mm-
   
   poet au aucb Wt fl - Ul<£C K ;. krcaIly ,...M&li&\bIe. By t bo above, eod> K E C • "'i~. On tire ", her band, aocb A E S • • union of diljoinl COIIIpoet au K. In 2: ]) and • ,...nocIi,p;IbIe _ (beet._ I' \0 ..... Iar), 10 A io "",!.alDed In. union 01. muntably IIWIY c-t. and .. I"M&li,p;IbIe let. The R"",," Nilaodym th«>r= """" pr
   Corollary Let fl, S. and I' IoI! .. in''''"".11";'''' tt. I .I . Ih/irK" .. "" S #t
   ",eo
   
   Wi.
   
   In ~icular, If 1'" 1" ..., r.gul&r -..reo "" 5, tben 1" + 1" 10 nocuJar, bta_ )1" + 1'.1 ::; )1', I+ 11'.1· lienee the n:gular "","",ureol 00 5 Iorm • .-ector I Ubel*)t 01. M {S,C ). Pro~ltlon
   
   22 .1 .2 LeI fllol!
   
   ~
   
   /oeoI1r......,...,' H.. vHorf/ '",""", I'
   
   ~
   
   ,
   .ri;ing""'" ",
   
   "'..,._ """ _mllf 5 in fl , .I. u.. &don ......".,.., " .I.." ope. mappillf frvm fl i ....... /oeoI1) """'pad Hu.'~"' 11 tJMI« fI' . DrfiM 5" • _"""so in fI' #tM;that .. - ' (B ) it......uiall) " .in' S. dl. fer all B E 5" . Theft I., 5") it " .,..;t«. MOT"'''' , if 1M comJllld ",""II 0/ fI' are .. (I"I ) ·~, IMn ""'1 it ~ """ J"i..u .... 10 u.. &Joft ............ (.1.).
   
   ... ..-..,* '"......
   
   Fnc. _me thal II it ;>oIIitioe. Write 1) 10< , 1>0 (~of t'- compooct acta K C fI ouch thal it cootlnooos. For 011 K E 1), \.here nit'- .. !Jeq~na: (lI.. )."" of S"«u IUCb lhat K is induo.!ed in u..~ , ,,-1( 8.. ). Since 1) is.l..oen. In-It eW> romptoCt 8U~ PIIOOP':
   
   .., of
   
   fI",
   
   n is .. u";on or <:»unLably ""'~ diijoill1 1)..cu ....t .. ,.-IqIi«i~ -.
   
   FlMlly. Hod> £ E S II .. diQoint "o;"n of _"!.abiy HW\y oornpAe'l &eW..-.d .. ,.·~~bIe Itt . ThIs impl'" tt..i (.... 5") io jl-Wiltd. Next, aNUme thal the compACt IUbtoeto 01 rr Me "'(II}-_"rahit. Let (K~).." A be .. IoetJlJo cou .... bie ~ of disjoint com11K' SOU ModI lhal N_ Ua€" Ka iI kJcaIly 1"'~i«IbIe, ud, For all (I E A. put "" _ 1..... ,. aDd .\" _ 1.01• . A. Sin<)t I dlrf.\,.) _ r id"" '" I' (K .. ). tM Radon ..-.ure .. (.I,,) II boundtd. f...ad> B E SO ill FMwl,. " fA.. }-i"~, and to f/{.I,,}l"~ beca_ ,,-'( 8 ) _ _ ntI&l11 .\,.. In~: ,r,o.oo.er,
   
   n-
   
   r
   
   rintqrablo.
   
   NQ. ttdI compooct subee\ l. ol ll' is II'CIo .. bfn,,~ it " flc,... lIMnllrabio UId flv..,) II bounded. 'l'betef<m, there er:~ 8,. 8, in g. such thai 8, C L C 8, and 8, - 8, io flc,... }-..-gligi\>le. Under ,I-. OOUoliono, 8, - B, 10 .. (A..}-'",«I;«;bI.! b«a._ r'{S: - H, l" .I.,,-oqI~1>Ioo. fl y n..o. rem 8.1.1. II ~ular and Ii_ rille to f/(.I,,).
   
   "c,...)
   
   Nell .
   
   ...
   
   •
   
   100- all 8 E S', to the famlty (-CPo))...,,, it ownm. bIe w;,b awn "(PI. 0.. the oo.ht< hand , the fMl11y (_ ('\")) ...,,, of Radon _ r e t iI.u",nW)Ie witb rum " (A). The rule! o I _ia1 inlqnUion wilh , _~ to ,""'" t_ ..."", .. c..) ud or{.Io) pr'CM, _ , t .... t .-(PI Ie rq>J\o.r and 1'1- n. to .-(.101. FlnaJlJ". powe to . he ",..... oJ ~. in which " if c:oonpie.. W~ ....>"11 1.-",)1 ~ or{bo l), ....(;.) if resuIar. Eoocb 8 e S' is e>l\lentwly .. (.Io)-lmq;rable. aod IIBd( ..{).» .. ).("-'{B)) .. ..(p)( 8). The Ijn~"donno 1 _ f 141.-(;.) &Dd I - I Ib(A) "" Q;"( or{ll'l» .. Q;( " (IAIl) ""' conl illUOllf anti t bey o.c_ 01> S I(S'. C ). So '''''r are ldmIi
   "u.) ..
   
   In tbol DOUtion of Propt:eitOoo n i l . aupP • nl("tri>Able spoooe Ie "u.}- ~ .. bene·...,. I 0 " .. A mappI", I from Il' into a 6ano.:h ' I*'" Ie _ntioJly " u. Hntovabie .. oo>e.~r 1 0 "If it _ioJly 1'.fn"'Wab!e; then f I n(;.) .. I I 0 "<4<.
   
   "._",...bIe.
   
   •
   
   22.2
   
   Baire Sets'
   
   Le\ fI be. locally comptlCt. H• ...oonr " p""". A ~u'-' E of fI .. Aid to be. G. if It;" tbe Intl'.-:tion of a n)lIUtable family of 01_ SCU. Au luteraectlon of ..... nt.ably many O• ..,., .. I_lf • Of; thl: union of. finlte family of 0 , ..,., ... Gf .
   
   "''''lin''''''' ~.tiGltoed !un.c:liorI on fl ~"" e E
   
   P m po,dtlon 22.2 .1 II I if Q R . 1M oct {z E fl : I (z) 2: e} II 0 o:k>w
   
   U 'f!f/ E "W (fll
   
   nc4"""O ~ 1 ~
   
   G•. II Kif
   
   I OM K _
   
   G {z E fI : I (z) - I} . " 0 """'pod
   
   tAm!
   
   PROOF: Let (V.I.;!:, be • dec., 'ing "'"'luoJX:e of o~ ..,., such that K .. n..~, V. , aDd ueume ~ V, has OOpeni... a
   
   DeHn !t lon 2~.~. 1 "Tho ''''';113 In fl gen<:rstbd by t he c' - of all compK\ G. II <:aI1M I"" rJaao of Haire..,.,;n fl. P ropaolt lon
   
   ~~ . 2 . ~
   
   E'Of:f'V ...... ,..." Bo"" 1<1" .. G,.
   
   PROOf": LetC he !hecJa.of aUoomptlCt. O. and denote by 4> the elMo oftt.o.e Haire..,., E with the bllowing pn>peny : there u:ilu . countable ",beta. C~ oIC ouch that E belonp lO the,,·ringvn II ."-ri,,,and II ~ equal lO the eta. of all Haire oeu. 1'1_ let K he • com.-ct n.lre 8oet- There u:IsU • 1"'1_ (K, l.>, of comP"d G, IU<:h that K hehW lO the
   fi' 7 (r . r ) -
   
   L 2-' I/,(r ) - 1.(, )1 • • ,!:'
   
   the relation r - , If d(z, r ) .. 0 .. an equivalence relation on fl. "The IIe\ of eI· will be denoted by fl /_ and lhe ranonlcaI prn~ion from fl!nto fl /_
   
   ",T If T(z ,),. T(w,) and T (..~) '"' T (,, ). tt.e-n d{r , . z.) :5 d(z ,",,)
   
   + d(". ,, ) + d(" , z. ) .. d("
   
   , ,,)
   
   aDd, by Aymmetry,
   
   'VI
   
   ale
   
   Tbt.do"., 4("" , J:l ) .. d(Vl , Yl ).
   
   It folloon tlw we I1l.I\Y define .. distl,,",,,, I, on !.be quotient oc-e 0 / ... Il)' I,(~ I .l~) .. <1(", I. "" ) if ~ . .. T (J:,) and £ . . . T{ ... ). Then T .. oontiD_ fl"()n>
   
   o <>nI.() 0/....
   
   A rub:oo:tol'O is I~ in,,",, image (uoo.,,- 1') of .. subeet 01'0/ ... If .. nd only If. .. hmew;r II cootalne .. pOint J:. it Uoo """",I", t~ eJa. of:r. Eaclo K. bat thlll ptvperty and tbe dais of.oll ~ i~ Is .. a·rlIIIt Now K belongs 10 I t. a·rI", SC,ot. Iltd t". lhe K. ; ~ lbon: ",,;.. , a $U.t.eI L 01 0 / _ ,udo thai K .. T - I{L). MorN.er , L .. T( K ) Is COIIlpaCt; ...... , sil\l:e """'>' (bicd ruboooel 01. a met.... G. , Ibm> aisu .. ""'I""""'" of open leU CV. ). 11;1 la 0 /_ willi L .. n..11;1 V•. Then K .. , I (L) ., I 'W.) is .. 0
   
   op-oe " "
   
   n..>.
   
   a,_
   
   .
   Pi"OpoAl t lou 22.2. 3 1/ C " .'" ooI/a;t;w 01 ope» te,,", r.,. 01 X , Iiot>' Iiot a-rinf ~t.aI b, C ..."toitu
   
   et'er)'
   
   II""mIkI rJo~
   
   B.rin:
   
   oa.
   
   P ROO'" LM.A. be the c!-. 01. aD 6n1te iDte' .. <:tioo<>ol olC-ett& Sy bypo<~ eooo;h 0I0t"I in n if the ""ion. olllOrDf: sube .... ol A We ~ .a.o. 1""1 <7{C) CO that K C C U• . Tb., do,e, K,., S., and K ~ 10 ,, (C), 0
   
   *
   
   a"
   
   s..
   
   PropoiIltloli 22.2.4 Le1 U lot " ...... in O. F(1r K &in< oeU V.,,", L -.'0 ~
   
   n..
   
   _"""I,
   
   n.
   
   K C U , tJwe uUt
   
   (G) K C V C L C U;
   
   (6) V "<>perl
   
   "owl'" 1M ""..... ~/. """....,. 01 """'pcad G j;
   
   fe) L ... """'poet Gf • E 1/">- (0 ) oocb that 0 S I -S I , J,K .. I , and I .. 0 on 0 - U. Ooli."" V .. (:c EO : I {z) > 1/ 2) and L .. ( ze n : I tz) O!: 1/1). Then. IillOO V - lI- (% , I{:r ) 2: 1/ 2 + lIn}. It .. llot uiiloo> 01. MW!_ 01 PItOOl', notre e>'*'
   
   COIIlpKI
   
   C_.
   
   I
   
   0
   
   If U ill Open and :r e u, lakins K .. Iz) in PI""";tion 22.U. _ lind an Open &Ire we V sud> that % E V on:! V C U . Illollow-oo lhaltlot eta. 01' Open BaI .... _;" a boosie ,.". llot ~ 01 n.
   
   P ropollllioll :n.2.~ Ld!Y anCnlt.J ~ ..t' K A ff.
   
   n-
   
   P AQOP: If F: and E" AnI compact BoJ.., IOtU In rr and rr. "",peaively, lben E' " E" ill. com~ C" roo thai. A' " A~ c .A. Now the c' - of all _ '" the form V' " V", w),e.., V' IU>d V~ all! open Hai.., oeu, ill a t.siII for \.be ~. By P1 <>j1<*ilion 2"2.2.3, \.be ....i"« ge"e.ated by ..4.' " A " o:muJ"" A .
   
   o
   
   22.3
   
   Product of Regular Measures l
   
   Lot rr and rr be two locally ~I tla""""'1f 'fW". S' and S" two .....;. Ii.... in rr and rr'. "",peaim" and Itt " . '" be two rqulu",......"", on 5', S" I x;""", .rith lbe Radon. tllC l -"f"(:IfI .I.' , .I.~. WriloC /' !or " 0 '" and .I.
   
   b .1.' 0 .1.". Pmpa.IUon 22 .3. 1 5 _ 1h4t 1" and'" .'" ~ 11 I if . 1" _nWleJ-:tiom""", {} - rr x rr ;nlD [O,+wl, ~ r I dl' '' rId),. PROOF: For all E' e S' and all E.:" E S". the te1; E _ E' x E" it ....."tlally .l.-int~ and .I.(E) _ ,,(E ); hence .I.· (E ) _ 1" ( E) !or all E E 5 (PlOp ' t;"" $.1.1 ). Each /" no:cli&\bIe "'" III locally .l.-nqligible '-"'1'Oe it is corualuco:l In • I'·""&!iglbie S· .... if E it wrroeuurabie ooeral.e, tben, .,.. isUI F e 5 with F e E ouch thaI E - F ill IVucsli«lble; by tbe p<-.li"", E ill .I..,~u:al>Io! and .I.· { E ) _ .I.· ( F ) _ ,,' {Fl _ 1" ( £ ). Nat, let J be • .... me" "",b/l: Ilrid I'.m<>dera~ fuOC\ion from {l inlO [0. +col. For eo.cb i~ .. 2: I. define J~ - £:,s os_ +' (.I: - IIZ- · IA•.• • wbere
   
   N_.
   
   A
   
   ....
   
   ."d ,.....
   
   _ { 1-1 ([{k - l l /r .kl r O If 1 < .1: < n· r r'([n.+ccll Ifk_n·r+1.
   
   r r r r r r r I dl'
   
   fi nally, let I be
   
   "I~P
   
   w.... "
   
   J~dj.l ~ "!p
   
   I d.l. .
   
   urable from {} ;,,10 [0 . +cc>r. We bave
   
   Il ,. do< ..
   
   ' Tbil-uoa, _
   
   I. d.l. -
   
   bo <>mi ......
   
   11I;dJ.
   
   ~
   
   Id.l.
   
   r
   
   for all l"meMu,..ble and I""modol~ leU E, ",hidJ ~~ that f d., ~ IdA OIl the hand, if K' and K~ .ro """"po.ct in lY And!Y' , ..... 8p<'ldiw:ly. then K .. }(' )<. K " ill p·rnoderat.t And/-' /1 If d), = /1 If II>. ..
   
   r
   
   
   rfl"dp.;S~tlw>rfdl'.w~rfd)' S
   
   Prop",Jelon 22.3.2 uj Flo: .. B..,,<J(/J I I
   fpG«.
   
   r
   
   " ,dp.
   
   4v.) c
   
   Then
   
   0
   
   C}(),) , and
   
   J
   
   PROO'" LH f
   
   r 1/ -" .tV"
   
   e 4u.)· For.,....". ~:> 0, Ih
   9 E ~V.), whicb
   
   pro'"
   
   r
   
   < ~ by Proposition 2'2.3.1, fUl'\her, e 4().). The linur mapplngtl f - J / dp
   
   11 -91 dV)'
   
   that /
   
   &om ~V
   aDd
   
   f - / /.v.
   
   P""lX"'itlon 2'2.1.3 Eodt. / e ')of'(O) .. ..II K C 0, V,,' (K ) ~ V)" (K )
   
   ~ 00 SI(S, F). 0
   
   •.."J./{Jr all _"""I
   
   "._~...w
   
   PROOf" r""Bai ........ in n' ""'1" . ......... rabIoand lbe Bai ... ..u in 0" ..... 1"'. _unbIe; bene.! the Sal ... II> {l are ".measu. able (Pn>J> O."d isequ.a/lo o for c S O. It follo:>wt t hat I ill ~measurable, and n<)W, by Pn,\M'ition 22.32, I
   ;..u
   
   f
   
   a,
   
   J
   
   V>.' (K)-
   
   inf
   
   '.~.
   
   J1d.V>.w
   
   '~'M
   
   ,.Inf... Jldl' l' I~' ~
   
   and V)" ( K ) 2: V,,' ( K ) for all compact K . Howt!ver, by definltlOll of
   
   r IdY,.,
   
   bo aU fllllCtiono! I fr'OOl
   
   r f)
   
   IdYl 'S.
   
   r
   
   IdY,.
   
   Im.1) [0.+001. lienee VA"(K ) 'S. Vp"(K ).
   
   [J
   
   Theorem 22.3.1 Th 0, lhere "";,,u an ~ ..... U with rornpact ""un: f!<><:h that B C U and VA(U):S:~ . II follows lhat B is ,.·nqligible.
   
   u_
   
   Next, \eI. A be a kw:&Ily !IeqIleDCe ( K. ).~E ~n:,l;gibie.
   
   ~.De&li&ible!let.
   
   at rompACt
   
   ... '-1.& of
   
   If E E S , I"",,, iI ....
   
   u.. K.
   
   E sud> lhat E -
   
   Bm , IiDoe
   
   V ...(E -
   
   ".,'hap to 0 .... liCl>ds to
   
   ,oct En", iI Locally
   
   K.l ,. V ... (E ) - V ...(K. ) _
   
   I ' ~ ( E )-
   
   ,.
   
   V~ ( E _ K. )
   
   -kIO, E -
   
   VA(K.)
   
   U. K" lulso 1'· De&lCible. On tbe «bet
   
   rinoe An K . io ~."'"Ili«ible for all n ~ I . it .. &lao 1'.~lW!. ThiI pnJII'I!I7 that An E ill'·De&lCibie. and finally that A .. locally ... ·nqlCible, Now , \eI. I E and \eI. (f.). ~ , be "'''qt>iCllOO In H (ll , C) thai. ""'''t" .. w I locally ~ ...t . Then (f. ). ~ , coo.tlp to I locally p-1U. and each I• ...... ..-urabIo; thUi I io 1I·"""",urable. We """ II>
   4,(")
   
   compIeteo: the proof.
   
   0
   
   Wh
   22.4 Change of Variable Fonnula To clarify ideal, _ give one application of Propoailioo 22. l.2. We retain tl>< now.ion of Section n .3, and PfOI"" II>< followl", ... ult. P EopotIltlon 22.4. 1 SZlppm IM.t G{,,+) And G(b-) etilt in it. Lei J _ (G(,,+) •G(~ » .. An illl."...j incn..ttd ill G( / ), Mill mdpointl G(,,+) and G{"' ). 1/ I : G (I ) - F .. • uch IMt (f " G)g iI ~.~, IN:n IIJ .. ".iIIlqro/lk. 011/1 f(f " G)gd~;, "'l""llo fflJtW or 10 - ff!JIW'"
   
   G (
   ~
   
   G ("') or G(
   PROOF: f'"1X %. e I. First . MBl7me that the ... prtffium !of for GIIo• .>} io DOl. at~, 00 that. C (... ) _ AI . Ltt c be & point in [%0 ,.) ouch thai. C (/J) :!: G(~) (0, all (J e fc, .), For every (J e [e ••), the fu"",ion U/ ( " G)g l [.0 .111 io ~Intecnble; 1><""" (fll lClq).Glj!J1 .. z,o.integrr.bIe and
   
   J111
   
   1[0( .. ).<>(6/1 d<- -
   
   J(III
   
   0 c)g( (q JII
   .
   
   (10 _ 1m., ~ f by 1~ .8) io Proposition 11 .3.1, and l7I7I! tbe remark j<* K>iIowiDII Ph"p....ltlon 22. 1.2). \.e(ting (J cooverp: to . , _ _ Ihat 1/ 1!(C(.. ).GfH) isz,o.i~ For.u (J E [e ,. ), tile function IllCl .. ).G(M1 io domiumd (In r.bIoIu\e ...Jue) by I/ ll !GI.. ).Q{.... n. 00 f 1! [G( .. ) .G(j!J) "" f(f .. G)g! (.. JII
   J 11{G( ...I.Ql....
   
   "tW - J(f " G)' I(.. .>} d~. ,
   
   Similarly, if t"" infimum", foc GII<.,o) is oot atw<M>d , d",n 1 l(c (+- I.(1( .. )1 is ... inteoyable and
   
   Ne>::I , MllUme that the bounds '" and /If for GII.. .+I are attaiMd ILl poinu ¥" and Id ('). ~l W" liOquoenoo in (%O, ~) <:<>II'\'erging to ~ For .,..,h .. E N , 11/0(..1.(1(10.» ..... int~ and domiMted (in aboo/ule val",,) by 1111IOlo,),G(o. ll· Thuo f II JG1 .. I.(l(" I] dv ID f lI ICi .. ) .0(.... 1) dv as n _ 00. If G(zo) :S G(6- ), t he toet""""'" <Ji the
   
   z"
   
   <:0<""'''''
   
   /I" -
   
   hall the _
   
   JU
   
   0 G}gl",.o.Jd.l. '"' ±
   
   limit .. the teet""""" of the
   
   J
   
   (f oG}, ' I« .... d.l. ,. +
   
   J
   
   ll lC( .. ).cc.. )] <W
   
   f 1 1(G(..I.C(" )1 d", hence
   
   J
   
   l '(c( .. ) ·(l(H) ""'.
   
   On the atlw hand , if G(6-):S G(r o), then
   
   JU
   
   0 GJg'I....1
   
   d~
   
   '"' - j
   
   111<'(.... ).(1( . .)1 dv
   
   RtopIacin& I%O ,~) by (0 ,:<01 ."d quin&"~. _ obcain oiml)&r r$Il~ Now tho: pt<>p G(b-J; t' '"' +I ifG(a+ J < G (z:o) II.tId c' _ - I ifGla+) > G(",,); ..... +1 if G{",,) < G(6-).oo c" ~ - I If G(: G(6-). Then
   
   r,
   
   d (c (-+) '<:(._1) = c'1(ct.... ) '<:("'1
   
   + "liGl'"
   .....t..,.,. ew:rywhet-e, ~ 1 1(0(. +) .cc.... )) io ... integrshle. and j(f o G)g dp -! j 111C(-+)'<:IH) dv.
   
   o PropOeilioo 'lH.\ ... seneroJ. "cha..ge o f varlabIe" Iormula.
   
   23 Haar Measures
   
   n.. l rodl.iou.!
   
   01 Fow'i.or ..... ""'" I'bo.,.;,pl>o>rlq _ 01....,..1' ' . MM" • • , ....u .... \ with the _ 01 H .... ~~·w'........ .-WlIu.ion of_utmPflC1IP""p. ~ .... Inttod.... ph , " ' ,
   
   01 , . n. " Ir ......J:yMo
   
   wit~
   
   -.
   
   _I'
   
   :tU !Jo ,b. -'ion ... dofino jrrwula»~ relati-elr I~~ and quu;""'''*'''m 'E 7"_ ..... ""'*"r -..ptId ......1'. 212 0.."..,.,. h "ly OIIaIpacl .!>ere exiou .. Ieft . _~ " ' re. Thill A "n;o UAiq .... Up 10 .. "",ltiplio:atl ... _ _ (1 Zl.2.1). 23.3 We drIi... 11>0 lDOdu .... 'uDCtion 011 a localJ;y. pro F<"'p. 2lA Q.......In.......' _ u _ "" .. locally """'pact "","I' ..., all eQmpooct P""P ...... H .. d , 1 . . ~ ol G . W ith ewry OIl tbo be g ... ~ 51 uA .I,' OIl (T1..." ..m 23.5.2) ,!oat po TO ia"""'iII&: Jl"'>PO'"I .... 23.8 w.. otoodr I~ rib ....1""1 to A' ( , 216.1 U>d 7
   
   ;,c.,..
   
   -;:_::,:A
   
   G/II ""' ...............
   
   G
   
   ""-'lion 7.J.6.IJ.
   
   23.7
   
   W. ' t< • .,litU" .I, In-.l.' (P' opooi\;o" :alA).
   
   23.&
   
   n... "0Dl:f".,...<~ of quo>-"''''- _ ..... onGf H ( 1 ........ _
   
   23.&1
   
   _"82). XU In _ ... in....w"' _ _ ..... on GI H .... .,. I( \bo mod~1&.. funotiono 6., and t:./f 01 G and H ......<' .. Oft H (Tb"",,"w n .i .2). %UO 8 y _ qn, of EIi_ '-:"be the \..-lant" ,.u...... tho "","I' SO( .. + I , R ) of ..,...._ of R ~ " (1""'>'< , it;..., nllU).
   
   ""'*'"
   
   ,.
   
   23.11 \\;, """'Ibe fb..... _ _ _ or R " · ' ,
   
   ....... "" ,ho l""'P SH(n
   
   + I , R) or
   
   In I bis chalKer and liIe foIlo..oing one, all locally compa.ct t.aIoen to be Hall$dorlf, 'mh I otberwioe stated.
   
   23.1
   
   by_boIic
   
   ' pIIOf!&
   
   ""jll be
   
   Invariant Measures
   
   I.e\ C be alOpOlogical group Mth uni~ t and X " loat.U)· comj)kt ' po«. A IefI actio<> of C on X is a <XlIItinUOU$ mapping (g,:1') .... ~ from G" X into X sud> that
   
   tl ) U "'" lor olI:r: E X ; (b) (.1):1: _ . (LI:) for aU . , 1 E G Bod a11 :t E X .
   
   l'hw G It tDl to "'" Qn X from tho loft. The ""lion .. oaid to be transit;"" w~, for all :1:. l' EX . there ~xi$t, , E G such that"., ..,. D.Do>le by T(O) tho mapping:l: "" f% from X jOin X . We ha"" 1(,t) _ n1(1) , In puticuI&r, for every • E C , T(' ) .. a bo~hism from X on\O X. Fo< map pi"ll I from X Into " oct Y . .... pm 1(.)1 - 1 01('-')' 90 tha, b(.lI)h hl.,) _ 1 (") k>r all., E X . At.>, If 1' '' '' Radon measure on X , ... ohaJ] dcnt!to: by 1(' )1' tbe image measUIe of /-I under 1(' ) , "" that
   
   n.)
   
   "'''''Y
   
   lor all 1 E "N( X. C)
   
   f
   
   /(", ) 4(n ' )I')-
   
   f
   
   1("' ) ';/-1(" )
   
   (1).
   
   1nHe&d of d b t.)I') .... lIhall """"",,,,,,,, wTI~ dll('- ' .,), then (J ) .-.,..ds
   
   f IC")d;.«.- ',, ) - f
   
   l (tz )
   If A 10 A 1(' )/-I·jntegrable BeI: , lben .- ' A _ ( o- La , " € A) It 1'.ln~3b1c and h (o )l.)( A) - 1' (0-' A). Dellnltion 23. 1. 1 I.e\ I' be" RAdon measure Qn X . I' is ot.id to be
   
   (I ) invariant under G (Le.. under the OoCtion of G l if 1(')1-' _ I' fo<
   
   ~
   
   . € G',
   
   (b) .... lati.-eIJ- irn.....-ianl. under G if 1(' )11 if; il'"QporIlonal to 1' , lOr e\'I!7)' • € G; (e l quasi·invariant under G If T(' )II is equivalen' to II, for """TY • € G
   
   (Cb&pter 20).
   
   2.1.1 I............
   
   ~_
   
   477
   
   If '" ill iDYarianl, Ibm ~I, Re{,..), and Im (p.) are invariant. If '" ill • .....J ~ _un:, tbm ",+ .. (1/ 2)( ""1 + "') and ",- .. ( 1/ 2)(1",1- "' ) are ~.
   
   Aaume thaI", ". 0 ;. m&ti...-ly invarianl.. For ~ • E G Ihtte exioto • unique X(.) E C ' such thl.t ""1(' )" - X(.- ' )p, and tbe function X ill. goup homomorp/I;'m from G into C ' , called tile multiplier of "'. FormulA (1) then ~
   
   f
   
   / (..: )<1"'(0:) - X(· )-'
   
   f
   
   / (z )
   (2),
   
   and ,,(.A) .. XI.),,( A) foe ~ ,....in~.bIe tel A. The A:lAtion ""1(' )" .. X(. )-'" can .... be writle:ll <4>(..:) ~ X(.) <4>(' )' S~ ""1(.)(1" 1), ............. " 00 X ;. q\l8lli-in....wn if and 0II1,y if ""I ill quul-in>ariant. Thill amounto to saying that the claM of" Is in>ariant. under C. A _un: " 0<1 X Is quaoi-invarie.rlt if and only if tile collection of alIloa.lly ""ne&liIlbie ru~ of X is in,wiant "nde' G, or, ~.-aIently, If ,K ill "'DeI~&ible foe ~ ,....nq1qpble CXIIIIPK' rut.et of X and '"'""'Y • E G (Theorem 20.3.1 ). If '" ;. quasj·invarilnl undn G, Ibm the IUppon. of " Is tn....nam under G. In particular, wllenfM' G ICU t""";ti""ly 011 X , tlo;' ouppon. is eithM empty (lor " _ 0) or d ee «iual to X . Let X , Y , and Z be 110_ top<>Iop:aI . paaw. If" : (z, ~) _ II(Z, r ) is • contilllJOl)l mallPi", from X" Y into Z. Ibm, for ewry :t E X , we ohall .s.IIOIe lor ... tile mappi", from Y Into Z defi..ed lor "~ ( r) - II(:t , r ).
   
   1.. .e,)",I-
   
   P ropoolition lS. I . 1 Lei X , Y, nnd Z 6e tII_ IOJIOIogimIlplU%I. A...."'" ~ Y ill«»I/r amo,....' and It:!,, : X " Y _ Z k ..... ti .......... ~. /d 1 .. a .....tin....... mIIPJling jrrnn Z into .. B.......,.., "...,., ..w. "'",....-I
   
   ooncI /d " 6e. _ 011 Y . 1/, I()r" aoefl' :to E X. IMrr. ailt. a
   
   U..... 11;' (5 ) 1141 """"MId
   
   s,
   
   ~
   
   V 01 "0
   
   mX
   
   ndI
   
   ~
   
   . 1oAn i .. Y , ~
   
   fe) I()r" .....,. r E X . 1 0 ", iI ..... tin ....... lllitil .......,.,."...",...,. in Y , (t) r -
   
   I .. 11.. (" , r » c/jJ (~)
   
   iI <>mIin ....... on X .
   
   P lIOOr , T he ueertiou (I ) is obriouo. We prtM! (b): Ii""" ..,.,tinuity;'. local pn;>S>eny, _ "'-Y " ' _ that U.Ex .. ; '(5 ) II contained I... """'1O*ct ",t.et Y ' of Y . Si""" tbe furoction (r, , ) _ I (..(:r, r») ;. continOOUl on X " Y , 1 011. OOflH''''' to I 0 " ... , uniformly on V', as '" tends to "'0; thlll 0 II. ) ter>dII top(j Oll .. ). 0
   
   "u
   
   We retain tbe . 1KM! notation. Pr~don 23.1.2
   
   1«»1"
   
   A _ tIw Gil" """'pod "'0 .... Let" ". 0 100:. nlnfl'todr inNri.nt ' " ', .... on X . Tht:Io it. '"1Ilti,&r Xii" .....tin_ ",notion on G.
   
   P ROOt": Let II; 1t( X, C), S ~ the oupport of I , "" • poItll of G, .nil V • <0<1>1*'1 neichbooo
   !!din,s On a IoctJly oornJ>ftd "P" .... X via the OQnUnOOUl ~.tloa (., , 1) _ :u fn>m X ~ G [nto X , with r(1I )" (:U)I for all. , I I; G, z E X , and zt .. :r for &II :r E X . Denou br 6(.) the homeomorphism z _ from X onto X . For ~~I)" mappl", I fmm X into a IIt\ Y, ~ PIlI 6{.}/ _ I 06(.- ' ), 110 that (6(.)f)(6(,~) .. 1(" ) br all., E X . Alto, if " II. RwIon --.u", on X , ..... shall denote br ~(.)" I~ i"",«", ~~ 01" under 6(. ), 110 thal g>'OUl'
   
   :u-,
   
   10< o.U I / 1(:r) d(6(.),,)(.,) .. / 10
   
   E 1'(X ,C)
   
   I("'- ') ~")
   
   (3).
   
   ,",d01 d(I(. )... ), ... .ru.u $Otuctjmes writ.! <4«:U); lhen (3) .-It / I (.,) dj.(:r.) .. / /( :.. -' )d"lz).
   
   If A ... 6(.),..inoqnbk _. tbell A, .. ( .... : .. E A) (6(.),,) (,4) .. ,,(A.).
   
   iII ,,·ID'~.nII
   
   In>"Vi.ant, ~lativdy Invariant,.nII quasi-invariaJll ~ (wider G) on X are t\tfill<:d in • oImilar ""y. If " ". 0 .. reiatioely 1A.-.rl&aI on X , we t\tfi"" lw I1lU!Upller l( by the formula 6(.)" .. .:d .)" for all. E G; then (3) ~
   
   / If.,·)01...(1') .. x{, - ') / I (z ) 4,,(.,), and P(A,} .. ri.),,(,4) roo- e'Vft)" ".I~ td A. ~ rdatlon 6(.)" .. X{.)" caa abo be .rit tel> <4«1" )" ;d.) d,,(:r). 14 0" boo t be ,~ JrOUP thalo hat the....,... uoo..rtytn,s oet .. G, the; ~ ~, and w"'-e ",ultlpiieatloa ill "'""" br (I. !)'" 1.1 ("the..",....;~ CJdft G (ope,oJiI'I; on X rl'l><\o:r C" (Operatin;s on X from lhe left) • • ith the MmOI multiplier l(. Fin.tJly,1et. G bea k> itet:U br left lranelallono .Dd rla;ht tTVlllatiom., ...... dinc to the bnnulM .,(.):r .. ,., Ill!Id 6(.,., .. .,. - '. We ha"" ..,(.)6(1) '"' 6(ln(') (4).
   
   If we.ppIy tilt pr..:edin,s mrwb , 1IO'lf, we ha"" on G tilt DOtIom of left In...-iant not"", ..... , ri&J>t im-arianl. _,....." ' ut'fOl reIatl .... y \.........,. with t'fOIped to left (or ri«ht) t"MI·'Iooa... (~., ""'" SortJ,on 23.4).
   
   I from G i!lLo & .eel Y , '"' de6"" I t". I {z) .. I (z - ' ). Abo, k>r ~ ftodoa _ ... I' on G, P is the lma&e ~ ... 0I/, ullder the ""'-"'_0 - Z .... z-'. Hence PUl ., /'(/) br I E "H(G, C). In ",her For ... ef) mapplnf;
   
   _0»,
   
   f
   
   If "" agree
   
   1(1
   
   f(z )df<{z j ,.,
   
   f{z- ')<4« z )
   
   ([0 ).
   
   write d/'tz- ' ) IDStead of d/«z), then (5) ~
   
   f 23.2
   
   f
   
   l (z) <4<{z -' ) ~
   
   f
   
   l {z - ' )<4«z).
   
   Existence and Uniqueness of Left Haax Measure
   
   D.llnltlon 23.2 .1 1A:t G bo: IlloarJty «)Qlpooct group. We w i Io:ft I1aar",.,. sure on G ( I .lpo:ll..,ly, rilht I1aar ,.,......,., on G) any pooiti.., """=ern RAdon E"u ... on G that II \eft invari.anl. (... ptoCIioe/y, right inftriAnt). m
   
   T"",,",m 23,2, 1 0......." 1<>CId4r """'P""1' G. ~ eriN ~ I~ft H".~ ,..- ... ('"-'P"'tiwlr, A .0,.\;1 H ...... ........ _ ) , "nidi .. ""..... "J' I<> II ..."w. plioo!i... aml/nt.
   
   PIIOO'"
   
   ..t. £riotOlClO
   
   - r,
   
   o..Ip&le by 14 lho _of d PUt 11:~ _ 11:.. - (0) .1f K .acompectsubM ofG, we deDOU: t". 11:~ ( K ) ,he 11K of 1"'- I E 11:':' whoee ""ppoft • c:od , E 11:~ . tbm!: ~ numben~, 2: 0.. .. . Co 2: 0 and eIeIn.eac.a " " .. , .. of G such IhIlt I s; L'«S~ e,--rt .. )v, 1""-1, tbm!: exlsto Il ~pty op<':D. rubeet U of G ruch IhIlt lnfoE lI gI.) > 0, and ,he suppM of I till 1M: WOti OO t". finil.ely m&rIJ' oelI ')'('W. 'Thm, writl~ U : I) b- tho ln6mum of tho Dum""'" L, S' S" Co , b- all ~ (e , •. . .• c" , I , • . .. , .~) of poiIili.., III1
   (a) (-r(.l/ ' I ) -
   
   U:
   
   I) for
   
   I E 11:... ,
   
   E 11:; •• E G ;
   
   (b) ( A/' I l - AU:g) k>r/ E 1'I.. . ,E 1I:".A 2; O; (e)
   
   (U +!'): r l s; (/ , , )+ (/' : g) for I E 1'1. , !, E 11: • • ,
   
   e 11:.. ,
   
   (d ) (!:p)2:ouP/loUI'I foc /e 11:+. g E1I:; ;
   
   (e) (! : h) S U: ,)(, : h I for I E 1'1+. 9 E 11::;', h E 1'1 :;' ; (I") 0 < 1/ (/0 ' f) S; (f:
   
   ,)I(f~ ,
   
   gl :5 (! : I. ) for /. 10. 9 I" 1'1:;' ;
   
   ••
   
   (d Ii""" I , f, h ill 1(+, wil" h{.) 2:
   
   I Or> I t... ~uppOrt of 1 + /" and Ii,,,,n .. compooCt nci,sborhood V of ~ ouch lhal., I'or """' Y
   
   r ;> 0, ther" ezi$Io 9 E 1t:;. {V ),
   
   (I, ,)+(f: g) Propert;e. (.0.), (b), and (c)
   
   ~
   
   s: ((I + f
   
   clear. vt
   
   L
   
   I :s:
   
   ): g) + r(h : 9)·
   
   1 E 1(+
   
   N1d
   
   9 E 11'<. ; if
   
   en{ ••)9
   
   l~'~~
   
   .. lth Co 2:
   
   o. ...., ho.,,, oup(f)
   
   s:
   
   L
   
   ",g($,")
   
   , :;;:;. for an • E G: Ih ... 1UP(f)
   
   s: (
   
   L "') .up(g). 'S'S~
   
   w!>ence (d). For (~), Let 1 E H ... , 9 E 1!:;' , h E 1!:j. ; if
   
   I :s:
   
   L c,-n •• )g
   
   gS:
   
   's's_
   
   L
   
   's,s..
   
   d;"I'('J)h
   
   (with Co 2: 0, d J 2: 0, I" Ii in OJ, then
   
   I :s: L
   
   c,d,.,.{.,I;)h.
   
   '0
   
   (I: h ) :s: (I : g)(s! : hI· If "'" .. ppIJo (e) to 10, I, 9 and wI, 10. 9, we obt.o.ln (f ). fi nally, let I, /" h boa in H ... with h{.) 2: 1 on t he IUpport of I + /" and let . ;> 0 t... Ii""". P m F _ I+/' + (1/2)<"h; It... fu~ ", &nd the ""woo t of I + /" .. ad that vanish outside t bis ""pport, ~ w H +. For """'l' II ;> 0, there o:;.to .. " " _, ~borbood V of ~ NC:h tbat 11"1') - 'j>(l li :s: II a.od I.... C.) - '1>'( /11:s: ., for . - '1 E V . Then Iol, 9 be NI "",ment of H';. tV); for ...u 0 E C , we ba"" '" . "I'(')g :s: ('0'(. ) + IIh(.)~. Indm , 01 poinUI ..here 1C')9 '"lWisheI, this is clear; t b"" Ihis inequality bolds oul8iQe .V; in IV. '" 1"1.)+ II. Similarly, .... . "1'(.), ('1>'(. ) + ")"1'(1)9 . Now, if _ let c" ... ,c" k pceitiYf: nom""'" .. nd ., .... , ... boo eiemento of G """h thllt F :s: L,s..s_ C(")'(I, )9, "'" _ lhat
   
   s:
   
   I _..,F S
   
   s:
   
   L
   
   '1'1'"1 (0. ), S
   
   L
   
   "' ("'(If) + 1I),{•• )g
   
   ry'
   
   ate
   
   and similarly lor ' 0 ht'nce
   
   L
   
   (J : , ) +(J':,)s
   
   C, ("'(8;) +",,'(8, ) +2'1) S (1+2'1)
   
   L
   
   Co .
   
   +.,.
   
   sillCe V' S I. Applying tbe definit ion 01' F and. !leXt. (b), (e), and (e), "'" collclude thAt (J :,) + (I : ,) S
   
   (I + Z'1){F:,)
   
   < <
   
   (1 + 2,,) (((J + f) : ') +~~(II : 'l)
   
   , ((J + /'), , ) +
   
   2"t{II , , )
   
   + 2'1((J + f) , 11)(11 , , ) H'lfll , , l· F\lrtbe",... e. if "'" hA..., cboooen '1 00 thAt ,,12«(J
   
   """'n (, ).
   
   + /'l ' II) + fl
   
   s (1/ 2jf:, ""
   
   tluou.ch tbe _ of """'pact ~bborboodo! 01' ~ , tbe leU 'It~ ( V) form. filter bMif B on 'It~ . Let Y be an ultrafilter on 'It: liner than B. Alto. fb 10 E 'It: and. ro. alii E 'It: . 9 E 'It: , put When V
   
   fIlliII
   
   1 (/l _ (J : , ). • (Jo: , ) By (f), limFII- I.(J»- I (J) c:iIIts in tbe compact_
   
   [ (Jo I, /l , (J' 101). By (e), "'" hr...., l (f + /') S l (Jl + I (f') . By (, ). "'" hr...., I (f) + I (/, ) .s 1(J+/,) +tI(II1 for ew:ry t > 0, If II is K, : rttT than I on t""aupport ol f + I. It '=>ilow"l tlw 1(J + /') .. 1(J) + /(J'). By Propo.itioo 1.3.3, I can be c:UDded t.o. ~'-' form on 'It(C , R ); t his li,-, form is a P'l'Itl." _aero Radon !77U~ on C, and. by (a), is Idt. iTMUirur& . This is l he deaired left H...,. ~ Applying lbe f",~ng 10 lbe OJ>P(IIsite group. "'" <:led""" lbe c:iotenoo of • ri&M H...,. ..-sure. m
   
   8.
   
   """'rm
   
   U~
   
   lei. " be • loft HI&5' "..uu~ and lei " be a ri&bl H...,. meuure. 'Ibm !I io left in..-.riant. We oDow 1hr.1 "and !I are proportional. Tb.ior will 1"0"", ill fod., dial 1_1dt. H...,.
   
   _
   
   urs are propo:: lion.ol.
   
   Let I E 'It ... be....n IIw ,,( f)" O. By Propoaition 23.1.1, tbe function DJ defined
   
   015
   
   C by
   
   ,.
   
   ;" """'lD"""", Chooee, e H. , n.., fun<;tion (,.f) _ / (, )g(c.) is "",,,illOOUll 011 G >( G. with cnml*'l ... pporl. Ii..,.,.
   
   ,(f'",' - (f I("d" ") (f "" """,) .. /
   1 {, )g(II)<w(I)
   
   ..
   
   / dv(l) / I(.)g(c.)dl'(')
   
   .,
   
   / <w(l) / J(r ' , ), (,) dp(, )
   
   .. / 'C .)(/ J(rl.).tv(j»)
   
   01'1'(. )
   
   .. ,.(g,.C/lD,) , aod 011)" (D,I')', "'11 PI'<"""" IIw D, dotoL DOt de~ 00/. Fot. if J' e H.. II .uch tW ,.(f') 'I- O. 'heb D,I' .. DJ' '' ' and D, .. D,. locally W &l1OOIIl ~ .. ho:n:; 6nt.1ly. D, .. DJ' e>-erywhere. beca .... D f aod DI' are 0'><1,;00008 aod the .U~ ol I'll G, put D f .. D. Formula ( I) gt_ I'(f)D{e) ,. v(J). Ne>:I , If J e 1( + II soch 1M, I'(f) ,. O. lhen J .. 0 ~he<-e, &nd lho! fonnula 1'(f)D(e) .. iI(f) mnai ......Jid. Si""", iI 'I- 0, ..., M ... D(~) >!- O. Tho. 1'10_ that " and (; "'"' ~rtion&l. 0
   
   The"""".
   
   Corolla ry Ever, left ..........n.I ( ......P<'<'/jtod~, "9iLl "'''''....nI) """"'.Na maL' ...... ~~ G ;',..,..,porlioroaI,.., "'" IefI H""T ......""" (...... l)o. to u~ rigA! H....r WLlJGlIU't) .
   
   """'i...
   
   In the al>o:MJ proof, l he uiom of dIOioe ..... UIIed. It ill I"JII8ible . looM:~r . \0 """"'met. Icfl. HMl ~we "" G ..[thout """""1113 to this &JIiom ( _ H. Cartan 'e proOf in C""'J>!c-...MU de /"A",dtmie du Sa c""", rk Pon., 1940). On 11M: addil i.., group R ' , ~ "",... we ~~ o. • H&.u _~ On 11M: multiplicatl"" STOUP Ri. , d~(%)/% is • HMl mtaSure: 1I:c).to.'er. the ho>rnot>mo.(:t)/%into .\. On lbo \OIUI T , the """""'''' " , lhe ilMl" 01 ~/" ,I] tmder % _ ~~, . . . 11&&, _n'. Thflo.., m 23.2 .2 UI G Joe a koctJIv o:.>mpod gro~" and lei,. /Ie G kfl HGlJT _1U't M' a rigAt Ho.ar ..........., .... G. G ;, """Cf'l:k if and onl~ if I'«( e}) > 0,
   
   G ;, com"",,! if .",,1 <>nIr if 1" (0 ) < +00. P ROOP': The
   clearlJo liCe 1"')". fot SLlf!icienc~, let V be. com""'" ~b<::ori>oo O. V io. finite ..... ,.(V) < +00; .. G it 11 ' "',,,,,<11', I~ io dl!ocmc. Now ,uwc-e t l>at,. "(G ) < +00. We ""'1 _u""" that I' is Itll. lIM-riM!.. Coruidcr the c:oIi«tion T of finite 1I\Lt.~ j,,.. ...... } of G IIlICh tlw O;V n kj V _. lOr i #-- j; """ h.o.~
   
   ".,( V) - 1'(., V u ' , . U t. V)
   
   """"woe
   
   =s I'" CG) ,
   
   ••
   
   23 .3 Modulu l'\m«ioII on G
   
   < ,,' (G } " - ,,(V) '
   
   We can, tb.. ef""" , choose" maximal eJ. ....ut {" " " ' '''' in F . For e"o'eT)" • E G , lhere Is .... iII
   23,3 Modular Function on G LeI. I' be .. J.ft Itu.r me""re on G. For e"o'eT)" • e G, 6(. )1' Is _ill J.ft In.ariaat , beaup 1'{1}6{. ) .. 6(. h{1} for aU t e G. So there exiou .. unique number ~(.);> 0 IUd! that 6(. )1' '' .6.a(' )I'. By """"""m 23.2.1, lhe ILUDlbo1- .6.a(. ) doo!o "'" depend on the dw>ice 011'.
   
   Dell.olt lon 1 3.3.1 The funCl"'" ~ : . _ .6.a(. ) 10 coJled the modular fUne> OIl G. If.6.a " 1, ,he poup G io aaid to he unimodulat.
   
   t"'"
   
   Notice that" io rdatl.o:ly in..n..nt under ri&ht t,...",I"Ioo>I, with multiplier .6.a. ThUl.6.a io" COfltiDIJDUI ~ from G Into ~ . In particular, .6.a(~. - II- I ) .. .6.a (~) .. I for all., lEG; lherem. if lhe oubgroup G' 01 G luwutfld lit the mmmuUt.ton Is de .... in G, tl>ra G io LLIIimodular; thil Ihows lhat aqy COfl~ oemisimple Lie J1'OUp ill unhnodulat. If 'I' io an ieo" ... phisrn from G onto " Ioc.aIly """'pod poop G', ~ " 'I' .. .6.a. In partic:ular. ~ <> 'I' - .6.a if 'I' Is an "uton... phisrn 01 G. H~r.....M Ih, we 8e1. 6 .. Slr.oe
   
   t:..::.
   
   6(. )( 6 - '1') .. (6(. )6 - ' ) (A(.h.) .. (6 (. )-'6- ' ) (6 (. ),,) .. 6 - 1I' for aU. € G, 6 - 11'
   
   _ II' Is" ri&ht H..... _ure. BUI
   
   1(')'" - ("1(. )6 - ' )1' .. 6 (. )(6 - 11') _ 6 (.),,' . lienee, lor .-.y ri&ht H..... m ' I mre II, 1(.)11 .. 6 (. )11. Sir.oe (J io" ri&ht H..... m ure, i< _ ...t:.-' I' forllO O. W" ded..,., that I' .. a(A- 'I')' -<>Ai< " her.oe .. .. I , and li...Jly i< .. 6 - 1". Similarly, II .. 6", In Ihort , we "".., the l>Ilowiq ....WI.
   
   ""j<;
   
   Tbeo..,m 13.3 . 1 £,01 G .... fooait, ...."pacl ~. 6 ito III=) .. f / (>=) oofp(:o:).
   
   1.. "'1··..
   
   ,,...ml,,..
   
   J 1(:U) «
   
   r )" ~('J-' f 1(:£) .....(.1:), M.,....,....... , J isa-' p-i"I.~bIe.'>d f I (:z - ' ).c. - '(:rj ds«r) - f 11,,)4,,(,,). Limi8lO, lor Ul)' ... i~ mapping / from C into ... "",j &nad>~. ite left lran!latel and right. lr&ll8lBtes are ... intqnt.bIe. f f (.U )d,,(r) • I/(:z) dv(% ), &nd f !I. r).u..(r) • A(, ) ! / (r ) " (:<). MOftlOV
   and
   
   Propooltlon 23.3.1 1/I/u:It erUl.J in G .. """'J'OCI ndgIo~ V 01 e til,,1 .. in",,,,,nI ""d.... in~ alll""""l'h ....... u.e.. G if .. nimodulor,
   
   PItOQf' : Let I' be .. left. II"",
   
   ole ' pee /)n
   
   ,,(V) _ I'(, - 'V. )
   
   G. For ewry l eG, g
   
   o.(a)I'(V),
   
   .. be,""' I. foIlows.hal; A(. ) _ I.
   
   a
   
   As a ~. if G is O)mrnutali"", or discrete. or comPKt, It is u .... modular. If G is diocmt. the mo eu"" on G lor ...·hlch ...... poi,O\ I... ~ I is plainly ... Haat m E' I" .... nII de> not ~ when G is IInite; in . his """" . ..., shoJI ..h....}... """"if~ the _un'- tt..t ..., <:all "normali:oed II ...... """""u...• :
   
   or..,..,
   
   23.4
   
   n-
   
   Relatively Invariant Measures on a Croup
   
   Let. G be .. Ioe6l1y oornpect group. Prop<>lli.ioo 23 . .1.. 1 LeiI''' 0 I¥ " """"""" "" G, rdolftooe/v i" ...""", .....ta k/ll"'n.Uo~"" ..w. m..ui~ X. [Ix l iJ "conImWtol Ao~..,.", frtym G;"I<> C ' . tMR ~ _ X, I' ;, rrl4!iq ,,,"'.......: .... /kr k/C 1""..l
   -y(.)(x,p) =
   
   h {fh,)("'1(.),.j (X,(. - ' ):n j )[ (' - ' )p (x' X)(,- ')(x ,I')·
   
   a Theorem 13.4.1 Li!1 p /Ie " kit H.... r ."...,"'" on G. A. ."...""" " cI 0 on G ;, re/4J,tId, in"";""l urukr k/C I"'~ if ""d m>Jy " '" ill <>1 ~ form "XI' , ~"E C- aM X ;,,, omtin....... hom""""""'ilmJn>m C in/<) C ' . In /Ail ",""" ito mllltipliu;, X.
   
   P JI()QOI : By P 'opo:ieiti<>n zaA.l. lho <:<>nd ition is suffidenl. On lhe ot ..... band. If ,,;. 0;' ....wJvely in,'llriant under left u A!lSlatioot .....lth multiplier X. then t.:ft ,nYaJianl (P""p..... tion 23.4.1) and he""" ;. of the fo
   X-'" ;.
   
   oomeG E e· .
   
   0
   
   in"""",,,
   
   Ptupollltkm 23 .•. 2 E~ m....."'" on G th4t .. rdoo ..... lr kJ! ~ ;, <W<> """Ii~/y imo..i ..", ~nbr r>g/LJ. tn.noIaI;."...
   
   ~,,
   P!tOOt': In the ootatiotJ of Tboo)rem 23.4. L
   
   o
   
   for all_ E G.
   
   On ~tof PlOJ>OtIition 23.4.1• ..., may ""fer "ruunhi~y to ",Wi""ly InYViant meMtl ..... on G. C,vm .. n:lal.i.-ely ,n,"&riant _ un:" y. 0 (In G, ..., mu!t diltj~il;h It.!; left multi plier X OUK! right multiplier defined W 1('''' - X(. - 'l" and 6(.).. - x'(_j". We Iut...., x' ~ X~, by Propc:Ctlon 23.4.2. In the noU.l;"" of n.o..nrn 23.U." _ (lj:j;. _ "(X", ,,,- I},,; tt.cr... m, " 1& ",lal~y i",wiant, with left multiplier X- ' lI. - ' and .itb rigbI:
   
   ,to
   
   t.
   
   "wi _
   
   rnIIltiplJer x- I . By Tbeorf:m 23.U, nesliglble. IouJly ...-di«ible. "","",ura.bil!, and locally In~bIe functiolll ..... the..., .. ro. all """zero n:1t.1;.-..Il 'nvatiant-.
   
   Propollit ion 23 •• .3 Ll:1 /' 11< " ~ft Hau """"'"' on G. In oni<:T tA.aI. .. ......... rt " ,. 0 (In G /Ie quui ....... ri4'" ~Mt:r "'II tronolali...... u is ....,..,..,., .... oi" ~ IMt if 11< "91'itoGknilt> " . Su~<::r
   
   is QIMous. For r.... 'tl."" let " ,. 0 be a II mn: on G, qo.IMi.ln...nant U!Idc< Ieh. t....".jat \ono., and ohow lhaI; ,, ;. equivalent t(J II. the " - ",here ,,;a pOliti"". We n.eed only B he a <:"Oml*=t 1JUbee1. of G. By Theorml :20. 3.1 , It ~ 10 obow t hat lhe condiliotw P( B ) _ 0 and I'{ B ) .. 0 An: "a\enI. If A;' .. OI)mpact ... b&et of G, the fu,,,:bon (>=. , 1_ 104 (>:)1B(>=, l On G " G 10 ,,® lI+integtable, beca ..... it .. up ..... oemia)rltln"'-""l with ~uppOrt <x>ntained in A" A- ' B. lIe~,
   
   PlIOOf':
   
   ".,oi
   r..c.
   
   / d..{,l /
   
   IA(:r)ls(:trld;4:r) - /
   
   oi"o<{>=}I ~(>:) / I ,,(>:~) oi"v(r)
   
   (I).
   
   F\Qt. _ .... tlut.t v(B) ... O. By bypoothooi:is, v(.,- ' B ) ... 0 for """I")' >: E C . and 10 the rlf;bt<-1wId. lid<: of ( I ) ill 0, t h.u ill, t~n: eUilll a ... QClligibie !lei. N~ IIUCh IMI, for,
   lA (.,)ls{%,) oi"o«.,j _ LI.{,)- I / IA("'- ') I s("') d,, (>:)
   
   (2).
   
   ,
   
   If K 10 a 0>I1l~ 1U\:JieC 01 0 IUd> tIw; o{K ) 'I O. _ may IW for A lhe "'" BK -I . a ie'" pOint ~ e K for .. hldl (21 boIds. Since I A(:r,-' IIB{ZI .. ho!:r) for oJ):r e C , thlo pt''''eltha, p{B ) _ O. A _ ...... ,hat p(B ) _ O. For....."., c:om~..w:- A oIC, the Ieft..Iw>d $idf: 01 ( I) is 0, and he,.."., the fi&ht-4nd side is ahto O. AI a """""'I~ t!J.e«o niWl a local", J...~I~bIe "'" AI wch that f 1.e1:r,Idv(,) _ 0 lor :r M . Since" 'I O. _ conclude ,hat <,(:r-' B) _ 0 lor """",:r E 0 , and finall), that
   
   n..r..
   
   t
   
   ,,(B )_ O.
   
   [)
   
   Wt can appI)' P'Oj)OOIitioo. Z3.4.J I<) CO. S~ 0. - 1P ioI a ~ H&&r OJ( F'rt fur eo«]' left H&&r m' ; t v", 1', rI&IK Haar _ and left H&&r mc.uu ... are f<juivalen1, ''''nce a II' ' ''P'rt on 0 ioI qua>l;'in..n&nt uoo.r r;v.t tr&nlllatkm$ if and onIJ if it 10 q\Wl.lnvui&nt UOOt. lefl tranolationa. In Ihiol ' -, ~ Iha.lI simply ....y Ih.u it ill quaa-lm-ari&nt .
   
   23.5
   
   Homogeneous Spac::es
   
   L« X be a 'OJIQio&Ioo.I~. R .... equivaitnat ~ltt.tiort on X . and " lbe caDOllieal pro)eo;tiolt from X onto lhe qllpOIuu. TMn a... I-ry and wttkitn, <:Ondit lon tha, a 1Ube6 V 01 XI R be open lit ,bIo-, ,,- ' (V ) bot open in X. By dIoIinlliort. ,be rdaliort R ioI opttI ... bot"" ... tbe .. t un.!A! So.t(A) _ ,, - '("A) of ~. 0pefI !$I;\. A lit <>pen. ProJKlO'itloo 23.6 , 1
   
   X" X.
   
   X/ R ..
   
   q
   
   XI R .. H4tU
   ~~.I/u.e~Iii)a. H~WtIO'r/!.
   
   R .. "I'U'
   
   ~
   
   fl"Jpi>
   
   ~nde;,
   
   e <>I R
   
   ..."', ,.. in
   
   do,,., iro X" X . ~n
   
   PROOr. elo Iht in..,.,.., im&«C tha, (V " W I Thcu 1l{V) n,,{w ) ..... ..-b1d> pM.. tha; Xl ll is Halllldonf. [)
   
   of,
   
   nc _ e.
   
   e
   
   A...""", Uta.! X ;, /«all], o»mpNv i/Sa,(K );, dc,, ~ in X /()f" ateI')' Q)D>pad,1Ib.Id K 0/ X . Provo-itlon
   
   23 . ~ . 2
   
   if
   
   PROOI'c S\I~!hat the """"ilion boItIo, and let (a.6) be a point In lhe cloo ""'" nl 0 , let. V (... pcor:tiw/y, W) ht . a:>rnpo.ct ".,.bborhood nl" I". (>flO>,hod), .• nci&hborhood 0( . ) in X. SiDCe e n {V " W) Itt l iiI)I)(tnpt)', W n Sat(V) 10 -.empey. ~ b ib ... Sat{ V). Nei&bb«hood V of o. So
   23.5 1100 ...",. -..
   
   w
   
   S-
   
   eonveroel.y.... ume tlw C .. <~ in X" X . lei. II' be acompod lU*tof X aod Ga point in tho clotw-e olSat{K). Foe......,. nei. bborhood V'" G, tbe ~ v n Sat(l() aod also Sat(V)n K .,." DODemJXy. 61tH F OIl II' ~ ~ tbe...u Sat{V) n K hall .. e1U1;ter poiQt ~ E II' /lJlnc in tho closure of """11' M E '1. If V , W .,." ".;ghborl.. ..t. of G, ~ in X . tben W n Slt.t(V ) <1-'; 'hut C n (V " W ) is .."..,.,rnJXy. Si..,. C ill d . d. (o, ~) la in C • ..Iw!noe it folio... tlw a belonp '" Sat(K ). []
   
   n...
   
   P .... posltlon 23.5.3 A.....me tJuoj X ill /oQUl~ a>mpcI<:j, R ill open,.ruI XI R ill Hautlorff. TMn X I R ill /oc:GI1y """'p,'d o...t, I..,. ""'"l' a>mpscl ...Net K' G/ XI R . th..e ea M. a """'pac! nd II' 01 X ,1Idl1lW 11" _ ,,( K ).
   
   PIIOOF: Tho fiHl .... rtion is obvious. N..,.. . lor every ~ E K ' , Io:t V (sI) be .. """'p"t. ~hood of .. poin' of f - '(,). 110 that r (VCr» .. .. compoct ......bo
   II,,,,,,.
   
   P~ltlon 23.11." Lol f "" .. ¢<m!in..., ... mapping Y ...1<> a ~ 'J>A
   /rom a
   
   IopOl"9ico1",..,. Gil! "'ld' ,/mt;
   
   0 jiU ... "" Y arul , € Zit ..
   (oj If F ..
   
   tJl.r.fi/W "" Y .:uo..t. E Z ill .. Iim~ poiRel 0/ f eU). the ... U a Iimil,..,.ReI , n40 /MI fCr ) _ o.
   
   ()) lf U ill an ~
   
   (e) / ill cIc«J "n4. /..,. ~ . E Z . f - ' (:) ill """'pad (ut n......·rU~ H"~).
   
   PROOF ; To _ that (b) lmplk'fl It), ~ ~ a c~ . I ...' - A of Y and .. point % In the closure oI/(A). When V "'''' t hr<Xllh t he collection 01 all IIdpbocIloodo 01 • • the""'" A n r '{V ) .......... itute .. filter boo$io B. If U it ..., u1traJiltc. fiDeI' than B. t~ ex;", a Urnlt poin' ~ 01 U IIUdI tlw / (, ) _ •. Si""" II h In A . % I;... in f(A ). Tn _ lhat (e) imp/a (a) ..... WIbider .. ftltc. F on Y ,um lhAl f ('1 hAt .. c l _ poIIIt %. Foe _ h M e F , f(lJ) 01 cbed and contalns f(M ). n ... erOOt. % lice In /(TJ ). Now . ... hen M ",no lh"""h F , tho ..... J1 n (zl oonstltutc a filw buia. 5i""", r ' {. ) is <:OlIlpact. . this 6]"" ~ hAt .. cluster pol", _ e r ' (z). n", F a.dml" ~ all clU$l."," point. []
   
   r'
   
   Deft"ltwm 23.5. 1 Under t be conditions 01 PrnpofIilion 23.5 .... be propft .
   
   f
   
   ill laid '"
   
   If I is p~r and K io. c<>mpoI(:l ,ubeet of Z. Il>en I-'(K ) ill """'JIft"t. h'? Fd, .. hen r'(K ) ". I, every u1tr.filu:r on r ' (K ) I..a • limit point. H ~th , ..., Io.t /I l>e • locally (Onlpea group operating COJItiDuoosIy (""" !l>e "«ht OD • k>caIly """,pea _"" X . and ...., ",,"unw: that 11 acto propo!fly on X , that ill, that the mapping /I : (;t, e) .... (;t, .re) from X ~ 11 itl1Q X" X is proper . Thm the i ...... of X" 11 under /l is c~ in X" X . In other words, if we C"OnSieD the graph o f R ill c~. ~ ""IuivaJeocc d ' w , !Or thio ... lIItion R ..... called the orbi\.8. FIx ~ry o pen sui:w:t V of X , St.1( V) " V/I '"' UHH V e II! opeu in X. This me.". lhat R ia an open rdatlon. By P rtIPOI"k>ns '235. 1 and 23.5.3, XI H .. XI R illloc&Uy compect . We shall ~ by .. In. canonical projection from X onl(l XI H. romp.::!. ""'-'- of X I II ill the lm&ge u nder .. oC • """'PftC\ out.et oC X . If K , L ..... t'M) cnmpllet IlUbeo:to of X , tloen P(K, L ) '"' (e E /I , Ke n L 'i- 8) II! compllet in C oi~ , sinoIJ. /I is p"""",", /1- '(K " L ) is <:ompac-l in X " /I I &ad Ito im&&'e P(K , LJ woOer tbe &AUi>d. pm;ection from X " If 001<> 11 ill a.IiIo compo.et. Let X be. COCItinllOOll bomomorphism of 11 inl<> ~_ If .. mapping g f""" X iolQ • real W>OIOr space oatloMo g(z{) .. X(()g(,.) lor &II ,. E X and &II ~ E lf , ito 8UJIp(>rt S II! im"llriant unmPftC\ IUt.et K of X (depending on g). "Then S _ . - ' ( .. (S n KJI, and"" S ill the _urate of .. compect!/d. In .,hat &>lion.,..., fix .. loft Il aar,~ fJ on H.
   
   e
   
   ,.e,
   
   E...,.,.
   
   Theonm 23.5.1
   
   wt I : X _
   
   maIO
   
   01_.,
   
   rAe IAIItl'"!IU
   
   C I>t. G """tin........ jun<;ti<>n, com....1 • ..bOd 01 X in a
   
   ~
   
   <»III""'"
   
   l"I'P"rl
   
   S
   
   I<:t.
   
   (a) FM" ~:>: e X , ~ ~ticn { ~ 1(:«) /)0/0"91 1<> "H( H ,C ).
   
   IH
   
   ~} ~ ftm~ri"'" 1": : % I (:r{)x({j - I d;ll({) i. <»nIin~ ..... , ... nWou .,..1$Uk 5 H , a .... ~;,fi
   (e) Ilg : X _ C ;, any """' in ...... /undi<m ... IU/JIirog g(:>:{ ) - J( (()9(:>:), IA< .. (f9)' - I ', (.n-./ ' (~j .. f f(~()"~W·
   
   (of) PM" ~" e H , (6(,,)1l~ _ X( IJ)~ H (")- ' IX . PROOF: Let:>:o € X . 1Id let V be .. c:om.pIICt nrighborhood of :>:0 in X . The ""' of \hoee {e H sod> that V{ rnectI S is oW!o t ho: IOet of I ....... { E H 1....:1> that V e """'" Sn V H. Hence il ill ""'P"C!, oinoe s n v H ill oompecl. Thill pro>"fJI • Il
   (I g). (:>:)
   
   ..
   
   k
   
   1 (:>:( ), (z{!1t:(W ' dP(()
   
   23.5 Hoo""..
   
   -.
   
   s,-,.,.
   
   .. L
   
   / (:r()g(r )x({)x{e)- I dO(! )
   
   .. /' (r )g(z )
   
   (6(,,)I)' (z ) ..
   
   k/(Z{")X(~)-' d8(n
   
   k/(:relX(~,,- 1
   
   ..
   
   4 H(,,)- '
   
   ..
   
   X(~)4H ('1)- 'I' (:r).
   
   )- ' dRU )
   
   o PropOIIlt loll 23.5.5 Lei K 100 .. """'pa.ct ,111/,." of X aM'" r ewry II e W (X , C) """- npport ;, <»tU4i..... ;11 K H , Utsol(g) Iiu i111(X,C ) aM' (1IU» _ II.
   
   ,,"!,t"') ....
   
   PROO~:
   
   For all z e K , .... ha"" "' (1'» 0. H~".,. Inf •• "'H,,' (r ) _ Inf20EK " '(I') il lt rictly posit;"". The functlod h, eqUlllO ,1.. ' OD KH &lid ..hIc:b ....isbel OP X - KH , lie! in 1t~ (X, C). MOi ..... ~r, (tJa)~ _ ,,' h _ II by pan, (e) of 'Tbe<:>n!m 23.~.1.
   
   TbUl f .... f' If" &lid" then, for each I
   
   [J
   
   mt.~
   
   1t(X , C ) onto 1t'(X , C ), &lid W (X ) onto 1(~ (X, C). to 1( ~ (X) &lid if II e 1t"( X ,C) "'Iis&es 1 , 1< If + If . <;!> 2 • .., define Pi u tbe functiou eqUll to
   
   be"""
   
   Jl(r) g(z). m r ) + mr)
   
   (I: +If )(z ) > O. &lid O~"'bm!. Then II. ~ F I "'~
   
   al puinlol z e X . uch lhal to 1(~ ( X, CJ . 1 1Io1:S &lid
   
   1.'.
   
   11 - ', + 9:1 .
   
   If I ii " li....... 1onn of fini te van.tion over 'W (X ,C ) (Section 1.4). tbon 1" : I .... J(J' 1 10 • linur form o f finite van.tlon """. "I(X , C j (bo:o<.... 1/"1!> II I" lor e\'U)' I e 1i(X , C )). Henoe 1" ill " Radon mFFFu.., OD X . The ""'!>Pi", 1 .... 1', iI an injeoT.lon. The - . . m I 1" «> obc.aloed ..,., be
   
   eh
   (a) 7Mn"";'11
   
   
   IintGr 1_ J of foUU"""""""',,- "I' ( X ,C ) n.cA tIur.t
   
   1(1' I .. ,,(Il/or oUt I
   
   e H(X . C ).
   
   M
   
   ~(()"
   
   .. .l ({) -I AH{( j"
   
   ~ III! ~ E ll .
   
   (e) Fl>r III! / . , m H(X , C ), lif.fg' ) .. I'tr ,). (II) 1[ [ E 11(X , C ) U
   
   ~~
   
   PROOF: If OODdil;c.a (al II H(X .C).
   
   III., ['" .. 0 , IIIf10 ,,(f) '" O.
   
   _if~,
   
   u ,~(eM ..
   
   then, for
   
   e\'tf)" (
   
   E II and emy 1 ia
   
   (~ce-' )/ . ,,)
   
   .. I«~{C I)/) ') .. ..
   
   t (:d W' .o. H«( )/ "' ) X«( )-I .o. H(( )(f , ,,);
   
   6«()" .. X(O- ' .o.H(O". AIoume .hal condition (b) . eatis!kd.lf I. 9 lie In H{X. C). lho: ful>C"tloM (%, ( ) - I (%)f(%{) a nd {%. ( ) - / (%{. - ' lg(%)X{£l.o.H (()-' are conti ........ OIl X .. H and o:ompao:tly wl"\X)rtPe, 1y OIl X ). "Theldi:u , ' " __ .haI.
   
   Ix
   
   L '"' h L .L L .L
   
   <41(%)/(%) k r(%()dIJ(() "
   
   dP«() L l{%)g(%( )<4I{%) IIP(C)
   
   l (tC '), (t)l: (O.o.H((l-' dp(t )
   
   1I,,(%)g(r )
   
   1(%( -' h (O .o. ,,{o- ' d6«( )
   
   1I,,(%)g(%) /., 1(.t{ ).t!W'
   
   .,hldo
   
   ptO¥<:II
   
   ~(!) •
   
   ,loa ,,(jg' ) '" !'(!' ~) .
   
   If condltioll (e) holds and 1 E H( X . C) is ouch ,hoot 1~ .. 0, then I'll g') .. 0 """"'-' E H(X . C). Cboooinc, ... I .....u pp(/ ) (whicb iI ~l* by ~tioll 23.:0.5), ' " """"Iude. thal !,(/)" O. Finally, "'ppoM that OODdi,lort (d) boIdo. Then , btn: ~ a li ... a:r bnn t OIl 1("'(X , C) oooIt d w ,,(II) .. f(IIX ) lor all II E H (X . C ). LK 1 E "K! ( X , C). 1"~ .... irl • ..."..,P""'\ .,,' I :t K of X fUdt tlw / ~O<> X _ KH . I~ tho:
   
   tIw.,' ..
   
   DOUol ioil of Propoeidon Z3':'.~. b I!f'i"I!rJ 9 E 11'(X.C l _isfiInc III!> b.~ I~)I ,,(f ). ~, ( 1/ {g)1 "' 9 E 11·(X . C), 19!
   
   if bouDded.
   
   s
   
   l,,{o6(,l)I:
   
   I, ~
   
   s t} 0
   
   X .. \. "Then / _ / 0" iii • _tOo-e "'*PPnc from H(X / H ,C) onto H '{X . C ). The preDl!di~ .....ulu can boo .... Iormulated at Now
   
   "'....
   
   _ume ,ha,
   
   'I'
   
   ate
   
   Le\ / , x _ C be a a>minUOOJll funcUoo whole IUppcn ~ the oatUrale of ~ oompo.ct subl;ec. of X !.n a oompo.ct ~ . "The formula r(J'j .. iN / (z()d6{{) (whore :i ... :-{ -ll ddinoo a coru.inUOOJll IUDCCion from X! H Into
   
   (I. (, 0 ".))'
   
   C . If I is any a>ml""""" tn_Ion from X! H InloO C, l ben
   
   r,o For ~ 'I
   
   ..
   
   € H, .... b&ve (~('1)/) ' ... t..H( ,,)-l r. If / belon&a 100 1't(X , C j, 1'lies in 1't(X! H , C) . The mal'Plnl: / .... >ie!>ds 1't(X , C) onto 1't(X! H , C ), and II. oendo 1't+(X I onloO 'W(X! H I ·
   
   r
   
   r
   
   Theorem 23 .5.l
   
   fa;
   
   F.,.. ft'e'1' inUI
   
   C ..
   
   >MOl.".., ~ "" XI H , rile/l<1Idi<m4J / A ........ " ' "
   
   ~I
   
   .... X, .n
   6 (Q~I
   
   - ~(r ) from :H(X , C ) .. t..H (()~' for all (E H .
   
   ()) Ccnoerxlr, 1ft I' kG ................. X nodi th4l 6({)1' '' t..N «( 11' /.,.. Ail (E H . TIIen tAere ezi.to a l£nique ........ "'" ~ .... XI H nodi /JW 1' '' ~' .
   
   o ~I
   
   10 re.ol ( ...pectlve!y, l'O'itlve) if and only if ~ Is. ~(r) can be "";"en
   
   ( f( z )d)..'(z) " (
   n,. formula
   
   ~IU)
   
   ..
   
   1(~)
   This """"In is Impropft. lxall'll: iN I( %() of8({j 10 o:onsidered a funccion of :t, and OM a funct.ioa of %.
   
   r_; k t
   
   /.10 "/.. mil~ of f"l'>2i,.. ... a ... "" XI H . ~ lamilJ (~J..., ;, !Immdtd ..... in M ( XI H ,R ) if.nntr if IN. fAmil, (~)"'I ;, k>vnded" .".,.,. in M (X , R ). TIIen $011'>(Al ) .. (aul'> ~ )I.
   
   PJ
   
   ( ~).."
   
   F"or...,., mol............,
   
   ~
   
   on X / H , (~+)I .. P ' )+ an
   (~ - )I
   
   ,. (~I)-.
   
   (e) F.,.. ......... """'pJu ............ ~ '''' X I H , I~I ' -I~ I I· PIIOOF: S"wooe that the [.".;Iy (A;).u 10 boun
   eon"".....ly, suwo- that
   
   (~:). is boun. Sl ..... 6«(1,l.l - t..NW~ lor ~ ( E H , clearly 6((~ .. t..H «( )'" II"""" Il>ereexisu a rMI m" m ... ,.' OIJ. X ! H sucb that " .. 11'. From,.11 :!: Al for .11 I, .... de
   'I'
   
   ate
   
   ",_"Ion (b ) f<>llowt !mmo:dWely_ FInally, If A Ie • oompielC mea8u~ On X I H, tben 1.I.j .. ouP. JU(.,~) . .. here (I ""I
   1.\/' .. supjJU(I.I.)]' .. sup Rq{"~)' 1 .. sup Re{o.l.' ) ... 1.1.'1
   
   •
   
   •
   
   •
   
   o
   
   b,- , -' :'t ion (a).
   
   Defl"it loo 23.5.2 If" Ie a meuuM on X ouch lhat ~ «()P " 6H (!)P for all { e H. thl: m,;q"" " : ,,'re A on XI H .ueb that I' '" A' is eoJltd the qU(Ol.~ of I' br {J and is wriltero 1',1"
   
   23.6
   
   Integration with Respect to ,\:
   
   Let H be • locally compo.cl. group ... hid> fOCU from Ii... r40;ht. coDtin........Jy and ",Ope' b". (lD • locally R\pIId. 'JIOo')f: X . and let {J be • leA. Haat" ',, ", (lD H _ Write fJ for the ",appll!&" {z .O _ (:r . r{) from X " H into X " X_ Flx:r E X_ If A it.,. d IUt.etof H. tben (z ) " A lec~ in X " H . 1"'bo,do.e. (:r),,:rA _ fJ( :r) "A ) lec~ in X " X , &no:!:rA itd d i.D X . 0.. the «he< band . lot all ~ e X. t he IICt H E H , z{ .. 1'1Ie <:<>
   LfM JfJ.u.)" Slno!
   
   l (z{ ) d,6({j - /, (.. )
   
   (1).
   
   r liN in 'H(XI H . C), forll>lllt. (I ) Plo... that l he mappi"'l! " ; .. -
   
   /J..
   
   Ie ~Iy
   flQftl <1<1.
   
   £
   
   XI H
   
   inIQ ,\,j" (X )
   
   Tbeo.,.,,, 23.6 .1
   
   Ld .I. 10< a OI>mpb:t....,.......,.,.,.
   
   XI H.
   
   (a) // 1 it. ).1·...eGI~nlW. nwpplng Jr- X irll.r»Ioulan.l ....ui...~ . ).I·m"""'''u~ ",, IM:n th~ .., 01 rMu: i in X/ II ,1Idt IIw { - /(r{ ) .. "", fj·ltllJAhrWk if A.l>t9li#k.-
   
   X- (0 . +oo[ it A' . .........rWk ....... A' •......un.I~. Ihe
   (6) 1/ f ; i _
   
   j,x.f(:r).q ~' I(:r) j,' -
   
   X/ H
   
   oI\).I(J) (' J{:r{) cIP({). i H
   
   'I'
   
   ate
   
   (~)
   
   If f
   
   .... ~I.~ ""'PI"ng from
   
   x into .. &nIlCh
   
   tpflf:It.
   
   tile .. 1M.
   
   oj tMH :i: E X ! H ,uch !hat { _ / (zO ;, 1I<>l.8.illtq>dle ;, J.. - s"giWt. Alolre" " 1M InIIpping tkfined. A·~t nt'~okre &to r (:i:)- IH 1(:r{ )d.8{{) • .. }t·"'t.ble QIId
   
   .<1
   
   r,
   
   I
   
   j"IH
   
   rdA - ( Id).'
   
   (2).
   
   i"
   
   PROOF: Asoertio"" (al and ( h) follow fn)ttl P~11on 2M.7, and from Pl'OpIl8ilioo 19.U and Theorem 19.4.1. Asoertioo (e) followll from The<:>I'fttI 19.4.2 and the ffod that A can be "';tt<:n
   
   o Pf'O~ldOll
   
   23.8. 1 Fu:.I. , .. tXlmp/t;:E m......", 0" X I H .
   
   ra; uS 9 II< "...t onI,
   
   Q
   
   a>mpl.er VGlutd /l
   if,,, .. ;,
   
   (t) Ul N "" .. -ml 01 X / H. N u t.:.:aIly .I. • ..."iigibk II /oaJIlr .I.' _nqIigthk. (e) Ld II .... ......... n>bI.!
   
   "",ppm, /rom XIH if "nd olllt
   
   "It " ..
   
   into c ~ iI .I.' .~.
   
   if aM. <mi., if ... - ' (1'1) 'PO'" Then" ;,),-
   
   PROOF: """me I"*'I , ,, .. is locally .I.'.inlqrable. For ev.ory I ;n 'H{X . C ), J . (g " :r) iI .),I·jntqrabie; thus (J . (g " .. ))' .. 9 iI A.in~bIe (n..,. 0I'ftII'Z3.6.L ) and fx/ Nf'd).. .. 1,,1-(g 0 ... )".V . Si""" / n>apII 71(X , C) onlO 'H{ XI H, C ). we .... thai 9 .. locally .I.-intqrable and that (g.l.)' .. (g Q Coovenody. jf , illocally ,\·intqrable, thm. by Prop
   r
   
   ,,}.I.'.
   
   ,0.
   
   j,0/ (zll, o .1(z) "I~'I(%)" j,0 ~
   
   r
   
   ~/ H
   
   f( ..llgl(.. ) dl~I("l < +00
   
   (ll.... e'" 23.6.1 ). H"""", g o . illocally ~·.io~bIe. Thil pw .... (&1. If N ill locally ~·!legli&ible. then .-I{N) ill locally ,\I''''-'&liglblo! (P r-opo. >ilion 2 1.2. 1). Cooverllely. If If- ' ( N ) It locally '\·-nqlipble. 'hen ( 1",,\)' _ ( I ~' 0 If)'\' _ 0 &I>d 1",'\ _ 0, 110 N It loetJIy .\.."",liglbIe. It 10 It .\..mcuurabIe. lheo h Olf iI.V·meuurabloe by Propooilion 2 1.2.1 . ~raely, .... ume th&\ 10 o . iI ,\1·meaeun.bIe . and Ioe\ I(' boo! • compooa out.cl. of X / H . Cbooooo: I € "H. (X ) 110 tlw. 1 DO K', &I>d Ioe\ K ., aupp(f). Since outside :r(K ). K ' iI included in :r ( K ). There exiN
   
   r "",,,,,",,
   
   r_
   
   • pN1l1ion of K ~11lI of. A' ·negli&ible let M and • llequtotf: ( K . )..ll oC <:<>onP'd ett1111Ud1 thal. A .. 'II:/IC. Is contlD-. br oJ) n. As 1M rNtliction oC Ao 'll: to K , u·· ·U K. Iu:ollt;n........ tM ....rielion oC A to .. ( K ,U ·· · u K~) it con!.inllOll&- ~ l,,- P tho: tot of li>oee pOinti of K' that do not I)o,Iong to (K.). "Then .. - '( PIn K II inclOOal in M. which _)"I that /1. - ,(1") iI A" Mgligibie. From TI.eoc.", "236.1. _ deduce thal
   
   u.
   o 23.11.2 1.'1.1 A 4114 A' 100 _ CI01IIJIlt;z ......,~ru """ XI H. :n...... ill <>nIer Uwtl A' 100 .. p - . . . .,w, baM A. il ;, nc« .....,. 4114 ~I t1I4I A~ 100 a ....... _.,w, ~A' . Pro~ltlon
   
   PIIIXlr: T"IUt IoIIoor1 ftoat (.) and (b) of Proposition 23.6.1.
   
   0
   
   Propw.ltion 23.6.3 UI A.,. 0 .. 4 """,pia"eo .. e """ XI H . .. ;, A' · p'"",,' 1/ an4 tmIJ if H ;, ......"..,t.. In IA;, CGK...(A' ) . 8( /1 ).\.
   
   ,ha, .\ .. ",*,1_ Si""" " : u -IJ. Is l,.odequatt. for every
   
   Plloor. Suj)pIIeO ,EO 1f·(X / H ) we ha""
   
   r
   
   (go .. ) <1.1 ' -
   
   L,H L
   (go "')("Odl1(O"" /r( H )
   
   f
   
   g(t )tU(t).
   
   He""" 11 iI lo'.proper If.oo on/y If sr(H ) is finitt or. tlqu ivalemly. jf H is CO
   23.7 Reconstitut ion of >'}8 1 From \Qp(>Io()', """' kI>oooo thal. locally cornl"'Ct Ipea! io po.:acom»Kt
   
   if and
   
   .... Iy II it can be partitioned jnt<) .. -comP'd <>po:n __ 23.7.1 Ld X .... "-iJv ......p<><' """",,, and "'" R" An <>pe7I .,..;""""""" rdatw. "" X .1IdI 1Ioa: u.. ""'"',....1 'J>CI« ;, pa ........PGd De· .......... 'II: 1M QI...,.,m ".o;itc:lion fro.,. X <mli:> XI R. The", ~ .. .....ti>o....., ","<:1Wot F "" X ....iJI IMloI.k>lIIiltg ,-n>J><mu.' Pro~ition
   
   ,.....ti...
   
   (.) f' ;,
   
   ....e >d.mIitoU)' u:ro ....
   
   ,.) F.". • ...." ...... poId I~Nct K " - ' (K ) " ~
   
   Gil)'
   
   ~I
   
   
   x l R, 1M ;"ur_tiqn "f_uPP(F) <>n4
   
   PROOF: With eadt U E XI S, .." •. w... te. funcbon I. E ""' (X ) !Iun doe! not Yanish identically.,.. .. - '(u). Then u liM in ,,(n.), ",hen:
   
   n. _ {Z EX: I.(z) :> 0).
   
   """""Ill!:
   
   Sil>Ot .. is open. thOIl1CU1 .. (n. ) form an ~n of X I R. "There iOOsI. an opeII _illl!: (V,) ... " loealIy 6niu, whlch reS".,. the 00'(, ill,(!: (:or(n.))., and • anillOOUl ~Itlon of unily (g,)"".,.. X I H wbonllnalft 10 doe OO"'illJl (V. )...,. For n.cb i E I , we ( hooeo: .. poinl ", E XI R 00 thal Vi C ..(n .. ). The function F, _ fJr< 0 "l/., helo"9 to 1-{"'( X) and Iw IiIppon. II included in ,,-'(V.). no.. ou.pporW 01 tbe F, tlwo Iorm .. \oaIl1y linite Wnily, and f' _ E .... F. II cootinll(lUS. For ~VCfy" E X/ lt. tbcs-e ",,;u,o an lnda i IOtlo thai. JiI; ("j '" 0, and " IM:il in V,. Then then: II .. point :r: E lot .. hIcb o.-(z ) - u ..... f •. (z) :> 0 and 9;( .... ):> 0, we F;(z ) :> 0, and , ev~lly, F (z) '" O. "T'hif poO..,. lhal F ..,isfoc,s (a). finally, let K be a a>mpolC\ .... t-I. 01 X I R. TMre .... ,,.. a finite ou~ J 01 l lUdo lhat, for i E 1 - J , Vi n K Ie empty. Th... ,, -'( K)n oupp(F.). ' Iot..u i E I - J . Sin<.'( U'EI sUI'p(F,) II do"ed, It II "'Iual wluPP( F ). 1'bm
   
   n...
   
   loa,,,
   
   ,,- ' (K ) noupp( F ) - ,,- '(K ) n
   
   (U OUI'P(F. ») ., a
   
   Apln , icl H boo .. locally <:CoIy, on a locaUy comPKt""""" X , and icl fj be a lefl Haar MFPI'reOfl H .
   
   P .... pOIOit ioo 23.1 .2 A.."",. thaI XI H is ,.. ......."'pct. Ld X De. <0lIl111 • ..... ~"",,,,,,<>rpiI;"'" from II R~ .
   
   ,"I"
   
   r.}
   
   :n.~ crim a ...IIIiIl........ p.IldiGII I'
   
   from X ;1\10
   
   10, +ao[ .1!dI
   
   tJw
   
   p(:r:() • x«()p(z) fM" 1111 z E X and 1111 { E H.
   
   (6) 1M l704PPirog, .... ' /1' iI .. Ii..... '~;,.". from "~( X, C) <>Ill<> "H 'CX ,C).
   
   n.eo..",
   
   PROOF: We apply 23. ~.I, toking £0. / • p
   Pl'Op .... X , 1<JLo", lUppQrl m«". tJu: oWurnle 0/ ttIe"1' ..,."pael ",I in a """1"'<'1 "", ",0\ U\nt 1>' .. L
   
   n.oor.m U .51 , with X .. I, t" kiD« for ! .. jIOIIitive fuDctioo th.at 00... DOl. VILDi!h idf:utlc:..Uy on any orbit. For ~ " E X . / ' (z:) II strictly jIOIIitiYfl. N"", I"'t I> .. 11/' . 1'htn, 1>' .. I 'll' .. I, and "" h' .. I. 0
   
   P IIDOF:
   
   W~ apply
   
   P ""pOeitlon 23.7." S_ ", 11>= """III> .... in Pro"",iti<>n !.t.1.'. litt), ... a "";1;...........""' .... XI H .
   
   (a) TIl.: poUr ( ... h) ;, ),I· adapkd OM /x h(z:)<.(.) d),I(:z) .. ),. (b) .- ;. I',oP'"
   
   I'" 1M rn=owo: 1»,1 ,
   
   and ..(Il.V) .. ),.
   
   (c) A ~Q k fr= X/ H ",I<>" mtI Ball4-Ch •....,.,;. )'.inI<'yrnWe" and onI) i/I>{ * ",-) ;. ),I .int j,rihh. P ROOI': Let / E 1f(XI H, C ). Then h · (f 0 ... ) belongl ..
   
   (
   
   1.'1/1
   
   - 1.>,.
   
   t()
   
   H (X . C ) and
   
   d.l.{t)!{i: ) ( h(z{) dP({ )
   
   il<
   
   f (tjd ),(t )
   
   en-em 23.6 .1 ). This prOVaI ell). (b) can be proved simil&z-ly. If I< ;" ),.. integrable, IbeD h· (k,, "l III ),1 .lntegrab\e e ~ 19,4,2). If h · (t 0 "') ia ),1.!1>legrabIe, Theorem 23.U ~ thlll. (h . (t" ... ))' = h't .. t is), •
   
   ,._•• _ -
   
   D
   
   23.8 Quasi-Invariant Measures on Homogeneous Spa<.. Let G be .. k>coIly G I H . On the other hanel , Il ""u f'OP"=,ly, from lhe rWtI, 011 G. &od tbe qu::otlem!pllCe ill none other than GI H.
   
   P ROOF: Lot V be .. oymme1.rie 00Inp&Ct noeW>boThood 01 e In G and Go .. u..?:l V~ tbe subgroup G gM!eU le<.! by V . Sir.:e Go " open in G , ,, - ' (Go') ., a.,,, - '(%) H iI Of"'" In G (fur z E G/ H ). Hence ~ orbit GQz ill open In G/ H . On tbe otoo band , Go' is the union of l he a>mpod. lIeU V~ %. 0
   
   or
   
   can apply 11>0 mnoll8 of 11>0 preoedi"3 ...-:\ioDs. witb X .. G , 10 obtain \.he rn&ppi"C' / ffDrQ 1(G. C) outo 1( G/ H , C ), and). _ ).' from M (G/ H .C ) Uno M (G,C ) (onoo .. lek H.... meMUre fJ on H hal been fixed). The fld. tbat G acu mm.inllOOl!y on G/ I/ leads to _ additlooal reJUll8. w~
   
   r
   
   If / brioncI to 1(G. C ) and • t;a, in G, lhen "lG/ H(.)r .. h o (.)!)'; indeed, fur~r E G,
   
   ("lG/ H(. )r )( ..(r »
   
   ..
   
   r(.-'.. (z»)
   
   .. r( ..(. - ' .. ))
   
   '" k .. khe(.)/)(r{)
   Thcn:fnre, fur every A E M (G/ H .C ) and "'""'1 • E G.
   
   (le / H('» ).)' '" 1e(.)A'
   
   (I ).
   
   PropooJtlon 23.8.2 Ld A f. 0 60! a m_", .... GI H OM I' II k /l II...,. ~ .... G. The loIknmIg """"u;';"" ""' "",iwkflt:
   
   (. J A if 9"I!Ui·in"""nl " Mer G . flo)
   
   The /oaolIW A. "'!1Iigiw" ,1Ib.. t. 01 Gf H "'" IAi>M. ,1IbHt. .4 01 Gf H lor d.it:Io ,, - '(.4) iI /otloil~ " .~.
   
   (e) The mM ' b.e A' iI I!qUiVGknl Ii> " .
   
   tMJ u-e.........,;tioru /ioU, C · rodt tMJ A' .. />1' . T'Mn. for mil. E G , if 8. iI" ~ 01 "1O/ N(' »). utA _ I to A, _ h4'" 8. (,,1') " 11I. - 'r)/l1Iz) /otloil, ".ot/mMt .4_
   
   ew" .....ure. P ROOF: (e) impliflo (b), by P'O!XMilion :23.S.1. If condition (b) hoido, the c_ 01 all locally bnec!~bIe subeeu of GI N is In''' riant under G, and 10 A .. q ...... iovariaot. Now ....""'" that A is qUMi-invariaot. For e,~'1 • E G , ""re(. »).1 and A' ~ equl.-.lent (Pf'O\)OIlt ion :23.6.2 and formul. ( I». TIIOC, dooe A' is equi..!tut 10 " (J>tw<-;tion 23.4.3). By P~tion :23.6.1 , (/I. 0 .. )A' .. (/1• .1.)' ; be"""
   
   (1. 0 .. ).1.' _ (le / H{. )A)' - le(. )A' ...
   
   ,
   
   be(' )' )" _ le(' )' A' ,
   
   o
   
   Theorem 13.S. 1 T..., ........ m> fl'GJ'i-in"' .... nl mtk ..1h rupeet Ie
   
   u.- _'~m if.M <>n4t iJ ".- '(A ) ;, 1ocrIIJ, n
   P ROOr: Apply Prt>pOSitioa 23.'.2.
   
   on G .n.I P .. 1«dI/JI1'mi." tit ......,.In>-'-I/..ad"'" '"' G. ",. /uu W 1_ A' if.....t ,,101. if, ~ tftry { E H , p(:r() .. (~H({l/6c (()p(J:) 1
   C. PItOOP: By'f'he\l«m 23.4.2. W b&o lhe form l' i(.nd only If. Ior ...clt {E H ,
   
   6({)
   aut
   
   6(O(w)" (6({)p*({)/i ) .. 6o (O(6({)p)I',
   
   wbel>o8 the
   
   (.)
   
   ~
   
   o
   
   "",WL
   
   _
   
   cunn....,.,.,
   
   jooncm.v p :> 0
   
   on
   
   G I'adI 1M! p(:rO ..
   
   )1'(.1:' '
   M
   
   "'"'-'U'O.
   
   (e)
   
   oM
   
   '"tor in.........t.
   
   n..... "",,u .. .....n......... t-nt:fil>nx}rom ex (GI ll ) into JO,+ooI.tod> I/olli x(, ... (:r» .. P(~)/p(':r) for all'. :r on C; ",o.toIo<' , "JoIN{')'\' ''
   
   x(r' ,.).1, /IH' """" ' E G P ROOf': Awenio;ln (_)10110<011 frQII. PtOpotillon 23.7.2. A.M,tion (b) ill then ........ seque""ll 01 PropOOtlooI 23.8.3 wd 23.11.1. Ftnally. _rt;oo (e) 1OIw.... from Proposition 2311.2. 0 In oAorl . the... 10 one and only one clatoi ol_>etO q"...p.ioY&tiAnt
   
   onG/ H
   
   ~".u~
   
   23.9 Relatively Invariant Measures on G/ H HeDcefonb, G will always ~ a locally ~ group, If • (10,,00 sub",",p of G, IJ a loti. H..... ~ on H, &lid /J a Ich H..... lI>I>II6u~ 011 G_ Propoe.lt lon 23.9. 1 U! A 1: 0 '" .. m.eII Gi ll oM X" c.",tm""", Mm"""""",imI /rr¥rrI G ml<> C' _ The 1~lDwIing prop'" lia ..,..
   m...,.,..,nt <>II GI H ,.,;/h m\J1ipiW X. io~, m""",,,,' <>II G,.,;tII kft m\JtipIitT X-
   
   (Ill) A;" ...1o.tit>dJ
   
   M A'
   
   (c) A' 11M !he
   
   r_ ..xp
   
   (~1Il
   
   E C' )_
   
   PIIQO": Condition (_) __ ". tbal. 1o/ H(_l A .. :d.)-'A fur e.,-..ry , E G, t llat (T<;/ HWA) ' .. X(.)-' >.' , 0.- that T<;(.)A' .. :((.)- ' A'. H.".., roodil ...... (a) and ( II) "'" equivaleot. By ~m 23.4.1 , rooditioos (II) fLOrem 23.9.1 kt
   
   J(
   
   he
   
   G
   
   """,tin""",
   
   hom(Jmory>AimI /rr¥rrI G into C · .
   
   (.) The ... <:rim, <>II 01If , .. ............, """tiod~ ;"...ri4'" ...... , ~ ... ..,;tl ...\J. h'l>li
   In t1W m .., 1M m _ io unique "" 1<> .. mlll«piioa.tm: """'t4r>t; ........ ~, il;, J'I"I'POrl;"""" t<> (xpll~ '
   
   P IIOOF: In 0«Ier tbat there eri$t "" GI H a """'AlfO re!&ti-my im-ariant meuun: with multiplier J(, a II« NY and anilide"! condition Ie thoot Xp IIaoe the form ).1 (ProplIr>n immediately from Prop
   There are ,;mple enmpJeo in ... hl
   Theorem 23.9.2 In...-dn" Uwt!Mre
   o
   
   For il\lit.arlCe. if"", take for G the JrOOP SOCPI + I, R ) 04 rotati<>D$ 04 R -H , equlpptod ... ith lto.....,.,leaI topology. and lot H t he lub@:roupofG ......... l itqll
   
   ol l ",*" E SO( .. + I. R) "hid> fix "HL .. (0 ....• 0. 1). tho:D H ..., ~ idwIi l'led with SO( ... R)_ The mappins" - u{"" ~ll i6 I homeomorphism ol SO( .. + I. R I/SO(n, R) (lnt.o the Spbett S". ~ VOOp SO{n + I, R) o.cc. OD S~. fro:m I~ left,,, i;)IIowt; ~ . % '" .. (%). Up to. multipl~i"" ocmt&!lI . II>@. I' . . 're cIS" defined In So.:IIon 1].6 (or I ~ __ia!.fd Radoo ~) is tM unIQue _ _ com,*"" II "" oure on S" "'hid> ~_I"" in"";.,. untltt SO(n-+ I, R ). Theorem 23.9.3 Ld C "" .. "-'I, ..,,,,pod ,...,..,.. G' .. cloMl normoJ fbb.. '""'" of G. G" 1M ,.,..,. GIG' . .. : % .... :i the OInonico>l pi uja:1i<>n /rom G ~14 G", .,.., lei o . 0' .IUI 0" "" J4t HOIIlr "'....."""• .,.. G. G'. 0"",, G", !""tIptIdilO!l, .
   
   (oj Up'" Io»rLl' OI>nIIont mtJlipIe of & .
   
   L
   
   I (%)
   for ~
   
   ,.)
   
   ~«)
   
   o~
   
   ., 010' .
   
   T1!.tn
   
   f(7" do"Ct l fa- f C:r(jdo'(O
   
   I e 11(0, Cl·
   
   '" ~(() /O
   "is!.
   
   G" .. m FI ",.., itlVlri&nI UIIdo::r O. namd)' 0". So. if we apply TI>!oOi ( '" 23.9. 1. "' obtai .. (I) and (b). 0 P flO()f'! 'I'bef'fI
   
   (In
   
   23.10 Haar l\'ieasure on SO(n + I , R ) Civet:! .... iJ>~ . .. Sidcred in Se.:tlo.. 12.~. WriU p,.
   
   for the ..... ppillj!
   
   ""
   
   ")
   
   (' ,. _.. . 9. ) _ (1 ( Z - 9, . , - I, ..... Z - 9. fr-om R " "'..... S o. Th ...
   
   ..!we, lor ..... I 5 j 5 n, the (j
   
   + l )at a><mIinale Df p..(' , •.. •," ) is
   
   0111('.) '" ..... (6, H ) t:o::.(9;). If J'('l "' lhe .....1rU. of (Vp,,(6)fl •.... Dp,.(6)e •. ""(' I) I.. \he ",,_"'-I . . of R - ' . Ibm
   
   23.\O H_M ....""'.,.,SO( .. +I. R )
   
   451
   
   Pul A" .. [0. b [ ,,[0 • ~]_ -I . Clearly. p,. is ODe-I().(Ioe fl'ODl [0 . b ( " [0 . ~(_-I onlO S- -{ r E S~ : r , .. 0 .... ~}. Fnr r .. p.(9) (with 9 in [0. 2~( " 10 ... ("-I) and for 2 I)fth pole of Si ) and ( I ~.,)- , "(r' •...• r / .. ,)' Likewise, tbe roI.&Iion of R~ wilh an&Jc - 9 , ""'1"1 the north pole (O, I) n£ S ' to (I - x1 - ... - r~.,)-'''(rl.:t~). FUrtbermorl:, dS" iII tbe Una&e me.MU", of lin"-1(9. ) oin"- '(9._. )· .. lin(fl., )(d8, . .. d6'. )/ A. undo:r p., /A•. For aU I :S k :S n and for all a E R . defi"" a rotation 'l(a) of R "'" ..
   
   xI+> _." -
   
   "'''"'' (b) 1be .... 'ri"' ion 0/ 9 to tru. plane (oriented "" that (el,el .. ,) is direet) is tbe rouUon with an&Jc - a . (e) Tho: rmriclion of 'l (a) to tbe on.""«nnaIlpo.o! (Ik, + Ro>H ')J. is tbe idmtity. We identify A .. n l~ '~_ A • .. i,h n
   ber")o",or'l. "'''''"'
   
   ,<" .. g,(r,),.,(r,) ... 9'(r,)
   
   (I).
   
   Propo.dl lon 23, 10 . 1 q: w _ q(w) ""'PI' " onto 50(71 + I, R )_ PftOQF' Aosu"", ,lIat 71 2: 2 and thai. ,"" p~lion is lrue for 50(n, R ). (;i ..... , E SO(n + I, R ), lei. (r" ... ,II;) be an elemem of "'. ouch Iliac. g(~+I ) .. ".(r, .... ,II:). By illduction on 0 j 71 - I, .... ..., Iliac. (g,,_,(I1:_ J) 0 ••• 0 g,,(tt;:)j (eo .. ') is ~u.oJ to
   
   :s
   
   sin(~ ) ...• in(tt::_; )c~_ J
   
   +
   
   L
   
   :s
   
   sin(tr.: ) , .. lin(9i'., ) C08{9i}o!H l'
   
   o- ,s ' ~_
   
   0,
   
   0, .. ,..--')0' .. og{l) .
   
   Tbeat£ot", W"']- I fix... e~+I' By bypotheoio. [g<-'[-' .-ben!....:h ,..oJ ill .. in (I), and "" 9" 9' · ) 0 ,.. _ _ 1) 0 ' , ' ,II).
   
   a
   
   P~ltlon 23. 10 .2 Lt. '" .. ('!'hSJ:S ':S~ ioc "" olcmcnI 01 A stodI !hot OJ ..""'" I<> )0, .. [ j 2: 2. 171m, .. 91 ... ) ;. 1M ;""'1" u..do'r q oj • .."ipe.1emet>t 0/ A. MD,eo"" .
   
   u'w:".....,..
   
   [,..-' 0... 0,(H'f I(91'0. I») .. (p,(r, .... ,r,). 0..... 0) 1M 10111
   
   :S k:S n.
   
   yr
   
   ate
   
   PI\OOt': Lee. -.I .. ("jJ , SlS os~ I:IfI an tleml:nt of A ,<><:h tbat q(,J) _ ,. Given I :5 I< :5 .., supp<Me it hal t-n __ bliahed th'" ~ .. 9'j fe, all k < I < r\ and .n l :5 i :5 1. Then
   
   ['~I <> ...<> gl""'ll n' (g{,,,,,j)
   
   ,., [gI.I., ... .,gI'J](e.. ,) .. , tol (e.. ,)
   
   .. ("" (r, ... -.'n.o... .o)Similarly,
   
   I ,., ,
   
   0"
   
   " '''1-'(
   
   g{~",, )
   
   ' 0 ,- -
   
   I _ ("" (_.
   
   V , . "
   
   Si""" gl0 .. ,ttl 10< all I.: < I :5 n, _ _ lbat "" ("~ ""' 9': ) ,., P.(r" ... , ~ ). But, II bdonp \(1 10 ....[ 10, 2 :5 j :5 1<. the first two
   
   '1
   
   ~ ... of her" ... ,r. ) do not """ith limultanenly; thus tr; ~ to 10 ,"[ 10< 2 :5 j :5 1.:. SiD(lt Pl .. _ _ OIl (0 , b ( )( [0, .. (0_ 1, _ ax>cludo tbal ctrj),S)s. .. (tt),S)s. ' So tho prOp(Ieitlon c:a.n he pc., ..,<1 iad"""lvely. 0 DoIInltlon 13. 10. 1 Under the hypotbNio of Pr-o:>JQltion 23-.10.2, tM "" .... ben t; .'" c&lled the Eu'" anp. al tho I'OUlioc,. Pro_
   
   itlon 23. 10.:1
   
   vq;... ,,-,
   
   Sf Ih~
   
   ®
   
   "-t,-
   
   ............
   
   am-'n'(IJI("'1,,,;)
   
   1 (.j")I I SIS ·:$~ 1
   
   "" .4 , ~
   
   c..., ..
   
   n'l.:> ~'" (r (I u.., ..... ".~Iized HtIV
   
   '"_ ' " ,,,. SO( .. + ), R "' ... q{... j.
   
   II
   
   1M. 'PINg<
   
   PROOf' A..."", tbat .. ~ 2 and
   
   ," Mlk,..
   
   0/1'00. 1 ~n.4tr lAo ....,.,m.g
   
   u... I'."'-;tion is ' ''''' for SO{n , R ). ~
   
   br K tbflsub&rouPoI SO{n+ I , R ) ~i",of t"'*' _ionl tbat lU ~&+I'
   
   Write II for thIIlDQfm.li·ed H..... me_ "'" K and .I. b. thIIl unilonn m " 8"'" on 9 ft w;lh m.u8 I . that it. .I. ..
   
   r(c.. + 1}/ 2) dS~ 2.... · ·. , )/·
   
   .
   
   For NCb oonlln...,.. axnp!a·va!ood function / on SO{n+ I, R ), define lho mow",
   
   r OIl Sft br
   
   for all g E 50( .. + I . R }
   
   ,
   
   ••
   
   T,."
   
   J .. J
   
   r (p.(". .... ,'.:»
   
   ..
   
   1{g(O){ j d,!l(el j (gl"), ("-I)
   
   "-',. ,
   
   !or al l (8;' , ..• r;:) In A._ Now US" ill t be; .., n
   
   Ullder p., .
   
   Jr"
   
   --,(1) 4<" ( r.l>:SJS~S.- I)
   
   imB«e ""'MU~ of
   
   ... .
   
   T~fun::,
   
   )( I J(I'
   
   " p., )(/l'j ... . . ~) &;""- '('=1' , ,oin(9;') (.w," . . . .w,:),A.
   
   / / C, (olgl° - l) , . , ,
   r
   
   Sinoo '\" : / - f J,\, !OI tbe .... mll.li..,.;l Maar ......"'" on SO(n (P,oposIlion 23.'}.I). u.., p«>oC ill ~
   
   +
   
   I. R ) 0
   
   Sy P ,O!>O'i.ion 23.111.3, tbe Euler ~ of g are deli""; lOr alm(llll &Il, in
   
   SO(,, + I. R).
   
   Noor, b SO(3, R ), 1I'fI c:i~ tbe. ~ric moMinl of Euler aIIlIB.
   
   LeI. t be .. unitary W!CWor in R', and let lbe plane Ii ortboJsooal to i: be ~ with tbe orie<>wion for .. hi<;b (e, . t " k) .. dif'ClCl. if Ct,. t . ) is .. di""", orthonormal ~ olll. For ~ reo! numb<:r 9, Ibm!" .. unique "... .. Ion , E 50(3, R ) 6xi,,« t, ... hooe rooAMion 10 H ill I"" rotalioo> 01' ... itb ~ I. II .. o:aI~.be roution around II: with ~, and it wrill"" 1'(t,' ). Let g E $0(3, R ) be luch tI'3' g(o. ) ;. •• and g{Ol ) .,. - .... Denou I»' ~ the unique _ _ ortt.oconaJ to " and 91 ... ) """" thai. 0"1 _ I and (" " g( ... ). II) ill d["",,_ Now ,bon: am 0 S '" < 2~ . 0 S '" < 2.. , &nd 0 < 8 < >t, 0I>Ch Ihu rC .......l "...,..., to a, r (II .' ) """"" " to , (t.), and r (g{t)),t/l) Itndo II \0 f(~, l- SiD<.'~, \0 g(c, ) and~. \0 f(~.l. II 10 "'loW to ,. From 1M ",1&1"'"
   
   r{r(e,), ,,) - (r(4, el 0 >ie. , 'I'll 0 r (e" ,,) 0 [r (a, el c r(4 , <;,)1 - 1
   
   g _ r (c._
   eo. 7 l .... ntl,r. 9'f -
   
   - 'P,
   
   r«(" 9) 0 r«(" "I.
   
   Ii - -e. and 91- - " ..,.., tbe E uler o.n&I<e of g_
   
   Haar Measure on SH (n, R )t
   
   23. 11
   
   Le\ E be. ruI ve:torop""oflinitedilneDSion 11+1 (... ","", n ~ I ) and ~ be • ~e ,ymmttric bili,-," brm <)I> E" E. w,",*, signature Lo (n . I). Write ., 1<>< the quadratic f«m E iI :< _ . (:<. :<). et- &II ~ bo.eio (.. , .. . .. .... ..., ) of E 8UI:h tht.t q(oJ} _ I for IS j ~ n and q( ......, ) • - I. Write At for.he matrix of ot in this '-II...
   
   ....
   
   M_(~ _~\ )_Art.
   
   Flr>&!ly. let, be &II ",,\oo,"yhiom of E and g' ita edjoinl with ~ 10 .... The matrix U _ (a;Jl. S.... Sa .. ' of gin lhe ha!!;, (.. , . ... , .... ...,) can be written ..... matriI of malr\coe$:
   
   D)
   
   u - (C'
   
   B
   
   .. here A . {... .j) . ~;.jS . and B = (a •• ' .... ' ). Then lhe matrix of g- in tboo """" ("" - -.....+. ) io
   
   U· .,M- '- ·U . M.,M(1(..,I,g(G/l))
   
   (q(..
   
   --;;)
   
   019 " g) in (0, •.. _...... ,1 and U· ·U;" the
   
   '9.1"'·'"
   
   ..... "'Iuivaient:
   
   ~~D
   
   . Tl-.... er......
   
   the foIlowin« condition.!
   
   (a) g III a unilal)" 01>"'-'- witb .... IW'IIO . ,
   
   (bl (g{", I, ·.· . g{ ..... ,I) io an orti"'W"'1&ll:uis of E /(Ir I :S j:S: n and .(r< ...... ,I) __ I. lei U- . U _ J•• ,. (d)
   
   U · U - _ /~
   
   ...... l hat
   
   .,(g(,,;) _
   
   I
   
   .. ,.
   
   i:laI&nalc ~' 0(. ) 1M ullitary liOUP relati.-e 10 ... . r ...- every , lo 0("'), det('U · At - UI _ det(M ), and .. det{U ) io I Or - I. Si""",
   
   a,•..... , _ ha....,
   
   10. .. ,.-... 1 ~
   
   From tM
   
   + .. , + a ....... ' - a ,... , .• ..., - - •.
   
   l.
   
   reiatioCl U· . U _ I ... , . "'" ded""" that
   
   det(U )· 'U·
   
   A
   
   -D) B
   
   _
   
   
   -
   
   dct(U) · (q(G;)q(";)oiJ)
   
   -C
   
   '51.15·· '
   
   2311 H...-Me.ooureoa5H(n. R)
   
   .... ,
   
   If the maaru: of oofOoCl.On of U. He""" del(U), o ~., If the oofodOr of 0" .. ' ..... ' . and det (A) . 0".' .... ' b&6 the S3trIe sign fIJI det (U). Let h be the mappi"" _ (l&l>(detA ). I!«D(o"+l .• +l ») from 0(. ) onto (- I , I) " {- I . I ). If, and ant t .... uniwy on E, with matne. U and U'. ' ....... tively. then
   
   """raton
   
   t
   
   If thIe ooelllcient of U · U' oituatf)([ in the (n ODIumn. N_ . from the l""'lualitia
   
   I&l> (
   
   L
   
   0". , .. o~ ,,,. ,)
   
   + I)st
   
   row and the (n
   
   + I J-t
   
   .. "«'1(0 •• , .•• ,) l&R(o~. ,.... ,1·
   
   'S ' SU I 'Ibenfo .... h(w l " h(g)h(g' l. and h is .. sroup homomorphism. Next . let (a'" ... •0:. .. ,) he an ",her Otthoco><W buill of E .udt that q(aj ) '" I for I !S j !S n and q(a:' .. , ) .. - I. and wtJds to for all 1 '$ j '$ n + I . Thoen. lor II:Vt'f)' , E 0(. ). the matrill of , in the buill (
   a.
   
   From _
   
   ai
   
   011 . _ take for " the bilir.ear form
   
   (r, ~) -
   
   L
   
   r; ~; - r " .,I/oo. ,
   
   tS, S"
   
   on R " · ' " R n '. The ke ......1 of h io called the IfOOP of hyperbolic rotationo
   
   of R··' and iI .mil.... 5 H(n + I . R ). Fe..- ';mplicity. _ aIoo ... rlIe G u,-, .. d of SHIn + I, R ). If , E G ~ In the stabilizer K of e. . , and U '"' (o;Jk, is the .... ,rill 01 , in the o:a:>Il.""" 0 • • ' .. .. , .. I and 00 ,,,. ' .. 0" .. , ,. .. 0 for alii '$ k < n . Therefore. tJ>. topolopeal FOUP K If canonically ;,ou.. ph;" to 50(... R ). Since the mappi"ll: (r' •... . z . ) _ ( I + r~ + ... + r! )'" ill infinitely differ.... tlable ill R ", IU pph
   
   H,, " ( Z E ft "· ' : q(r) _ - land .. ". , ~
   
   II
   
   4511
   
   23. H... "
   
   " •
   
   .. a 00CI....:ud oubma.nifold of R ··'. and 0; , " _
   
   ("' " .••
   
   :r.1
   
   ~
   
   &
   
   chart 01
   
   H •. E """, 9 E
   
   0(. 1map" H. 0010 H. I:rE R·" : q(" 1 -
   
   or
   
   - I and r.H":; -I} .
   
   Define 0 +(. ) M 1M IfOOP oftboee 9 E 0 (. ) IIw map H~ 1/ E 0"( . ) liel in C If and ooly if dec.(g) '" 1. For evo:ry r E H •• equip the I INIOI' (~)" -
   
   bE R· ... '
   
   (WO
   
   H •. CIurIy.
   
   ,to{~,'I "O}
   
   (d.o;I -' (e,) •...• (d.o;)-I (e~ » iI diretl (d .... .. lhe mapp/Ilfl f,..,.,. (Rz)O onto R· ,..botoe Im-e~ if 1)Iy-'C.,...,)). TIle matri>: of ((d."')-' (e, ) •... ,(4.",)- ' (eo). :.-) with ..... p«t to (e" • ... ~~.,)
   
   '.
   
   ""C71
   
   '.
   
   - "..' . . , -..!.. ...
   
   "·• •
   
   '
   
   .
   
   ·
   
   '.
   
   ~.,
   
   .. 1X'*1Ive. Tbuefote, If (" "'" ...... ) ill It.IlJ" dlretl '&Sia of ( R.:.-)' . tl>erL ("" .... .... , :r) .. diretl br 1M \ISIlal : oll/ In tbe IlI$I ( .. , . ... , .... , :!'). Then AB ill lhe puotri~ of (g(",I, . . . , pf .... ).g(:r) wilb tftIf"lCI. 10 it" ... ,'H ' )' Thereflm. (pf",), .... g( .... ), g(:r)) II diretl in R ' +' If and only if det(A B ) _ dd(A) det{g)" pooili .... , IhaL II., If~) _ I. In tMn. 1/ ia OOCIlpUibie wilh lila oriet::ou.llooiO of (Rx)· and (R9(" I) 1f and only if it 1100 in G.
   
   .u % , r In Ha , If (a" ... ,aa) and {b" ... , 11,. ) _ di...u on""· ow t..ee. 01 {azy and (Rr)", ~vel,., lhen lhe eodomorpbism 01 R a +l that f'bo"
   
   ,,""-:r to r aDd eMil <11 to., I:oo:Ioop to G. Henao G.as t raMiti...,],. on H~ .
   
   "l"bt maJ>Pinl .. , j . - At"a ..,) from GI K onto Ha 10 _ 110001II. OIl the «bot Iw>d, if, lor NCb z E Ha . .... de/lue .{:r) U lhe cndomorphism of R~'"
   
   lhat roends Ca .. , to :r and cl 10 ( I -
   
   "U"!",)-'12(d.,
   then $::r _ .(., );. QORtlnuouoI fmm Ha into G &I>d. % (~.,)) _ .,. n.,.ebe, % 10. h..." ..... uoophbm. All K and GI K _ ""","",1M , I~ 1",,,o!apc&IlfOUP G 10 con~ (J . DieudonDt. Thootix on A ~alpU, 12.10.1 2), and flO G IIt~ torIllCded (:OmIO" " I of 0(. ) SIruct. pOOoiti...,,, :" "", on H~, Im-arian1 uodtt G. FIx thla, .., DeO:d t~ i:>Il<wiDc ...,u1t.
   
   Lem ma <13.11.1 Lei A /roe ........ ~ I"inf, AM "', M{X " . ... Xa) /roe 1M !INl:rW fa +(X, Xj) ' 1 jJ5" ...tiI .... tne. m1M riIog AIX, •...• Xa1 Q!; '1"';.&. n",.
   
   dH M(X , ..... X~) _ 1 + X~+ ... +X!.
   
   x, X,
   
   X•
   
   X,
   
   ...
   
   X. _ ,
   
   X a _, 1
   
   Subt. act.in& from the ~m. row 01 M~(X,. '" Xa_ rl I~ product of X , and t~ \Nt """, .., _ that
   
   oll n , .., obWD I~ matrix b
   
   .mc
   
   I + X~
   
   .. .
   
   X , X,
   
   ·• 1 + Xl.. ,
   
   X. X ,
   
   X. X ,
   
   ·•
   
   •
   
   1+ X!_.
   
   ...
   
   X.XJ _1
   
   X . X ._.
   
   X. X j
   
   XJX. df:lM~ _,(X •. "_.X, _., X j ., .. _.. X. _.) = Xj X._ the ooIo.ctoc ol Xj X~ in tbe matri~ M(X , ..... X.) iI - X;X._ Expo.ndin( deUf{X ." " . X.) aloG.!: the last column of M {X , .. ... X.). _ _ ,ha, The,~lore .
   
   delM{ X, ,, __ . X.)_ (i + X!)dc!.M{ X ,, ... , X~_,J-
   
   L
   
   XJX! .
   
   ,S.S"- ·
   
   T.kIna; b- A the IOeId C ol CIl
   - A(XJX. ),s.uS.) ~ dH (I. + (i.,rU, i.,rU ' )'$i"5:0) - i - ,,( Xr+" ' +x!l
   
   lOr all ,1, :> O. In pwticular.
   
   ~ (I. - I + ~1 +1 ... + ~~ (:r,z,)'s....s.) - (I + :r1 + ... + :r!)-' for all rtJaI DWllben :r" .. .. :r • . For all :r e I/o . • rilOl m{:r) lOr- ,"" ,.,.., ri<:tlc>n ol 41 to (Hz)" )( (IU:, • . 80 tl:J.u z _ mCz) .. & ~mannian m<:tri<: on R. " ~
   
   m("",,) ,., m{,,) <> ({d.~r ' )( (d• ..,)-')
   
   OIl R")( R" (... bote (,) if ,"" ""uaI. ....,.. prod..o:o. of R O ). The ..... trix of m{
   ,
   
   ...
   
   I . - r.(",:rI )"S'''So-
   
   +,,1
   
   +
   
   HeDCI! lhe diocrimin&lll of m(tpz) II ( I + . -- z~ )- '" f'b f e G and deaote "'" ,. I "" mawilll:" _ g(:r) (""Dl 1/. onto Ho . fOr all :r e H•. _ havc ",(, )<> (f/(Ro .. " ' /(Rot' ) ,. me:r) ("''''''''' , • g(:r»). &nd ~
   
   m{n)<> !O(fl·.., - ·)C......
   
   »( D(w'..,-,)(tpz)]- m{"n).
   
   ",brnoI lt foIlooo1I tboo. 1M diecriminaN d~( m(~:r)) of ;;;(11"<) is "'Iuai to
   
   dis(m{'PlI»
   
   -I
   ate
   
   la Mon,
   
   -
   
   , IdttD(W\O-' )(1"%)I" -' ,
   
   ~.,
   
   :r ~ H
   
   (I +:rt + ... + :r!j- Ind;o, . . ·dr. on R " iIIlnvario.m under ..,1",-' . Therefore. dH" , lho i..,. T (J + :r1 + ... +:r!)- '''''d;o, . . . d;o. under ..,- ' , is invari.lnt under ,..
   
   EF
   
   " , , , ,
   
   Ql
   
   Deft n it lon U .Il. ! dll. ill called tho Riemannian .ulunoe <.>l H•.
   
   We n:t1Oln tho """,bon of Section 23.10. RIO"..a. !n~' n 2: 2. Itt f~ ho tho mappOn« from R~ onto H~ thai. a.endo e->, (I, ... ..9. ) to ($Inhll. )p,,_I(I , •.... I~ _ I).COIh(I. ). AIIG. let I, be tho "metloa' .... (sinh'.""""" ) from R 0010 H,. For ~ n 'a: I and. "'" every R ' , write r(9) be lbe matrb: of (DM'~' . · . · . DI.('~•. I.(f » iD. tboo ~k:.1 I..- of R~·" . If .. 2: 2. write KO (, ) lor t be n "n ..... trix wbooe Ii... row "' obWlII!d Il)' multJplyllIfI the lirst row 01. r (f:. , ... , 9~ ) by OW 9, . and. "''''''''' ith row, b &1.1 2 S j S n . ill "",. 0( r (f:. .. ... '~). Similarly. ...nte L"(I) !Or the n" n malrix whole lim row isobWlII!d by multiplylD,g the lim. row Ql J" (... .... B.) by $Ia9" and. .. boee ith row. for all 2 .s; < n, I..,qual to the ill> row 01 r (... . . .. B,,). Now .
   'Ii.
   
   ainb('. ) ain('._,)
   
   sln(~)coe(fJ} del
   
   KO(f ) + ainhll. ) ain(B~ _ I) .. ·$la(6, ) $Ia{', ) dol L "If ),
   
   B)' loollC\ .... on n .
   
   de!. r(B) _ ainh" - I (B.) Iin· - ·{B~ _ I ) ... sin(9,).
   
   e
   
   RIO" ~ n 2: I and. for ~ E R", let J (".. I.) (B) be the Jacobio.n matrix of", o!. u,. For all I S. j .s n+l and l .s j.s n+l. -Itt ..,,, 1M:tb.t enl". of r (,} "!...aled In the IIh row and the jib column. Thoen, for n 2: 2. tbe Olfaclor 0( 0 • • ,.... I .. OOIh{f. ) In rc,) io de!. (J (",. I. M ' l), wbe.- the I'dertor 01. "~+I.. _ eiah(9~ lie - del (sinh('.)J' (9" . . .• ' ._ 1 Dovelopin« dHJ" (') aJoua; the 1M. row 01. r (, ), "" ~ lben:b-t. thai.
   
   ».
   
   deere' ) _
   
   -.nh·"' (' .) lIin·- '(I. _, )· ··ein(9, )
   
   + <:Oi.bll.)dH (J(",. I. )(B)
   
   .., ddc (J("., " /.)(')) ..
   
   ~('~) ( t + slnh'('.») oinh"-l (6. ) ,ioO - ' (' __ I) .. -lIin(9, ). F inally,
   F'n::Kn thill IMt ....uaIity k>l1ows, for ;nlltaoce, that f. ill • dilfO'!C)l'Jl()lp/li8m from P - jO •2"1• ]0 , "co-' .. )O. +<>or Q DI.(> H . - (:z E H . : lo , .. I) &Dd
   
   r. ~ OJ .
   
   n... Invert!! dilfw" .... pltllm .. a chan. of H. t b&l. pi
   
   ''''' 1M orientation of H •. More Important,!be 1'«" __ calo:ulusof dtt (J{

   
   linh"- '('.) ';0"- '(' . _1) ... ain(9, )(d9 , ' .. dB.),,, UDder f •. F,..- all n ~ 2 &Dd k>r t.l1 .. E R, defi ... h., (a ) 118 tho! byperbol;" rotation 01 R - '" whole mau;" with ....P"CC lO (~" ... ,"... ,) ill
   
   '... 0 0 ) o ( o
   
   58 B ..
   
   ODIh{a ) oInb( ..) $loh(,,) (:(JIIh(a )
   
   .
   
   n({.... "'SlS. s . ) B: , ..here at .. (0,2101. 8~ .. fII .") loT 2 ~ j:S
   
   lnf(~ . n - I ). and B: .. ,.10 ) 0 ••• ., hll) , ... be",
   
   ]0. +0<:>[. roo- tech '" .. (e: )'SJSO£" in B , put r (w) _
   
   hIO) - QI(r, j···g.Cr.J
   
   forI S k Sn-i
   
   g._,
   
   ,,(. ) - 91(r,) , .. (~_ , )h,, (9;; ) A'IUi"l n ac<.ty .. in Sodion 23.10, we _ that r : OJ _ r (... ) _ B 001<1 SH(" + I . R }. ,,",010,_. if ... .. ('1 hSJS:'S: ~ io an ~lemeut of 8....::b tlw EJO , +[ .tId ~ eJO. ""' few 1 5 j ~ inf(.!:, n - I). t hen 9 .. r(... ) is tIM: Irnap I1IIoIift r of . unM!"" ~ltaoent of 8_ flnaUy. few .. ~ 2, doll"" "~.,, as l he m F7'Ul'e
   
   s:
   
   I Jinh~ -l cs: ) sln~- 2 (s:_ .) . . . sI11(~)(tte;' ... <w.:)/ II. J ~ ,,~ "" 8, _IM:n: 8. .. 1<1. h[ " (0. ""1"·' )( [0 . +0:>[ and "" 11M been defined In So:c\.ion 23.] 0. By P,..,....tion 23.t.l , IIM:;~ IrF""~ol "HI ut>(\l:or r (w) io. Irft Maar - . " " on SH In + I . R ). We shaU show In tIM: nal chapter that (C, K ) ... ~f.oo pai' , ",1M:""" II foIkM.. thai SHIn + I , R );' UJlimoduLar_
   
   OJ _
   
   'I'
   
   ate
   
   ErercUu lur Chapter fJ 1
   
   Let a be " locally ,be Iof'I wtilorm otrO>Ot"", of G (&no, ... w--tloa!. A 10 ooml*").
   
   2
   
   be " lDcaIIy """'~ """p. I' "- WI H...- "coo.", on a, A aDd 8 ,_ ouboo
   a_
   
   Let
   
   a
   
   a, ""'"
   
   I. S u _ tJw
   
   a A 10 1'.;n\.OVohOo_ 'Then. b- &II o. I E a,
   
   I,,"(oA n 8 ) -
   
   I'"(IA ('\ 8 )1
   
   So 1'" (CoA <"I 8 )6(1,4 ... 8 ))
   
   o..d""" from du. iDoq..,w'y and ~ <. . I .haI. with .... ;oet to .ho loft u.oilorm .. ".,,,,~ of G.
   
   s. I'" (...t6IA).
   
   I io w>ilotmly .,.....In,,,,,,,
   
   2. Suw:-.Ioa!. 1'" (11) 10 linite and 8 10 1"_~,abIo. Let ( A.. },,2' boo. 1 , Ii", ooq . _ of I' . in~ _ oonl-'nlq A."""" IIoa!. I'"(A) .. itof I'(A,, ). Gi""B C ;> 0, ...... e N be oudo tJw I'I A~) 1"(A )+c/3. S...... ,10M (I So l'(oA~ <"I B ) - ,,"(o A n 8)! c/3. n.... Iht lei'! unilomt otruc:\u,," of a. l. Su.wooo.1oa!. II 10 and 1'" (8 ) 10 hi ... o..d""" from part 2 ,hao 110 o.nIlormJ,. _in..,.. .-I\b ~ I<> \.boo rl&ht uuifcmI otnoel""" Od.e,- (0•• ) _ tOl' .• - ', and ,1I.o.1
   
   s.
   
   "._utabIo
   
   =.
   
   J" ~
   
   1(,)11'1'(' )"
   
   s.._ ,hao A - '
   
   J" J" dl'to'
   
   b (_ )I A- · (I"' )41'(1) .. p(A - 'J...c8).
   
   10 I'-i~ _
   
   I" (B) io ftn;..,. Let (8~},,~ , ... "" , dnS of """"nlq U ""'" U>&11'" CBl iaf~~ , 1'18~) . S...... ,Iw .."(o A ('\ 8 ) _ inl I'/.A ('\ 8,, ). Thou, I""", _ 3 and 4.
   ""'l'I''''''' ,...,"""'"&1>10 .... f
   
   e..
   
   r
   
   ,,' (A- ' l _ +00).
   
   7.
   
   ""'beI'
   
   Su_.1oa!. A 10 1'"...... uraI;oIo, and .hal A "" B 10 k><.!I;;r I" slioJibio. 0.0.. &.u* C of 8 _that(l < ,,'(0) < +00_ and e.. u.. fuO>Otioa, : . _ ,,"{.AnO) io ....,un..,.., (II> G and ",4 " 1"(A - ' lI' "{O). Moc80 .... C A -' oJ_ Oed ..... Ihlt 8A -, _ "-II. I.nIft>o< poi"" IIo ,hao SA and AB -t. 100.", ...
   
   67"'"
   
   co..
   
   Ia\.orioo p
   ,
   
   Write T lor "'" ..... lr.q,I ..... "'" _of~ o.wrobon wit.h ~u1 ... I and" lor the _ .lj~ 110..· ..... """ (II> T . Leo f, and" be ,_ ~ ~u.mben""" thal the 0Q1>"-li000 "" , +..", _ p bao DO ...11« "''''ioa n" p ) i .. ,Iw>. io, 0, 0)_ Dec "'.. t., B lbe "'bpcup { up (lio1" ," +".,.) : n, E 72, n, E z }
   
   C""
   
   of T , t., C ,be ...
   
   * ....,B of T . and poo. A _ 8 uC. ~
   
   z'
   
   I.
   
   "* doli... "" T .. equn.u...o.
   
   w.: ( _ ( if Mel oo.Jy If C', 1... iD A. I,.et E boo 010111.0< ofT ho."';"C """ and only one ole_ in.:lo eq~M"'" ~ Show'~ EB and EC p&rti.io;m T &hd.~ B
   
   ODd C onde_ ift T I_ SecI ... 4-31· 1. Show t ho.< tho fu"""ioon • _ ,,' CoE n Bb1 from T
   
   InC<> R · io "0 ) . bon
   
   d iocoon.",- lhouVt ~' (£) and ~' (B£)"" hi .... CcmpOl'< .~Io ....11 writh p&I13 01 ENto e.. 1.
   
   •
   
   ld G boo 0 IoaIly .....1*' _po. ". 0 Idt Ha... _ ...., "" C. and A 0 ,... I~ oubeet 01 G .""b .hat ,.( ,01.) > O. Denot~ Ilr H I A) .~ ... o f . _ 0" G b wbid> ~I AI " ".(A n .....). i.
   
   Show.hat H IAI 10 •• _
   
   l.
   
   Le. 8 boo.
   
   ....
   
   ........ P <>fO (_
   
   compact ........ of
   
   b . E HI A). 8 -.1.8 ....
   
   Ii
   
   A
   
   0...,.
   
   pOn
   
   1 of e:....,d08l) .
   
   ,.( 8 1 > (lt2)I'fA). Show .hat. boo d"loint. C-roo. ,hAl H I A) io..-.
   
   _~.hat
   
   Let C boo. ,....."... " ...... k>eaIIy _ : .~, writ_ oddl.i
   For...:lt. Ii G , .... A' .. A \J( B + 0) Mel B' .. (A _ . )n B . Show . hat I'fA') + ,,(8') .. ", AI + I'fB) MeI.hat A' ... B' C A + R 2. 11...,."" . ", . u _ that A io """"""I" -.I 0 lloo In B. Conotruct • 0Ui_ ((A., B,» . ;!O "i.b . ho foIIoori. "'.",.". I
   
   la)
   
   Ioral k 2: O, A, _
   
   B • ....
   
   _"'1
   
   ,...~'u"""'<>fG.-.I
   
   0 ......... to 8.: (bl ("-o.8o) .. ( A. 8 ); leI lot all t i!: 0. t.IooeR.u.w .... ;" A. ,\id:I , .... A.... , .. A,u( B, ........ , I ....:I B•• , _ ( A. = . .. , )nB" (d ) ".(B... ) S 2" ... ''''>f A, ".(1,01., _ ol n B.) . B_ _ U.~ A. and n..l'" B •• --"W:V. .... • hat 1'fA.) + I'fB.. ) .. JOtA ) + ..cB). FlnaUy, _ tbat ".«,01. _ _ . Jn B.. ) .. 11(8,.) too- - , . 0 E A...
   
   Not. do!fini . A. _
   
   s.._ that
   
   11(8.0) > O. B1 - " 1 &hd ~ 01 &..!,d•• 2, tho l u _ I : . - ".(C ...... - .) n 0..) .. ,,(A... n(o + B.. I) "' -.lln""'" "" G....:I 1$ .. ".(- 8".)II( A.. ) .. ".(A.. ),.(8.. ). M~. I \aI;to tho val ... ".(8,.) <>II A•. P.- 1Il00. I .aboo <>Illy .boo -.1_ 0 _ ".( 8 ..), Mel I ..... C .. /" ( ,,(Boo) 10. d.",en ... _ 01 0 , ~ that ".(C) .. ..cA_l Mel that C io .boo 01 A ... 4, Su_ that ..cB.. ) ::. O. W$ <104'\"" I) ... ,hoo'" 0( oil 0 E B.. oudo
   
   l.
   
   f
   
   ""'*'" that."" __ I;'" 01 B.. wi\h - , . iL ,'P!Whoo>d 01 0 10
   
   _~bIe.
   
   So D '" ... pp( l , .. " j n 8 ... """'" tho • ..cD) .. ",8.. ), If 0 E A. ....:I I E I), ___ ...... (. + VJ n "- 'I • Jot -.y -..""". ~hood V 011. Co"d ude IbM " - + I) C G ....:I . 1101 0 + 0 c: C . Conoidet HIC) " (. E G, ,,({C _ .lnC) ~ ".(C)}, For -=1>. E HIC) •
   
   .,..,..,.hal
   
   Ellen '
   
   ~.
   
   b
   
   Cb&pter 23
   
   tbe """,p' .... 01. (C - . )n C in C io empt1 aDd that ... ,k , ..... E G tieo in H{C) if ODd 00LI,r if C t , io iDcluded in C. llenoo D io Included 1ft H(C) aDd C t H(C) '"' e. Dod""" from pw\ 7 01. Ewoe" 2 ,hat H(C) 11M _"'''''" 1 _. Coo>cludo tho., H (e) io botb open ODd <>:) 0, If . tieo in e , ,boD [e + H(C ))n(. _ D) '"' • _ D (bee',. - D c H{C), 110...100.,
   
   ,,(A.., n(. _ B ..») a ,,(e n (0 _
   
   D ») .. ,.(. _ D) _ I'( B .. ).
   
   e io i ...h.ded in A t
   
   B. From ,be fact t hat ,.(C) + ,.(D ) .. I'(A..) +I'(B.. ) a ,.( A) ... ..(B }, aDd.me.. D C H(C), clod ..... tIIA1/'lC ) i!: ,.(A ) t ,..( B ) - .. ( H(C)) . DodIl!Oll that
   
   8.
   
   If ..(B .. ) .. 0 , ' - ,hat ,..( A t B ) 2 ,.(A ) t ,.(B ).
   
   7. S u _ _ that A, B "" lM>ta, pty. W. do _ ~ulr< lhat B ' . toi"", 0. 0..:..-1 E B . Then ",( A t B ) .. "' ( A + (B -I)) aDd /'l A} + ,.(B) '" /'l A ) t,..(B _ I). Dod""" from pula 4, 5, and 6 .hat
   
   (. ) ei.her ",, ( A + B ) 2 ,.(A ) t P{B ) 0< (b) .bono _ ... opeD compoo<:t ... bf;roup H of G Iud! .hat A + B ........... ada. 01 H in G, ODd ". ( A + B ) 2 /'l A) + ,.( B) _ ,.(H). ..
   
   8
   
   Aoo..-.hat G io """,...,.... and _ <>:)
   Leo A (-"iwly, B ) be . l>o ... of rUIIlWmbon z ... hMO I""JIft d)'lldk ex, per ....... Z ":Eo t r:.~,(z.f'2·) (.n.b :Eo E Z ) io IUd! .hat r . .. 0 for" n a ! (-"ioely, for n odd ),
   
   I,
   
   >0
   
   M
   
   Let ~ be ,he ""'Wi.. (z. ).~. _ r:.~ L (Z.f'2" ) from n _ ID,IJ 0<110 [0 , 1[, and let ,. be tbe .....:I product _,,","" the locally ...... ~ 'IIO!CII 0 . If A d d,,"' . ~ _""' .... R, _...,.u.hat .. (p) _ l lll,.). Pro.oo . hat A n 10 , II io lroqlisiblo, """"" ,hat A io lr"'"l!li3ibio. Similarly, B io lrDOllicjblo, and)'fl A + B a R,
   
   R ... _ _ ...... ,be field Q 01. r1!tion.ol ""mbon, ...... ,hat A U B """""... bMio H 01. R and that '" .. U.... Til io .\..nea\ilible. 3. F'o< ~ .. E N , _ _ t." P. tl>o ... of t ....... rUII."m ...... % ... hid! loa... 2.
   
   ~i",
   
   at _ .. _-.."""",' 00'...... i... to.l>ob!lloio H . GiveIce 1M ru.....1010 Z _ A( " + E) n E) io comin"""" 011 R t." Ea . . .. 2, it .... _ ....iIl> Ide",k&lly .... (Q - (0»,.. Pro.oo IILN tiLl. """"iM" • comrod_ .nIb
   
   ,1>o_oI,_ ..
   
   our
   
   n.u..
   
   0I"iii.... _ump
   3 that, lor """'" ,ui, obIe iDtqer k ?: 2, ,1>0 _ 1'! , ... , f\ , L ~ ~ oocIilible, but f\ io _ ~ _urabiIo. Put Pi, D - f\ " , ODd _ lho.! + D io _ ~-....bIo, .....n I~ C ODd D ~ lrnea\iliblo.
   
   • . Dod""" from
   
   pw\
   
   e
   
   e-
   
   yr
   
   ate
   
   23. MM' M tt", tt
   
   4M
   
   T IA G bf, t . _lIipko,tl ..
   
   ~ of _ricoo
   
   (:
   
   ~ )""""
   
   that • ;> 41
   
   0:I>d , E R. W• ...., ...... i(r G wi. b tho """" ... ;"i..., of .ho
   
   ~"P
   
   H 3 " _ClZ+~
   
   N<- _ idaU,plaDo U., b): . :> 0, bE H J ., 1', ~ ,iI&t {I/.'Id , hat ( L/ .. }dd"/ ~ ;, .. rip< Ib&r IDNOU.., oa
   
   a
   
   w~
   
   0 , p..".. that
   
   ""- "r
   
   ~ ((:
   
   !)) _II..
   
   !) (..,6, c.
   
   oJl """ricoo ( :
   
   !) out!> ,11&.. ..
   
   ol
   
   
   (EdV6<~iJ" ~M)" If. br 50(2, R ) 1•
   
   ..,bp<>oIpol 5 L(2, R ) "
   
   d, ...
   
   .. , . . . . bpwp ol O L{2, R ) _ _ br Wlh.."dl1iar. w~ ..m. I' Jot t loo lWf·pIaAl
   
   ~
   
   0
   
   if t .... ol
   
   _~ 0:I>d GIl - be .. I, Since SL(2. R )
   
   ,.
   
   (OID1IUlw..n
   
   (> E C : Im(»
   
   of Gt,(2, R ). it It
   
   :> o} ,
   
   ,11&. SL (2, H) "P'""". 0<1 P from •• Itft ria , . mawi.. (
   
   (
   
   ) uH ' J' ),I -c.+J
   
   0
   
   •
   
   that 5O(2. R) io .be otabilliltr of; ....,.;.. "'""' ... 1',
   
   2,
   
   It,
   
   OLf2, R ) (rM!-='h'dy , SL{2. R )) 1110 IIIUIIlpboatlve _
   
   .... 'ricoo M _ ( :
   
   I , Show
   
   Jot.n G:> 0 _ H
   
   Writ
   (i> _ - I), 0:I>d liI&t 5L{2. R ) KU;
   
   '1'.,.. ( I/ "'IId. * d~),,. ....
   
   1', ria ( :
   
   : ) 1.
   
   SL(2, R io aad Itt .. I:>< "'" _ •• LtC > - (G> +b)f(a +<1) _ I' ontG P. SIoow ,hat dot (Poo(>1) _ . "". + cIJ' Jot &II > E P _ that A It \1>"Wiant
   
   under ... 3.
   
   H_ _ u"' "" 50(2. R ). Fb< _ry I in 1f: (SL(2. R). C),~ ... • '~""Ion p by ~ by " • • ,",""
   
   of
   
   t
   
   r ""
   
   Oo.d""" from Pt OjX4i.;..., '13,0,1 .hat A' : 1_
   
   ~n
   
   io a H.... a
   
   .""" ow
   
   SL{~. R).
   
   r-.
   
   -..pee.
   
   I. i l K .. _7 .......-i.. _ _ . IocaIIJ' &old. it io P*iblo ",dMcrll:>< lI_ 11".. _ SL{ ... K)( N._-'fUl,. ' r' ,~Cbpt .. 7),
   
   'I'
   
   ate
   
   24 Convolution of Measures
   
   <JI _~_ "" • Ioeally __ pert ",",p G io 01 flnt Impan...... ia ...,.jyoIo.. l.' {G) beioIc. B·n..... ",.. bea.. ..... QIl owIy..,.,...oI u..",o_ portai .... 1",10 Ilwe oIS '''' ''· B"I - , ;. .. ""'l' inten.ti"fl to 1100 I0:Il ftJUlar '"Pi ''OCioAclC""L' (C). Conwol~lioa
   
   detno,,_
   
   """'.Ion of _ .t,' .... Two " ol**Yt..,... ....... -m_' , "1 oad """"It;""" 10<. "",."to aI>d. fun.> -,_ oad
   
   110.1 W. d
   ~
   
   7'
   
   ~_
   
   .....
   
   ~ (~24. U) ,
   
   :U,l 'ioa ...
   
   ,o:J!ic;rn\
   
   too • I'OIvabIo (Ph .
   
   2U. ~
   
   ~U,l).
   
   24...J If,. II." 'TZU'" "" G {~. a" ' '",," with __ pert SUppOrt} ud If ,10. _ ' - fu"1_ '"' C ";th -..I*" .... wort (.... I~T.' CODtl_ timet;".). , _ II &lid I .... coo.on.bIo. 24.4
   
   Fe>.....,. _ _ _ W'1!,. 01> G"""", i>r.-,.!\metlon ,110 U {GJ ( .. it~
   
   IS, S "-). ,. &ad 1_ ........,t.abIe (~_ 7U.1 """ 24,0).
   
   u..s Focl :Sp<+oo.u...' 1-,. " (~..... 26.&.2).
   
   '1"';'m,- (J.. ' ''''~ (G)iotloot"n'i
   
   1U W. OlIO<\! """oo!,,Ctiono
   
   5 '"
   
   (n....._ 2-1.7.1
   
   .-124.1.2).
   
   24.8
   
   Volt,:
   
   -n.
   
   GoIfuod pooiro (im--.- in bantioAIo; analyoio). w• . - , bat ODd (SHe.. + I, R ). SO( .. ,RJ) ..., c.lfaad.1*n-
   
   (50(.. + I , R ),SO(w, R l)
   
   In Ihls chapC.cr. all locally compo.« 'PH'" are .uPI'"
   
   t
   
   1 to be Hallldorll'.
   
   24.1
   
   Convolvable Measures
   
   Definition 24. 1.1 lA:l ( X,l itl bola 6ni~ family of 1oc&I1)' comp8Ie if ~ ;, ,,' P<9P'"'" (DdinitiM 21. 1.1); ;., thill ..... V" 'PCP) II wled tho: ron¥Ollltion (or oon¥01y,ion prodllC1 ) of the family (1'. )0(1 (re!o.liV1l to "..) and is wrill"" * ~(/';),u 0<" * "" " , .
   
   *"" '" <;an bol uo.ed only ..ben Ihe<e is IIOamhigu.ity 00.(l'.l.f., io _voMbie if and only if (1,..1).. , ... In 'ltil CIIJIO . 1*;0:' ",I s Of~ I~ r>OW.iM
   
   * ",,11',1; ""1<-':' . ("l)"" 1..,1S 1",1.
   
   ill",
   
   :..,.01..... blfc
   
   lor
   
   &11,,-.,..... v, ouch ,hal
   
   N.,.. lee. ( X. )iE' be a finite family of locally compact _ _ (I "" I ). and lee (l1);H be .. partition of I ioto n,one,npty Od& For"""'; E J, lee
   nit"
   
   -. For
   
   '*"
   
   n,v
   
   j
   
   E I .....
   
   p" .... ",.
   
   a..-ute on X,.
   
   PropoSit ion 24. 1.\
   
   (.j
   
   ( *"'"
   
   Va.)•." ;, I;>r"",,,,,",,,,hk, for
   
   notry ; in J. GM Ih.at 1",1),(1 .. ~. Q),.,.' ,,/U 1'11<:,. (l'd .. , .. j,> ........",.tvII/e Gnd
   
   S"PJIC'«
   
   III4t
   
   . .(" * (1'. ).(/ .. *
   
   )
   
   * (1'. )0(1, ' u
   
   (I).
   
   ~) Suppo« Ih.aI.
   
   'P ."" W
   _~
   
   Under the !lypOtheo.is 01 P IQPO!ition 24.1. 1. part (b), i to ... n b$ ~n. for fuoed j E J, lhal II ...r oonYOl .... bIe .. bcrwlver 0 ... ,,,,. ". it """,","",. Ik1oocbrth, Iee. ... be .. in Mnilion 24.\.1.
   
   u-.}..;"
   
   ,
   
   PropotIltiop 24. 1.2 Lei " . /of 0 IIoottndtd ....,..,.. ee on X , (~ I E / ) W..;u " lor ®It, ,,, , 1h.aI. ... ;. "._y...w... 11w:n "").. , ..
   
   0"" np,.....
   
   II * 1<1.' ",II ~ /Joe '" _ ......rn.e.
   
   I""""""~ .. u
   
   s n ,u lp;l. In lad,
   
   II * ,(1 ",I -
   
   n.... 11'0 _
   
   24.2 Coom>I ....... of" M
   
   P ROOF; Clearly, 'jO;'
   
   """ ODd " Flme
   461
   
   ,..proper and U* ~I I<;R'S l' I of( 0 , 11<;1) ,. n.. , II'
   wi
   
   P ....p.-l tlon !it",1.:! SUPP"k 'jO" (101Itin ........ I'< E M' (X" C) (""'-i E / ) omtl, giw:n j E / , kline F ... ~r MI. M'(Xj, C) """""'Vin... """",,,I,
   
   .....",...,., M' (X" C) ~ " j _ j. Then P(F) ..... t'Cf9U no..,.".,.,l, /.0 * ~I j<; .
   
   to Pj. Ld" ioe
   
   ;+
   
   ~
   
   *;~,"",
   
   ....... v, ... I'< /.,,-
   
   P ROOr. Ld / : y _ C be bounded..oo contin"",,". For _h z J E X j. ~ by /., !he fw>ctioa (:r;)o£l _U) -
   
   (f 0.")((,,,)... )
   
   from
   
   Xi -
   
   n x,
   
   into C.
   
   "" - I!)
   
   fu.l E Xi' Now. fur"""')' t :> 0, dim: erl8t1a ~ ... beet K 01 X j .uo:h tb.t lvl(X; - K) s ("/( t D/ U) (wt..re" ,. ® "'HII I'<)' For " I 0 - ~
   
   1
   
   dvl
   
   IJ
   
   dvl
   
   to AI, ..., ha... l x(f., - /" ) 'S t/'l. hence; (f., - I.,) 'S t. In ohM. g: %J tb; II oontlnlX>lM (and bouJ>ded ). We <:OIIdU
   '"
   
   J I.,
   
   ,dv,
   
   ",) .. a
   
   aIooo1: F.
   
   P.o.,.,..ilion '24.1.4 ".,,- 411 i E 1. ter I'< ioe a .......""" "" X ; "';111 Apport S"l'P"H- tJu.l.." u ".,.,uin""... on.! ~ ,.."Iri<;ti<m 0/ 'jO /.0 S ,. D.u So u ,iC; (lk/i"ui"" n .s. I). Then (P;)"", if 'P·"""M . l te """ IUpp( * ~II'<) is MdlOdtJ in th • .w.... "I '/A.$).
   
   s..
   
   PIIOQF: If 1< is " oxnpoo:t subeet of Y , then 'jO- ' (1<) n S III """"pod. Si""" $< 1II1"""lILl«i1* (w!>Oft " ... 00f l "') , ..." .... that
   24.2 Convolution of a Measure and a Function Let G be " locally <:OnlptlCt VOUP oai,,& from tbe left (ID a lo<:aIIy UJ.ly
   ,.
   
   or
   
   Undo:r \hil J>OU.\ion, we Ieo."" to the reader the t ra
   t""'"
   
   E M (G,C). ~n
   ....
   
   ,
   
   ~I"
   
   P l\OOr: We may "'1'1"<* thU jJ. ...d., NO! pOOIill .... Let 1> ; X _ (0. +00) blfliower ..micontilll>OUS. 5.illOl! tbe fuDCtion (. ,r )_ lie,.,) 101 ItW\>l)Ull ,
   
   /1I11fJ< . ,,) .. j"IId(,. . ,,) .. r
   
   ll(n ) IIfJ< I!I " )('.z)
   
   .r
   
   Io(n)d{JJ. ~ ,,}( •. z )
   
   .r r dl'{.}
   
   F\U"\hcrmore, I
   
   _
   
   I>(,.,) ct&-(z).
   
   r lI(n)""'{z) .. lower oemIco!1l.in........
   
   Now let. K be. 0)IIIp&<:1 subo!et ol X , and let. g E "H"CX } boo """h tbat g
   r
   
   gd(jJ. . ,,) ..
   
   r
   
   (g - lK) tI(;. · ..) -
   
   .....
   
   Sinoe, -
   
   r r r r ""(1)
   
   g{uj dv(z),
   
   4(.)
   
   (g - IK)C", )dv(Z ).
   
   f g{u)dv(z ).ad • - f Cg - IK)(-':) " (r) jIKd(JJ. o,,) ..
   
   NO! jJ..ln~bie,_
   
   jgd(jJ. o" j- jl.9 - I K)d(p o ,,)
   
   ., / .,.,C..)/
   
   1,,{ttZ) dv(z)
   
   ..
   
   1",l{-l{,j,,).
   
   / djJ.('" /
   
   _un:
   
   But . fllreac!>. E G . ... (.) ... 101. wltlt ~ fJ. H~. if K iII~ne&llciblt. it ;. aIoo 1' . "'Deglipble. n.;. Ili a'eo tIw " . ... if &" : 5$1·.-e ",ith bwo 8 . 0
   
   o..HnlLIon 24.2.1 Let p be. "",&sun: orr G and let. f: X _ C be locally .6-~.t>ie. p and laM oald to be cortvolYl!.bie (with...spoed 10 8) wt:mcv.:r
   
   /J and I~ aM conYOl ...hIe. In this ca9C, ""Y ~1III;ly of 1"
   
   (J~)
   
   w;lh I 1*<1
   
   (J" ..ll!w! a conWllUlion product of I' and I (relali...., to 11), wrillCn I' ~ I (or I' .. I , wbeu no "",~iIY on ~ ill JlO"Iiiblo:) . loG
   
   ~ I ..
   
   .8-<>e&!;,g;hl~
   
   set. If oupp(11) .. X and if lhere ....... a convolUlion prodlX:'! of I' and lihat iI conli""""". l bio Iast_ .. Wliqueiy do:!enaiM
   defi,,...;I
   
   "P
   
   loG • Io<:aily
   
   (reWi...., loG fJ). u.nuna 24.2 . 1 Let 0 k a Iooill~ o:m>poa If'O'i'1> aeling ctmli.."""", /rim> t.V IqI .... 10<»11, """pad .po.tt X , OM let fJ j 0 joe " "",;titoe m...... re ... X , pAOi.u."""",," ,.,..!c~ G. Aloo. leI X joe II ami .......... ",,,,,1icIl from G .. X into )0. +«>[ .wJo thaI 1"x (. )fJ .. :d'-" ·18 1M ."" ,h • E C. 111 ...., 1M "'" mm /J .,.. G, ® ~J io tk i~ ......... re 01 I' ® p 14 ~i.mI ('. zl"" (., . - I Z) from G " X "",to G ~ X .
   
   xv.
   
   "I.
   
   P ROOF':
   
   I
   
   Let F Iii H <-(O
   
   ~
   
   ....w-
   
   X ). Then
   
   F(.,. - IZ)d{p.0t1)( •. z )
   
   I I .. I I ,. I I .. I ..
   
   dl.(' )
   
   F( •.• -l Z)dtI(zl
   
   d,, (.)
   
   F( .... ) dj,-( . - ' )")(z)
   
   tIp(.)
   
   F j,. z )x( •• zldtl(z)
   
   Fj,.z)x(•. zl"(jo 011){. ,:oI·
   
   o Propoooition 24. 2.2 LeI " boe " """"",re om G "M leI I : X _ C joe 10I rIu>!. ..... I(.-I:O)X- (.) io ~Ir /J-"'tqnobk ~ , ~ fur " koo:aIlr " . ...,."t~ 1<:1 til ,..Juu 01 " , lInd tJw. Io\e z .... J 1/ (.- 'z)l x-'(.) d);Il( .), de/ind I«
   "" _o";,."·i"1 SlOw... 5..,.,....,
   
   PROOf';
   
   "".(;no..
   
   w. ->" SIIppoo!1e t hat /J and I
   
   aM pooIiti .....
   
   Let II Iii H+ (X). W t Pf"O"'" thaI ( •. z) .... h(IZ) is .... lIIlallY ln~ for /J 0» Uti). or that h(.,.)!( .,) <4«1) "PI,,) <: +0::0. W~ """"" ooIy ohow ,110= exioI.J II 2: 0 Neb that h(u )/ (,,)l K(')"/J(,)dD(z) S" a lor any oompACI-
   
   Ir
   
   If
   
   SllboetoCG. Sh>Ot 1' 0 tI is tbe ima&t measure cI. (x-' Ii) 0" under tbe l>ooiofiClll>Ol"phjom
   
   r.,.,).... (', . -1.,),
   
   If
   
   h(u )/ {z)I K{.)otp{. )dD(.,) ..
   
   If
   
   h{.,)/ (, - J.,)IK(·)X- I(·l d,,(.) " jj{.,j
   
   (I).
   
   N_ (.,z) _ lI(z)/ (.-' z) I K(' )X- '(') i!I "iii p.mea&unble. wit h compact support.. ~,~ ria:bt-Iw>d" of (I);" equt.! to
   
   r
   
   "11(z )II(z )
   
   r
   
   I{.- ' z )h "(')x-'(.) d,,(. j .
   
   r r
   
   • nd -'0 .,.. than
   
   1111
   
   d/f{z ) Is(z )
   
   /(. - ' z)X-'(·'''''(· )
   
   o
   
   (where S '"' wPP(II)). Tht proof:' Qlmple\e.
   
   C ond Id I : X - C k I«GIIr ~I ~ 1"" 0"'Mf ",,,diliQ,u MlIb.:
   
   Proporoltlon 24 .. 2 .. 3 UI" k . _ _ 8 .. ;nlCf' ,6'e
   
   (I)
   
   I "
   
   S..,.,.,K
   
   1.\0;1 _
   
   ~N
   
   (ONl in ........
   
   (~) G GCU J"op< "~ <>II X oM f t'Gn~. ~tIII""" G "m:" .. oj """"~r ""''''
   
   _JMd
   
   KU.
   
   (c) " .... .........,
   
   ...
   
   ~
   
   •
   
   ~_
   
   oj <>:>llAlUO, .... n' _,...:' flu .
   
   .. euolllHllr " .int;"W. aupf for. 1«:01/, 8 .. n.c:giigolok Id of oah
   (p~ f) {Z) -
   
   f
   
   J (. -' r )x- '(.)./jJ(.)
   
   (oj
   
   I«GI/J /I . obn<Ml .... ' roohcre. f'1tOOF"": le\.10 E "H(X ,C). Sinoe (I,Z) _ II{n)/{z ) :' Fli'O'iaUy p 0P. int.egral*, the function ( •. z ) _ h(z )f(.- 'z)X- ' (') :. _i&lly ,, 08InI.e(rebIe. In face , under oor>
   beeo" PF, '" the!irK ..... ;t is corotin"""", and in the ~ ..... it ftl1~ 0Il1$lde. unloo:>. of QOl,mtat>!y IIWlY ~ _ Theil, "" Fllbini'l tbeoz.""
   
   111I(n)<4I{') d(JJ1)(z) -
   
   111I(r )J('-' r )X- '(') <4«')J8{z)
   
   ,. I f dp(z)
   
   h(z)J(.- ' zlx- ' (·) ....C.).
   
   We _il>
   •
   
   g-,, · I .
   
   ,.
   
   ....1'1"* t hat I.< ;. ca:ried b)' .. union S of OO (M ') _ h( ~ l/ {r' " )x - ' (e)l$(') io , : ,nt"-lly 1.< $6lnt~ aM vanloIlel OUUlide ... union or OOWlUbIy many o:.>mpKt --". bmc:e Next . "
   
   II 141 1.<8tJ.1nUp"&!)le. sm.:.. 1' _ 1.1"1' • ..., ~ oomplHfl l hur..,ment "" bfofore.
   
   o
   
   24.3
   
   Convolution of a Measure and a Continuous Function
   
   Let G. X . f'. and Pro~llon
   
   x bt; "" in Section 24.2.
   
   24..3.1 Ld I' k .. mou"", an C ...111 ..",.pod...".,.,., ... 0:1' Id
   
   1 : X _ C .. .....,u,,"""'. 171 ....
   
   r.;
   
   I.< no:l'
   
   (t) 2 ....
   
   I ""' ..,,,,,,,",We;
   
   Jf(.- '"lx-'(,) d/J{o)
   
   " •
   
   ~in""'" k nril, ~/I.< '
   
   (It!) "':111
   
   .....,.." Ii> fJ.
   
   P ROOP: C t...., -=0 In X , I~ fu nction' - f('- ' " lx- '(,) O)D,~ unllonnly on .... ppV. ) to. _ f(' ~ '.tolx - ' (6)" " - -=0 . aM I (' - ''' )X- ' {' )o:r;.(.) tmdo 10 f I (. - ' " .)x - ' {, ) dl' (' )' Tberefonl, ~ ,., 0-0 f l (r ' .,)x - ' (. ) dI'(,)
   
   J
   
   .....,;,......
   
   No. , by P """",idon 24.1.4 . I' and I fJ .,.., coo.voIvt.bie. Mew OJ~·. br P r<:p<> .idon :U.2..3, ,10. densit, of 1" (161 .. itb ' .. pa' to fJ. 0
   
   Otsterve that. If I.< 10 . _ _ on G with ('(lm~ , uppOrt and I bdonp to 'H{X .C ). then I' CIlmptoCt ....ppOrt.
   
   =1 va.allbeo oollide (.... JlPI.<)(IUppJ), "" that ,, · 1 """
   
   Propoot.!tlon 2"',3.2
   
   SqptJl
   M IG,e) IMI iIl N(X,C).
   
   r..) I' .0Id 1 (t) :o: _
   
   an!
   
   ........"",w..
   
   J 1 (1+' :o:)x+' (. ) o:I'l'(')
   
   ;, A omtin"",," 0:I'~1. 01 1"
   
   (It!) 1IIiIII
   
   I'UpOtt Ii> JJ.
   
   P IIOOP". Ci""" ... in X , ~ V be ... oomptoCt ~ cl ,... 1be !let K . cl u.o.. • In G IIUd: that , - ' " b.:Ioop to ,uPP(/ l lor at . . . "'" :::: in V, III ~ FurthE.m,n , f(. - ' ,,)X- ' (' ) coo.",fP u niformly on 1< to I (' - ' -=O)X- ' {' ) " '" - " 0 ;n V. ThUSl J !('- ''')X- '(.) ..p(. ) 0J
   f
   
   1(·-'%)X+' (· ) dI'(.)
   
   'I'
   
   ate
   
   II continUOllll. Nat, I' &00 I {J ~ (:OII>'OI,'&bIe ( P f'OI)QIiliou 24. U) &00 9 is a deMity cl I' • (fP) wil h 1eupee! to (J ( P IOpoiiilion 24.2.3) . 0
   
   24.4 Convolution of Ii E M (G, C ) and
   
   f e 0 (/3)
   
   LeI. G, X, {J. &00 11: be U ill. Section 24.2. For all p e 11. +001, _ writ e N~ r.;". tbe....w tomioorm 00 O (p) .. ~(Pj . Noot Ioet P &lid 9 (whe~ p 2. I) be 1..-0 conjuptt cxpOOcou.. Villi! funher 00(;';;", Wi! IUypOII<: t hoo\ P "' finite. fbr ~1IOCh • E G and Hod> I E O (p). _ lot 'I~r.( .l/ be the function" ....
   
   /(.- I%)X- I/ , (. ) OIl X . 5i""" 1(.11 : % .... "~"('II II po,,-urabIt; moo CO"'.
   
   1(.- %) II X- I(.)pomcasurable. •
   
   r 1/('- '%)X-" '(·)I' dtl(%) .. r 1/ (. -1 %)1' d(,.(.){J)(%) .. r il l' d(J. H.".. --,., ..(. )1 I.,. i"1'1(pJ and N,b~ ..(. )J) .. N,{J). For Hod> • E G , Wi! deooI.e t". ,.... (.) the .:ondn"""" mdotnorpbism
   
   -
   
   "~A'l/ of L~{fJ) . T he mappi",
   
   I
   
   ....
   
   'I~~')
   
   "'
   
   j ....
   
   a repr_nt ..ion of G O
   Lc{fJ)· P~ltlon 24.4.1 For all I e O (f1) . 1M. ""''';''9 ..... 'Iu(')/ '""'" G illto L~(fJ) ;., omtm ........
   
   P IIDOP: ~. I UppoM that I e 1t(X , C ), and put K .. ,upP(f). C i..;:n ~ e G. Ioet V be .. comPKt nciahborl.:ood of ~ in G; 1... (. )1 con",,'i"t \0 ""I':u(..)J II> 1t(X , V K ;Cj, aDd 110..... in O (Pj, u. _ ~ in V. n",.do. e, ..... 'I...(.)} . ....ti ........ No.. Wi! ~ to the ce",,,a.I __ Si .....
   
   N. ("h"(')1 - 'I~ ..(.)g) .. N,{I - I I for a.ll I e "H(X , C) aDd
   
   oJ] • e G, \he mappi", ..... 1",(')9 convercee to unllormlJ l1li compKt U I E "H (X , C ) (:OII' U p to I i" prOp(:eih:'n foI\oq. 0
   
   . .... ,....(.l! 0(fJ). The
   
   Similarly, lor -=b. • E G aDd
   
   .eu..
   
   ...ao
   
   I E o (fJ), ..-e lot '1. (' )1 be lhe
   
   fuooion " ... /(,- ' '' )X- '(.) on X , to that l.{.l! lio:Ii In 0 00 and N, h ' {') /) .. X- I/·(.)N,{J). W~ det>oXe ~ ,.. (.) t be ....tinuous endomor-
   
   .
   
   phlsm I -
   
   ~
   
   "~(')I cl
   
   L'c(fJ) . Then ..... ,.. {I) "' .. ~ulion of G on
   
   L~(lJ). ~"e<. 1or oJ] I E ~(IJ) . • -1, (,)1 10 contln"",,>,
   
   24.40 ooIu'ioIooll'EM{G.Cj aDd/ EZi{,8)
   
   ~13
   
   Theorem 3~.~.1 Ld I' "" .. "'OOPM ... "" G ndL thCIl X-I'• .. ,, · in.t~.
   
   end kll (e)"
   
   E
   
   171ft).
   
   n-
   
   0"'" I-~;
   
   (') ,,t!.olmqJI 0(lfJ) .w .. 4enmr " • I E (O(p) ...110 oj QI ~ ~/rmr>Wt.
   
   ruped to
   
   p~tion
   
   fl, ,n- 1«:01",
   
   U.I.3;
   
   (e) N~(" • I) ~ N~(f)( f X-I' . 011,,1). PROOF: X- I, o", and., ", iI arriod by a counl.able union S 01 com~ II!U. Let 11 E H (X ,C). Sin.ce ,, @/1is the lmqe rroe.~ of (x-',,)@ /1l1nder (. ,:r) _ (•. :r j, 'ho function (,, :rJ _ 1(,-' z )X-'(') ill ,, @t'-..-urabIe. Then (.,:r) _ I1(:r)/(,- ' :r)X- '{') ill ,,@lJ.rroe.urabIe, and
   
   ,_I
   
   r '"' r
   
   1"(:r)/(·-' :rlix- '(.)1,(.) d(l" I@t!){ •. z)
   
   -r
   
   011,,1(,)15(')
   
   r
   
   1I11:r)/(·- ':r)l x- ' (.) dDI:r).
   
   r .r
   
   1"(:r)f(,-':r)lx- 'I.)dD(:r) ..
   
   1"{:r)I(·-':r)l d(7(.)fl)(:r)
   
   1I1('~)/(~)ldp(lI)
   
   ill Ie. , .....
   
   r
   
   "'.II)N.(7. -,(I1»),. "'.(f )X-'/.(.)N.(I1). Therdore,
   
   1"{:r)/ (' - ' zlix - ' I') d(l"I@fJ)(" z) S N.(I) ( / X-I', dI"l) N,{I1),
   
   and
   
   (. ,:rJ _ II I:rJf(.-' :r)X- '(') ilL .-ntially ,,@t'-im~.
   
   ~umtly,
   
   (',,) .... I1(.~)f( ~) .. _otially I' @ t'-lo~ \0 short, " and It! _ con¥Olvabie. Next. , tho mappiILS • - "r~ (.)i r..... ' G into L~1ft) iI cootinUOUl and N.(7. I.)i )
   r
   
   '-
   
   Denot.e by (,) I"" conlinllOUll bi~...,.,. mapplILS (j, h)
   
   L~Ift) " Lb lft) into C , and pul
   
   """
   
   g ..
   
   .... 19" d{J
   
   f h~ {.ji)dp(.). Let
   
   II E 'H( X,C).
   
   / h "(gfJ) _ / ph "fJ _ {j, h) .. / ("r, (.)/ ,h) " 1'(. ),
   
   / IldUP)
   
   -
   
   / dJo(.) / h(z )/ {.- ':r)x- '(,) " fJ(:r)
   
   ..
   
   / ",,(.) I s{')
   
   from
   
   / I1(z )/ (· - ':r)x -'(·) dfJ(:r).
   
   ID
   /1I d(gf1l '"' / II{Z)I(. - IZlx- I(' II'(' Id(j. ®tJ)(•. zl
   
   -f - f h(",)d!J'®ff1l(',~)
   
   h(.,)/ (,)ls{.) d!J'®.Il')(•. ,j
   
   ., !J' ' /.Il')(I1 I·
   
   o Wri~ C'"'(I') roo- Cf!{fJ'), &nd C>(X ) (relpecti~. C"(X ) foe the let of 1 ' - ~; ....... futldio)Qo from X IntD C that ~ bouf\ded (respecti...!y• •hat .....h InfiDilJ'l.
   
   a'
   
   T~m 24.4.2 l.d" k 0............, "" G n.dI tII~ ie, I E C""(.Il').
   
   X-'
   
   "".int",
   
   I' •• ond
   
   (a) I' 0"" I QF"e ~w.. OM " . IfJ htU. oktu1lJ I' 0 I E C""{p) """" ren«1 '" 8, ,;un IocoIIJ 8·111_ e~~ 6r /tmntJ4 (oj of Prr>, . AAon !4.U; "'0' '''''' . N...!J' . !) S N ...(!) ( f x-' d\l'1J· (&) (c)
   
   1/ I EC"(X ), tlltfomc_ 9: r - f I (.-'.:a:)x-'('Id,,(' l I;e. inC-(X), 1/ I EC"(X ), ,. I IOu j" C"{X ), 0
   
   PROO. : By d ... argun>COl.of Tiwo>rem 244 .1.,. &nd / aN: coovo/vabW. P ropo> llitiol> 24.2.3 tbc>w1 thal 9 : r _ j{.-' .:a:)X- ' (. ) <40(. ) if delioed locally fj. ahna", C>'try",ben, &nd II a density of 1" VB) with respect 10 ./1'. If I' : X - C 10 III>do that I' - / ioaJIy fj.tJIOOsI evayw-bere ..00 JI'J N""VI, then (J. . Il{z) '"'!P' f )( z ) - f(, - ' z)x - '('l doo(,) Ioc&lly ,8.&1_ evayw-I>o:~ 1Iet>tt: N""{,, 0 I ) S N..Ul f X- ' "lui. Nen, ~ that I t..1on.p to C>(X ). Clearly, , _ 1('- 'z )X-'(') ill ",Intevabk, I'or ~ z E X. Let Zo E X . Ci,.... ~ ,. 0, deli .... K ... a ~ ....' - of G BUCb thaI fc_"x-' 4 1S (IN.V), uIot.s a nel&hbochood V of %0 ED X turiI lhal Z E V Impla II(' - ' z)x - ' (.) - j{. - 'z,,)x - '(') I (/(lIol(K» "" all
   
   J
   
   s:
   
   J
   
   n-e
   
   s:
   
   .€ K , Tben,lo.-rE V ,
   
   f
   
   / (' - ' ''')x- '(') djl(') -
   
   f
   
   IV' zo)x - ' (' ) dp(,)]
   
   s:
   
   ,
   
   ~ly.,: z _
   
   LlIol;K) tiI,,1+ 2k_It
   
   N...{!)X- '
   
   djpl
   
   ~
   
   f / (. - 'z )X- ' (. )dp(. ) '" oontin.........
   
   ••
   
   Finally, _ume t lw I E C"( X ). Lo:t r and K be . . .~ Abo, let H be • ampo.c& 1Ubee\ of X auch tlLat 1/ (, )1 < € lor ~ j H . Fix % j KH . SiDO! , -*%f- H £or, e K , we have
   
   11<,,)1 S IK rx-' dipi + k
   
   10, ...,,",,-
   
   _K N",,(J )x -' dlPl S €
   
   (1 + I X-' dlPI) , a
   
   at infinity.
   
   Propoaltloo 24.4.2 Ld 1' : G _ C · lit: G 1lOftti1l""'"
   
   ~
   
   AIoo,
   
   leI ~ e M {G , C ) and I' e M (G. C ). II p), aM {JIJ g~ ........"tooOWo (cilh .....,...1 to> (. ,1) _ " ). lAm ~ aM P " .... ..... ~ """ p() '1' ) _ (P~) . (pjo).
   
   PItOOI': Lo:t I e H(G. C). Since (., 1) - (Jp- ' l(" ) io (I'~) ~ (WH~, tbe fIon<:tioo (I , I ) ~ 1($1) io ~ ~ I"iu~ H~ ~ and p ..... OOI'IvoI.,.b\e. FUrtbenoon:.
   
   (phw)(J ) -
   
   II 1(·')p(·)p(1) 4~(.) 41J{1)
   
   -
   
   II(Jp)(") d~(') dlJ{l)
   
   =
   
   p "p)(JI')
   
   lor all I E H(G. C) , "" p~ . 1'1' - p(~
   
   ' 1' ).
   
   a PTa~ltlon
   
   24.4.3 411' : G -
   
   It: lie. (OnIin""", ~ ...... WnU
   
   10<" 1Ac ..., 01 IJuHc m .. , _ P "" G ncA ""'II' ;, 1J ·;IIIq;, dl •. For all p E M~(G), _ ptoI. lI'l~ - J 1''''''1. :no.... M~{G)
   
   (a) 1_ ammo,., domt:rW 01 M O(G) a ......... ..,,'"We; f') "'" lAc "..,...,oI1IIi<m """ lAc """" p -
   
   lI'l"
   
   M~(G) ;, •
   
   BonWi..t,.:m
   
   PlIOOf': If ~ e M '(G) and p E M O(G), tbtu p), and W ..... OOI'IvoInble (P' ........ 'ion 24. 1.1 ), 10 A and I' are convol ... bIe: II>OIw ......
   
   IA • 1'1, - (P(A• 1')11 L - II'A . w i L S IpAI,Iw l, - IAM IJI, · Since tbe mappilLll P - III' io alinear ioomctry of M '(G) om.o M ' (G), M~(G) "' • Banad> space. Finally. t". Section 24.1 . tbe "01....oh.tion in M~(G) io 7'
   
   -i-' ive,
   
   Clearly, tbe _un! t o. doIi"fd • of G, io . " " it [" M '(G) .
   
   0
   
   t". pIAci"l tbe unit . - &I tbe unit elenoeut
   
   Now let G, X , tJ, x be loa In Section 24.1. e.nd ~ p, f (wil b ! :s p !f +0::» be t...o COI\iup$e 0.»01 .....1&. F<>r...:ll ,.. in ." _ M"-''' (Gl. _ ~ l>f "r.v.) the O)lIlim"'" ~"m
   
   j .... p-:""! of L~ (tJ) (Theorem 2U .!).
   
   p ROOf': The _ un foIlo."s from !.he floCl thaI (.\. . p) . (ltJ) all .\. £ A., ,.. E A., e.nd 1 E VIP).
   
   ~.
   
   1$1 . / P) for 0
   
   N.,.- ~ lhoI..-e in ..·hid> X _ G , IM ... tl.oo ol G on X 10 (1, 1) - ~I, and tJ II l left U.... _ "' on G. lben X .. I The mawilll "r" (which DOl depeod on the "h
   oo.e
   
   of M '{G) on
   
   24.5
   
   'epi_. . .
   
   Vc,(tJ).
   
   Convolution a.nd Tra.nsposit ion
   
   r..... G, X, tJ and x he at< in s..ctio<> 24.2. E M (G. e ), A~d .....ue " ''''" the inoog< m_", 01 ,.. ouwIer . .... ~~ ' . l.d 1 : X - e _"" g: X - e 10< lotuI!r ,8.inl'F"We. Sy,o.ot llilll Propollitlon 14.5. 1 l.d
   
   fa) ,. _nd
   
   1 .....,
   
   """ '~ !,
   
   1«811)' tJ·dlm<>#.
   
   (/»
   
   p
   
   ,""""'4 (-) 01
   
   "k and
   
   'W' ,~,
   
   Q
   
   "" ..
   
   P,o".,.,1iQn ~ . I. ' dtfi ....~,
   
   •
   
   ..,n.,.... """"""' ,.. 0I ;
   
   xP
   
   ilrI'Op<>Iiti(>n 1... 1.' (oJ."" p .. J( j r, ,,' ~ xi' _"" I ~ g) d.ji...... , 1<><»11)' 8·.w.w,1 eo." .~, a
   
   ..... DOhdion pn>
   ,1$1 , Il
   
   ",nod
   
   II
   
   (I,Z)_
   
   IUp. II _re _fUll)' 8·,nl.;. dl..."" 0
   
   JI b:fJ.O g)d!J- f g{J. · lld(J PI\OO'" P ut "" I.i and I ..
   
   z --- r(:r)
   
   p." ,c.tic>rI
   
   f I {. -'
   
   (I).
   
   f II d{p '$} P) . 'Tho: funetlon
   
   ")X- ' (.) <1",.(.) ..
   
   f
   
   g{:r)/{. - ' z )x-'(.)\?(.)
   b de/iMd IocaIlJ' ~"""'t ~here. By Fuhini'. I~m. it b .....Iialiy ~ intosnblo, and. f g{p ' 1ld!J'" 1.
   
   2.1.5 (;o ...... ull....
   
   0Dd~....
   
   417
   
   I,, ;r) - g{;r )f (nb:{. )v l. - l ) '" h(. - lZ) ill i> 0 JJ.iolqrable. ~ro:£o"" (. ,z ) _ , (z)f(n)ol>(, - l) is (x") 0 .8-inl~. and (. ,z)_ g(. - ':r)J{ZjV(. - l) ill [email protected]·""d>le. ~Io .......... , ~,
   
   1 ",
   
   II
   
   g(.-l r)!(r )w{, -l) djl(. )dP{:r).
   
   From FUbini'. theorm>, ...,
   aDd It..t 1 _
   
   f!W ' ~) rI,!;I.
   
   gl
   
   ill .....cntially .8-inlqt:abie
   
   a
   
   uf p
   
   and q ~ t100 a>njtlgctc _!1D1lJ (1 :S p :S +00) . ALN, kt 1' ... . .............
   Prop
   / ! 1xj; · g)d{J- /
   
   g{p . !)rIP
   
   flotJ <1Ioountably many com~ O!!U. Sl~ I' ill ,uode'ate, II It carrW:d br ft a>Unlabio unl"" S 01 com,*" fleta. W~ put ~ _ I,. ~ fUl>(:!;\on h: ("z) ....... g{z)!V ' z)X· l (.)¢( . ) II I' $ jJ-n
   
   r
   
   1(.- ''''J.e '(.) if. Tbere£ore,
   
   mrabie, iii""" (,.:1) _
   
   h d!p 0.81
   
   ~
   
   r
   
   hdll'0 's1
   
   -r r ., r dPC,,1
   
   If (,,)f(· - ',.)Ix -' (.)-f>{') d(pl{')
   
   19I(1pl ' l/l) rIP
   
   o
   
   ill finite, aDd .., moy o.ppIy Propooilioa U .5.1.
   
   If I :5 p < +00. PropoeiUoo U 5.2 mcens lhat t he endorro>
   .
   
   ~
   
   w ) • g 01 Lh(.8) ill the t.anspr::toIO of the endomorphism ! Now ,upr- It..t X '" G.
   
   ~
   
   I' •
   
   I of
   
   L~(.8).
   
   P ropOrrltlon 2....5.3 uti a.... ~ b< , .... "-I.lrfJ·~fundi<ml from G into C. A/IeI, let I' ... a.........,.. "" G. S _ u.er. airtl S C G, ""' " ing 1', nell /Jw ( •. :z) - / (r )gf,r ' r )ls (' ) .. I' 0fJ·~ The ..
   
   r r dlPl(')
   
   1! (Z)g{. - ' r )1 d,8(z)
   
   -r r '" r d:l(z)lg(z) 1
   
   !/1.zl lx(.) dIPl(')
   
   1!(:z )g(· - 1:z ll l s (.ldjp0 P!(u ).
   
   ,
   
   1/110;' n"""'" if /inil.e,
   
   ~
   
   IN. /tt..m- xl ' j : , .... / /(:r)g(, ~ I .z ) oIl, ".&tm.at ..... " a' £I. """ .. UK1ItwJl, "',1lI"",We;
   
   (aJ
   
   ~) {I. . I ; :r - If("b:{.)dp('j '' tI'fi~ loc..U~ 9P·~ ~ ... opJi""" tOM iI <Woet\li&llr ,/J.illlqraWe:
   
   JI.xI • jl "I' .. {(P • I), d8.
   
   (e)
   
   P IWO'. ,.. (I, Z) .... p(z ) II (xl')@do .. ' '",.Ilk, tho: fuD(l;iOdetaU: for l',ot .....1 011 • E G, aDd ..... I/ (z)P{,- 'z)I I,(. ) dP(:r) it 1"">O
   r
   
   r
   
   0lil<1("
   
   f' 1J(:r),V'.zIldP(z)
   
   '" r r .r r .r ~lt,)
   
   If(.~)g{·- L:z)I I,(.)d8{:r)
   
   d!i<11' )
   
   I/(z)g{·- '%)1 15(.)d8(Z)
   
   I/ (z)g('-':zll ls(.)d{iI'l® Pj( •. 'zj .
   
   Nett. II. ,,) .... J (.:r)r(:r)I ,{.1 II "~rable and ~ for ,1>0 1..... U ' ''", w l3 fJ 01,. p undtr (I. zl .... ( I .•-'.,). and
   
   eo
   
   r
   
   I/{.z)g(, - ' %)11,(.) 4< s 81(',%)
   
   '" r
   
   1/ (.... )g(z)l ls(.) "lUl'l 8 81(•. "j.
   
   1Wf ~ ,he liM : .: .ilion 01 ,he ou,temenl. f ilially,
   
   l~ppQII:
   
   r
   
   tbat
   
   I/("IA'- ' zllls(')d!I' ® tII( • •,,)
   
   "' finite. tlw ii, (1,:1:) .... ! ("'lft. - ' z ) l $(') II ,. 8 tJ.i"t~. For I'.a/mc:JoIt all , in S, .: _ / (.:jg(,- ' .:) it 8- i~bk. Thuo, fur locally I'.al~ all • tn G,..,'" / (Sb{.- IS) ill P.!~ MOlen..,. , the functio"
   
   X/ · t,.-
   
   f
   
   J {.:)g(.- '.:) d8(:r)
   
   ;,; -eo1i•.I1r ".~ a.nd
   
   / (xl .
   
   j ) dp - / /«.)g{. -',,) l s(.) 4(p @.8X.,:r)·
   
   On th8 other Iw>d, (~, ,,) - l(u)g(z} IS(')X(~) ;,; ,,0,8-intqn.bie. Honoe, for b;allr.8-aJ"",* all z E G, ..... l (n W(z )ls (' )X(' ) 101 ".lntq;rBbiI:, and ..... I(unc,,)x(~) ~ !7m'iaIly ".~. Now .... I (u )x(,) 101 .-:nI.iaU)' ~ln\.eVlo~ lor locally ~,8-al",,* aU • E G. and . ... f l (u)x(')J,J(.) 100 _ially ,,"I~ Furtherm
   / (p .
   
   f)~~m .. / -
   
   ! (u )g( . )l s(' )X(.) d(" ®.8){.,:r)
   
   /1(:r)g('-' :r) I! (,) 4(p e.8)(.,.).
   
   o
   
   24,6 Convolution of FUnctions on a Group
   
   ... e<,w~ by th8 bmula
   
   Heoc:dorth. let .8 ewher .. (J > 0) be a ~la1i-..)1y inv&r;.nt n>MBIII'f: \>II. a locally """'I*" group G. We .. rite X (.... pealYd)·, x'J for tbe left multi· plier (","""",Ivd)', tho, "Iht multiplier) of 8. Recall llur.l tbe ptOPffly, for a
   
   2~.
   
   -180
   
   Co,.."ohtloa of M r . •
   
   oompltx·.ct.lUoN! function on C, to be kM:alIy 6-ln~.bIe doftI not depet>d "" the <,,*8 01 fJ. We let t (G} be the lOt ollUIICIM.Kw .-jIb Ihlt """",ny. When "'~ rqanl l ho! oompOSitioo lew 01 Gas. left act;.,n (reo;~IIVEly • .. ",hi. act""') 01 G onto lLM!lI. we can de!iDe. lor " in .M(G. C ) and f in t (G}, the con..-olvabi!i\y of" and / and the producU" . / ( respectiVEly. lhe
   
   •
   
   con-..ol..bililJ' ol
   
   f and" and lbe prodUO:I>I 1 ~ ,,).
   
   For f E t (G} and g E t (G}. the rel&tioo6hlp " / fJ and g8 "" convolvable" DOt ~pH>d. on any partk:ulu d ..... ol fJ (Ph........ l..,., 2U•.2). 1 and 9 an Ibm said t(I be conYOl.-.bk. (/P) - (gP) can be .mlt<:n 118. with II E t lG) •
   
   .x...
   
   •
   
   h ill determiDed up to lbe locally P-ntPlIibk ""IS. h .. wrilten I ' 9 and .. ClUed .. COIm)/ution product of 1 and !1 with re5pt<:! 10 6 CP" omitted whw no ambiguily " ~bIe). II ", " .. conl'DUOUIi bomoomoorphiom from G Into ~ an::I il" ;> O. thon and
   
   •
   
   •
   
   •
   
   I · ~ .. (/{J) . g .. I · (~Q). If ODe ollbo con\'OlU\ion produo:u ol f and g", conlillUOUi. it it Wliqucly ~. &tid It II called tboo conv
   Ld / E C(G), ~ E tlG} , gnJ z E G. TIle I"ndi.." • -- g(r' :r)f (')X-'(.) it UWltWI/r fJ· intnIy if. I (:r.- ')g(.)x'- ' I.) iJF .......u.:.ily 6·in',;, Mt. In tAit
   J
   
   r(·-':r}/(·)X-'(·)d9(.) .. / f(.u - ' )g(. ),i- ' (. ) dP{e).
   
   PROO', Reteillhal. x' .. X.o., ... hen!.o. io the modul .... lunclion on G. I'Itnber· ........., X" D"" ~ H..... _ .... 011 G and X -'fJ io .. ",ht Hu.r ..-.un!
   
   ooC 11 f
   
   _ gl ' ~
   
   ''' )/(')x-' (. ) is ..en1iallJ" ,6-lntegt.t:lIe, then
   
   / g('- '''l /(')x-'(.)dfJ(.) .. / g(u)l(.-') d(X':'fJ)(.)
   
   . . Jg{n)f('- ' )~- '(')X - '(.)dfJf.) .. J g{.:r)/ (.- ')x'·'(. ) dD(.)
   
   ..
   
   / g(n )/ h - ')d(x'- '{J)('J
   
   .. / g(' l/ (:r.-')x' - '(.)d,/J(.).
   
   ••
   
   Simll'rly, if • _ g(' )/ (:r.- ' ):(-' (.) it _ial11 ~jntqr&ble, \.h<:n
   
   / g(')I (:r.-' ) r'(.)
   o Pl"Opoodtlon 2.... 6 .2 ~II E C(Oj " M 9 E .e{G). S.."",,,, WI! """ "I thU<'! /IIIIcfiI>ru" amtill_.,.. ...m.Ilu ov.,; .. " vnimeIion I . , II Ii...... '-Ilr fj·Ill"",.1 ~ 6y
   
   (f . g)(:r) -
   
   / g(.-' :r)/(')x-'(,)d,6(.)
   
   _ / 1(:r.-')g(.)r'(')dP('j
   
   {I}.
   
   o
   
   PItOOP: This &:.IJcrA.s from Proposition 24.2.l.
   
   .s
   
   The-re m 2 .... 6. 1 ~'" GIld q lie 1_ amjtlgGu ~nU (I p .s +coJ. If Ix-' /' E er(C) ..ow: , E O(G), IN:n I GIld, ..... ..,.,.....m.w.., I • , .. ,;..... /oca.!lr (J·lllmQM. no
   o
   
   pROOf'": Soe 'T1>enre1Dl 24..U and 24.4.2.
   
   U.6.3 lllx -' E zr(p) and g E C-(Gj, .,.. if I E ('II(e) "nd ,:(-' e zr(fJ), tAn. I .,..j , ..... """"'" 1... 01, .nd fonn.Ja ( I ) cleJineo, e.....,.. .......... "" G, .. co"ooIvli(Pn """,,,,,I I . 9 t},,, j Iiu ;" ('II (e). o Propood tloP
   
   Wbeo V
   
   nmtI
   
   tluough tbe filter of neilhbotboodo ol~, tbe..u
   
   {(:r, r )
   
   E
   
   G "C : :r- 'r e V)
   
   (' $ pa;tlvdy, lbe..u ((:r,r) € G" G: r :r- ' € V }) form" lilt... t.sla on G" G. 11>0: filt<:< ~t.od by t bit t.sla .. a unifonn Irlru<;tu", (1<1 C, called ,be left unilorm 1I.ru<;tun: of G (.... ped;vely. lbe ri&ht uJ>ifonn 1I.tu<:t_ 01
   
   G).
   
   Jllx- ' e £T{,,) aM g E C""~ , 1_.Ja (I) ufinu. ''''''- I .... "" G, .. ..... IIOItiIKm """,,",I I • g Ukol .. Ionncl.cd "M ""il<Wmlr .....tm ....... /.,.. 110. rlfIU .1r1oeu. ... 01 G. Tb...." 'm 201.6.2
   
   .....,.,rm
   
   'I'
   
   ate
   
   E lilft} and {x-'JJt .. ),r'JJ, II.. function ix'-' lies III ZT(tI}. Gi- . % E G, 6(%- ')(}x' - ') bdonp 1<1 CT{x' - '(:<)6): thus . .... I (r. -')x'{'- " ~ 10 zt{,6), and • _ I (r , - ')g{. )x'(r') is' ' :.AWI,. tJ-In~bk, ~ by I ' g 1M function PROOF: Si,...,
   
   /x- '
   
   r_ and by" 1M _~
   
   f l {n -')Iri,)x' (· - ') d8(, )
   
   Jr 'JJ,
   
   For.u%, r'EG.nuw
   
   IU · , )(:z ) -
   
   f I/ (r. -') - / (:r'. - ')I f ., f .. JIb.·.·,Il- 11.w,
   
   {I . ,)(r')1:S N..{J)
   
   f I/ ln - ' ) -
   
   I (r',- ')I dv(.) ..
   
   dv(,),
   
   1/ (:10)- 1(r" II cW(')
   
   lI(n'-' ,) - / (· )I ,a.{·)
   
   lu · g)I:z) -
   
   (! . , )Ir')1 :S e .. Ao(lIl . . r'r ' 101 clofe ~gh 10 ( (f'ropoollion 2~.oll ). qll".III,., I • , is unilormly ton'i ...." ... for ,he rl&b, IlIIiIorm , tl'll<:C ure of G, Nut. I ' ll is.~;oa pn>duct of I and g, and I ' p ~ to C""Iflj. by Tl>eoc .... 2U,2, D
   
   H""""
   
   eta
   
   Theorem 24.6.3 Ld, aM 9 "" two ~~ ul"",mll ( I < p < +<>oj, Su".... WI fJ ;, It:fI ........... nL Ltl I E l '1flj ~ 9 E li~) . 1lw:n I ""of , _'" "",~.LdW., Forni. . (1) dtfl._, G , d ..... ...,,;''100I0 ,IIt"IIOd"c/ /hal kWng. '" CO(G) ..,..j ~;,fiCl N"",(! .g) :S N.(!)N. {j),
   
   £0,................
   
   For.,..,,')' r liI G, 1. {j) liM IlIlih.J:I) .. 0 (.8), to . _ g{,-' r ) beIoop to li(tl}, and. - f{,· 'r)f(.) io _ntial/, tJ-!rn.osraWe; , _, PROOI' :
   
   J
   
   19{· - 'rl /(·) l d,l}{,) :S N,(!)N.("I'. (Jj) "
   
   n ..ulott. I
   
   N~(IlN.{j),
   
   and g '"'" con...,n..,b!e (P " ' p", itioon 2,1.2,2). and formula ( I) Ind",, proo:Iuct Iud> tIw II ' , I(r) S N.U)N.(~) for
   (P,<>!"*tion U.6,2), For
   
   I., I,
   
   Obten-e tlld!. III gmonI,
   
   I',
   
   ~
   
   in ?t(G. C ). I . , liM la 'H (G.C) (PI'Op<>' IIiUon 2U,I); lhut. be I E v f?J and, E Zi(/IJ. ,lot C<><mJIu
   does DOl ~ 10 COIG) for
   
   I
   
   E ZI(tI} and
   
   , E ~1ffJ,
   
   'I'
   
   ate
   
   24.7 Regularization LH G, X. 8, and X be M ill Section 2U. and let p, q be t....o CClljugat.e ""poo>ellUJ (1 S P +00). Deli"" F ... filter OIl an index OIet I . Finally, let (Pj)~' be a family of """"""'" 011 G witb tbe foUo...iug JIf'OI>eI'lies:
   
   s:
   
   (Il) )C '/~ ill ",~;nt.egrable, for all i E / .
   
   fbl
   
   ~,. ISlIP""
   
   I x - llo "1",1111 finite.
   
   fel / X- I' Odp.; ~ to 1 along F . Cd)
   
   -,
   
   For~' C'Ompect neighborbood V of ••
   
   Iv. X-I ,. oil,..1OOD~ to 0
   
   PR()OF : Fb: , E CO(e). Gi...., • > 0, let V be ,. rompect noighborhood of e IIUCh tbal. h!<:Ej - g(e)j ~ for dl :t E V . and d _ A E F ~ th.at and /v. X-II'dill,1 ( £or all ; E A. Then, for ; E A , _ ba""
   
   s:
   
   11- /x-,'tdp.;1 s. g(l) -
   
   f
   
   gd{x- 1/,j.<, ) - ,(l) (1 -
   
   + (
   
   lv
   
   every
   
   s:
   
   Iv
   
   (g(s) -
   
   X- · /,
   
   d/>i)
   
   g(.»
   Th~
   
   o Theorem 2".1.1 S"I'J"')H IJI4I. P ;, ""''''"l'U 10, in Zi(p) AJO"l F .
   
   pl'lOOJ':
   
   w. ma.y ouw-
   
   I ,,'" dp.,.
   
   Ii"it•. and
   
   It:t, E Zi(8).
   
   171m,.. .,
   
   that g is 6--moderuL For eadI ; E 1, put '" _
   
   For~; E I ,
   
   N.(P; . , - ",,)"
   
   s
   
   (
   ,
   
   Si""" I _ I" is <Xm-.a OCI
   
   Ill, +00 [. J..-..'. iDOquali\)' ('Thtoo~ 3.7.3) sI--.
   
   'W
   
   r r
   
   N,u<" ' - c"Y' :5 0. - 1
   
   d,8i;J)
   
   1~,-I;J)X-I"(,) - g(ZII' d(X- I" !/I,l)f,).
   
   Gh..".. , ,. O. ther>e exiou. oompoo:t nci.ghl>o
   N.(-r. ... f.)g - 1....(e) gt :5.
   
   for all •
   
   (P.opasitloa :M.• . I). Noor <:hoot.! A E T .. thai i e A. Then, ror evuy; € A,
   
   i Since
   
   I., ~)
   
   eV
   
   Iv. d(X- I/" " ,1) :5' rc.r all
   
   d(x-If°Il<;I)(, }N,("1:. ... (, ), - i ...(e),)' :5 O€
   
   "l' (r)x~ 'f.) is X(x- "'", ) ® ,s.lntegrable and lhe lDOture ~ Ia tbe """"F of (X- I',,,,) @ ~ unok< h,,.j - (.,.- ' ,.),
   
   ....
   
   xU-" '",) e
   
   tbe. function (., ,.) .... ,{.- ' ,.)X- "'(.) liEol I.b £~(x-I}'", )
   
   @11) • .. ...,U
   
   ... (• •,.) -. g(.- ':o)X- "'(.) - g(:r). Th.... br -rubln!'1 tbeorem .
   
   J'
   
   d8(,.)
   
   r
   
   Ip{·- ',.)x-"'(.) - 1/(" )1 ' d(x-"'II'.I)(.)
   
   -Jd(x-" O[P, [)(.)N.h•.•
   
   N,(p., . II - G;1Il' :5
   
   "'t+ o"-'2"N,,,l"
   
   (· )g - "I. ,,{. ),)'.
   
   for all; e A.
   
   Tbe proof II completlO.
   
   0
   
   14.1.1 ld. , e £oo (P). Along T, ,.. . , "'''''''''9''-' .......tI, 10, in c.-(f1) (c.-(f1) ...,........ .. 1M d....J 'JII'« 0/ "(f1)). Pto~ tlon
   
   PROG,., FIx It E " (fl). Fl:x- eadI , E C ,
   
   (-r. (. - ' )It ., ) ..
   
   xl.) /
   
   It(u)g(,.) dtJ(,.) - /
   
   It{,.)g(.-',.)d~(,.),
   
   and ,. .... It(:rl!l{.- ' :r) Ia ,s.inlesrablo.
   
   ,.
   
   For ~ry; E J , the functiorl (I,~) ~ 9(,) ill X(x-l~)0 /J-meuun.ble. TInIIF (I, ., ) _ 9(.-'.,) ", (X-'~ )0/J-meuurable. Since
   
   r
   
   1h{z )9(, - ' r )1 4(x -1 1~10/.1}(., r )
   
   -r r 4.8(r )I " (r )1
   
   Ig(l- ' r )1 4(x - '1#.1)(.)
   
   - (!hI, 1~ 1 ' lgl), tt.eor..m,
   
   (h ,1'; . g) -
   
   f
   
   4(X-'I'; )(')
   
   f
   
   h(" )9(,-' r )4II(%).
   
   Therefure, if V ill a comc-t nei«hbocbood of ~, (h, 9 -
   
   (1 - J4(X-'~») (II , g) + J
   
   4(.x -'!';)(.)
   
   -i ,
   
   f
   
   U<; • g» is equal to
   
   " (r )(9(., )- g( . -'.,j)dP(.,)
   
   d(x-'I';)( ' )
   
   f
   
   ,, (.,)g(.-'.,)48(:r).
   
   Gh..., £ > 0, "" can choo8e V 80l hal N,h, {~)II - 1', (.-')h) S • E V. H~",,"
   
   f
   
   h (.,)(9(:r) - g(. -'.,j)dP(" )
   
   -
   
   l(h,,)- f
   
   _
   
   1(1'· (t)h -1"('-')h ,g)1
   
   X(.)
   
   £
   
   for all
   
   h(.r )9(:r ) d,(l(:r)
   
   "' ' - than £N..,(g) for , E V . "Then , takillj! A E :F 8O lhal.
   
   1-
   
   f
   
   4(x -' I'; ) S
   
   I
   
   roc all ; E A, ..., have
   
   1<11 .9 -
   
   v.. •g»1 S 2<"1(II ,gJl + ""Noo(g) + I N, (II)N .. (gl o
   
   PTo.,.,.ltlon 24.1 .3 LeI f E C"{ X ). 1kn # , ' 9 """1I<'fl'CI 10 9 olanf:F. tmi/ormlr "" tad! """" ",d Id. PROOP: #"g liellr! C" ( X ). by l'beoJem 2M.2. Let K be cornpace . Given t > O. there e:rIau • """"pace nei&hborbood V of e I ud! lhal g(" >1 S t
   
   1,(,->.,)-
   
   £or &II. E V and", E K . C t - A e FlIOthat 11 - j ld(x- 'I'
   (1 - Iv .I",) X-I
   
   +(
   
   Jv (g(%) _ g{.- '1'»):.; - ' (.) .11',(. ) - l(y. ,(.- ' >=)X- ' (. j <4<.(1). I, (z) - (p" ,)(r )1s 2<1, 1 + ...: + tn,l,
   
   a
   
   whleb romple\al the proW_
   
   Similarly, il g E C'{X ) and if, for fM:fY ! > 0 , 111m: exisU .. oompB<:t nel&hborbood V of ~ e..m that I'{'- ' %) - g{z)\ :5 t for. E V and :z € X ,
   
   ,hen ", ' 9 con"'..,s unlk>nniy to 9 aloo&: T . f'ropwi tlon 1 .... 1.4 SKpJJt>U. ~ erilu G ..,m»Od .<1 S C G ",mt4i"ing U;f;I suPP("'I_ 1/ g e e(X , C). lit.,. ,,",' • g om"'.,.' I<> , along T, ""i/ormJy "'" ..w. """JlGct Id PI\OOI': By P rop<:lIIIlion 14..3.1, 1'< ' g • cootin........ for all i E / , Lot K be compooct. Givo:n t > 0, then! nisU" compo.c' ~ V of e ouch that 9(' - '%) - '(")1 < t lOr all • E V and toll" e K . A E F 110 tho.t 1I - d(x- L", )1 :5 t and d(x- '!I>;1l :5 t for ; E A. Tbt;a
   
   J
   
   a-
   
   Iv.
   
   g(.,j - ().. • gl(z) - g(zl
   
   +f
   
   lv
   
   TIl"" £ot ol1 ,
   
   (1- Iv d(x- lII,))
   
   (g{:r) _ g{.- ' ;oj) d(x- ' ",1(. ) - (
   
   lv.
   
   g(,-' z) d(x - ' ",1(') ,
   
   e A and foe all '" E K ,
   
   Ig(%) - (1.; . g)(:<>1:5 lr.up Ig{.)1+t il + t .~K
   
   sup 19{:)I·
   
   'fS- ' 1<
   
   o lI.,..,.rorth , I" G. AIt<>, _ aIIi € l .
   
   -..e ... _ """lUI'l>e
   
   tl\&t X .. G aOO {• •:.:) _ .u '" tho: IRulUpii.r.o,tion tl\&t",., +00 aOO /'; is .. """"""" l.fJ wilh ba8e fJ for
   
   Pr"pOIdllon 24. 7.5 l.d K e G be o:>mpac~ 1/9 E C""{fJ ) ;, o>n.Iin ....... 41 .,.c;. "';n.l "/ K , Ihff> / • • 9 """""'Y" t" 9 GIong F . omi/<mr>Iy "" K_ PROOf': For all i € I . " • 9 ill bounde:I and continuous. by TIoeootm 24.6.2. Gl~ ~ ;> 0, Itt V and It be • In P""I""'ilion 24.1.3. Then, for all ; E A
   
   and for 0.11 r E K , g(r) - &<. • , )(r ) can bt ...Titten M alxwe. \\,'r ite j for tbe IDIIppi", ~ .... from G onlo i~lf. SillOl! 1906..... oj l :5 N_lJl .I!I'IOII eYer)"W1Iett "'" (j "6,, )(P), _ ba."" :5 N",,(g) locally t (z ),lJ-al"""" t\U)"~' """celoeally ~ben:_ Tberebn!.
   
   ,_J
   
   .11-_'_
   
   Ii>c&Ilr
   
   Ig{.-' r)1
   
   .,
   
   Iltr ) - (I• • g)(r)l ::;: 2€~» Ig(:)1+ ,,, +tN... (g) .
   
   o Pro_lt;,;,n 2".7.6 S _. til,,1 J, E 1'/(G. C) for 011 i € I . MOOWOCi, AI.um. """'" "':'ta, for I!CCh """"""'" ntigJll>arll<»t/ V oj c, "ott A E:F • ..:11 tllc!t V "" t .... U'(AsuW(h )· ul 9 E l,(PI be: """""""'" Gt ..dI poi'" of .. ti_ ......JlGCI od K . :no.". I • • 9 """""'l"" t<> 9 along F , '''"fo>rml~ "n K . PROOf': OJ,,,,,,. > G, lei V be u in PrupositiQn 24.7.3. C , - A E:;1IO that V (JQlIlaI ... supp(f,) OJ>r all i E A. TbeD, for .u i € A &nd for all '" € K ,
   
   u., ..
   
   ....
   
   ,(:0:) - (I. ' g)(.,) - g(z )
   
   (I - f X-I
   
   d ... )
   
   +
   
   J
   
   (g{z ) - g(, _I ,,») d(}:- '", )I') '
   
   o /;ow , ..... 4"" PI" "''' the usual .... ulu mati"" 10 tbe ~izuioD. of ,,- • ••..,... (III • Iox:alIy compACt. Next, ...;: «i"" an application aI re:n 24.7. 1.
   
   """p.
   
   n-.
   
   Tt-o ....m 24 .7.2 1d G be: G iooIJly """'Jl"Ct ,...,..,., aM id 8 (""""" fJ :> 0) /10. II re/lI.lmel) in...rieI", ......... ~ "" G. Efvi, L ' (/1) ooith 1M m..uip&&li.... (/ ...) -
   
   ~
   
   I ' g. Tho< .. doNd _lor '~qIIa: W of L'(P) N II kft ""'-l oJIM. ~
   
   BaMCh · '501", L ' (fJ) IV .
   
   ,E
   
   if GM <>nIr if n')'
   
   ~ IQ W
   
   I'" all •
   
   E G
   ~
   
   P1tO(I1': First, lUppa!e tlw 1(')11 btIon,p to IV for all • E G aad all , E IV ,
   
   -f
   
   Let I' E M '{G, C ). Sineel'7 g (1~ (.)g) <1;«.) beIonp to IV fur dIg € IV , _ _ that IV iI. \eft idUI of L (Pl .
   
   'I'
   
   ate
   
   ~y.
   
   1UW'* W io.o.left idooaI 01 L'(p). Let U be ... buio lor tl>$liIter 01 oeIp.I>otboo'''' oft. Fo.-~ V E U, chootte Iv E 1(+(G) eo lbat Iv _ 0 on VC and J Iv$ &IlerolMdior>lof U . Fix; E IV .nd lEG. By Theorem 2~_7.1. Iv • (~ • • ,) (>,>ILk' .... 10 L • • 9 in £T(,8). alou& F. BUIll" (t • • , ) .. Uv ' ~,) . , ~ 10 W . T hUfl g _ xCI-'h.tg) ~
   
   abo ~ In W . We ~udIo II.... ,,(1)
   
   24.8
   
   bei0<J9 1.0 IV .
   
   t.·
   
   0
   
   Definition of Gelfand Pair
   
   0 be ... locally com~ ,"",p Bnd K • oomPKt subgroup of C. Tbeoon1111 __ oomp!e>.-vaI\Lot>d fUL!oClione on 0 1K .,., in o;w..to-oae _''"'10«'''''''. ~ 1_ I " " (whe-rt ,, : G _ GI K .. ,he e.nonleal mawl"&.! with the cD<>liD\IOUi <:(JIIlp!e>..vaIL1Od fwtoct~, OIl G...,h lha, g(.I ) .. g(.) for all • e C I.e<;
   
   and oJII E K , <w ~ul'4lently _ h l hat.
   
   6(rl)g .. g
   
   (I E K)
   
   (t ).
   
   We ahaIlldeDtify .M _ l paoe C(CI K ) of all o:mtinuoUi fUll(:Iio ... OIl GI K wI. h . 1>$ IUbllpaoe (I/. fuoc' ionfI , E C{G. C ) that f&t4(y (ll. Wkew ..., _ Ihall ~ify C(K\G) wtth 1M IUbolpo.ce of C(G) .. CtC . C ) oonIiKln« '"
   
   an ,,,_ ..... that ....ilfy
   
   ,(I)g .. ,
   
   (I E K )
   
   (2)
   
   (<w ~"';.-.IenI1ji g(to) .. ft.) lot all • E G -.J I E K ). We da>ote b1 C{K\OI K) tl>$ ~ioa C{OIK) o C(K\G), .. bic:b ~ of tl>$ ODDUnUOUll fun<:tiom tbat are OOt on tadt le 0Dfld K IK. Sintt K io COIIIpIa, tbe i n _ ~ ,,-'(A) of t'Vft}" OI'XI,~ .... btet A 01 GI K II t(lmpow::\ (Se
   bI}fdion oflbe rublpflCt H(GI K ) '" qC! K ), oooalating of thole eootlnuous fu...wo...on CI K with ODR'PKt-lUpport. onto ... _ _ ... ~ ofH(G) ~ H (G, C). We$h&ll t be.ebre ;det.tify H(CI K ) with tb" IUboil*"", ,.bich <:&II be .. rltt.:n '" H(G) n C(GI K). 1 II I.;,e, _ denote by 11(K\G) Ibe Inter_tlon H (G) n C(K\C). and t.,. H(K\G! K ) .be inlmoclio>n
   
   H(C I K ) n H(K\C) .. H {C) n C(K\C! K). Equip 0 wit b ... Left II...,. _u'" "'G. H {G) is .... u~ (with ftIIp«:\. to 0DI\>'(I/L111on) of tbe ~ ~(C) (b«aLIM I ~ ouwon of "'G is t IM: whole '" G)- II foIlowt from (1) and (2) thai H(OI K ) ;. a lefl ide&! and 1i(K\C ) • ... ticf>t kIeal in HiG). Tho: 1nl('ll'(t;oo H i fl.'\GI K ) .. a ... ~a of H(G) (and aIoo of LhiC) . LeI. "'K be t he _maliud U.... rmMu~ 011 K. If _ put
   
   IIi') _
   
   j'i(
   
   I {III') dmK (I ) ,j"'K{t' )
   
   K •K
   
   I'
   
   ate
   
   lot all functiono I E CIC ), ,he mapplq: I_I' is 8 prOjoocUon 01 d~ _ _ _ C(C ) om .. tho: _ ....- I ~ C{K\CI K): , hIo IoIIooft dimody from lhe Itft. and ~ iDvariaDoe..r mK . Furwmore, ,"" projediolll - P majlII 1I:(G) on\.<> 1((K\GI K ). Definition 24 .8 .1 Ir ' '''' uIled.,. Gd&nd pair.
   
   alKd>ra 1I:{K\G/ K )
   
   is COOlmuUti>"e, (G ,K) "
   
   PTopa.;Uon 2".8.1 S _ 141 (G, K j .. a Gel/aM ""ir. 171m C ......i· .........T.
   
   pROOf': Let.o. be tho: modular fUDCI.ion on G. A• .o.{K ) it ,. t"OmptlCt IlUtwoup ot ~ , .o.(K ) _ (II , and.o. _ I on K . For eood! I E 1I:(G),
   
   11'dm(; - 111
   
   - 11 '
   
   (tot'j d"'<; (' )dm K (t )dm K(r'I
   
   1
   
   dmK(t )dmK«()
   
   B~
   
   l
   
   (tot' )dma(, j ' -
   
   1,
   
   I (tn') d",<; (' )'
   
   (sf'j dma (' )
   
   .o.(r ')
   
   1
   
   I (. )dma (' )
   
   - /1-
   
   1I' 1I d,lOG 111 / drtIG -
   
   Similarly,
   
   1
   
   p.o.-' dma
   
   -
   
   (ut').o. -'(')dm.o (' )dm K ( l j dm K (r'I
   
   - 11 u - 11 l
   
   .o.- ' )(tI() dma(. )dm K (t )dm K(t' )
   
   dmK (l j dmK (t'j
   
   -
   
   lu
   
   .o.- ' )(u t'l dmc(')
   
   l(f.o.- ' )d111Q.
   
   Now, po I in 1I:(K\GI K j, let 10 in 1I:(G) be such that 10. .. I 00. K tupp{f)K. Then 11 _ 10.' beIonp \.<> 1I:(K\GI K j and , _ I 00. JUpp{f) .
   
   This p o.'tO! 11oa\ f I dmo .. f I tlt. - , 04"1<: 10. ~II e '11.( K\GI K) and. In fad, by lbe 6$ part of lbe proof, for all I E H(C). The~. me .. ll. - I me . and ll. .. L. 0
   
   P I'O_IUon 24.8.2 (Gelfand) Stq>JlQH tMn: ~t.I ~n I1Iwjul<>r)o_ pm-' olC otodo Nt, far _,." E C, , - 1 lIt/ooog. to K I(.)K . n.e.. (C. K ) ill. Gdf-nd pair.
   
   J' .. l
   
   P ROOF: F... flVft)' I E 1'1 (0), pul TbenI e:Uo O.u.cb tbat l("'
   of . WIG (bta._l{me ) ill left invaria"t). l. ThO! impl ieol \ha' J' . "" II ' g)' tho, other band.
   
   (f . g)"(%) .. (f . g)(%" ')
   
   .. ! .. ! g(,- ')/I"' - " I drnc(')
   
   g(· - '% - ')/ (· )dmo (l )
   
   .. {§ 0
   
   I ){"'I
   
   gT=b · 1-
   
   r
   
   I .. fo< all I e H(K\G/ J<J. all 9 e 'H(K\C / K l. (f . ,)" - 9 ' /" g'
   
   ~
   
   /., -' .1.......
   
   10
   
   .
   
   ~"'~, for all
   
   l in
   
   J' -(g . 1'1 .. (g o ff: o
   
   be sbowo.
   
   No.. _ I b a t X _ G/ K it equipped with an i","8riant diM.".,. d:
   
   FOr a1I:l" e X &00 lor all. eC.
   
   d{U,I,) " 4(% ,,)
   
   (d COpatible wit" U"'IO\O)!o)U of GI K). ThfI ~ or G on. X '" Miod to be doubly l ransiti¥'< if. lor all (%. ~) and (r . 1'1 In X " X III>tiefyllli d{% ., ) .. d{:r' .1"). tl>er-oo ......... E G oudt l bat :r' _ u and ,; .. . ~. Prop
   PAOOf'; Let :to .. ~K. For ~ lEG , 01(%0 ,""') .. 4(%0, , -' %0), By tho, dou~ traneitlv~y. tl>er-oo exi'lt.l I e K IlUCh that .-' ~Iortp 10 IAK. Th"" I _ I belongt 10 K .K. &Od ... may &pply Proposition 24.8.2. 0 Now..., Ill¥'<
   
   1_ enmpleo of in~ n ~ 2, "";IOl
   
   Gftfaod pai ....
   
   t""
   
   G lor SO(n + I , R ) &nd K ro,IlabI.~"". of .... , .. (0, .... 0. 1), 80 tbat GI K it hOllnt:'OiDO
   lUI
   
   2U Ddlnitioe of c..1faDd "-0
   
   491
   
   (z ,, ) be the usual ....1• • product in R"~' , r or all z . II in S· . put d(z ., ) .. ltCOlI(z , II). We PfO'ii thal d is . dist.a.nce on S". Let z , ,. ard % be pOinU in S' suc:b tbat • ;. z and z ;. II. Dmote d(z .• ), d(1/. ;), .00 d(" ., ) by 1I , II, and ,., ""'p<'Cti",ly. z _(" . • ) • .00 1/-(11. z) z alii the ortlqonal prajmiool
   ..
   
   (" . 1/)
   
   .. (oin(a ).. + (z , %) • . oin(lI)tI + (1/ • 0)' ) .. coo{a ) ooe(fI) + $in(,,) sin (fl)( u •tI), be...,. cce(,.) ~ cce(1I + 11). T hill""""" that ,. ::; 1I + fJ • .00 t hat d iI ..,.uaUy .. dilU:lce. Next . Iel " . 1/ .00 "0, ~ be pOin1Jl in S' ruch tW P«< u...t z . 1/ alii distinct , and define ,. "" <1(" . 11). Then
   
   ~ - .cooh)z.. belorl&l to (lLto)J. and .In(,.)
   
   11 -
   
   .00.{,.1z
   
   bekIop to (Rt ).L
   
   ""h)
   
   ...
   
   I ~ - ~h)zOJlU..
   
   I.
   
   . ",(,.)
   
   11 , -·roo(' ~ II_ OInb)
   
   ,.
   
   There exiota ~ E C tho! .. nds "0 to
   
   (II - oo.{,.)z)1 sinh). N..... 1/ - cooblz .. g (~ - "'.(1lzo) .. g(~ ) - "'.b lz.
   
   he...,. II . gil'll ). We conclude t hat the lIC\ioo niAn oubmanifo!d S'
   """ on R "'" " R ""' . r o< all r . II in II., put d(" ,lIl" uscooh( - . (" ,11)), We prowe that d io .. distsnoe on H • . called tbe hype , bulk dilunoe. Let " . ".00" be poinu in H. l uc:b thaI ,, ;' ,. .00 z ;' II. Dmote d(" .• ), d(l/ . I ) . ...d "(" . 11) by 1I , fJ, .00 ,.. ""'ptrtivdl. Z + . (; . _)1 io tbe ore'·:'&O".1 prajmioo
   , (z
   
   q(z ) + 4>(z . • )' q( I ) + 24>{: . :)' .. • (Z. • )2_ ]
   
   + . (z . • ). ) ..
   
   .. sinh'(a ).
   
   yr
   
   ate
   
   + .(p,tl~ III; th
   •
   
   . _~ (1' + +(1' .•h) < • .0 t t!lop<>:t~ .
   
   on (IU)·, and
   
   and
   
   Then q{_) _ no) _ I. NDo-
   
   + (1',,)
   
   _
   
   . ( tinb(o J.. - ~1',. )., tinh{ft~ - .(~, .).)
   
   -
   
   Jinb(o ) oinh{ft}+ (... u) - a>lh(o)c:ooh{ft),
   
   h
   1. (... u)1
   
   COIIIhl :5 -'>(o ) ...tt(.8l + oinh (o )oinh{ft) _
   
   0>Ih("
   
   + ,8),
   
   and &naIJ}'., :5" -+ /I. Thll ~ tlw II ill ... tuoJly • dO" • ....,. "" H~. NUl. let 1' , II and 1'" 110 hoe poln'" in H~ .... h that
   
   <1(1',,1 .. 11(1'0."')' Su~
   
   tlw s, ~ ~ dittirrt, and &.Ii .... ,. .. d(s , ~). Then
   
   •••
   
   -. There ..... , e G thal; omdo 1', to l' and ('" - oooI>('T)1't)/oi nh('y) to (, - CGdo('y>:O)/oit>hh). SlDoo. _ g(",), tbe ... t lon 01 G on H~ III; doubly trantlti.." and (e, K) ..... Gelfand p&ir. When H~ if! equipp«i with the Rk
   d if! tbe p»' k dist&n<:1l 00 H. _In fod. if l' and ~ arI\ tWO distinct pOin'" oi H. and it..., IM't 'J _ ..-g COllI { - . (r . wl). t~ <:DIU ... wolqu.e IJ'O"' t ' P ' (0 , 11- H. sud> that P(O) ., s and p(t).". , is «i_ by
   
   P(I) -
   
   'VI
   
   ate
   
   I
   
   l.d a bo a ......u,. """"~ pOUp ..... Ie< fJ be • IeI\ RMr " u ... "" a. B7 abo.. of __t a, _ idftItily " tilolly tJ.'nt :,,&bIo comp!e>;.-aluod fuDttiono OD.a ..... tboir d · 1IIIL.b (P), ';' wlk lor ,be LC(fJ}.
   
   I. .... A bo a """'in _ " " ''' 'Iob;'''' of Q,{p) . """ that A(J . ~.) .. A(J) ••• b&1l f E L:, (P) ..... &1I. E a. ~ .hat, 1oreYft)' ,E ~(a, C), _ b.o.... A(/ . , ) .. AU). ' ( ..........hat . _ g(. )A(J" ' ) io a tJ.il>t Q, (P). Oed""'".hat.here . . . . a uaiq ... bou ....... _~ $I OD. a 0IIdt .hat A( f) .. JI' / b ..u / E Q,(JJ). &lid ....... • hat IAI-IJlI (_ .... com~_ 01 .... ...,;. boJI 01 M' (a .C) bo .... ..... '''i''''no ....., .. d•• _ioo> '" Theo... a 'Uo.1.2. 0DfI0idef .... lin>i, 01 A(lv ) oloo>c!I .... w.n/Uter 6_ ........ be alt« 01 -'iono 01 U ).
   
   """*'"
   
   2.
   
   ....
   
   -
   
   $I
   
   _
   
   E
   
   .hat .be """,1nuouI
   
   M·
   
   a io """"~ ..... $I io paoi.jww. ~ that. b """" "' .......... i,"""" eo' ,.u.) : /- $I . /
   
   >t"""
   
   Suppc.o
   
   :I.
   
   to
   
   IJII.
   
   I " ~ " -+01>. . be
   
   01 ~(fJ} io eq..al
   
   4. ,...... b a.be c,-clic pOUp Z/3Z. l.d .. be . _U ... OD. G. -.nth "'wort lhat; _ "('1 /11'1(0 doe! ""' ...... 00"11&", val_ S'- that 1110 """" '" ..... """,.....;.." ,.u.) • 1 - .. . I of ~(fJ) if 1Ir1c11y ... I.b.o.>
   
   G.""""
   
   W. bl < , <+oo.
   
   1 No<&tloo io that '" Ed, . 1k I .
   
   S_
   
   that. ............. • . ....., .. OIl G io ..... that 1'0\ (,. . f) .. ,v,(f) b.n / E L.b(fJ} 1f""Pl'Uo) if . poi ....... 1101 .. 1 (Shoot I ..... _ _ f~1 .. f lfl ~l b eYft)' f E 1f(G, C); lira. dec\..,. ,hat .. .. t\Iol • .. bert • • • """,pie,c ""mber wilh "",""ullll L. and 1Iel
   ....... If
   
   If""''''''''
   
   S Fer -=10 .. E N . ....... be .be _""''''' (O. I ) _ that .... ({Oll ... .... « 1) .. 1/1, Write .. 100- ®'2! ... . .. ' b .... 1_'" '''''" 0I .. ....de.-
   
   (~')'2! and .... lot ,be
   
   imoco n,,,.,,,, of ..........
   
   Shoot.hat ~ and .... _ _
   
   .L,. ;:::: .
   
   01 ........ _
   
   ,""''ISh ..' . .. " .. 1".11"
   
   (~, ... : " ....
   
   ..... "" Il).
   
   ,.
   
   .
   
   .. C n-. t € N ,wrlte,\,oorlhIob<.-a p' , . va/uod -""""- fuDction "" R' , ouch . haI io bQuodod (I>,- M ~ 0) i. R ' &lid ,,(w)'" • I . Rc 011" > (I, write 110 10< ,bt functio
   IlII" 'I" Ml
   
   f
   
   ~[~) r 1/ (%+.1-1(%) 1," • i "I!.'
   
   ..~y J.-. I/(. - ,)- f(%)I"v S~
   
   "".uo<" S o
   
   •
   
   I.
   
   p.."..bat
   
   1(1 0!loll.) -
   
   1(:1">1
   
   s
   
   J.....,"2" (')"
   
   -,(1 )
   '1"
   
   ~M ( , ) .. , ;o(1)d,.
   
   1< .-1 "
   
   "berc ' (I ) • oup
   
   ,
   
   {I"("'II : .. € S'-' }. Dod_ .haI
   
   Iv •,..)(.)- 1(,)1 s ~V N_ Uoo ) -I- .. ,If / 1_' ...[ ;1:! cII. ,-~
   
   .... J.• ,H'
   
   ..,(1) III < ' HO'
   
   -
   
   [o){') iW
   
   4t -I- ' HO I J. -
   
   •
   
   •
   
   0I(1) .tt. ,,,,
   
   Iv . p.)(. ) - l (zll s . V N...(,,) -I- . VM(k -I- I )
   
   J.
   
   +
   ...
   
   01(1)
   
   , ..,JI.
   
   .~
   
   G I.- p. be. pooici>e bounded
   
   DO R ' , aDd lot % E R' be"""" that k'!~:~ l),.{:I" + s:.) _ ' ±zr 0 .. Q - 0 ill jO.~. In II.. _ .... 01.
   
   .
   
   " 'PC"",
   
   ...... ", . 110 )(:1") <0"'....... '" 0 .... _ 0 ill )0 , +00(.
   
   .... ... _
   
   ,be jmo . . _ ..... 01" UIIdet r - ~ - z . For an,, :> 0, lot .. be ,.(- . ); baIoe (p o ")(:1") .. t. Cr) dm(p).
   
   Write m b
   
   r-
   
   f
   
   01_ ~ :> 0, d _ .. > 0.., ,bat ( l / a'V )p(z + s:. ) S • b .... 0 < " 'S .. Nat , l'u.O <":S m.
   
   1/< ' ,.)(:<)
   
   2,,, ('1
   - /.
   
   "' · ..
   
   I(z)S~VN,,{,,)+ "M J(.... .,. II'I~""""U')'
   
   2. Oefi.... fwoaloa hou R' by ~(,) .. 0 fo< • E tr,. aDd ~(.) .. 1/11'1'" lor .. E (ly,.r _Sboor that
   
   J
   
   1Io{, ) 4m(,J
   
   ..
   
   / m(h "'[I ,+ooj)dl
   
   ..
   
   /(k +I).-("')«~)""
   
   .. 1.-
   
   (k
   
   ......., c:( ..) .. m(" "(M" ,•• IJ . 1.
   
   f'n>aI - ' 2, ..... """ tbat [
   
   (k
   
   c. "'- -'" I
   
   f. ;"! ... }" (1/11,1" ') "",(, ) .......... 10
   
   1:(t + 1)00 ' ''' ''n' V on. + m(s;.)o. _
   
   _( ' 0 1).
   
   3, dod ...... ,bat ", . .. Ie.:) ill ..........
   
   _.haI II' • ,. )("J
   
   •
   
   +0011
   
   + I ).. -1 '+')m.I!Y.Jcfu + mff(.10 -'' ' ') _ m(s:.)
   aod ... ,han
   
   Co....
   
   + I}.. - l" "m(w")",, - m(u..)
   <011. . . . . . .
   
   10 0 .. " _ o.
   
   Let I' be" boullded _""''''' R ', aod lotI' .. ' .... +(p - /Ao ) be.boLebeogue ........... tiooo oI,,";'h I , . 10 ..... [a d •• ,,'0';'" 01 En..... 4 . ..... 1o
   'bat.
   
   .us "'/(..) .... -0 ;,,)0.+00/.
   
   1
   
   La T bo. .110 _Ifipllcod .. """" 01 <:Omph ....-.. with
   
   _ul ... I lnd J.t
   
   "bo. . be _.....tb8d 11_ " In ... T. PO< e>'try ~ e Z· .
   T 3C _-2.... " D,(el ~ "' I L, .~ $ '
   
   'C"'
   
   L
   
   - oS' s_
   
   .~
   
   by D., ...
   
   (I _~ )" "1-1
   
   o. (...,*,;~. ll.. ) ;o
   I. p,.,.. tI•• •
   
   .0.,..... ).
   
   oi. (n ... Ilu)/ oIn(u ) 10< oJl t e Z' Md 100-
   
   .U~ E R - Z .
   
   l. 1'0<.,...,. R e Z· *"
   L wl>ett • - (,,-. /ZI)
   
   'Ie
   
   e It -
   
   Z , ",-,.I>&!
   
   ';o ((2k-+ I>n) _ : + I.
   
   Eososo .,..•••. Oood_ :l. ( ...., _ ....' • "
   
   .bat
   
   (f" + lIn)
   
   I n ... \ ) oif,f(u ) '
   
   ••
   
   J6,."" _ I . Oi ..... 0 < 6 s in. • ,ite V (il 10< ,1M ott exp (2;>'(-6.41). SI>ow 1M< h _v ... A.J.. ........ g ~O .. n_+oo ( _pNt21.
   
   &.
   
   Let ~ be. d
   
   l. 1"- .1Iao
   
   .,.'" 011 T
   
   and J.t ;FI' bo. tbe Fbo.a ..... Ifa"dom'
   
   Z". _ /
   
   C 'd"( )
   
   01 ... F\:I<.......,. n E Z ' , _ " oM!.
   
   L
   
   (P ·o,, )(Cl -
   
   TI'(t )('
   
   -·~ · s·
   
   (.00 . A.)(C' -
   
   (1/( .. +- II)
   
   L
   
   (p . DJ)!C)
   
   ·~s ·
   
   and,. .
   
   10< 011 , e T. ,. . D. A...., ealled tIM nth fl>wir< ...... aDd tho. ~.~ Fijet ... .,. 01 1' , .......... ~_ 6. .... , e q,(I') ( \ S p < +«». Dod_ r..,." puU 3 lnd 4 ...... , . . In fan, 10< I < p < +00. 11..,... boo ,' .... ,hat , . D. . -... ,
   
   ,
   
   to
   
   f i.
   
   ,~ ( ,,) .
   
   .u-1nIqr"** -n.o 0 ..
   
   G~! e 4,1..)•• M 1 . _ , : It " r - f (e'' '' ) 10 1".;..Uy I.e<. E R be ...... that (1/ ..) 1. 1,(%-+.) +,(z - r ) - 2f{z)1 4Jo ......
   
   .. - 0
   
   no 10 . +OO{ . !Nt , ........ W~ _
   
   tt...
   
   u . 4~)(() ""'".... to II()
   
   DoMe ~ _I H .. . ... ~, - 1r(" "' rl+r(,, - , ) - 'lg(z)I - , , J; 1o{.1tJ,oo R.. I.e< .. 2: 2 boo ... i~.
   
   I.
   
   S"- It..;
   
   !
   ,/.". ".
   
   ". ";aC..,) ~ ~
   
   fo< , II'
   
   to> 1/21-
   
   Oed .....
   
   IbN
   
   lu ' 4.._> 1m -
   
   J.
   
   io ....
   
   ,n .2- 1'nH(,) ~ . "tr I
   
   ~(')+ ~!!!!l
   
   ..
   
   / (()I
   
   If'
   
   2n 1/.
   
   I.....
   
   ,.
   
   > (I, ......u ..... O < f $. Ill....,.. ..... H(..):5 qo lor 011 .. II' (0.6)Cboooo " 00.11&1 ' /" :5 , . Show .br. lu . ~._ , )«) - /ml lo .....han G~.
   
   '1' , , /.'n, r, ') , (') '2 (....
   
   "H -
   
   I
   
   +- H
   
   H(,):;;q+2n
   
   + 270
   
   ,,~r
   
   <
   
   H(,),oiJI .
   
   w ._
   
   (1/_h'(CUP(2irl_ ... al)
   
   T . aDd ... be • poiDo. of T ,..:II .haI _'w'- (I .... teDdo to (I ill (0,1/21.
   
   tbat (,. . 4,,)«) """'bi
   
   to (I .... -
   
   !At p hoi • pMi';'" ....... uno _
   
   +00.
   
   .. 11')0 , +00(, put V• .. <.,.., (2;,,[_.. ,.ll. Lot n be an in. V' " 2: 2. s- 111M A.._,(C/€) :5 .. b 011 P'"D" { iD V' /._ o.a.... u.... jv d. _. (C/<),,*,({) """,upa '" (I . . .. - +00.
   
   For -a.
   
   I.
   
   ".
   
   t.._. {
   2.
   
   f'r<MI ,hat
   
   3.
   
   For.ol 11')0.+00(. PIli F, _
   
   J.
   
   T _V, / .
   
   for aU
   
   {c.·..· ,.. IE E.,
   
   6. _J(C/€)<40«)s/'"•
   
   ,.(F.).
   
   riDE. - [- 112 ,1/211/("'.-' ) 2: I} . Show IbN
   
   24. Cl: ,,00101;" of }I p
   
   ~ 9Ii
   
   ,.
   
   ..
   
   6. Ci ...... > 0, Irt 0 < & 5 In boo ouch . 1Io.I ,,(V.) 5 C1>o<>oo n ~ 2 to.hat II_ 5 ~ · p.",... , Il0.l
   
   at lor
   
   011 . EJO.6).
   
   L_~" , A._. (CIf) d;.(() 5 [,. I'lV.l(I/("'~·») ~~
   
   1.- 11'1( 11(2)o.')) .t~ _.Iw>.
   
   + o..d_ ,hat I/' • A._, )("" I II
   
   -
   
   (I/'~V",).
   
   lor ..... ~ ...
   
   Let. " boo. '-""' .... T ...... Poe<; I' _ /~ + 1'. boo ,boo ~ docompooiIiooo of .. with ' ........ to ... Flon> £»r:r<:_ 8....t 9. dod_ .hat, .". ... _ 011 C. "'. )(C) n ' ,aM ' to I (C) .. .. - ...".,.
   
   c.. •
   
   Index
   
   alw.oluuo COIItinuily. 201), 010 1 at:.oiuU! vol"". 7 abooIuta ....1"" 01 oompW; Ii,.,.,.., i:>rm , 16 &!.::>lUI
   o.b.o!uu.!y c:ontin\OOl>US fw><:tion,
   
   ".
   
   -.,""* ...
   
   lIdditive set funo;tiooD, 30
   
   oJ_
   
   e<>u ywhcre, 44 ~umy. 312 ..... mblnK ""1""""", 376
   
   .-~
   
   _ k m"5 , ,,,,=. 80
   
   Baire IOta, 419 bMd, II L
   
   ~
   
   /kppo
   
   ""po"';"", 86
   
   Levi'. t'-"em . 42
   
   BertllUin poIynawial , 161 bi_niallaw, 130 Birkbolf'. crpIic .tot... """ 237
   
   II<m>J ",.algebra.. 30 BoTeI ', normal numbo:< . 1>0" hm,
   
   '"
   
   Iknl-Canltlli ~mtnM. 31$, 123 I>ound'Ulatlott. 2ti6 Ce.otQt 10)\ . 88
   
   Ct.ow ....gular function, 89 O!fltBi limi t thtOi tlt,. :J.36 eb&n# of ......... blo . 236, ~2, 423
   
   dw'aOI'1Il.III nu~r, 3Z1 (OnditionoJ ~xp«t.ed val"",. 30M,
   
   '"
   
   """,di.;."w \0.... 365 ooojupte expOnents. ge OOII!upta ..-sure, 17 continuity wilb ~ to • polrr.mHef. ~ I
   
   801>1', th,,(H(w, 171
   
   _"'C"""LICe in I..,., 3Z7
   
   Boo",o oJ~&. 20 Bon!l IOta, 30
   
   QOn~
   
   in o>eMure. 105
   
   0>0\1' "'3''''''' In probt.bility, 312
   
   ..
   
   ....
   
   ,
   
   ~nce
   
   in tbe!MM. C.
   
   <:OIlw/utkK> produd, <1li6. 469, 47'9, 4l(I comroIvabie fUJ>ctiono , 479 ooo",*"bIe~, ~ W ~ ..... Irix, ~, 313
   
   _DI W ;"'ego.!, 68 ....",U.J prolongation, 81
   
   _",wly in\"Il"~ fu""liou.. fiB
   
   EllIer &IlII1fs. :ted value,
   
   au. ~
   
   txtremdy Dan ..U ",p'~ , U
   
   Fatou'.
   
   o.tboux 1IUmII, Jf.O De Moi~Lapl""", tbeor..m. SlS do:<:ompOsabie fund ..... j\I """' _"position in die«, ilIl • W'o"pOIIitlon lemma, .6 (4, ""position of inunai, 2!iti. Dedekind oomplet..-, 1Il deruill' of m '' ''IN), lilT, dIU deri....t]ve of intesral, 2&1 derivative of _un::, :u:! dill'ermdation under tho Integral $,gn. ~
   
   difr_ _ Iln!. ao
   
   '-...I, 496
   
   ditjoin; oompWnea.I , .6 dl&joillC. eiolDOllUI, 8 diljolot" ,,., ..... l()3,
   
   j(l{
   
   dil!joinl.!IeU, 8
   
   d b lribution function , 293 dominued <:OIl""~J>tt In "-"
   
   _ , ll2
   
   """'I""IM am'e. gence tlwooCDl, doubly
   
   lransili~
   
   ac;tkll>, 490
   
   dow~......d-
   S
   
   dyad><: ""I"''''ioo, !!7
   
   E..,...,.'. tbc:oi ..m, M emplrioeal rustrioolkll> fuDCtion,
   
   '"
   
   ~_
   
   emplncaJ median, M8 ~ ddinltior:t of ablIoIu~ O>nIinuill , m.flH equioalmt fuoct ....... , ~ equi~ maoumo, MY e~ ID&pJIiq, 239
   
   Apace, 2l
   
   !13.
   
   Fatou'. dlOOlcm. 49 Fq.. leer",,] , 496 F~tu"'" ~
   
   ""ier' . formul&, 2J.O 611 ... of lleCtiono, 18. finiU: variatioo, 1.6 F",,~ 1m ...1on lMoo,,,, . .m Fourier transform, 296 Fubini'l lheorem, lS3 funcbon of bounded "",I,,""' , 2Il6 G'pIIoCklng, G~, G~l&Iioo,
   
   directed STOOP, 5 Dirkftlet
   
   d~
   
   :u.s
   
   G."", . 419 p.mma function, ~ 2tifI Cebeoo.l>tt po/yDomiaI, 2J..8 (;elf. "" !*it, 48!1 Gliwnko-Cant.elli tbeorem, 3.\l Gooclw'ov'. thamm, 3.Y Gram'. dd~rminMl, i l l
   
   Gram '. onlooconaliution, ill Haar m 'F ' U"'. 429
   
   flal""", ...u, 30 Hamburfler'. """"",01. probiom,
   
   '"
   
   Hamel b&$io, 110 Hnuodorlf'. I ..,.,..,01 pro~ l.62 Hermite fundior>, 2J.6. Hermite poIynomial, :llIl HOlder'. inequality, ~ by~boIi<: distance, 491 idI!&!, II
   
   _""', :lll. <W9 lma&irwy port or _"«', U Im"le
   
   lDdU$loo-<e>:dus\oQ fonnu1a, 3.1
   
   lode. U"'" in~.
   
   001
   
   m
   
   locally at..Iutdy _lnllOUl
   
   ill 1~ ... t"~. lLI iDdllCled F71,re. Ill. 380 indlQ(! ",.,;"Co AI iDOqUli.y ol """'Diablo: CCWhex\.y. =
   
   .
   
   lno!quality
   
   of. u...
   
   _
   
   129
   
   produa '" -..... m " ..... 1JI lftle&nlblot fuDaion, iii Inte&rabIe WI., M illfinile inner r
   
   '''' oal.,
   
   4.} ;~ by
   
   ..
   
   1ocaI1)' count.t.bIe cl-, 3&:1 IocaUy imqrable fuDt'lion , 1D5, locally
   ""-'" dm .....u.... 2fIl
   
   ~
   
   lnw:cnJ. t.I.
   
   ~ .. tbeottta. 22l
   
   M...d..-Opee. ...... "'...
   
   - ."
   
   n'\aIn pt'(Iloo:>g..uo..
   
   plUtlion
   
   of.. rn : ' u.....
   
   ~
   
   PNU. 272
   
   Imwiaat wore,
   
   run.:.1oa, m locally -.1_ ewryw'-e, 62
   
   j2fI
   
   m.- FOOl............rorm. 296
   
   mappl"l Ib ..•......... l2! Mar"","" equality. ill
   
   JIICObi poI,_nial, 2.U J -a'. !~I)' "'lXI:pecloed val""", 3dl lneqllali.)' t~. I8
   
   m
   
   Jo,"",,',
   
   ~j+4J7"
   
   Ld
   
   cue
   
   '"
   
   K~'. cxistt
   'lln,
   
   jj
   
   u..ore.n,
   
   '"
   
   KoI""'COOoo '. _ _ La.., 319 ~'l lbeotCi ... 2S7
   
   A-aywum,29 I Ar" x', _hod , 2aI
   
   4g , LOO , •.••~'••• ""'0:1 pair, 392
   
   . . do n.ndcm ~ . .1l2 Lobtecue M,.,,~1on 01.
   
   I""Mifquale fllElll)'. 381
   
   functio" . 279
   
   Le~ detomp
   w"'.
   
   m
   
   ~n 39, l88 Lc~ tbeat "", lor funct!onl
   
   ,..dc:_ family 01 com~ leU,
   
   '"
   
   11_0 E1U ... bIe
   
   function, 113. Wi.
   
   "'-'" ,,·tnOdt. ''''''''lola, $
   
   ,..-p ..,rabje~ , 62 Ie
   
   wiLh fonJIe do:riwati-.
   
   lI'proper rnaPJII"l . !IOIl
   
   '"
   
   ~1a.. ,
   
   WI. rfCUIar it.... lIlatloA. '-1'1 lheoo(w, 302 Lindtlw:q l~rn, Jl2
   
   m
   
   _lied poJr , multiplier, m
   
   m
   
   L30
   
   ,.... _ural !lemiri"l! of .on !nlcrttJ, !l!I natural ~ri", of R· , 181
   
   Radon meu ...... l3! ftadon-Sik>dym tbeor=.
   
   """,Ii>e 01... _ . "!Vp1j~ p&rt . !
   
   nu>dom vaNbI., i l l
   
   f"'&Ifcl~
   
   funotioa, !l3 non_urable 1OtC, 90 I>(llmal law , J06 _mal ~! 1If>I>':f:. l3oI. nonnaln..! H..... meaaue. Ufo order dual, H Otdet 'deol, II (Ir't\er of .. u ' 29-1 order stalOn .... 3&S order ....mmabIe family, 18 order·boo'nded lioear Ionn. 14 «deled direct tum, to orde.ed V'O'Ip, !l Oidued _ 1I*>e, 9 wdlI.t.UoD. l1iIl
   
   .rt.
   
   ,, -
   
   ~ ,booow~ j)on
   
   ....u hnur form, III 1ft!
   
   p&tt
   
   U&Ula:r
   
   IIlCOI8U""
   
   poW _dina ...... l50II
   
   l'oIya'. o;riler>on. 3OB. poifItI~ neu~nt.
   
   II.
   
   poo.Itlve l i _ form, ltI politi.... _ ...,,"t. 34 poeitM pm , I prob&blUty "f*"", ill prod_ " IP ' ''''", 1& i l l prodllCt proboobilitr. 113 produt\~ m
   
   prod"", I'eft'Ilril\f;. U8 produ<:l. ""riq, at pro}ectlve '~ID, 230 prt>po< mappi"" II3I
   
   quadr.bIe ed. 150 quasl-ln~
   
   fu"",,ion , Z6. quaoj.lnvlrlant meloSlll'e, ~ qUAtl·.-unt. 32 qootiem of " t". (J, -"2
   
   lfifI
   
   n!gUlariulion, !W rqulauod function. M
   
   mati...ty Invariam. meuure. i26 R¥-.....perti len>rna. .:w Rletnann lnt.egrable funaioa, l.JI1 ltiemt.nn 150
   
   ill""'.
   
   rucman ....LtbeIgue lem...... .1IXI Rleoz rlemmp"'ltion tbtot ..... 6 9
   
   rue... """"",,
   
   R.!om· r ietbe, Ibtoo:."~ !16 ril#-
   III:rnct, 262
   
   of linear form, U
   
   -.,, '"
   
   29
   
   r~,29
   
   Pc '
   
   ""
   
   m
   
   "
   
   function, 34
   
   sampie. 3&I
   
   ...-0:1 mean Yl\lue formula, ru ..nri·'gt.... 30 "mirina;. Z! shih tranolomlAlioa, 319 .,.&in~ ,!>leo fuDOtioa.!l6 Mandan! deviation, m 1IIandan! DOI'1Il.&! ......""' . 29.'> StioMtp. _un:, no St~I)eo.' mOlDent problem, 1.IW StoDe ~oc.ation t~, 20 Sto~' 1 conditlon, .ill Blrlctly regul&1' meuu"" 1tl6
   
   m
   
   at"""
   
   law 0( Ia,. 1lIIlnbf,,., J2I)
   
   strooc 8emirinc, 39
   
   ••
   
   .ubdivilion
   '"' ' 'UW'fI
   
   IIIp!!i tidal '!lppOtt of
   
   
   funct ..... , LJ:i support of meantffI. llO .ynomccr\e dilfereoce. 2ti
   
   tall_, 3Ig t..u ""field. 31g Ifcbt ...... ill ToneUl's t~OOOI eHl, 1M wuJ ...... <11.. boo,~ _""'''''.
   
   .
   
   t~ion
   
   min&, JIll
   
   measurable function.
   
   upper Mri.... li~. 2:SIl upper _Iia! lnlqral. 6I:l upper !nte&ral, i2. 12 uP""lU'
   t.opoIr:v. W
   
   varisnoo,
   
   29~
   
   "";&ti<)c.
   
   1
   
   vari&tion 01. funo;lion. ~ nri&lion of o..udl ~. If! vari&tion
   tn.dlc. e:<panlI<m, 8:I I~Lar """y, nt uniform ot
   ...
   
   uDi,~ly
   
   of condilo:.n..l upeo:ted ' 'alue, loU
   
   """-e,,,,,,,,,,
   
   In ..... M..,..,. LIlr:i uniformly diltribllted 1flQ""1>OO . ill unibml), lntqn.blo ..... , lOS unimodillar ~P. ~
   
   YooidA '"Pi Illation t""""'m, 23 YooidA- Hewill J.wmpoeitlon tbello..", ill
   
   Copyrighted material
   
   S)'!llbol Index
   
   A"': com, ! metlt rL A (In"
   
   &I"""
   
   qM(O, C ): ..... 01 QUIIIt_1UtOJi U
   
   ~)
   
   M (n, C): 'I'N'"' 01 f)e,,'el J
   
   ~- "
   
   A - B : diBae..... A ll B"
   
   QM(O, R ):
   
   itA: ftIIIIkl .... 01 fu .....1on J to A (a ,.): Interval In R willi endpolou .. aDd 1[
   
   "D
   
   \0," : opftI ild.cn&l1II R :H{n, C ): _ _ 'pace town· me 01 oomp!ex· walued
   
   - "
   
   1I:(n. Ry. 1we d. '--.1""" fullC\ioQf III ?tIn,C ); 1.6
   
   'H"' : "'" oI""';dve ru"cUo.. in :H(O, R ); 14
   
   ~of
   
   ral quasi-
   
   -"*" "
   
   I,,: IryIk"," fund ..... of A (on"
   
   " ..... "" OJ
   
   _
   
   M (O. R ): of - ' Donlell , n IlreoJ: U M +, _
   
   01 po::.iti.... ~1 " :.-,..... 1.8
   
   1': CCIIjupte of c.be
   
   Oo,,!t!!
   
   " ' U.IWp; U
   
   LuI or
   
   v~
   
   abooiut\': ~. or ~kIo, 01. ,.; 16
   
   ,." and 1'-: poIitive.DC! ' .."'hoe pmU
   
   0111>'1 mil "
   
   ....
   
   II; U
   
   . (C)
   
   or C: " .rln& ..."aud "" C;
   
   "
   
   r hIed IT
   
   r
   
   st(S.F):
   
   "'*'" d. s..lmpko fuatUona from (lInt<> !.he >'IdOl"
   
   "*'"
   
   ()( (I . ,,): imfCr.t of IE £).{p): 4.S
   
   F ; l4
   
   St~ (S): aet of pCIIilioe
   
   ,
   
   ,,(n. ()( f I dp. ()( f l Iz) dp( z ),
   
   S-.iwpko
   
   /'I, (J)'
   
   ~...JuaI
   
   funa ...... 36
   
   QM (S .C }, n
   
   "*'" of u" on lbe ...mrinc q......
   
   r IJldV" brfuDclioa lhe to(~ I:
   
   £' {p: R}
   
   " "*'"
   
   QM(S, R):
   
   IDrMW'tII
   
   of..eal qUM!.
   
   f: 1 (1)<11: inu:gr.t..i!h ... ;:«, 10
   
   M·(S): oet of ~ WET'" on lhe lHIIirlnc S: 37
   
   d. fullC1lom from (l1DlO (0.+00\; 41
   
   r ldiJ"or r
   
   1(~) djO(~):
   
   \lPPft II\Up1l! d. d:le Pl"ilioe flmctao I ; 4'1
   
   "
   
   I(z -): Ioft.hand lim\<. ol f; M
   
   'it: rl"l 01. jO·i"\~
   
   ..
   
   01, ' .......IuecI
   
   I
   
   auoh N,(J) < +00; ~
   
   ful>Cl.lono
   
   £~(P):
   
   ,ha.
   
   . , ..... d. p-i-cr.t>lo fuDCdon.; 4.S
   
   L).{p): I[00I,",, ~ do £).{p);
   
   "
   
   J I.
   
   --=
   
   56
   
   d" of E: E
   
   R;
   
   ® F.: produ.<:t "'-'Ptn: 61 M : ~I_ 0I..u , . . _....ble jO' (J), or
   
   "r
   
   oet4;
   
   r I {z ) dp(s ): \IW<'f ..entIA! , .... fCI'7!I Idp. OI'
   
   01. lhe pooIti.., functioa
   
   able flmctio;Q;; ...
   
   ...JuecI fUlrtIoa J; 42
   
   .
   
   ,,(E): ..-ore
   
   '!'i(n, Ill@ F : "I*" of detom~
   
   r I/ l dYjO for the.-ector·
   
   M
   
   I {z +): npt.!>and Umlt of I ., :z:
   
   on S : 37
   
   "*'" 01..-..1..........-. on S: 37
   
   F~u<): III*'"
   
   48
   
   ~= '1 ru .....
   
   M {S , R ),
   
   N,(J):
   
   01 in~ eo·
   
   fUDCti
   M {S . C ): "IW"" of" rMUftlt "" S;
   
   jO' (1), or
   
   lei
   
   u;,dtd re.I-...I""'"
   
   S; 36
   
   :r:_
   
   "
   
   1:66 N;(1):
   
   FtIJo):
   
   r I/ ldV" lot \ he _
   
   "41""'" functlor> I:68
   
   •
   
   EI*'" olw:c\Or· >1!IuaI rullC1icm1 1 auoh tlw.
   
   1r,(f) < +00: 68 q:{P):
   
   
   of ·
   
   mo-n,-
   
   intecr-b18 functl!lM; 68
   
   ,,(f) , or j 1diJ" .... j 1(,.) dp{,.), ....
   
   f I dp . or J J {z ) diJ,{,.).
   
   V .p):
   
   _u!aI iDupl 01/ E 4(J1); 68 (It
   
   I" / 4 : 1/1" 4
   
   for / e c].{jI)
   
   L';.(P): quod""t 7'/;(1):
   
   Md JI E M . intosral 01/ ()\W the eet JI 68
   
   ~I: total __
   
   r I 4Vpol,,; 69
   
   .1: at 01 fUDCtioo:Jl /I
   
   with 1M 1oIIow'i"ll propetty: I bcre E 'H{Il. R ) ouc:b thal ~ to .1~ ;
   
   ex.".. , 11- ,
   
   Nld """'" Illt<'lnOl 01 the em:aded '-.allled I\mclioa I ; 12
   
   p ·(JI),ouwrll" I'~rl"dpol lbe eet JI; 14 " . (JI): In_ a
   
   ;.:. _n
   
   LIf'II!
   
   01 JI; 14
   
   pro!oap' ;'" 01 p ; 81
   
   It rit1& of •
   
   ~(,,):
   
   oI,,·.-utabIe ~ In ~(jI); 119
   
   IpA<'e
   
   lbe _ _ ...rued fUII(\"'" I ; \01
   
   C1'(,z< ):
   
   of F "!o,lty boundod fu.J>aiozzo; IOl
   
   _pooa!
   
   L'P'(P) : quo!.lcot"1*'!' 01 ';'(... );
   
   '" C(A.,.; F):
   
   IpKe
   
   01. funak>M lhal
   
   ..... ,....-uabIe oa A;
   
   '"
   
   ... pp(f): IUwon of tbe fullCUoa
   
   [I" l~dVpl '"
   
   for p E /O.+OO[and potIlfve/:
   
   "
   
   .
   
   N,(n: tf" I/1'4V"p" 1or
   
   the
   
   _,..h.rl func\.loll / ,
   
   .r;(p): "P"<' 01 vector·.alued fu~
   
   1 IUd!
   
   N,(I ) <: +00 +00); 98
   
   4(P): ~ 01
   
   I;
   
   i'I,..(J): eIf.. ([II). b
   
   I; l~
   
   fl.: T',·!aI pn>Ionpli.ooI of ,,: 8'1
   
   N.(n:
   
   lor tbe vector· .alued fUDCtion
   
   r,:(P): fP'!« 01 """"""....hoed flEDCtioo:Jl ouch lbal 7'/;(f) < +0<>; 99
   
   nt;"ny lntqrable
   
   -,81
   
   " If" III' 4V...P" "
   
   ro
   
   I" / dIt. and I. 1dp: UP11ft' IntosraJ
   
   "'*" 01 ' o;.(J.);
   
   1(U, F): I I*'!' of """'1*Uy
   
   ",pported C07ltlDuouo fU7>Cliozw from n Into F;
   
   '"
   
   'H(U. K ; F ): 1.-.. of 1"'C07lI;IlUOUl functlollll from U inlO F thal ......Iob ouUi(le, 1M ~_K; 137
   
   thal
   
   (I S p <
   
   ,...""zr.....1:*
   
   fuactk>M ia P'(P); 98
   
   'H +(U):
   
   01 <:Ompeto:\!,- I UP. potted c:omJD ...... fu,n¢. t.lonI fmm U lnIo R" ,
   
   110\
   
   '"
   
   .1'"
   
   let
   
   01 ~hoe ~muoo. (undO.
   
   ,,' $
   
   1"':
   
   (on 0 ); ]38
   
   M (O. C ), ",*", oi ..... ,..... R..s..
   
   M· (O): _
   
   138
   
   aI pOeit~ R..... Pllrw; 138
   
   "
   
   J"'''; ]41)
   
   _,.4
   
   it,- v..,lGE.4 oi peSti"" UloInoCl. 077A' ..... 1~
   
   q",(JO):
   
   ..-e alloeaIIy
   
   .,.;]"'"'" h"y,;'-'; 196
   
   '''' ~ wit.b deosity 9 E
   
   of the bouDded ,.J.. o.oIued fuuctlon J; 147
   
   I" J dV'" lower IUtmaruI 10"".01 CO(O. F):
   
   J; ]47
   
   ~
   
   <>l bouncItd
   
   ~(JO ) ~I"" to '" 196
   
   cbI/ d.p: dtaoily of" with [ ....... ' to II; li7
   
   " <
   
   Il l. ", ... WIon of mulual
   
   ."IuLari1y; 203 .. (,,): iruace
   
   ~F; I~
   
   meNU ...
   
   <>lFhe
   
   .t.trloCl. - wnue II ~. ; n5
   
   C"(O. F): opere olltK.e cnminUO\ll fuoetlo... rnxo 0 inFO F Fhat ...... iIIh .. t Infiolty; .~
   
   M ' (O, C ): '1*8 01 buo""""d RIdooI " : "',,.., 1~ M ~ (0 ): let aI bounded pOeiti""
   
   --...;IM
   
   .., Fbe
   
   S' ,,9", p....hott oI lbe zmlrinp S' -=I 9"; 17&
   
   I@I"
   
   II: ~ioa of at:.olute
   
   CCIGImwly; 200
   
   coadnOOllt fu~ from 0 I~ tbe 1'IOt1IIed
   
   " IT': ~r PI ,~ i!ld1OOell bJ " kwUinc 1': 17&
   
   ~
   
   illFqrabio: 001II ......
   
   I" J tl\,I" upper lUo;o un II>7.e&f.ol
   
   aI
   
   ~
   
   !: I'e' IUID oilhe IWOmabie ram..
   
   ouppVo): oupport aI lbe R......, "
   
   -..... "",., and,...., 17'9 " ' pure; ISS
   
   ..-e ol,.J. R..s..
   
   -EF"US;
   
   01 ,be u.trw::l
   
   .\0, .l:rdilllmlioo&l
   
   _urw; 138
   
   M {O, R ):
   
   prod!!CI
   
   '"rei"" (2'." ' _ I (2'}g"'{:t"'); 181
   
   nS,: Infinite product of tbe
   
   ~I
   
   zml-aLPrM S; ;
   
   m
   
   .'
   
   ®,.,: Infinite prod!!CI of u.troct Dp(r ): deri_ive of the -. 7 mff: " &I.
   
   r, 243
   
   O(n + I, R ), sroup of orUqoarJ Ira
   
   ha •• 1C'7 ID
   
   R~" ' ;
   
   '" e , ~ aw"'"""", S~ ;
   
   '"
   
   VIM, J) or V(J): variulca 01 M ~J: 2M
   
   Var( X ): YVianceof X i :l.J.J
   
   ®P, : prOduct 01 ~ pooo."biliUei
   
   lIN: Stldlja" 7'N,,'" ·"ed "IIilh M i 2:1D
   
   -.:,
   
   f .. /: Une In\teraI; ru
   
   a(( X ,)",,); a·aJpa gme,.ud by l be n.
   EUo): man oI"tlle '-771 ,,. p; 295 D{p):
   X~
   
   P, ; 3.l.J
   
   _ I' &ad X. _ X : coaYft"_
   
   I"fICe !n law; l2:Z
   
   m
   
   N,f.... .,,): DOtmollaw ... R 1I"il b _ '" and .~ a';
   
   '"
   
   £ (Y IX ): conditioaal expo!Cle::l
   
   value 01 Y aI-- t.be ..,'(Ioo"...-iabIe X ; 3!i3
   
   F,., n.urier I ... ; .... 01" lbe """"'*I mE ' .'" /'i 29G
   
   "F", ;".,.""", 1';
   
   Fourier Itl.Dlll'orm 01
   
   I'Y
   
   or
   
   \D
   29G
   
   FI: Fourier lei
   
   I'I Y: Radon "-In:
   
   tn." .....
   
   01" tbe
   
   aue IaupabIe
   
   fu~1ca
   
   1: "": I\I.IIl oI"!.be.....uroable family u... J...- A 01
   
   .... A
   
   _
   
   I ; 2'91
   
   ....;.w
   
   R.oo..
   
   "FI; i Q _ FOo.:rier t....worm 01" 1; 2'91 <:b.(.u); ~ fww:lioa 01
   
   /'; 303 A':
   
   1. . . .- ... 01
   
   N~ (""D):
   
   I· " or , ,,: Radon _ure with
   
   ..,
   
   tbe OlIalPix A: ~
   
   """maIla.. on
   
   R~
   
   deaeily, relative to
   
   with
   
   "'""" '" and .,...."..;....
   
   ....trix D; ~
   
   I' .1
   
   Ir.
   
   ..,1M1ca 01 mutual sin&Warity lor Radon -U/1II;
   
   Px or p(X ): Ia.. oIlhe ...:dom >Wi&bIe X i 312
   
   E( X ): ""pected ..J\III! 01 X ; ill
   
   D(X): ~ ma:.rb: 01 X ;
   
   '"
   
   JI;
   
   40.1
   
   ..
   
   .-(p): imact" 75'", WIder "I: 01 tbe Radon ""'''PI", P i 1'@lr.prOductoi R..s.., " . mrs; ill
   
   r-gtlted ma8"ial
   
   MO
   
   Symbol I""""
   
   ,.(.): t.......u.tioo X ,. r _ .., (G -.:Ii,. bun tho! left. OIl X ): 4'26
   
   8 \ G:
   
   ~lIp&a!of~
   
   "'{. )/ : fl.metlao r - J(. -' r ); 476
   
   SO{n
   
   + I, RJ: gou~ of rowiooo
   
   u -'
   
   "'*'" of G modulo H ;
   
   '"
   
   In R M
   
   , 4'9
   
   ' .
   
   6{.): ric~ tn "'letlao r (G actinc "..., lbe riChl OIl X) ; 42S
   
   S H { .. + I, R ):
   
   6(.J/: (nDeCIao r _/(u); 428
   
   G L( .. , K ); ,roup of lbe .utomoor-
   
   IfOUp of t.ypo,rbolic ..,. •• ;.... ill R· ·'; 455
   
   ph ...... of
   
   x: left ...uhiplltt
   
   pvm
   
   of •
   
   orUtr ..
   
   ore; 477
   
   m
   
   /: function I _ 1(.- ' ) OIl G; 429 jl: i=,1" T
   
   "
   
   u",
   
   Uoa prod"'" of m
   
   6:0: m<>duLar fuHCtIao OIl G : 433 cI X ( H
   
   ),' : _ ' " _ ••• ; '~tod
   
   wllh~ ;
   
   441
   
   or: znll~
   
   cI /J bf
   
   fJ;
   
   G / N : ,p&ee of Ief\
   
   twtt.I
   
   aoodulo H; 446
   
   cI G
   
   / : """"",,ulion product of 11>1: - . m t P &nd lhe
   
   f • p:
   
   r Ct ): mean of 1_. ftt-; 441
   
   .,
   
   1"
   
   "'111;
   
   funcUOOI f : 46Q
   
   act\Qc OIl X from t be rl&bI); 438
   
   1'/. : qootitm
   
   tho!
   
   *.u 1'0 ....t 1', . "' . 1'00 : ..........01...
   
   ...
   
   of I' .......,.
   
   ~
   
   {m.ti~ 1.0
   
   "~Id K ); .s4
   
   . _ .- ' ; 4211
   
   X / H : ",... iont
   
   m- p1JUp of
   
   SL(.. , K ): ..,...w
   
   rtl&Il ody inv&ri&nt
   
   K- ; 464
   
   aavo/ulioo produet cI ~be function f and lhe meNu", /J: 479
   
   / • " cou>1)luu.... prod"'" cI lbe functlao / &nd tho! {unction
   
   g; 480
   
   UlliVCl"3itext
   
   ,
   
   ' " l.
   
   ,f)
   
   klllIrl i'!<>o
   
   • r-I"
   
   a_pin- c
   
   , ' ,, ' O, L .... _
   
   ('1
   
   ' . ' .", ... ' ' ' ' . . - . ,
   
   1\Mo __ d ' ,., _ _ • • ' . " .... 1 _ .. h , _ _ _ . O " ., _ _ .. ' ..... I
   
   . , ' • ...",
   
   ,,!~ ,
   
   _
   
   ,
   
   ,
   
   .....
   
   __ "-
   
   ,C, ' P
   
   SO_ .... I. .. , I ......... ... . . -. . . \.up Do "'0' .... '
   
   T
   
   ,
   
   , ... Ia'b ....... _
   
   . - . ....... k l.
   
   " .... _ t
   
   ..
   
   10.'1:: '
   
   ••
   
   ,.
   
   ••
   
   n
   
   •
   
   •
   
   I
   
   This book s "ntefldad to serve as
   
   I:lI
   
   e book for a course In
   
   measure theory at a gra.dua e level lin pore or apphed mathe ma1lcs. Th inclusion of some more developed matenal makes i also suitable for post-graduate stlldents. Apart from some ease 1n malhema ical reasor'lIng , 11 [s assumed only that the reader IS acquainted with the basic: results of topo ogy and func lonal analy,s ls.
   
   The approach adopted here is based on the concept of com plex Daniell measure. This a lows the developmen or a 9 neraJ theory ha e compasses both Radon measures and (real· or comp ex-valued) abslrac ' measur~s on families of subsets of a g van se . While the setting chosen by the autnor is very gen-
   
   eral. the material Included In he main theorems IS eaGY 10 undersland and 10 use. All the essential lools or applications of measure theory In analYSIS and probabil" y theory are fully develo ed. Numerous exercises help he reade to clarrly he heory.
   
   111
   
   •• ISB
   
   ~387- 94D'M-'6
   
   9 760387 9 6L43 )