Measure and Integration Theory (De Gruyter Studies in Mathematics)

xo > 0, that is, differentiable in JO,+oo[, and

(16.7)

(x) = -

e_2(1+")).1(dw)

for x > 0

and via the substitution t = w f this reads (16.8)

cp'(x) = -Gx-1"2e-z

forx>0

where G designates the integral (16.1) that we are trying to explicitly compute. Its existence is already fart of the preceding analysis, but can also be inferred from

the majorization a-' < e-t, which holds for t > 1. From (16.6), (16.8) and the

§16. Applications of the convergence theorems

95

fundamental theorem of calculus

V(x) - V(a) = GI t-1/2e-° dt = 2G 41. e" dw, for x > 0 and a > 0. Upon letting a run to +oo, we will get (16.9)

p(x) = 2G

+oo a-", dw

J,rif we notice that V(a) -+ 0 as a - +oo, which in turn is a consequence of the inequalities +w2)-1A1(dw) = p(O)e-0

w(a) < e-° f(i

for all a > 0.

Because cp is continuous on R+ we can pass to the limit x -+ 0+ in (16.9) and get

it = p(0) = 2G

r+ e-"'2 dw = G2,

J0

using the obvious (on grounds of symmetry) fact that f °. a-"'' dw = f0+00 e' dw. G = . That is, Since G > 0, it follows finally thatfe2

dx = r

(16.10)

or equivalently, in the form seen in probability theory,

2a.

(16.10')

This derivation goes back to ANONYME [1889J and VAN YZEREN [1979]. A par-

ticularly short alternative one is made possible by Tonelli's theorem (cf. Exercise 4 in §23).

Exercises. 1. Which of the two functions below are integrable, which are square-integrable with respect to Lebesgue-Borel measure on the indicated intervals? (a)

(b)

f (x) := x-1, f (x) := x-1/2,

x E I:= [l, +oo[; x E I:= 10,1] .

2. Show that for every real number a > 0 the function x H e" is A1-integrable over R+.

3. Show that for every real number a > 0 the function x

- a_°x [sinX x13 J

96

1 1. Integration Theory

is A'-integrablc over JO, +oo[ and that

rsinx13 A1(dx) x J

Jo is continuous Oil 10, +00[.

§17. Measures with densities: the Radon-Nikodym theorem Again let (12, dd, p) be an arbitrary measure space and E' = E'(f2, sd) the set of all W-measurable, non-negative numerical fimctions on 12. In 12.4 we defined the integral of every function f E E* over every set A E id'. We are interested here in how this integral behaves with respect to A. 17.1 Theorem. For each function f E E`JA the equation

v(A) :=

(17.1)

f du

defines a measure v on sd.

Proof v(0) = 0 and v > 0. For every sequence (An)nEN of pairwise disjoint sets

from W with A:= U A nEN

IAf =

IA, f n=1

and so by 11.5

v(An),

v(A) n=1

the final property needing to be checked in confirming that v is a measure on 0. 0

17.2 Definition. If f is a non-negative .d-measurable, numerical function on 11, then the pleasure v defined on .0' by (17.1) is called the measure having density f with respect top. It will be denoted by

v=fiz.

(17.2)

Concerning the relationship between v- and µ-integrals we will show

17.3 Theorem. Let f,, E E', v:= fu. Then (17.3)

1

§17. Measures with densities: the Radon-Nikodym theorem

97

or, written out,

Jd(f,i) = f Wf dµ -

(17.3')

An id-measurable function V : fl - R is v-integrable if and only if ,pf is µintegrable. In this case (17.3) is again valid. Proof. First suppose p =

a,lA; is an sad-elementary function. In this case (17.3)

holds because n

n

f ,pdvaiv(A1)a;f lA,fdµ=Jcof d µ. For an arbitrary p E E' there is a sequence (un) in E such that U. T V. Since then un f T W f as well, (17.3) follows from 11.4. Finally, consider any id-measurable numerical function p on Sl. By now we know that

fco+ dv = Jco+f dµ = J(caf)+ dµ and

f

W- dv = f V f du = f(f ) dp.

From these equations and the definition of integrability follows the second part of

the theorem. 0 It now follows that the formation of measures with densities is transitive:

17.4 Corollary. Let f, g E E', v := fit and P := gv. Then B = (gf )µ, that is,

9(fµ) = (9f)µ

(17.4)

Proof. For every A E id

g(A) = f gdv = A

f

lAgdv

and furthermore, according to 17.3

f lA9dv=

f

lA9fdµ= f(9f)dii.

We thus obtain p(A) = fA g f dµ, for all A E W; which is what had to be proved. 0

On the question of uniqueness of density functions we have

17.5 Theorem. For functions f, g E E' (17.5)

f =g

µ-almost everywhere

= f p = gµ .

If either f or g is µ-integrable, the converse implication holds as well.

98

IL . Integration Theory

Proof. If f and g coincide p-almost everywhere, then so do 1A f and 'Ag for each A E a(, whence JALgdp

for allAEd,

which just says that fit = gp. Now suppose that f is p-integrable and that fit = gp. Since g > 0 and f gdp = f f dp < +oc, g is also p-integrable. Let us show that the set

N:={f>g}, which lies in 0 by 9.3, is a p-nullset. For every w E N, f (w) - g(w) is defined and is positive, which means that the definition

h:= 1Nf - 1N9 makes sense. The functions 1N f, 1Ng, being majorized by the p-integrable func-

tions f, g, are themselves integrable. Because fit = gp, they have the same itintegral. From this we getr that

J

hdp=

r Ir fdp- /Ngdp=0.

Since N = {h > 0}, this equality and 13.2 tell us that p(N) = 0. With the roles of f and g reversed, this conclusion reads u(N') = 0, where N' := {g > f }. Since if 54 g} = N U N', the desired conclusion, namely that If 34 g} is a p-nullset, is

obtained. 0 The converse of implication (17.5) is not valid without some additional hypothesis on the densities f and g. The next example illustrates this.

Example. 1. As in Example 2 of §3 let fl be an uncountable set, 0 the a-algebra of countable and co-countable subsets of (1 (see Example 2 in §1). But the measure p will be defined on 0 by p(A) := 0 or +oo, according as A or CA is countable. If f and g are the constant functions on ft with the respective values 1 and 2, then indeed f p = gp, yet f (w) = g(w) holds for no w E ft. Of course, it then follows from 17.5 that neither f nor g is p-integrable. Before turning to the principal problem of this section, we will examine another characterization of a-finite measures which is important for what follows and is of interest in its own right.

17.6 Lemma. Let (fl,.ad,p) be a measure space. The measure is a -finite if and only if there exists a p-integrable function h on Cl which satisfies (17.6)

0
forevery wEf2.

Proof. If It is a-finite, there is a sequence in a0 such that p(An) < +oo for each n E N and A7, fi Cl. Choose positive real numbers gn satisfying both r) < 2-n

§17. Measures with densities: the Radon-Nikodym theorem

99

and i p(An) < 2-n, for each n E N. Then the function 00

h := L?In1A n=1

does what is wanted. It is measurable, 0 < h(w) < 1 for each m E 0, and f h dp < 1.

The converse implication is already known: it is contained in the second part of 13.6.

In the light of 13.2 this lemma has another formulation: For each or-finite measure R there exists a real, measurable function h > 0 such that the measure hp is finite and has the same nullsets as A.

We come now to the main problem, already alluded to: On the v-algebra sF of the measurable space (S2, 0) two measures v and p are given. We pose the question

of how to decide whether v has a density with respect to µ, that is, whether there is an .W-measurable, non-negative, numerical function f on St satisfying v = f p, satisfying in other words

r

v(A)=J fdp

for allAE.d.

a

For an affirmative answer it is necessary, as 13.3 shows, that every p-null set in a be a v-null set as well.

17.7 Definition. A measure v on W is called continuous with respect to a measure it on 0, for short, p-continuous, if every p-nullset from 0 is also a v-nullset. In the case of a finite measure v there is a condition equivalent to p-continuity which clarifies and justifies the terminology:

17.8 Theorem. A finite measure v on jzf is p-continuous if and only if for every c > 0 there exists d > 0 such that v(A) < e. (17.7) . A E O and u(A) 0 if A is a p-nullset. Hence v(A) = 0 and v is thus a p-continuous measure, even without the finiteness

hypothesis. For the converse we will show that if (17.7) fails, then v is not µcontinuous. Thus, for some c > 0 there is no 6, which means there is a sequence with the properties (An)nEN in p(An) < 2_n and v(An) > E for each n E N. We set

A := 41.s .up An := n U An nEN m>n

and have a set in ap which on the one hand satisfies 00

A(A) < µ( U Am) < E p(Am) <_ m>n

m=n

00

m=n

2-m = 2-n+1

for every n E N,

100

II. Integration Theory

whence p(A) = 0, and on the other hand, due to the finiteness of v and 15.3, satisfies

v(A) > limsup

E > 0,

nix

which proves that v is not p-continuous. 0 Examples. 2. Let 12 be an uncountable set, W the or-algebra of countable and cocountable subsets of .W (Example 2 in §1). As in the preceding Example, consider the measure v on .i which assigns to a set the value 0 or +oo according as the set or its complement is countable. Let is denote the counting measure C on at (from Example 6, §3). Since 0 is the only p-nullset, v is trivially µ-continuous. However, v cannot have a density with respect to p. For from v = f p with f E E* it would follow that

0 = v({w}) = f f dp = f(w)k({w}) = f(w) W}

for every w E S2, making f = 0 and therefore v = fit = 0, which is not the case because Sl is uncountable.

Let (R, 0, It) be the 1-dimensional Lebesgue-Borel measure space (so p = 'V) and denote by A" the system of all p-nullsets. Then is an example of a or-ideal in W1: The union of any sequence of its sets is another, as are the intersections of its sets with those of ,5d1 (cf. Exercise 5, §3). These properties insure that 3.

v(A)

-

10 +oo

ifAE-4 if AEJO\.X

defines a measure on 1 (cf. Exercise 6, §3). From its definition it is clear that v is p-continuous. Here however (17.7) falls, since for every b > 0

jp([o,ap = s and v([0,ap =+oo. Thus the finiteness hypothesis on v in 17.8 is not superfluous. Example 2 shows that for the existence of a density f E E' with v = fit, the µ-continuity of v, while necessary, is not sufficient. All the more noteworthy is the theorem of Radon and Nikodym which we will prove, after a preparatory lemma.

17.9 Lemma. Let or and r be finite measures on a o-algebra ii of subsets of 11 and let a := r - a denote their difference. Then there is a set S2o E W with the properties (17.8)

(17.9)

e(fl0) > LOW); @(A) >0

for all AESTOltW.

Proof. Let us first proof the weaker claim: (*) For every, e > 0 there exists 0e E 0 with the properties (17.8') (17.9')

N(1l) >- 9(f) ;

g(A) > -E

for all A ED, ft a/.

§17. Measures with densities: the Radon-Nikodym theorem

101

We may obviously suppose that p(l) > 0, since otherwise SlE := 0 does what is wanted. If then e(A) > -e for all A E .sad, it suffices to choose 1 := Q. So we consider the case that some Al E ad satisfies e(A1) < -e. From the definition of e and the subtractivity of the finite measures a and T, e(CA1) = e(fl) - e(A1) ? e(1) + e > e(11) .

Therefore, if e(A) > -e for all A E (CAI) fl 0, we can set S1E := CA1 and be done. In the contrary case there is a set A2 E (CAI) flsat with e(A2) < -e. Then because A1, A2 are disjoint

e(C(A1 U Az)) = o(Q) - e(A1) - e(A2) > e(fl) + 2e > e(n) and the preceding dichotomy presents itself anew. If after finitely many repetitions of this procedure we have not reached our goal, then we will have generated a sequence (An)nEN of pairwise disjoint sets in gd with e(Sl \ (A1 U ... U An)) > e(Sl)

and e(A.) < -e

for every n E N.

Because of the finite additivity of a and r, this would have the consequence that n

e(A1U...UAn)=Ee(A,) <-ne

for every n E N

i=1

00

and entail the divergence of the series 1 e(An). But the latter is untenable, n=1 because when the a-additivity of a and r is applied to the disjoint union A U An it shows this series to be convergent: nEN 00

00

E e(A,) = 1: (r(An) - a(An)) = r(A) - a(A) E R. n=1

n=1

This contradiction proves that the construction procedure must terminate after some finite number n of steps, with the set QE := C(A1 U ... U An) then satisfying (17.8') and (17.9'). We now take e = 1/n in (*) for successive n E N. The sets (1 can be chosen with

the additional property of isotoneity. For if Sll D 121/2 3 ... 3 Sll/n has already been realized, we simply apply (*) to fll/n as a new base space in the role of Sl, that is, we consider the restriction of the measures or and T to S21/n fl dd. Finally, the set Slo := n Sll/n will be seen to do the desired job. For since 01/n j Sla, nEN

(17.8) follows from (17.8'), and (17.9) follows from (17.9'), which insures that

e(A)>-1/nforallnENandeveryAESlofl.od. O As indicated, this puts us in a position to answer the important question we posed earlier.

17.10 Theorem (Radon-Nikodym). Let u and v be measures on a a-algebra .srd in a set Q. If µ is a-finite, the following two assertions are equivalent:

I l. Integration Theory

102

v has a density urith respect to A. (ii) v is 14-continuous. (i)

Proof. Only the implication (ii)=(i) is still in need of proof. To that end we distinguish three cases.

First Case: The measures µ and v are each finite. Form the set 9 of all d measurable numerical functions g > 0 on Sl which satisfy gµ < v, that is, which satisfy

for allAEd. The constant function g = 0 lies in 9, so 9 is not empty. 9 is moreover sup-stable, that is, g V h E 9 whenever g, h E W. Indeed, setting Al := {g > h}, A2 := CAI, every A E d satisfiees

J

gvhdµ= 1

Ana,

r

gdµ+J

ArA,

Since f gdµ < v(Q) < +oo for every g E 9, the number

ry:=suP{ f 9dµ:gE9) is finite and there is a sequence (g;,) in 9 such that lim f gn dµ = -y. Due to supstability the functions gn := gi V ... V gn lie in 9, and consequently ry > f gn dµ >

f gn dµ (since g,, > gn) for all n E N. Which shows that lim f gn dµ = ry. As the sequence (gn) is isotone, the monotone convergence theorem can be applied,

assuring that f := supgn is a function in 9 and that f f dµ = ry. All this proves that the function g H f g dµ on 9 assumes its maximum value at f. Now we prove that v = f µ. In any case we have f µ < v, since f E 9, and so

T:= V- f A is a finite measure on sat, evidently µ-continuous since v is by hypothesis. We have

to show that r = 0. So let us assume contrariwise that r(Sl) > 0. Due to the µ-continuity of r, this entails that µ(11) > 0 as well, and we may form the real number

Q:=2

(M}>0,

which satisfies r(Sl) = 20µ(Sl) > Qµ(St). The preceding lemma applied to r and a:= Q3µ supplies a set flo E 0 which satisfies

r(flo) - lµ(ilo) > r(1) - $µ(!l) > 0 and r(A) > Qµ(A) for all A E f o n 0. The .sat-measurable, non-negative function fo := f +,81n. therefore has the property

ffodiz=jfdii+I3(QonA)

jfd+r(A)=v(A)

§17. Measures with densities: the Radon-Nikodym theorem

103

for every A E sV. These inequalities put fo in 9. Since r is p-continuous and r(S2o) > Qµ(S2o), we must have µ(S20) > 0, leading to

f

fodµ= ffdµ+ap(no)=7+i3µ(Slo)>7,

an inequality which is incompatible with the definition of -f and the fact that fo E 9. The assumption r(S1) > 0 is therefore untenable, and r = 0, as desired. Second Case: The measure µ is finite and the measure v is infinite. We will produce 00

a decomposition SZ = U On of S1 into pairwise disjoint sets from d with the following properties

(a) A E 1o fl at (b)

n=0

either µ(A) = v(A) = 0 or 0 < µ(A) < v(A) = +oo . v(S1n) < +0o

for all n E N.

To this end let 2 denote the system of all Q E 0 with v(Q) < +oo and define a:= sup{µ(Q) : Q E _l} . This is a real number because the measure µ is finite. There is a sequence (Qm)mEN

in .l with limµ(Qn,) = a. Since 1 is evidently closed under finite unions, (Q,n) U Q,n is then a set from std satisfying may be assumed to be isotone. Qo mEN

µ(Qo) = a. We will show that 52o := CQo satisfies (a). So consider A E Stood with

v(A) < +oo. We need to see that p(A) = v(A) = 0, and since v is µ-continuous we really only need to confirm that p(A) = 0. Since v(A) < +oo and, as noted already, . is closed under union, each Q,n U A lies in 2, so that p(Q,, U A) < a, and consequently µ(Qo U A) = lim p(Qm U A) < a. "t-400

Since A is disjoint from 1o, u(Qo U A) = a + µ(A). Conjoined with the preceding inequality and the finiteness of a this says that indeed p(A) must be 0. Finally, to take care of (b) we merely define S21 := Ql, and u n := Qm \Q,n_1 for all integers m > 2 in order to get a decomposition of S2 with the desired properties. Now let An, vn denote the restrictions of µ, v to the trace a-algebra On fl 8d, for n = 0, 1.... and note that each vn is a µn-continuous measure. Moreover, for all n > 1 both An and vn are finite. Case 1 therefore supplies Cl,, n 0-measurable functions fn > 0 on Cl,, with vn = fnµn Taking fo to be the constant function +oo on Sto, vo = foµo also holds, thanks to (a). Finally, "putting all the pieces together" gives our result in this second case. Namely, the function f on Cl defined to coincide on each Cl,, with fn (n = 0, 1, ...) is non-negative, sad-measurable and satisfies

v=fp.

Third Case: This is the general case: only the a-finiteness of it is demanded. There

is according to 17.6 a strictly positive function h E 2'(µ). The measure hp is therefore finite and possesses exactly the same nullsets as does A. Consequently

v is also (hp)-continuous. By what has already been proved there is then an

104

II. Integration Theory

0-measurable function f > 0 on 1 with v = f (hµ). According to 17.4 v then has the density f h with respect to A. 0 The question arises whether, in the situation of Theorem 17.10 the density f of v is p-almost everywhere uniquely determined. From 17.5 we at least get a positive answer when f is p-integrable, that is, when v is a finite measure. But more is true:

17.11 Theorem. Let v = fit be a measure having a density f with respect to a a-finite measure p on 0. Then f is p-almost everywhere uniquely determined. The measure v is or-finite exactly when f is p-almost everywhere real-valued. Proof. First we show that f is µ-almost everywhere uniquely determined if the measure p is finite. In proving this we may assume that v(St) = +oo, since its truth is otherwise a consequence of the second part of 17.5. Furthermore, as we now find ourselves in case 2 of the preceding proof, the decomposition of St into %J11,... employed there lets us confine our attention to Sto, as 17.5 takes care of the remaining Stn (n E N). So it suffices to treat the case ft = Sto, that is, to assume that p and v are linked by the alternative: A E srp

=

either p(A) = v(A) =0 or 0 < µ(A) < v(A) = +oo.

The constant function +oo is then a density for v with respect to p and what has to be shown for uniqueness is that f = +oo holds p-almost everywhere. And for that it suffices to show that µ({ f < n}) = 0 for each n E N, which in turn is a consequence of the above alternative and the inequalities

v({f


f
fdp
coming from the finiteness of A. We will use 17.6 to reduce the general case of or-finite p to the case just treated.

That lemma supplies a strictly positive function h E 21(p). The measure by = h(fp) = f (hp) has the density f with respect to the finite measure hp, so f is (hµ)-almost everywhere uniquely determined. Since the measures p and hp have the same nullsets, f is therefore also uniquely determined p-almost everywhere.

Next, suppose v is a-finite. From 17.6 once again we get a strictly positive

function k E 21(v). Then kv = (f k)µ is a finite measure, that is, f k is µintegrable, consequently also p-almost everywhere real-valued. Because k takes only non-zero real values, this means that f itself is real p-almost everywhere. Conversely, suppose that f is p-almost everywhere real-valued. We want to 00

see that v is a-finite. First of all, there is a decomposition St = U On of f? n=0

into a sequence of pairwise disjoint sets from 0 each of finite p-measure. Set

An := in - 1 < f < n} for each n E N and Ao := { f = +oo}, the present

§17. Measures with densities: the Radon-Nikodym theorem

105

00

hypothesis being just that p(Ao) = 0. D = U (1li n AJ) is a decomposition of fl i.J=o

into a (doubly-indexed) sequence of pairwise disjoint sets from sat. If each has finite

v-measure, this proves that v is a-finite. Consider any i E Z+. Because p(Ao) = 0

and v = fµ, we have v(1l, n Ao) < v(Ao) = 0. Because v = fit and f < j in AJ, we have v(12i n AJ) < jp(ni) < +oo for all j E N as well. Thus all is proven. 0 In the generality presented here Theorem 17.10 was proved in 1930 by O.M. NIKODYtM (1888-1974). H. Lebesgue proved the theorem in 1910 for the case where At is the L-B measure A1. J. RADON (1887-1956) pushed things further in a fundamental work which appeared in 1913. So 17.10 is often also called the theorem of Lebesgue-Radon-Nikodym. The uniquely determined density f in 17.11 is called the Radon-Nikodym density or the Radon-Nikodym integrand (of v with respect top). A beautiful proof of 17.10 by elementary Hilbert-space methods was discovered in 1940 by J. VON NEUMANN (1903-1957) and appears in many textbooks, e.g., in RUDIN [19871, p. 130-131.

The history of the result to be presented next, the Lebesgue decomposition theorem, runs somewhat parallel, Radon and Nikodym having also made significant contributions. We need a concept complementary to p-continuity, namely p-singularity: 17.12 Definition. Let (Sly, sat) be a measurable space, µ and v measures defined on sat. Let us write v << p if v is p-continuous. v is said to be singular with respect

top (or p-singular), written v J p, if a set N E sl exists with µ(N) = 0 = v(CN). It is obvious that the relation v J p is symmetric in µ and v, so it is also expressed as p and v are singular to each other (or mutually singular). The definition

of v 1 p expresses the fact that for a suitable p-nullset N E W (17.10)

v(A) = v(A n N)

for all A E d,

as follows from v(A) = v(A n N) + v(A n CN) and v(CN) = 0. The condition that v J it thus says that the measure v is "carried by a p-nullset". From v << p

and v 1 p together follows that v(N) = 0, and so v = 0. In this sense the concepts p-continuity and p-singularity are diametral or antipodal. Relative to L-B measure Ad every Dirac measure ex on

d obviously satisfies Ad 1 ex.

17.13 Theorem (Lebesgue's decomposition theorem). If p and v are a -finite measures on a a-algebra sat in a set 12, then v can be decomposed in just one way as v = v, + v, with measures vv, v, on sat that satisfy v, << p and v. J p.

v, is called the continuous part of v with respect to p, v, the singular part. The Radon-Nikodym theorem is applicable to the part vc. Proof. We will carry out the proof in detail only for finite p and v and indicate in Exercise 4 how the reader can then handle the general case himself.

106

1 1. Integration Theory

Existence of a decomposition: Let ,, designate the system of all µ-nullsets in W. Since v(A) < v(Q) < +oo for every A E Of, a := sup{v(A) : A E Xµ}

(17.11)

is a real number. Since .X,, is closed under countable unions, there exists an isotone sequence (An) in .A', with v(An) T a. Since v is continuous from below, it follows

that

v(N) = a for the set N := U A E .A',,. We will show that via nEN

vc(A) := v(A n CN) and v,(A) := v(A n N) two measures are defined on W that do what is wanted. Evidently v = Me + v8, and V. 1 p since N E .NN. To prove that ve 4 it, it must be shown that v(A') = 0 whenever A E -A;, and A:= A n CN. As a subset of A E Xµ, the set A' and then also the set A' U N, is p-null. Therefore v(A' U N) < a by definition of a. But A' n N = 0 and v(N) = a. Hence

a + v(A') = v(N) + v(A') = v(N U A') < a, from which follows v(A') = 0 as desired, since a is finite. Uniqueness of the decomposition: Suppose (17.12)

v=VC +v,=vC'+v,'

are two decompositions of the kind described in the theorem. The measures v v, are carried by p-nullsets N, N' in the sense of (17.10); which means that (17.13)

v,(A)=v,(AnN) and v,(A)=v''(AnN')

Setting No := NUN' gives a set in

for all AEd.

so that from vi,, u< K p follows

v,,(AnNO)

=0

for every A E.Qd.

Therefore (17.12) and (17.13) give

v(AnNo) =v,(AnNo)+v,(AnNO) =v,(AnNO) =v,(AnNonN) = v.(A n N) = v,(A),

for every A E Ad.

Analogously of course, v(AnNo) = v,(A) for every A E 0. Thus we have v, = va. A return to (17.12), recalling that all measures are finite, gives v,; = v'' as well. 0

There is a short, elementary proof of 17.13 that does not make use of the Radon-Nikodym theorem; see Woo [1971).

Exercises. 1. Show that the Dirac measure e., on Rd has no density with respect to .1d, for any x E W'. (Physicists occasionally work with such a "symbolic" density d5,

calling it the Dirac. function at the point x. The correct mathematical object is nevertheless the Dirac measure es.)

§18*. Signed measures

107

2. Show that the relation << on the set of measures on a a-algebra d is reflexive

and transitive. The relation p - v defined as p << v and v « is is then an equivalence relation. Two measures p and v stand in this relation just when they

have the same nullsets. For a-finite measures p and v on d show that p - v is equivalent to v = f 1L for a density f which satisfies 0 < f (w) < +oo for p-almost all (or even for all) w E Q. 3. On a a-algebra 0 in a set 11 two measures a and v are related by v < A. Show that if further it is a-finite, then there is an d-measurable function f satisfying

0< f
4. Lebesgue's decomposition theorem was proved for finite measures p and v. Show

how to infer its validity for a-finite measures from this. [Hint: For the existence proof use 17.6. For the uniqueness proof choose a sequence (An) in 0 with An T Sl and a(An), v(An) finite for each n, and consider the measures vn(A) := v(Af1An),

AEd,nEN.]

5. Let v = vi+ve be the Lebesgue decomposition of a a-finite measure v on d with respect to a a-finite measure p. The singular part V. has the form v,(A) = v(AfN)

for all A E 0 and a suitable p-nullset N E d. Show that if N' is any other pnullset with this property, then u(N 0 N') = v(N A N') = 0. 6. Let (S2, .mot, p) be a measure space, v = f 1A a a-finite measure on d having density f with respect to p. Show that this density function is p-almost everywhere uniquely determined and is p-almost everywhere real-valued. Show that if f is strictly positive, then p itself is a-finite. 7. Let (11, d) be a measurable space. For every measure µ on s0 let .M,,, denote the a-ideal of its nullsets. Show that for any sequence (Pn)nEN of a-finite measures

on ae there is a finite measure µ on d for which /V,,= n N,,, nEN

8. The set n := 10, +oo[ is a group with respect to multiplication. Show that the measure on SZ f1.1 defined by p := han with density function h(x) := 1/x is

invariant under each self-mapping x H as of fI (a E fI). p is thus the Haar measure of the group f2 in the sense of the remark immediately following 8.2.

§18*. Signed measures It is worthwhile turning our attention back to Lemma 17.9. The measure concept in this book is that formulated in Definition 3.3: Measures are premeasures p on a a-algebra sad, and so are non-negative a-additive functions on d satisfying the additional condition u(0) = 0. In Lemma 17.9 we encountered a real-valued, aadditive function p which is the difference of two finite measures. Similarly for any f E 2' (p) the function A H fA f dp on W is the difference of two finite measures, for example f + p, and f -µ. We will call a real function p : sr' - R on a a-algebra a finite signed measure if it is a-additive in the sense of (3.2), non-negativity not being required. From

108

I l. Integration Theory

a-additivity applied to the constant sequence 0, 0.... follows immediately that (3.1) is also satisfied, that is, g(0) = 0, because g is only allowed to take real values. A second pass through the proof of Lemma 17.9 will convince the reader that this lemma is in fact valid for every finite signed measure. As a corollary we immediately get the following theorem on the existence a Hahn decomposition of g, a theorem that goes back to H. HAHN (1879-1934).

18.1 Theorem. Let g be a finite signed measure on a a-algebra and in a set Cl. Then there are sets Sl+, St- E of with Cl = Sl+ U fl-, Sl+ n fl- = 0, and g(A) > 0 for all A in the trace a-algebra Sl+ n 0, and g(A) < 0 for all A E Sl- n dd. Proof. Set

-y:= sup{g(A) : A E 0}

and choose a sequence (An) in 0 with limg(An) = y. By applying 17.9 to the restriction of g to An nad, we may replace An by a set Pn E 0 satisfying g(Pn) > g(An) and g(A) > 0 for all A E Pn n 0. We will then have

y=sup{g(Pn):nEN).

(18.1)

The decomposition of Cl that is sought can be realized by

Sl+ := U Pn,

S2- := S2 \ Q+ .

nEN

Indeed, all A E H+ n .ad satisfy g(A) > 0 because such an A has the form

A = U Bn nEN

with pairwise disjoint sets B. E P. n ad (by the disjointification procedure used in the verification of (3.10)). From this representation of A and the a-additivity

g(B,) > 0. Thus p assumes only non-negative real values

follows g(A) _ n=1

on Sl+ n .sad, that is, the restriction of g to Sl+ n 0 is a finite measure. Moreover, because @(P.):5 g(Sl+) < y and (18.1) this measure satisfies

y=Q(sl+) In particular, y < +oo since p assumes only real values. g(A) > 0 cannot hold for any A E Sl- n .sat, for otherwise g(C+ U A) = g(Sl+) + g(A) > y. Thus, g(A) < 0

for allAESl-n0. Measures (in the sense of Definition 3.3) have occasionally been interpreted as mass distributions on the underlying set Cl. A finite signed measure can be analogously interpreted as an (electric) charge distribution smeared over Cl. The foregoing theorem justifies this metaphor by showing that as with charge in electrostatics, there are two disjoint sets, one carrying all the positive charge, the other all the negative charge.

§18*. Signed measures

109

From this theorem another important feature of signed measures becomes evident: The difference p in Lemma 17.9 is more than an illustrative example of a signed measure - it is the typical signed measure:

18.2 Corollary. Every finite signed measure p on a a-algebra sat in ] is the difference of two finite measures on sat.

Proof. Let fl = S2+ U S2- be a Hahn decomposition in the sense of 18.1. Then evidently p+(A)

p(A n St+)

and p(A) :_ - p(A n St-),

A E sat

define measures on d, which satisfy p = p+ - p-, since each A E sat is the disjoint

union (AnS2+)u(Ancl-). 0 With this result the circle closes: finite signed measures are nothing more than the differences of finite measures. It is however possible to dispense with the finite-

ness hypothesis if a-additivity is handled with sufficient care, but we will not go into this further. In the final analysis it is because of the preceding corollary that we only consider measures with non-negative values in this book. Often to emphasize the distinction with signed measures, what we call simply measures are called positive measures.

Exercises. 1. Show that every finite signed measure on a a-algebra is bounded and assumes a largest and a smallest value.

2. Let p be a finite signed measure on a-algebra d in Sl, and St = Sli U f1i , fl = fl2 Uci be two Hahn decompositions for it. Show that ii LSl2 and Sti OS22 are totally p-nulsets, meaning that p(N) = 0 for every N E 0 which is subset of either of them. Conclude that to within such totally p-nullsets there is only one Hahn decomposition for p. 3. Let p be a finite signed measure on a a-algebra sat in Q. Show that the specific representation p = p+ - p- of p as the difference of the two measures on sat which was produced in the proof of 18.2 is characterized by the following minimality property: In every representation p = pl - p2 as the difference of measures pl, p2

on 0, pl = p+ + 8 and p2 = p + b for an appropriate finite measure 8 on sa7, and indeed if 11 = Sl+ U S2- is any Hahn decomposition of S2 corresponding to p,

8 = (ln+)p2 + (1n-)pl. (Conversely, of course, every finite non-zero measure b on sat generates in this way a different representation of p.) Infer that the only measure v on sat which satisfies v(A) < min{p+(A), p-(A)} for every A E sat is the identically 0 measure. [Remark: The representation p = p+ - p uniquely determined by this minimality condition is called the Jordan decomposition of the finite signed measure p. As with functions, p+ and p- are called the positive part and the negative part of p.]

110

1 1. Integration Theory

§19. Integration with respect to an image measure Along with the measure space (it, .0', i) a measurable space (W,01) and an jW-d'-measurable mapping

T : (fl, a) -a (ft', d') are given. Then the image measure

p` := T(p) is defined in (7.5). The connection between p-integrals and µ'-integrals is elucidated by:

19.1 Theorem. For every s/'-measurable numerical function f' > 0 on 0' (19.1)

Proof. The non-negative function f' o T is d-measurable, by 7.3. The integral on the right-band side of (19.1) is therefore defined. To prove the equality there we first consider only d'-elementary f': n

f

ailA s

i=1

(with coefficients ai E R+ and sets A; E d'). For such f

f'oTa;lAi e=1

with A; := T-r (A;), so this composite is an d-elementary function. Since

T(p)(Ai) = p(Ai)

(i = 1,...,n)

holds by definition of image measures, (19.1) follows in this case. For an arbitrary s9'-measurable f > 0 there is an isotone sequence (un) of d'-elementary functions for which u;, T f'. Then (un o T) is a sequence of s(-elementary functions for which u;, o T T f o T. From the validity of (19.1) for the u;, and Definition 11.3 of the integral in general, we get (19.1) for f'.

19.2 Corollary 1. Let f' be an sf'-measurable numerical function on W. Then the T(µ)-integrability of f' entails the p-integrubility of f' oT, and conversely. In case of integrability (19.2)

1

§19. Integration with respect to an image measure

111

Proof. From 19.1

f (f')+dT(p)=J(f')+

o Tdp and

J(f')_dT(P) = f

(f')- oT d1 z,

and of course

(f'oT)+=(f')+oT

and

(f'oT)-=(f')-oT.

Both claims therefore follow from the definition of the integral 12.1.

19.3 Corollary 2. The mapping T : S2 -+ S2' is bijective and d -d'-measurable,

with W'-d-measurable inverse T'. Further f' is a numerical function on W. Then the T(p)-integrability of f' is equivalent to the p-integrability of f' o T, and in its presence equality (19.2) prevails.

One has only to note that the integrability of f' o T entails the measurability of f' o T and therewith that off'= f' o T o T -1. The content of 19.1-19.3 constitutes what is called the "general transformation theorem for integrals".

As the behavior of the L-B measure with respect to Cl-diffeomorphisms is known from (8.16'), the transformation theorem for Lebesgue integrals follows at once:

19.4 Theorem. Let G. G' be open subsets of W', cp : G -> G' a C1-diffeomorphisrn

of G onto G'. A numerical function f' on G' is Ad-integrable if and only if the function f' o cp I det DWI is Ad-integrable over G, and in this case (19.3)

IG, f' dAd =

fcf' o' I det D,,, I dAd .

Proof. The Ad-integrability of f' over G' and that of f' o W I (let DWI over G means the AG,-integrability and the AC-integrability of those functions, respectively. According to (8.16') ' (Ac) = I det DWI Ad ;

furthermore, the Borel measurability of f' is equivalent to that of f'o
f f'dAc,_f f

f

f'o,pIdctDWIdAd,.

Because of Theorem 19.1, equality (19.3) holds as well for all non-negative, Borel measurable, numerical functions on G'.

112

1 1. Integration Theory

Exercises. 1. Let (0, dal, p) be a measure space, T : fZ -+ f 1 a mapping which together with its inverse is an d-d-measurable bijection. Show that for every f E E (St, .ad) the image measure T(f p) has a density with respect to T(p), namely f o T-1.

2. Let (0,.', p) be a a-finite measure space, T : ) -4 i2 an alf-d-measurable mapping such that T-1(A) is a p-nullset whenever A is. Prove the existence of a measurable function q > 0 such that

r

fA-'(A) f oTdp fqdu

-TJ for all dat-measurable numerical functions f > 0 on fl, and all A E d.

§20. Stochastic convergence Let us return to the study of p-fold integrable functions begun in §14. Our goal will be to replace the almost-everywhere convergence concept that underlies the theorems proved there with a weaker convergence concept. It is suggested by a simple but very useful inequality.

The setting is once again an arbitrary measure space (el, 0,u). 20.1 Lemma. For every measurable numerical function f on 0 and every pair of real numbers p > 0 and a > 0 the Chebyshev-Markov inequality p({IfI >- a}) <-

(20.1)

af

Iflp dp

holds.

For p = 2 this is also known simply as Chebyshev's inequality.

Proof The set A6 := {IfI > a} lies in d and

p>j Iflpdp2j apdp=app(Aa)

fd

a

a

which is what (20.1) claims.

Therefore if f If Ip dp is finite, which when p > 1 means just that f is p-fold integrable, it follows from (20.1) that (20.2)

lim p({IfI > a}) = 0.

a-r+co

One can also study the dependence on it E N of the measures of the sets { I fn - f I > a} when f, fl, f2.... are measurable real functions. That leads to the aforementioned new convergence concept.

§20. Stochastic convergence

113

20.2 Definition. A sequence (fn)nEN of measurable real functions on 1 is said to be (µ-)stochastically convergent (or to be convergent in p-measure) to a measurable

real function f on S2, if for each real number a > 0 and each A E d of finite measure

nlim tt({I fn - f I > a} n A) = 0.

(20.3)

+oo

In this case we also write

µ- lim fn = f

(20.4)

and call f a (µ-)stochastic limit of the sequence (fn). Remarks. 1. For a finite measure p we may take A = 52 in (20.3) and in this case stochastic convergence of (fn) to f is equivalent to the requirement

lim µ({lfn- fI>a})=0

(20.5)

for every a>0.

The more complicated condition (20.3) is dictated by the desire to treat infinite, and especially a-finite, measures as well as finite ones. 2. For a-finite measures p the stochastic convergence of a sequence (fn) to f is generally not equivalent to (20.5), as the next example illustrates.

Example. 1. Let St := N, 0 := .9(N), It the measure (obviously a-finite) defined on sad by the equations

µ({n}) = n

for every n E N

and the requirement of o-additivity. With An := {n, n + 1,.. .} and In := 1A., for each n E N, the sequence (fn) converges stochastically to 0: For every a E 10, 1[, { jn > a} = An, and since An ,. 0, it follows from 3.2 that lim µ(An n A) = 0 for every A E Af having finite measure. On the other hand, u(A.) = +oo for

every nEN.

Remark. 3. Let f be a stochastic limit of a sequence (fn) and consider any measurable real function f' on 11. If f' = f p-almost everywhere in every A E d which has finite measure, then f' is also a stochastic limit of the sequence (fn). This is because the sets

{Ifn-f*I >a}nA and {Ifn-fl>a}nA differ from each other only in an (n-independent) nullset. The converse of this is important:

20.3 Theorem. For every o-finite measure p, any two stochastic limits of a sequence of measurable real functions are µ-almost everywhere equal to each other.

114

1 1. Integration Theory

Proof. If f and f* are stochastic limits of the sequence (fn), then from the triangle inequality in R

{If -f*I2al C{If.-fI? a/2}U{Ifn-f*I2! a/2}, whence

p({If-f*I >a}nA)a/2}nA)+p({Ifn-f*I2:a/2}n A) for every n E N and every A E d. Letting n -3 oo shows that

p({ If -f*1 >- a} nA) = 0 for every a > 0 and every A E ii of finite measure. Then however, f = f* "-almost everywhere in every such set A, since

If 54 f*} n A= U{If - f*1 > Ilk} nA kEN

is a p-nullset. Upon taking for A the sets in a sequence (An) in 41 which satisfies p(An) < +oo for all n and An t 0, the p-almost everywhere equality of f and f follows. D To supplement this fact we mention:

Remark. 4. Stochastic limits f and f* of the same sequence (fn) are almost everywhere equal without any hypotheses on the measure itself if both functions are p-fold integrable for some p E [1, +oo[. This is because for every real a > 0 the

set (if - f* I > a} has finite measure, by (20.1), and so f = f * p-almost everywhere in this set, whence { If - f * I > 0} = U {If - f* I > 1/n} is a countable nEN

union of p-nullsets. This just says that f = f* p-almost everywhere in Sl. But the next example shows that it may fail if one of the functions is not in any 2P-space. Example. 2. Consider the measure space (fl, Y(fl), p), where 11 consists of exactly two elements wo,wl and p({wo}) = 0, p({wl}) = +oo, fn = f = 0 for every n E N. These functions lie in every .2'P(p) and the sequence (fn) converges stochastically

to f , as well as to every real-valued function f * on 0. Every such f* which is non-zero at wl, however, lies in no 2"(p) with 1 < p < +00 and fails to coincide p-almost everywhere in 11 with f. The considerations with which we began this section lead to an important class of stochastically convergent sequences:

20.4 Theorem. If the sequence (fn) in 2P(p) converges in e" mean to a function f E 2P(p) for some 1 < p < +oo, then it also converges to f p-stochastically. Proof. The Chebyshev-Markov inequality tells us that

p({Ifn - fl ?a}nA)
§20. Stochastic convergence

115

holds for every n E N, every a > 0 and every A E s+d. The claimed stochastic convergence, that is, the convergence to 0 of the left end of this chain as n -+ oo, follows because f I fn - f I' dµ -+ 0 as n -+ oo is the definition of convergence

in pth mean. 0 The proof shows that convergence in eh mean actually entails the stronger form of stochastic convergence in (20.5). The situation is different when the given sequence is almost everywhere convergent. (On this point cf. also Remark 5.) 20.5 Theorem. If a sequence (fn)nEN of measurable real functions on fl converges µ-almost everywhere in Sl - or even just p-almost everywhere in each set A E st of finite measure - to a measurable malfunction f on 1l, then this sequence also converges p-stochastically to f.

Proof. For every a > 0,

{Ifn - .fI 1a} C {m>p Ifm - .fI

1a}

n

and so

A({Ifn - fl2! a}nA):5 µ({supI.fm-f1 >a}nA) m>n

for every A E d. The present claim therefore follows from our next lemma, applied

to the restriction of p to A n sl for each A of finite measure. 0 20.6 Lemma. If the measure p is finite, then each of the following three conditions on a sequence (fn)nEN of measurable real functions is equivalent to (fn) converging p-almost everywhere to 0: (20.6)

(20.6')

Ifml > a}) =0

for every a > 0,

lim µ({sip Ifml > a}) = 0

for every a > 0,

p(limsap{Ifnl>a})=0

for every a>0.

lim A n-rao

(20.7)

m>n

m>n

Proof. To prove the equivalence of (20.6) with the almost everywhere convergence of (fn) to 0, we set, for each a > 0 and each n E N

An :_ { sup IN > a} . m>n

Obviously both n H An and a H An are antitone mappings; then k H An/k is isotone on N. If we also set

A:= {w E fl :limo fn(w) = 0} = {w E Sl : limas

op

Ifnl (w) = 0),

1 1. Integration Theory

116

then these lie in W. either by appeal to 9.5 or by noticing that each A; E W and

A= n U kEN nEN

Passing to complements,

CA= U nAnk kEN nEN

and so

n A ;/k r CA as k -+ oo,

and Al/k n 1

fI' dl "m

as n -00.

mEN

nEH

Consequently,

u(CA) = sup p ( n A,imk) = sup inf

(20.8)

kEN

kEN 'nEN

nEN

because the finite measure µ is both continuous from above and continuous from below, by 3.2. Thus (fn) converges almost everywhere to 0 just when the number defined by (20.8) is 0. In turn, the latter occurs exactly in case

inf p(AIlk) = Iuu p(An1fk) = 0

nEN

n-+oo

for every k E N. The first equivalence follows from this. The equivalence of (20.6) with (20.6') follows from the observation that for any numerical function g on S2

{g>a}C{g>a}C{g>a'} whenever 0 < a' < a. Finally, the equivalence of (20.6') with (20.7) follows from the validity, for every

a > 0, of the equality

a(( sup Ifml > a}) = µ(limsop tlfnl > a}) .

(20.9)

m> n

For the proof of which we introduce

Bn:= U{Ifml>a} and B:=llmspp{Ifnl>a}. m>n

On the one hand, Bn I B and consequently tim p(Bn) = µ(B). On the other hand, however,

Bn= U {Ifml>a}={sup Ifml>a}. rn>n

m>n

From this finally we get the needed (20.9). 0 The conditions involved in Theorems 20.4 and 20.5 are indeed sufficient to insure stochastic convergence, but they are not necessary for it, as the following examples show.

§20. Stochastic convergence

117

Examples. 3. Let S2 :_ [0,1 [, s/ := 1 n 91 and µ := an, a finite measure. With converges to 0 at every point of Q An :_ JO, 1/n[ E a, the sequence and so, either by appeal to 20.4 or by virtue of

µ({n1A > a)) = µ(An) = n

whenever 0 < a < n E N,

this sequence also converges stochastically to 0. By contrast

= n"p(An) = np-1 shows that the sequence does not converge to 0 in pth mean for any p > 1. 4.

Let (fl, 0, µ) be the measure space of the preceding example. Write each n E N

as n = 2' + k with non-negative integers h and k satisfying 0 < k < 21 (which uniquely determines them) and set

An :_ [k2-h, (k+ 1)2-h[,

In

n E N.

lAn,

It was shown in the example in §15 that the sequence (fn(w))nEN converges for no w E S1. Nevertheless the sequence (fn) does converge stochastically to 0, since for every a > 0 and n E N

p({) fnI 1 a}) < 2-h < 2r2 . In this example stochastic convergence can also be inferred from 20.4, since the example in §15 showed that (fn) converges to 0 in pth mean for every p E [1, +oo[. The connection between stochastic convergence and almost-everywhere convergence is nevertheless closer than one would be led to suspect on the basis of the last example.

20.7 Theorem. If a sequence (fn)nEN of measurable real functions converges ,u-stochastically to a measurable real function f, then for every A E 0 of finite p-measure some subsequence of (fn) converges to f µ-almost everywhere in A. Proof. For A E sa( with µ(A) < +oo, the measure µA, which is the restriction of p to A n.ad, is finite. It therefore suffices to deal with the case of a finite measure u; moreover, in that case we can simply take A to be St itself. For a > 0 and m, n E N the triangle inequality shows that

{Ifm - fnI 2: a} C {If,. - f I ! a/2} U {Ifn - f I

a/2);

thus by hypothesis µ({I fn, - fnl > a}) can be made arbitrarily small by taking m and n sufficiently large. If therefore (rlk)kEN is a sequence of positive real numbers with 00

E rlk < +00, k=1

118

I l. Integration Theory

then for each k E N there is an nk E N such that

forallm>nk.

{t({Ifm-fnkl?nk})<-nk

Clearly the sequence (nk)kEN can be chosen strictly isotone: nk < nk+1 for every k E N. If now we set k E N, {Ifnk+t - fnk l llk}, Ak then 00

00

(Ak) < E 77k < +00,

>

k=1

k=1

and consequently,

p(Ak) = 0.

lira

n-oo

k=n

From this it follows that the set A := lira sup An satisfies n-,00

p(A) = 0, 00

because A C U Ak for every n E N, entailing that p(A) < E p(Ak) for every n. k=n

k>n

The definition of A shows that if w E CA, then the inequality Ifnk+. (w) - fnk (w) I ? rlk

prevails for at most finitely many k E N. Therefore, along with the series E Ilk, the series 00 1: lfnk+l(w) - A. (w)1 k=1

converges (absolutely); that is, the sequence Y n& (w))kEN converges in R. In summary, the sequence (fnk) converges almost everywhere to a measurable real func-

tion f' on !l. By 20.5 f' is also a stochastic limit of (fnk )kEN. But, as a subthat sequence converges stochastically to f as well. Hence sequence of by 20.3, f = f " almost everywhere. We have shown therefore that (fnk )kEN con-

verges almost everywhere to f. 0 In terms of almost-everywhere convergence we can now even characterize stochastic convergence by a subsequence principle.

20.8 Corollary. A sequence (fn) of measurable real functions on 11 converges pstochastically to a measurable real function f on ) if and only if for each A E of of finite measure, each subsequence (fnk )kEN of (fn) contains a further subsequence which converges to f p-almost everywhere in A.

Proof. The preceding theorem establishes that the subsequence condition is necessary for the stochastic convergence of (fn) to f, since every subsequence of (fn)

§20. Stochastic convergence

119

likewise converges stochastically to f. Let us now assume that the subsequence condition is fulfilled, and fix an A E W of finite measure. Since every subsequence (f,,.)

contains another which converges almost everywhere in A to f and by 20.5 this latter subsequence must also converge (in A) stochastically to f, we see that in the sequence of numbers

(kEN),

p({Ifnk - fI -a}nA)

in which a > 0 is fixed, a subsequence exists which converges to 0. But, as an easy argument confirms, a sequence of real numbers whose subsequences, have this property must itself converge to 0. That is, the sequence of real numbers

>a}nA)

(nEN)

converges to 0. As this is true of every A E d having finite measure and every a > 0, the stochastic convergence of to f is thereby confirmed. 0 Remarks. 5. It is not to be expected that in 20.7 and 20.8 the reference to the finite-measure set A E W can be stricken. This is already illustrated by Example 2

if one replaces the sequence (fn) there with the sequence (f) defined by f,, :_ nl(,,,, ), n E N. This new sequence also converges stochastically to f := 0. See however Exercise 5.

6. The second part of the proof of 20.7 shows that for finite measures u there is a Cauchy criterion for the stochastic convergence of a sequence (f.): Necessary to a measurable and sufficient for the stochastic convergence of a sequence real function on S1 is the condition for every a > 0.

litre

m.n-ix 7.

The sequence formed by alternately taking terms from each of two stochasti-

cally convergent sequences whose limit functions do not coincide almost everywhere

shows that in Corollary 20.8 it does not suffice to demand that in each A some sub sequence of the full sequence (fn) converge almost everywhere. A particularly useful consequence of 20.8 is:

20.9 Theorem. If the sequence (f,,) ,EN of measurable rral functions on 11 converges stochastically to a measurable real function f on. Q. and yo : R -4 R is continuous, then the sequence (y^ o f )nEN converges stochastically to V o f.

Proof. One exploits both directions of 20.8, noting that from the almost everyto f on an A E 41 follows the almost

where convergence of a subsequence everywhere convergence of (,p o

f on A. 0

The general question of functions p : R -* R which preserve convergence, in the sense that (o o f, inherits the kind of convergence (f,,)iE14 has, is investigated by BARTLE and Jo1CH1 (1961]. They show how Theorem 20.9 can fail if the more restrictive definition (20.5) is adopted for stochastic convergence.

120

11. Integration Theory

Exercises. are stochastically convergent sequences of measurable real func1. (fn) and tions, having limit functions f and g, respectively. Show that for all a,,8 E R

the sequence (af,, + 13g,,) converges stochastically to of + fg, and the sequences (fn A gn), (f V g,,) converge stochastically to f A.9, f V g, respectively. 2. For a measure space (Si, d,,u) with finite measure p let d, be the pseudomet-

ric on d constructed in Exercise 7 of §3. Show that a sequence (An) in saf is d,,-convergent to A E 0 if and only if the sequence (NAB) of indicator functions converges stochastically to the indicator function IA. 3. For every pair of measurable real functions f and g on a measure space (Cl, sA, µ) with finite measure µ define

D,(f,g) := inf{e > 0 : p({I If - gI > e}) < e} and then prove that (a) DP is a pseudometric on the set M(d) of all measurable real functions. (b) A sequence (fn) in M(W) converges stochastically to f E M(d) if and only if lim D, (f,,, f) = 0. n +00 (c) M(se) is D,,-complete, that is, every Dµ Cauchy sequence in M(d) converges with respect to Da to some function in M(Ao ). What is the relation of D,, to the dµ of Exercise 2? 4. In the context of Exercise 3 define

If - gi

dp,

for every pair of functions f, g E M(ss). Show that Dµ also enjoys the properties (a)-(c) proved for D$, in the preceding exercise. be a or-finite measure space. Show that a sequence (fn) of measur5. Let able real functions on Cl converges stochastically to a measurable real function f on Cl if and only if from every subsequence (fk) of (fn) a further subsequence can be extracted which converges almost everywhere in 0 to f. [Hints: Suppose (fn) is stochastically convergent. Choose a sequence (Ak) from d with p(Ak) < +oo for each k and Ak 1 11, and consider the finite measures pk(A) := µ(A fl At,) on sW. The claim is true of each measure Pk. Given a subsequence 4 of (fn), there is for each k E N a subsequence of (g;,k))nEN of 4' which converges pk-almost everywhere

to f. It can be arranged that (g nk+u)) is a subsequence of (gnl) for each k. Then the diagonal subsequence (g;,ni ), EN does what is wanted.] 6. Give an "elementary" proof of 20.9 based directly on the relevant definition 20.2.

To this end, show that for each E E 10, 1[ there exists 6 > 0 such that fl f I <

11F}fl{Ifn-f1 :56}C{IVo fn-Wofl<£}for all nEN. 7. (Theorem of D.F. Ecoaov (1869-1931)) Let (S2,srd,A) be a measure space with finite measure p. Show that: For every sequence (fn)nEN of measurable real functions on Cl its convergence almost everywhere to a measurable real function f is equivalent to its so-called almost-uniform convergence to f. The latter means

§21. Equi-integrability

121

that for every 6 > 0 there exists an A6 E W such that p(A6) < b and (fn) converges to f uniformly on CA6. [Hint: Exercise 2 of §11.]

§21. Equi-integrability The sufficient condition for convergence in eh mean which is set out in Lebesgue's dominated convergence theorem can be transformed into a necessary as well as sufficient condition with the help of stochastic convergence. But we need the concept of equi-integrability, which is of fundamental significance.

In the following (S2, sz4, p) will again be an arbitrary measure space, and p is always a real number satisfying 1 < p < +oo. The point of departure is a simple observation. A measurable numerical function f on S2 is integrable if and only if for every e > 0 there is a non-negative integrable function g = ge such that

J I9} IfI dp <e.

(21.1)

For if f is integrable and we take, as we then may, g to be 2 If I, then { If I > g} _ { f = 0} U { If I = +oo} and thanks to 13.6 the integral in (21.1) is actually equal to 0. Conversely, if we have (21.1) even for just one real e > 0, then

f IfI dp=

f

{IfI?9}

IfI dp+

f

{III<9}

IfI dp<e+f gdp<+oo

and hence f is integrable. This observation induces us to make

21.1 Definition. A set M of d9-measurable numerical functions on S2 is called (p-)equi-integrable if for every e > 0 there exists a p-integrable function g = ge > 0 on 0 such that every f E M satisfies (21.2)

f

f I dp< e. III_9}

Correspondingly a family (fi)iEl of measurable numerical functions on f is called equi-integrable if the set { fi : i E I) is equi-integrable. Equi-integrable sets and families are sometimes also called "uniformly integrable". From now on, any function ge as described in Definition 21.1 will be called an e-bound for the given set of functions. Obviously, along with an a-bound g for a set of functions, any integrable g' 2 g is also an e-bound.

Examples. 1. If Ml,..., Mn are finitely many p-equi-integrable sets of measurable functions on S2, then their union is also p-equi-integrable, because whenever gj is an a-bound for MM (j = 1, ... , n), then gl V... Vg,, is an a-bound for Ml U... U Mn.

122 2.

1 1. Integration Theory

Every finite set of µ-integrable functions is u-equi-integrable. This follows from

Example 1 and the fact, demonstrated in the course of proving (21.1), that any set consisting of just one integrable function f is equi-integrable, the function 2 If I being an a-bound for every e > 0.

Suppose M is a set of measurable numerical functions on fl, 1 < p < +oo, and there is a p-fold µ-integrable majorant g for M, that is, every f E M satisfies 3.

µ-almost everywhere.

If1 < g

Then the set

M":={IfIP:fEM} is equi-integrable. Indeed, as in Example 2, the single integrable function h := 2gP is an --bound for every e > 0, since by 13.6

J

fIdµ < J

gP dµ = J

dµ = 0

{g=too}

{gP>h}

1f1P>h}

This example shows that Theorem 15.6 on dominated convergence is really about an equi-integrable set of functions. Of course, one cannot expect that conversely from the equi-integrability of a subset of .`" (t) there should follow the existence of a single integrable majorant for the set. The following example confirms this. Consider the probability space (N, .(N), µ), the finite measure µ being specified by µ({n}) = 2-n for each n E N. The sequence of functions fn := 2"n-11{n) (n E N) is equi-integrable: For the constant function 1 E .2o1(µ) the inequality 4.

fn dµ <

1

n

holds for all n E N.

However, the smallest function g which majorizes every fn is the non-µ-integrable function n i-- 2nn-1 on N. 5. Let (St, d, µ) be the measure space of Example 3, §20, and (fn)nEN the sequence of functions considered there: An := [0, [ and fn := n1A, for each n E N. This sequence is not equi-integrable, which wensee as follows: for every integrable g > 0 and every n E N

/

JIf-I>g}

If,.Idµ=J

r

ndµ=J ndµ-J A

ndµ>1-J

A

From the finiteness of the measure gµ and the fact that An 1 {0}, it follows that

liminf J n_+00

Ifnl dµ> 1,

{If..I>g}

showing that g cannot be an a-bound for any e E ]0, 1[. Here is a useful characterization of equi-integrability, which, for o-finite measures, will be improved upon in 21.8.

§21. Equi-integrability

123

21.2 Theorem. A set M of measurable numerical functions on l is equi-integrable if and only if the following two conditions are satisfied: sup

(21.3)

fEM

f If I dµ < oo .

(21.4) For every e > 0 there exists a p-integrable function h > 0 and a number 3 > 0 such that

< d=* Jill/iforallfEMand Proof. For every A E &/, every measurable numerical function f on 0, and every integrable function g > 0

f AIfI du=

f

An{IfI>g}

IfI du+ f

An{III
IfI du<_

f

{IfI?g}

IfI

du+f gdu A

and in particular for A := fZ

f IfI du <_ f

IfI dµ+

{IfI>_g}

f gdu.

Assuming that the set M is equi-integrable, let us choose for g an E-bound for it and then set h := g, d 2. Then conditions (21.3) and (21.4) follow from the preceding inequalities. Conversely, assume the two conditions are fulfilled and let e > 0 be given. Let h and b > 0 be as furnished by (21.4). For each f E M and real a > 0, consider the obviously valid inequality

f IfI du

4IfI?ah}

Ifl du > f {If (If I>_-h}

or its equivalent 1

J IfI?ah} h djo < -

If I dM.

The integrals f If I dµ here are bounded as f ranges over M, by (21.3). Therefore a > 0 can be chosen so large that

hdµ < b for all f E M. {IfIiah} (21.4) then insures that g := ah is an c-bound for M, which proves that this set is equi-integrable. 0

21.3 Corollary. Let M C 2P and the set MP :_ { If I P : f E MI be equiintegrable, where 1 < p < +oo. Then the set

M;:={laf+,0glP:f,gEM,a,,0ER,Ial:_1,1,01<_1} is equi-integrable.

II. Integration Theory

124

Proof. For every f E 2P(p) and every A E dd, I lA f l <- If I shows that 1A f E 2'(p) too, and so for all fl, f2 E 2P(p) Minkowski's inequality (14.4) gives Np(lAfl + lAf2) :5 Np(lAfl)+Np(lAf2), whence ///'

JA

If,Ip dp)

Ifl + f2Ip dp <

1/v

+ (!A 1f21P dp}

1/1 p

Applying this inequality to fl = a fl, f2 = pg with f, g E M a, 8 E R and Ial < 1, ICI < 1, and hearing in mind that 21.2 is (by hypothesis) valid for the set MP, one realizes that conditions (21.3) and (21.4) are fulfilled by M: as well as by MP, with the same function h in both cases. 0 We are now in a position to deliver the sharpened version of the dominated convergence theorem mentioned in the introduction to this section. That we really

have to do with a sharpening here is attested to on the one hand by Example 3 and Theorem 20.4, according to which stochastic convergence follows from almosteverywhere convergence, and on the other by Example 4 of §20, which shows that there are situations in which the dominated convergence theorem is not applicable but the following theorem is.

21.4 Theorem. For every sequence. (fn)nEN of p -fold, p-integrable real functions on a measure space (1l, sd, p) the following two assertions are equivalent: (i) The sequence (fn) converges in p`h mean. (ii) The sequence (fn) converges p-stochastically, and the sequence (Ifnlp) is pequi-integrable.

Proof. (i)=(ii): Suppose

converges in eh mean, to f E 2P(µ); thus

lim Np(fn-f)=0.

n+oo

In the light of 20.4 only the equi-integrability of the sequence (I fnI") has to be proved. By (15.2) the sequence (Np(fn))nEN converges to Np(f) and is therefore bounded, so the set M := (If,, 1' : n E N} satisfies (21.3).

For every AEa(andevery nENwehave by(15.4)

(fA

If,.Idt) "<-Np(fn-f)+(JA If1Pdµ\1/µ J

To every e > 0 corresponds an nE E N such that Np(fn - f) < 2-eel/p for all

n > nE. Therefore, if we set 6:= 2-'Pe and

h:=If1IPV...VIfn,IPVIfIP, condition (21.4) is also satisfied by M. (ii) .(i): From the stochastic convergence of the sequence (fn) and Remark 6 in §20 it follows that (21.5)

lim p({I fm -

n,m- .

a} n A) = 0

§21. Equi-integrability

125

for every A E W of finite measure and every real a > 0. We have to show that is a Cauchy sequence in 2P(µ), that is, that the doubly-indexed sequence of functions frnn := frn - fn satisfies rrr

= 0. lim fIfrnfll' do

According to 21.3, along with the set {IfnIP : it E N} the set 1190 :_ {lfnrnI m, n E N} is also equi-integrable. Hence to every e > 0 corresponds an integrable function gE > 0 such that f{f _g. } f dµ < e holds for all f E Mo. If we set g := 9E1 /P then g is p-fold integrable and the preceding inequality can be written

J

fnrnIPdu<<e

for allm,nEN.

Because

f If,.. I" dµ =

f{If.,,I>g} Ifnrn IP do + J Ifm,.I
frn lP dµ

it suffices to show that Ifnrn IP dµ < 3E

(21.6)

{Ifm I
holds for all sufficiently large m, n E N. Now gPµ, being a finite measure on so', is continuous from above. Since n {g < k-1 } = {g = 0), i'l > 0 can therefore be kEN chosen small enough that

fwnl

g" (11,<E.

Consequently we also have (21.7)

J

fIdµ J

g «}

gP dµ <

for all m, n E N.

The Chebyshev-Markov inequality insures that the set {g > Y}} has finite µr measure. According to (21.5) therefore the doubly-indexed sequence of sets Ann :_ {I fnrnl > a} fl {g > 7)}

satisfies, whatever a > 0 is involved, lim

m.n-4Q0

µ(A,,,n) = 0.

We choose the positive number a so as to have

()PJgpd1j

< E,

in,nEN

1 1. Integration Theory

126

The p-continuity of the finite measure gPp and 17.8 provide for an no E N such

that

J

gP dp < e

for all m, n > no.

r

for all m, n > no.

,,

Hence (21.8)

J

Ifmn IP dp <

gP du < e

A second application of the Chebyshev-Markov inequality furnishes the estimate (21.9)

JIfrnnV' dp<&A({g>r)})<()"f?d

<efor allm,nE14,

17

{Ifmk9}fAm.,

Amn := {Ifm,il < a} n {g < rl} .

By adding the inequalities (21.7)-(21.9) we get finally inequality (21.6), whose confirmation was the last outstanding claim in the proof that (ii) implies (i). Remark. 1. Theorem 21.4 does not claim that from the stochastic convergence of a sequence (fn) to a measurable real function f, the p-fold integrability of f and the convergence of (fn) to f in pth mean follow as soon as the sequence (if. JP) is equiintegrable. Rather the theorem guarantees the existence of a p-fold integrable function among the possible stochastic limits of the sequence (fe). The sequence (fn) does converge in eh mean to every such stochastic limit, as follows from the proof of the theorem in the light of Remark 4 of §20, according to which any two p-fold integrable stochastic limits must in fact coincide almost everywhere. But stochastic limits that are not p-fold integrable do exist, a fact that can be demonstrated with the aid of the Example in §20: For the sequence (fn) there, (If,, 1") is equi-integrable. But among the stochastic limits f' that occur there, f' E .`BP(p) for some p E I1,+oo[ if and only if f'(wi) = 0.

However, the phenomenon discussed above does not occur for a-finite measures. By 20.3 in that case any two stochastic limits are almost everywhere equal. Therefore we have 21.5 Corollary. Suppose the measure p is a -finite. If a sequence (fn) from. "P(p) converges stochastically to a (measurable, real) function f, and if the sequence (IfnIP) is equi-integrable, then f E 2P(p) and (fn) converges in pth mean to I.

Theorem 21.4 can be sharpened by bringing in a further condition equivalent to (i) and (ii) which is suggested by F. Riesz' Theorem 15.3. En route to this sharpening the following lemma plays a key role. On the other hand, from the sharpening that we are aiming for, the lemma can in turn be deduced, as can the theorem of F. Riesz, even with its almost-everywhere convergence hypothesis weakened to stochastic convergence.

§21. Equi-integrability

127

21.6 Lemma. Suppose the sequence of functions f > 0 from 2' (p) converges stochastically to a function f > 0 from 2'(It). If in addition lien

then the

sequence

f f dit = If dp, J

converges to f in mean.

Proof. We consider the sequence (f A fn)nEN. The inequalities

0< fA and Example 3 show that it is equi-integrable. Since

05f-fAfn<-Ifn-fI

(forallnEN),

stochastic convergence of (fn) to f entails that of (f A fn) to f . From Theorem 21.4 this new sequence then converges to f in mean. We therefore also have (21.10)

lim n>z

From this, the decomposition f + fn = f V f + f A fn, and the convergence hypothesis follows the companion result (21.10')

lim

If V f dp =

f

f du.

But then the decomposition

If,, - fl =.f V .fn -.f A.fn shows that the claimed mean convergence ensues upon subtracting (21.10) from (21.10').

Now we can get the sharpening of Theorems 21.4 and 15.4 mentioned earlier:

21.7 Theorem. For every sequence (fn) in 2P(t) which converges p-stochastically to a function f E 2P(,u) the following three assertions are equivalent: The sequence (fn) converges in p'h mean to f . (1) (ii) The sequence (If,, 1") is equi-integrable. (iii) lim f If,, I' d;i = f If I' dp. n-, x.

Proof. The equivalence of (i) and (ii) is contained in Theorem 21.4. We need therefore establish only two implications: (i) .(iii): Assertion (15.6) in Theorem 15.1 affirms this. (iii)=,>(ii): From the hypothesized stochastic convergence of the sequence (f,,) to f follows that of (I f I') to If 11, via 20.9. And then from the preceding lemma

it further follows that the sequence (If P) converges to I fI' in mean. Finally, Theorem 21.4 - with the p there chosen to be I - shows that the convergence in mean of this sequence entails its equi-integrability.

128

1 1. Integration Theory

For a-finite measures µ, equi-integrability can be characterized in a way that is particularly convenient for applications. The a-finiteness will be exploited in the form expressed by 17.6, that there is a strictly positive function h in Y' (it). 21.8 Theorem. Let (S2, dd, p) be a o-finite measure space and h a strictly positive

function from 2'(p). Then for any set M of dd-measurable numerical functions on Sl the following three assertions are equivalent:

(i) M is equi-integrable. (ii) For every e > 0 some scalar multiple of h is an a-bound for M. (iii) M satisfies sup

(21.11)

fIfI dµ < +oo

JEM

as well as the following: Given e > 0 there exists 6 > 0 such that

fhd6=JIfIdlA
(21.12)

for allAEdd,fEM.

Statement (ii) simply says that

s lim

(21.13)

JIfI>ah} If I du = 0

holds uniformly for f E M. Condition (21.12) is for obvious reasons (cf. 17.8) called the equi-(hit)-continuity of the measures If I µ, f E M. Proof. (i) .(ii): Let g be an E-bound for M. Then for all f E M and all a > 0

{IfI>-hh}

IfI dµ=

f

{IfI>oh}n{IfI>g}

< fj IfI>_g} I fI dµ+

IfI dµ+

f

f

{(fI>«h)n{(fI<9)

gdla < E +

IfI dµ 9dµ

fig >cth} According to 13.6, µ({g = +oo}) = 0. Since gµ is a finite measure on dd, it is {g>ah}

2

continuous from above. Hence the fact that

n {g > ah} = n {g > nh} = {g = +oo} a>o

nEN

is a set of (gµ)-measure 0 means that

k>ah)

g dµ < 2

for all sufficiently large a. Coupled with the preceding inequality this shows that indeed ah is an a-bound for all sufficiently large a, that is, (ii) holds.

§21. Equi-integrability

129

This can be gleaned from the inequality derived at the beginning of the proof of 21.2, ah being now eligible for the function g there:

JIfIdJLjIJI> an}IfI d1+a

for all f EM.

hd/1

21.2 affirms this. 0 Theorem 21.8 is of special significance for finite measures p. Then it is often expedient to choose for h the constant function 1. When one does, (21.13) assumes the equivalent form (21.13')

lim

a-++oo

J IfI?a} IfI dp = 0

uniformly for f E M.

This condition is thus - just as (21.13) for a-finite measures - necessary and sufficient for equi-integrability of M.

Remark. 2. In part (iii) of Theorem 21.8 the 21-boundedness of M expressed by (21.11) cannot in general be dropped from the hypotheses. It suffices to consider the measure space ({a}, Y({ a}), Ca) consisting of a single point and the sequence

of functions f,, := n 1. This sequence is not equi-integrable, although for every e > 0 and every strictly positive h, (21.12) holds whenever 0 < 6 < h(a). Let us close by deriving a sufficient condition for equi-integrability in the finitemeasure case which generalizes the introductory Example 3.

21.9 Lemma. Let p be a finite measure and M C Y' (y). Suppose that there is a p-integrable function g > 0 such that (21.14)

J{Ift?a}

IfI dp <

f

J{IJI>a}

9dp

for all f E M and all a E R+. Then M is equi-integrable. Proof. The case a:= 0 of (21.14) says that f If I dp < f g dp < +oo for all f E M. Then Chebyshev's inequality tells us that p({IfI ? a}) <_

f IfI dp < a f 9dp

for all a > 0, f EM.

It follows from this that (21.15)

lim p({IfI > a}) = 0

a-4+oo

uniformly in f E M.

For each e > 0, 17.8 supplies a 8 > 0 such that

AEd and p(A)
=

fdize.

130

II. Integration Theory

Putting this together with (21.14) and (21.15) gives us Jim

n++0oJ(Ill>o)

IfI dp = 0

uniformly for f E M,

that i4, (21.13'), which we have seen entails equi-integrability of M. O

Exercises. 1. Show that for any measure space (0, a, p) a set M of measurable numerical functions is equi-integrable if and only if for every e > 0 there is an integrable function h = hr > 0 such that f (If I - h)+ < e for all f E M. [Hint: For sufficiently large q > 0, g := r)h will be a 2e-bound for M.] 2. Let (S2, d,14) be an arbitrary measure space, 1 < p < +oo. Suppose the se((t) converges almost everywhere on 12 to a measurable real quence (f,,) in function f. Show that f lies in 2P(p) and (fn) converges to fin pth mean if the sequence (If,, I P) is equi-integrable.

3. Show that from the 2-convergence of a sequence (fn) to a function f E 2"(e) follows the 21-convergence of the sequence (I fn IP) to If I, for any 1 < p < +oo. 4. Consider a finite measure .t and an M C Y1(µ). For each n E N, f E M set

an(f):=nµ({n<_IfI
Show that M is equi-integrable if and only the series E an(f) converges uniformly na in f E M. [Cf. Theorem 3.4 and its proof in BAUER [1996].] 5. Consider a finite measure p and an M C 2 (z). Show that M is equi-integrable

if there is a function q : a+ - R+ with the properties lilri q(t) t0+00 t

_ +oo and

spu

J q °If I du < +oo.

(In fact we have to do here with a necessary as well as a sufficient condition, which goes back to CH. DE LA VALLEE POUSSIN (1866-1962). Moreover, q can always be chosen to be convex and isotone. Cf. MEYER [1976], p. 19 or DELLACHERIE and MEYER [1975], p. 38.)

6. Let (fl,.ad,p) be a measure space with µ(S2) < +oo, (fn)nEN a sequence of measurable numerical functions fn > 0, and set f* := lira .supoofn. Show that: n (a) If the sequence (fn) is equi-integrable (or at least satisfies condition (21.12)), then the following "dual version" of Fatou's lemma is valid:

lim sup f fn dµ < J f * dit

(*)

A

for all A E S1.

A

How does the corresponding result in Exercise I of §15 fit in? [Hint: Exercise 2 of §11.]

(b) Under the hypothesis f f' du < +oc, the sequence (f,,) is equi-integrable if and only if (*) holds. [In proving the "if" direction, argue indirectly.]

§21. Equi-integrability

131

(c) Result (b) can fail in case f f ` dµ = +oo. Try to corroborate this with a se-

quence (an derived by appropriate choice of (sufficiently large) numbers a,, > 0 from the sequence (f,,) in the Example from § 15. 7. Let (f), .x, µ) be a measurable space with µ(S2) < +oo, and let (v;)iE f be a family of finite and it-continuous measures on 0. Suppose this family is equi-continuous at 0, meaning that to every sequence (An)nEN in iA with A,, J. 0 and to every

c>0there is an nEENsuch that y;(A,)<efor all n>nE,and all iEI.Show that then this family is equi-µ-continuous in the following sense (cf. (21.12)): To every E > 0 there corresponds a 6 = 6e > 0 such that

and µ(A)<6

vi(A)<eforalliEI.

What does this result say in view of Theorem 21.8? (Hint: Review the proof of Theorem 17.8.1

Chapter III

Product Measures

In this short chapter we will investigate whether and how one can associate a product with finitely many measure spaces. And for the product measures thus gotten we will want to see about how to integrate with respect to them in terms of their factors. We will recognize the L-B measure Ad as being a special product measure

when d > 2. One important application of product measures is the introduction of the concept of convolution for measures and functions.

§22. Products of c-algebras and measures j = 1, ... , n E N are given. We consider

Finitely many measurable spaces

the product set

n

Q:= X11j=Q1x...xQ,t j=1

and for each j the projection mapping Pj : 52 -> S2y

which assigns to each point (w1, ,w,) E I its jth coordinate wj. The a-algebra in Q generated by the mappings pa,. , pn is designated n j=1

and called the product of the a-algebrns d1 r ... , d,,. According to (7.3) we have to do here with the smallest a-algebra s® in ft such that each pj is d-safj-measurable.

The reader may recall that the product of finitely many topological spaces is defined in a very similar way. An important principle of generation for such products is immediately at hand:

22.1 Theorem. For each j = 1, ... , it let Ag be a generator of the a-algebra salj in SZj which contains a sequence (Ejk)kEN of sets with Ejk T Q j. Then the a-algebra ®.n is generated by the system of all sets A(i 0

E1x...xEn with E., E 9, for each j = 1, ... , n.

§22. Products of a-algebras and measures

133

Proof. Let 0 be any a-algebra in Q. What we have to show is that the mappings p,

are all d-Oj-measurable (j = 1,.. . , n) if and only if s+d contains each of the sets El x ... x En described above. According to 7.2 pj is .V-Afj-measurable just exactly if p 1(E3) E 0 for every E3 E 8 . If this condition is fulfilled for each j E {1,.. . , n}, then the sets

El x ... x En =p11(El)n...npnl(En) all lie in 0. If conversely, E, x ... x En E s+1 for every possible choice of E3 E 4 and j E {1,. .. , n }, then upon fixing E3 E 8j, the sets

Fk:=Elkx...xEj-1.kxEi xEj+1,kx...xEnk,

kEN,

all lie in W. Since the sequence (Fk)kEN increases to

U1 x...x1j-1 xEj xflj+1 x... xOn =pj1(Ej), this set too lies in d, for each j. The claim is therewith proven.

13

Remark. 1. The restriction imposed on the generators S, cannot generally be dispensed with. Take, for example, n := 2, sail in which .QF2 contains at least four sets.

{0,111}, ell := {0} and 82 := W2i

A particular case of this theorem is the fact that the product dj ® ... ®srdn is generated by all the sets Al x ... x An with each A3 E . . Our further course will be guided by the following example:

Example. F o r each j E { 1, ... , n} let Std := R, . rt :_ .41 and 8j :_ f 1. The system of all sets E1 x ... x En with each E? E Jr' is evidently just the system .5n of all right half-open intervals in Rn. According to 6.1, fn generates the a-algebra R" of n-dimensional Borel sets. Taken together with 22.1 - whose hypotheses are clearly satisfied here - this reveals that

,qn = a1 ®

(22.2)

(& R1

(n factors on the right).

By 6.2, A" is the only measure on R" which satisfies

,\' V1 x ... X In) = V1(Il) . ... Al (In) for all I, i ... , In E .01. This remark and the example preceding it leads to the following question.

Measure spaces (f13, O j, pi) are given, 1 < j < n with n > 2, and for each dj

a generator 9j. Under what hypotheses can the existence of a measure a on

010 .. . (9 On satisfying (22.3)

zr(E1

for all E,ESj,I<j
be proven? The accompanying uniqueness question can be settled at once:

134

III. Product Measures

22.2 Theorem. Suppose that for each j = 1, ... , n irj is an n-stable generator of ao which contains a sequence (Ejk)kEN of sets of finite pj-measure satisfying Ejk f 11j. Then there is at most one measure rr on alt ®... ®x/ erljjoying property (22.3).

Proof. Let 8 denote the system of all sets El x ... x E,,, where Ej E ej for each j. According to 22.1, 8 generates the a-algebra dj (9 ... ® 04. Since each Bj is f-)-stable, so is 8, as the identity ?I

n

9=1

j=1

X Ej)n(X Fj) = X(E,nF,) J=1

makes clear. Moreover Ek := Elk x ... X evidently satisfies

E N) defines a sequence in 8 that

EkTf1,x...xf1,,. Recalling that µj (Ejk) < +oe for all (relevant) j and k, we see that the uniqueness claim therefore follows from 5.4. (Obviously it would suffice if U Ejk = f1j instead kEN

of Ejk T SZj were satisfied for each j.) 0

Under the hypotheses of 22.2, which obviously entail the a-finiteness of each measure uj, the existence of the desired measure it can also be proven. This proof will be carried out in the next section, first for it = 2, then for arbitrary n > 2.

Remark. 2. In closing it should again be mentioned that a mapping

f:S2o-4 SZlx..-xSZ of a measurable space (11o, ado) into a product of measurable spaces (0j, Afj) is measurable with respect to the a-algebra all ® ... ®as' if and only if each component mapping fj := pj o f off is d0-Oj-measurable - a fact which is immediate from Theorem 7.4.

Exercise. Finitely many measurable spaces (flj,.Wj) are given, j = 1,. .. , n. Show that the algebra in S21 x ... x S2 generated by all sets Al x ... x A,, with each Aj E .rrdj consists of all finite unions of such product sets.

§23. Product measures and Fubini's theorem

135

§23. Product measures and Fubini's theorem Initially measure spaces (521, .sdl, pj ), (522, sd2, µ2) are given. For every Q C ill x 112

the sets (23.1)

{w2 E ill : (WI, W2) E Q} {w1 E ili : (w1,w2) E Q}

Q111

Q,,,.,

are called, respectively, the w1-section of Q (w1 E ill) and the w2-section of Q (w2 E p2) This notation is chosen for typographic simplicity and will see us through §23, after which it is not needed. In case ill = il2i however, it presents obvious problems, to circumvent which, alternative notations like,,,, Q or Q4 for Q,,1 are also popular in the literature. About these sets we claim:

23.1 Lemma. If Q E sd1 ® sd2i then its w1-section lies in ad2 for every w1 E 01, and its w2-section lies in sd1 for every w2 E i12. Proof. For arbitrary subsets Q, Q1 i Q2.... Of fl :=121 x 522i and points w1 E ill

(!\Q)w, =!2\Q.1 and

(U Qn)

= U (Qn)., . nEN

nEN

Furthermore 52, = 112, and more generally for Al C 111, A2 C ill we have (A1 x A2),1 =

j A2 0

if w1 E Al if w1 E ill \ A1.

For each w1 E 121, therefore, the system of all sets Q C fl having section Q,,, E .ode

is a a-algebra in Cl which contains every product set Al x A2 with Al E .o'j, A2 E ode. But according to 22.1 01 (& ad2 is the smallest a-algebra which contains all such product sets. This proves the part of the lemma dealing with w1-sections. Of course, w2-sections are treated the same way. 0 Since now µ2(QW1) and make sense for all Q E 01 ®.02, wl E ill and w2 E S12, we are in a position to take the next step:

23.2 Lemma. Suppose the measures p1 and µ2 are or-finite. Then for every Q E sd1 ® . 9 the functions w1 H µ2(Q.,)

and w2 H A, (Q..)

on 121 and 122, respectively, are sd1-measurable and 02-measurable, respectively.

III. Product Measures

136

Proof. The function wl H P2(Qw,) will be denoted by sq. We will establish the d1-measurability of sq, for each Q E d1 ®sal2. The other function can be treated analogously.

First suppose that µ2(1Z2) < +oo. In this case the set ) of all D E .01 ®sal2 whose sD function is.call-measurable constitutes a Dynkin system in C := 111 x 11.2. This involves the following easily checked assertions: 811 = /12(122);

sf1\D = 851 - SD for every D E .9;

svD = ESD. for every sequence (D,6) of disjoint sets in .9. Furthermore 9 contains Al x A2 for every Al E salli A2 E sale, since SA, xA2 =112(A2) - lA,

The system if of all such Al x A2 is fl-stable and generates sale ®sd2, by 22.1. Therefore 2.4 insures that 01 ®ad2 is the Dynkin system generated by it. From 9 C -9 C Wl ®,42 therefore follows that .9 = .call ®.v i which is what is being claimed.

of sets from ae, each of If 162 is only a-finite, then there is a sequence finite 162-measure, with Bn T 112. For each n, A2 H u2(A2f B.) is therefore a finite measure 162,, on sate, to which the already proven result can be applied, showing is .aft-measurable for each Q E Of, ® 02. Now that wl H 112(Q,,,) = auP112,,(Qw,) nEN

because of the continuity from below of the measure 162. From Theorem 9.5 then the mapping wl -r 162(Q,,,) is indeed al-measurable.

It is now rather simple to construct the measure it that we seek:

23.3 Theorem. Let (f1j, dj, pp) be o-finite measure spaces, j = 1, 2. Then there is exactly one measure.. it on all ® .sate which satisfies (23.2)

rr(A, x A2) = p, (Al)112(A2)

for all Al E sli, A2 E sate.

In addition this measure satisfies (23.3)

it(Q) =

f

f

for all Q E sail ®d2

and is a-finite. Proof. As before, for each Q E sate e s12 let sq denote the Wi-measurable function on 121; it is of course non-negative. Consequently via

w1

ir(Q) :=

JSQdILI

a non-negative function it is well defined on 010 sate. For every sequence (Q,)nEH of pairwise disjoint sets from sat 0 szt2 the equality sUq = E sq, and 11.5 insure

§23. Product measures and Fubini's theorem

that

137

00

7r U Qn) _ F, n(Qn) n=1

nEN

Since so = 0 we have 7r(0) = 0. This proves that 7r is indeed a measure on .od1®a2. It has property (23.2) because SA, XA2 = p2(A2)IA,, whence integration yields 7r(A1 x A2) = pl(A1)a2(A2)

Proceeding analogously, we confirm that

ir'(Q) :=

fi(Qw2)iz2(dw2)

also defines a measure on s1® ® d2 having this property. But when Theorem 22.2

sr'1 and &2 := W2 it affirms that there is at most one such measure. Thus 7r = 7r' and (23.3) is confirmed. There is a sequence (Ajn)nEN of sets from ,rarj, each of finite pj-measure, with Ajn T 52j, for j = 1 and j = 2. Using these as the A1, A2, respectively, in (23.2) proves the a-finiteness of IT because is applied to 9d°1

r(A1nxA2n)<+ooand A1nxA2nTfu1 xQ2 23.4 Definition. The measure IT on 010 .W2 which is uniquely specified by (23.2) whenever (521,911,p1) and (122,d2ip2) are a-finite measure spaces is called the product of the measures p1 and 02 and is denoted by

Thus also the question posed in §22 is answered for a-finite measures p1, P2.

If namely ej is a generator of salj (j = 1, 2) with the properties formulated in Theorem 22.2, then according to 22.2 and 23.3, Al ® p2 is the only measure IT on 01 ® 02 which satisfies (22.3). The Example in §22 therefore entails that A2 = a1 ®a1. Similar considerations lead to the validity of Am+n = '\® ®)n

for any m, n E N, once the appropriate identification of 1R"'+" with RI x Rn has been made. We turn now to integrating with respect to the product measure 141 ®p2. Our notation for sections can be usefully extended to functions for this purpose. If f : S21 X 122 -+ 12o is any mapping, we define its sections f, for each w1 E 521 and f,, for each w2 E 92 as mappings of 121 and f12, respectively, into 11o by (23.4)

f., (w2)

f (w1,w2)

for all w2 E 112

f,.,2 (wi)

f (wi, w2)

for all w1 E 521.

Notice that if Q C 121 x 122 and f := 1Q, then these functions satisfy (23.5)

(IQ),,, = IQ.,,

and

(IQ),,,2 = IQ,

.

111. Product Measures

138

Note, of course, that these indicator functions have different domains, and, just as with (23.1), further caution is called for with (23.4) in case ill = f12. Equations

(23.4), and (23.5) lead us to call the mapping f,,,, the wj-section of f. It enjoys the expected properties: 23.5 Lemma. For every measurable space (W, d') and every measurable mapping

f: (11 x122,4110A)-(11',d') is sate -d' -measurable and f,,, is .11-d'-measurable for every wl E 11 i w2 E S12.

Proof. For every A' E W', w1 E 11 fJ,'(A') = {w2 E 122 : (w1, w2) E

f-1(A')}

_ (f -'(A')),,,

and similarly for every w2 E 122

(f-1(A'))w,, so the measurability claims follow from Lemma 23.1.

Decisive is the following theorem which extends formula (23.3) from indicator functions to non-negative measurable functions. It goes back to L. TONELLI (18851946), its corollary to G. FUBINI (1879-1943). Both statements are often combined under the single designation the theorem of Fubini.

23.6 Theorem (of Tonelli). Let (111,41z) be o-finite measure spaces (j = 1, 2), and let

f: 121x122 R+ be s1® 0 .sat2-measurable. Then the functions

w2' J f,n dµ1 and w1 H

r dµ2

are .sate-measurable and Ol -measurable, respectively. Moreover, (23.6)

ffd(i0u2)=

J(ffW2dul),02(dw2)=J(ff1due)µl(dw1)

Proof. Set Sl := Sl1 x 112, NY' := . at-elementary function f :

®.so42 and rr := Eq ®µ2. Consider first an n

f:_Eaj1Qj

(ai>O,QjEa,nEN).

j=1

Then a glance at (23.5) reveals that for each w2 E

f

f"

n

Eaj1il040, d,u1= j=1

aj14IL2 and so

§23. Product measures and Fubini's theorem

139

an iA2-measurable function on l2 thanks to 23.2. Its integration is therefore accomplished by (23.3) thus:

f(ff2d1) _

aj7r(Q7) = j=1

J

f d7r,

which confirms the first equation in (23.6), for elementary f.

For an arbitrary d-measurable numerical function f > 0 let (u(')) be a sequence of .say-elementary functions such that uini T f. Then, as was noted in the first part of the proof, is a sequence of dl-elementary functions, which obviously satisfy u) T fw2 (for each w2 E 112). Consequently, the functions

Ji4)dir

V(n)(w2)

which are

w2 E 112,

d2-measurable by what has already been proven, increase to the function

w2H

f

f)2dp1,

by 11.3. This function is therefore also a02-measurable and the monotone convergence theorem 11.4 says that

ff

fiat dµl)µ2(dw2) = suP f

w(') dµ'2

nEN

Again, by what has already been proved, f ep(n) dM2 = J u(n) d7r

for each n E N.

By the choice of the sequence (u(')) and definition 11.3

f f d7r = sup I

u(n) d7r.

nEN

Combining the last three equations gives the desired

J(ffdi)P2(dw2) =

f f dir,

and wholly analogous arguments establish the claims about the functions f", . 0 Having disposed of non-negative functions, the next step in integration theory is to pass over to integrable functions. For them we get

23.7 Corollary (Theorem of Fubini). For j = 1, 2 let (llj, a4j, 14j) be a-finite measure spaces, f a k1 0 p2-integrable numerical function on !l x 02. Then for µl-almost every w1 the function f,, is 142-integrable and for µ2-almost every w2 the function f,,,2 is µ1-integrable. The functions

w1 y f fu dp2 and w2 H f fw2 dµ1

III. Product Measures

140

thus defined p,-almost everywhere on fl, and P2-almost everywhere on f12, respectively, are pl-integrable and µ2-integrable, respectively, and equations (23.6) are valid.

Proof. Evidently for all w? E S2? (j = 1, 2),

(f+)., = (fWj)+ and

IfIWj - I fWjI,

(f.,)-

(f

so we will employ parenthesis-free notation. According to (23.6) the product measure it := µ, ®µ2 satisfies

f(JIf1I d142)µ1(dw1) = f (f IfW,I dill)µ2(dw2) = f

Ifl d' <

+10-

In particular, the ddb,-measurable numerical function w, H f Iff,I dµ2 is µlintegrable and so by 13.6 it is µ,-almost everywhere finite. That is (by 12.1), for µ,-almost every w, the section f, is µ2-integrable. Consequently, w1 14

f L. dµ2 = f f.+,

d112 -

f

f,;, d02

is a p1-almost everywhere defined function, which is x/1-measurable because that

is assured of each integral on the right by Theorem 23.6. In turn each of these integrals is µ,-integrable by 23.6. So our pi-almost everywhere defined function w1 H f fW, dp2 is µ,-integrable and

f (f fW, dµ2)µ1(dw1) = f(Ji-, dp2)pi(dwi) - f (f fZ dµ2)p1(dw1) =

f

f + dir -

J

f - dir =

f dir.

Of course, the roles of w1 and w2 can be interchanged in this argument and we thereby secure the rest of what is being claimed. The theorems of Tonelli and Fubini insure, in particular, that under the stated hypotheses the order of repeated integrations is immaterial. We can emphasize this by writing the equation (23.6) in the form

(23 6')

f f d(µ1(9 µ2) = f f f(W1,W2)111(dW0112(dw2)

= Jff(wiw2)2(dw2)i(dwi). That exceptional sets of measure zero cannot generally be ignored in the conclusions of Fubini's theorem is illustrated by the following example. Example. 1. Consider L-B measure A2 = A' ®A' on R2, the set A := Q x R E R', and its indicator function f := 1,1. According to 23.3 or 23.6 we have A2(A) = f f dA2 = 0, so f is A2-integrable. Nevertheless, for every w1 E Q, the section f,,,, = la is not A'-integrable.

§23. Product measures and Fubini's theorem

141

Remark. 1. For certain measures µ1,P2 which are not or-finite the existence but usually not the uniqueness of a product measure can be proved by other methods. See, e.g., BERBEIUAN [1962]. Even if just one of p' or 112 fails to be a-finite, the second equality in (23.3) can fail. Cf. Exercise 1, p. 145 of HALMOS [1974], as well as chapter IV, §16 of HAHN and ROSENTHAL [1948]. Moreover, there exist f : 91 x f12 - R+ which are not sail (9 02-measurable yet the "iterated integrals" on the right side of (23.6) make sense (and are finite). For an abundance of illuminating but elementary counterexamples related to this famous theorem, see CHATTERJI [1985-86] and MATTNER [1999].

A useful and at the same time surprising consequence of Tonelli's theorem is that it permits p-integrals to be expressed by means of A1-integrals.

23.8 Theorem. Let (S2, d, p) be a a -finite measure space and f : Il - R+ a measurable, non-negative, real function. Further, let W : R+ -+ 11 P+ be a continuous isotone function which is continuously differentiable at least on R+ :_ ]0,+00[ and satisfies w(0) = 0. Then

f

(23.7)

+00

co o f dp = fit ,

(t)p({ f > t})A1(dt) = 0

w (t)µ({ f > t}) dt .

J0

+

Proof. Consider the L-B measure A' := AR+ on the o-algebra R' := R+ fl9l. The function F : 0 x R+ -+ R2 defined by F(w, t) :_ (f (w), t)

is, according to Remark 2 in §22, 0 ®.4'-measurable, because each of its component functions is. Therefore the F-preimage of the closed half-plane {(x, y) E R2 : x > y}, namely

E:={(w,t)ESZxR+: f(w)>t}, lies in sad®.. Theorem 23.6 for the product measure p®A' consequently supplies the equalities

JJ

(23.8)

V

(t)IE(w,t)A'(dt)p(dw) = f f V(t)1E(w,t)µ(dw)X'(dt)

= Jw'(t)iz(Ei)A(dt) =

Jc'(t)({f > t})A'(dt),

since the t-section of E is just the set of all w E 1 which satisfy f (w) > t. As V is isotone, W'(t) > 0 for all t > 0. The continuous function gyp' is integrable over [1/n, a] whenever 1/n < a < +oo, and since [1/n, a] t ]0, a], and

f

oal

(t)A'(dt) = limo J

n

(t) dt = W(a) - n m V(1/n) = w(a)

142

!IL Product Measures

(cp(0) = 0 and Sp is continuous on R+), we see that V is also integrable over 10, a] for every a > 0. It follows from f > 0 and the preceding calculation that

p'(t)a(dt) = (f(w))

J

for every

E S1,

o,f(W)l

both expressions being 0 whenever f (w) = 0. We thus get o f dµ =

f (Jlo,f(W)l

= J f o'(t)llo,nw)d(t)A*(dt)µ(&) =

J

IV

which combined with (23.8) concludes the proof. D

Example. 2. The relevant hypotheses are certainly fulfilled by the functions V(t) := t' with p > 0. Thus for every a(-measurable real function f > 0 on S1 (23.9)

J

fl'dµ=p

+ 0

When p = 1 we get the especially important formula (23.10)

f f du =

r p({f > t})A1(dt) =

t})dt.

The reader should not overlook the geometric significance of this, which is that the integral f f dµ is formed "vertically", while the integral on the right-hand side of (23.10) is formed "horizontally".

Now at last we turn back to the general case of §22 and consider finitely many o-finite measure spaces (S1i, di,,a ), j = 1, ... , n and n > 2. The two product sets (f21 x ... x 1li_1) x On and SZ1 x ... x Sln_1 x Stn will be identified via the bijection

((w1,...,W,y_1),wn) H (L11,...,wn-l,wn) The agreed-upon equality of these sets leads at once to the equality of the corresponding products of v-algebras: (23.11)

(Wi®...®An-1)®-Wn=010...®An-1®dd/n.

In fact, by 22.1 the sets Al x ... x An- l with each Ai E jz(j generate rote®...OAfn-1,

and by the same theorem the sets

then generate (.Q91 0 ... 0 s0n_ 1) ®6dn as well as .c

® ... ®sOn_ 1 ®SF,.

§23. Product measures and Fubini's theorem

143

In a completely analogous fashion one confirms a general associativity in the formation of products of a-algebras: m

n

j=1

j=m+1

(23.12)

n

-'10

= j=1 ® 0j

(1<m
The convention (23.11) opens up the possibility of proving the existence of product measures on any finite number n > 2 of factors via induction on n.

23.9 Theorem. or-finite measures µl, ... , µn on a-algebras .d1, ... , jVn uniquely determine a measure 7r on safe ® ... 0 do such that (23.13)

for all Aj E 0j, 1 < j < n.

7r(A1 x ... x An) = ul(A,) .... µn(An)

This measure 7r is a-finite.

Corresponding to Definition 23.4, 7r is called the product of the measures µl, ... , µn and is denoted by n

®µj µl®...®µn. j=1

The question posed in §22 is finally answered in full, by this theorem.

Proof. In 22.2 take for the various generators 8j the o-algebra .dj itself, and learn that there is at most one measure 7r which satisfies (23.13). The existence question has already been settled for n = 2, in 23.3. We make the inductive assumption that 7r' := µ1 ®... ®µn-1 exists for some n > 2 and show how that leads to the existence of µl ® ... ®µn. Evidently the a-finiteness of µl, ... , µn_1 entails that of 7r', as in the proof of Theorem 23.3. That theorem therefore supplies us with a measure 7r := 7r' ®µn on (.W1 ®... ®.dn_ 1) ®.dn which satisfies 7r(Q' x An) = 7r'(Q')µn(An)

for all Q' E .d1 ® ... ® .dn-1 and all An E dd4n. Because of (23.11) this measure does what is wanted at level n, completing the induction. Again, a-finiteness of 7r is confirmed exactly as in the proof of 23.3. 0 This inductive construction of the n-fold product measure builds in the equality (23.14)

(141 ®... (&µn-1) ®µn = µ1 ®... ®µn-1 ®µn By now familiar considerations show that in fact a general associativity prevails in the formation of product measures: m (23.15)

In particular

n

n

(®µj)®( ® µj)=® µj j=1 j=m+1 j=1 xd

=

V

®V,

(1<m
144

III. Product Measures

In view of (23.15) induction can also be used to extend the theorems of Tonelli and Fubini to multiple factors. We will formulate only the analog of 23.6: Let f _> 0 be an s91®... ®.c 4-measurable numerical function on 01 x... x Stn. Then for every permutation j1, ... , j,, of 1, ... , n

Jfd(ii®...®in)

(23.16)

= f(... (f (f f(w1i...,wn)µj,(dwj,))µj.(dwjs))...)µjr(dwj.)' Every integral that occurs on the right-hand side is measurable with respect to the product of the appropriate Oj, namely those corresponding to the coordinates in which integration has not yet occurred. This right-hand side is often written in the shorter fashion

J ... J The simple proof of this theorem (involving induction), as well as the formula, tion and proof of the analog of 23.7, will be left to the reader. One more piece of notation is convenient:

23.10 Definition. For finitely many a-finite measure spaces (SZj, Wj, µj), 1 < j < +,

1l

1!

n, the triple ()( SZj, ®.Wj, ®µj) is called the product of these measure spaces 7=1

j=1

j=1

and is denoted by

n

j,

14Y

j=1

Remark. 2. Throughout the preceding the index set was finite. But there is also a theory of products of (finite) measures indexed by arbitrary sets, which is particularly important in probability theory; it is treated in detail by BAUER [1996], and somewhat more extensively in HEw rr and STROMBERG [1965]. For p-measures SAF,KI [1996] gives a short, elementary proof that uses only 5.1.

In closing we will consider the case where each measure µj comes with a real density f j > 0. According to Theorem 17.11, vj := f jµj is then a a-finite measure too.

23.11 Theorem. Let (S2j,.Vj, jAj) be or-finite measure spaces

andfj>0real-

valued w(j-measurable, functions on S1j. Set

vj = fjµj, Then the product of these measures is defined and satisfies (23.17)

n

n

j=1

j=1

®vj = F. (®µj)

j = 1,...,n.

§23. Product measures and Fubini's theorem

145

with the density function n

[ffj(wj),

F(wl,...,wn)

(23.18)

j=1

The function F is the so-called tensor product of the densities f1,..., fn Proof. As already noted, 17.11 insures that each measure vj is a-finite, guarantee-

ing that their product is defined. It suffices to treat the case n = 2 and refer the general case to induction. For sets Al E and A2 E s12 vl(A1)v2(A2) =

=

(jfid14i)(j12d142) z

Jf

I ._

lA,(w1)fl(wl)lA2(w2)f2(w2)141(dwl)112(dw2)

= Jf lA,xA2(wl,w2)F(wl,w2)1L1(dwl)122(dw2) From 23.6 therefore Fd(141 ®1L2),

v1(A1)v2(A2) = J

for all Al E. iA2Ed2.

, x A2

But then according to 23.3, v1 ® v2 coincides with the measure F (141 ®14z). 0

Exercises. 1. Consider 521 = 522 :=1R, 01 = 02 := ,41, it, := Al and 142 the non-a-finite counting measure on .41 (cf. Example 3, §5). Show that equality (23.3) fails to hold for Q := D, the diagonal {(w,w) : w E R} in 121 x 522. Why does D lie in jV1 002 =W2? 2. Show that the function (x, y) H 2e2xv - exv is not A2-integrable over the set [1, +oo[x [0, 1].

3. With the aid of Tonelli's theorem find a new proof of Theorem 8.1 along the following lines: Up is a translation-invariant measure on mod, 14([O,1[) = 1, and f >

0, g > 0 are Borel measurable numerical functions on Rd, compare the integrals

f

f()f(x + y)14(dx)Ad(dy)

and f f g(y - x)f(y)14(dx)Ad(dy)

and, finally, take f to be any indicator function, g the indicator function of [0, 1[. 4. Compute 00

2

I:= f e_x dx, 0

and thereby evaluate anew the important integral G = 21 in (16.1), in the folye_y2V2 lowing simple way: fo a-e2 dt = fo dx for every y > 0 and therefore

146

III. Product Measures

I2 = f °° (, fn f (x, y) dx) dy for the function f on R+ x R+ defined by f (x, y) yP-v2(1+z2). Applying Tonelli's theorem leads to I = 2Vr7r.

5. Let IxI := (x + ... + xd)112 denote the usual euclidean norm of the vector x := (x1,. .. , xd) E Rd. Show that the function x H e-Iz1° is ad-integrable for every a > 0. (Recall Exercise 2 of §16.) In case a = 2, show that the Ad-integral of this function is Gd.

6. KL(xo) will denote the closed ball in Rd with center xo and radius r > 0. Set ad :_ and prove that ,\d(K*(xo)) = adrd .

Show also that the numbers ad can be calculated by a2q = 4 9rq,

2q(2q

and a2q- i = 1 3

- 1)

a-1

(q E Dl).

[Hint: Use (7.10) and note that every xd-section of K,.(0) is either empty or is a (d-1)-dimensional closed ball. Tone1G's theorem then leads to a recursion formula for the ad. Here, of course, 7r has its customary geometric meaning.]

How do these relations change if we replace K,.(xo) by the open ball Kr(xo) in Rd of radius r and center xo? [Cf. Exercise 3 in §7.] 7. For every compact interval [a, ,Q] C R+ designate by R(a, Q) the spherical shell

K,3(0) \ K.(0) _ {x E Rd : a < IxI < /3} . Show that for every continuous real function h on such an interval (a, /3] C R+

f

h(Jxj)Ad(dx) = d ad f

.

a

R(a,p)

h(t)td-1

dt,

ad being the number ad(KI (0)) from the preceding exercise. [Hint: The function H defined on [a, p) by

H(t) := f

h(IxI)J1d(dx),

is differentiable with H'(t) = d ad h(t) td-1 for all such t.] 8. Apply the result of Exercise 7 to the case d = 2 and h(t) := show, using Exercise 5, once again that G = f.

tE

a-t2

in order to

9. Let (S2, d1. p) be a o-finite measure space, f : Il -+ R+ measurable. Show that

the set of all t > 0 such that u({f = t}) # 0, as well as the set of all t > 0 such that µ({ f > t}) # µ({ f > t}) is countable. Therefore in the equalities (23.8), (23.9) and (23.10), p({ f > t}) can always be replaced by µ({ f > t}).

§24. Convolution of finite Borel measures

147

§24. Convolution of finite Borel measures Consider the d-dimensional Borel measurable space (Rd,.gd). Every finite measure µ on Rd will be called a finite or also a bounded Borel measure, and the set of all of them will be designated by.,&+' (lR'). For every such µ the number (24.1)

lI,II := IA(Rd)

is called the total mass of A. Making critical use of the group structure of (Rd, +) a so-called convolution product can be assigned to any finitely many measures Al, ... , An E .K+ (Rd);

in contrast to the previously studied product measure, it is again a measure on the original o-algebra Vd, even an element of .,of' (Rd). What we do below can be carried out in every (abelian) locally compact group. We cannot, however, go into this generalization, but must instead refer interested readers to the excellent monographs of HEwIrr and Ross [1979] and RUDIN [1962]. Initially we consider

the product measure Al ® ... ® An defined in §23. Since W d = Rd ®... 00, this measure is an element of .,W+b (Rod) The mapping A. : R"d -3 Rd defined by

A,,(xl,... , xn) := x1 + ... + xn is continuous, and so Vnd-.mod-measurable. The following definition accordingly makes sense:

24.1 Definition. The image under the mapping An of the product measure -IC/+b(Rd), plo. .®Idn is called the convolution product of the measures pl,... , An E in symbols (24.2)

The theorems on product and image measures combine to yield the most important properties of the convolution operation *. First of all, At * ... *An is again an element of .0+1 (Rd) and

µl*...*µn(R")=µl®...®p,(R"d)=11µ11I ...

IIJUnII

so that in fact (24.3)

IIµl * ... * poll = 11µ11I ...' 11µn11

In studying the convolution product it suffices to deal with n = 2, because (24.4)

Al * ... * An * I`n+1 = (Al * ... * ln) * ltn+1

for every n + 1 measures from .4 (Rd). To see this, introduce the continuous mapping Bn+1 : R(n+l)d _+ Red by

Bn+1(x1, ... , xn, xn+l) := (XI + ... + xn, xn+l )

148

III. Product Measures

and have An+l = A2 o B.+1. Checking that Bn+1(p1 ®... OA. 0 pn+1) = A. (j AI ®... ®pn) ®pn+1,

and remembering that the formation of image measures is transitive, we get Al * ... * pn * µn+1 = A2(Bn+l (JAI ®... ®pn ®pn+i )) = A2((1.t1 * ... * A.) 0 pn+1), which confirms (24.4). Henceforth therefore n = 2. For any measures p, v E .4f+' (Rd) and any 0-measurable numerical function f > 0 it follows from T19.1 and 23.6 that

J

fd(E.e*v)

r

=J foA2d(p®v) = ff f(x + y)p(dx)v(dy)

(24.5)

= f f f(x + y)v(dy)µ(dn)

As this holds for f := 1B, they indicator function of any set B E fed, we have (24.6)

p * v(B) = J µ(B - y)v(dy) = J v(B - x)p(dx)

(Recall (7.8) that B-x = -x+B.) Consequently * is a commutative, and by (24.4) also an associative operation in .1/+(R.d) Due to 19.2 and 23.7, (24.5) are valid as well for every p*v-integrable numerical function f on Rd. Equality (24.6) is frequently taken as the definition of p * v. Evidently .,W+6 (Rd) is closed with respect to addition and under multiplication by numbers in R+. From (24.6) we immediately see the relation of convolution to these two operations: For all p, v, v1i v2 E .41+(Rd), a E 11 Y+

p*(vl+v2)=p*v1+p*v2, p*(av)=(ap)*v=a(p*v).

(24.7) (24.8)

The distributive law (24.7) even holds in the following generality: For every sequence

of measures from .4r+(Rd) satisfying E IkvJJ1 < +oo, the sum n=1

00

E vn is also a measure in .4f+1 (Rd) (cf. Example 4 of §3). Taking account of 11.5,

n=1

it therefore follows from (24.6) that 00

(24.9)

14 *(E14t n=1

00

Ep*vn n=1

for every p E A,(+(Rd)

Let us now compute p * v in some special cases.

§24. Convolution of finite Borel measures

149

1. We again denote by T. the translation mapping x H x + a of Rd onto itself via a E Rd, and by ea the (Dirac-)measure on Md defined by unit mass at the point a. Of course, Ea E -f+(Rd) and IIEa1I = 1. From (24.6) follows that Ea * µ(B) _ µ(B - a) = µ(T; ' (B)) for all B E mod, and so (24.10)

E. * µ = Ta(p)

for all p E .4W+6 (Rd), a E Rd.

Now To is the identity mapping, so co is a - and obviously the only - unit with respect to convolution. If, namely, E were also a unit, meaning that p = E *,U for every µ E 4. (Rd), then it would follow that Eo = E * co = E. For the special choice p := Eb, (24.10) says that (24.10')

for all a, b E Rd.

Ea * Eb = Ea+b

2. Let f > 0 be a Ad-integrable numerical function on Rd and p := fAd. Since IIµII = f f dAd < +oo, p also lies in W+ (Rd). Let us compute p*v for an arbitrary v E .,4+(Rd). From 17.3 using the translation-invariance of Ad and the general transformation theorem 19.1, we get

p * v(B) = J J 1B(x + y)f (x)Ad(dx)v(dy) = f f 1B(x +

y)f(x)T-v(Ad)(dx)v(dy)

= f f 1B(x)f(x

- y)Ad(dx)v(dy)

for every B E .mod. With the help of Tonelli's theorem it further follows that

p * v(B) = f 1B(x)q(x)Ad(dx) = f gdAd, B

where q is the non-negative .mod-measurable function x H f f (x - y)v(dy). This function is also Ad-integrable, since f q dAd = Ilp * vfl < +oo. Thus whenever p has a density with respect to Ad, so does p * v. We set f * v := q, that is, we make the definition (24.11)

f * v(x) := f f (x - y)v(dy)

for x E Rd.

The preceding result now assumes the more suggestive form (24.12)

(/Ad) * v = (f * v)Ad.

Naturally f * v is called the convolution of f and v.

3. Besides p = f Ad, let now v = gAd also have a Ad-integrable density g > 0. According to 17.3 and the preceding f * (gAd)(x) = f f(x - y)g(y)Ad(dy)

(x E Rd)

150

III. Product Measures

is a density for u * v with respect to Ad. We denote this function by f * g, that is, we set (24.13)

f * g(x)

f f(x - y)g(y).d(dy)

(x E Rd)

and get

(f Ad)*(gAd)_(f*g)Ad-

(24.14)

Here too f *g is called the convolution off and g. It is defined for every pair of nonnegative Ad-integrable functions and is itself such a function. Nevertheless, it might

not be real-valued, even if f and g each are (cf. Remark 1 below). Ftom (24.13) and the translation- and reflection-invariance of Ad it follows that for every x E Rd

f * g(x) = f f(x - y)g(y)Ad(dy) = f f(x + y)g(-y)Ad(dy) =

f f(y)g(x _ y)Ad(dy) = g * f(x)-

That is, the * operation between functions is also commutative: (24.15)

f * g = g * f.

Similar calculations confirm its associativity; that is, (24.16)

(f*g)*h=f*(g*h)

for all Ad-integrable, non-negative functions f, g, h. The distributive law (24.17)

f*(g+h)=f*g+f*h

and the homogeneity property (24.18)

f * (ag) _ (af) * g = a(f * g)

(aER.F.)

for such functions hold as well and follow immediately from (24.13).

4. For arbitrary functions f, g E 2' (Ad) decomposition into their positive and negative parts and appeal to the resusecured in 3. show that x +

ff(x - y)g(y)Ad(dy),

while possibly defined only Ad-almost everywhere (see Remark 1 below), is always Ad-integrable. One can therefore define f * g by f * g(x):= f f(x - y)g(y)Ad(dy)

but generally only for Ad-almost all x E Rd. Once again the expression convolution is used for this f * g.

§24. Convolution of finite Borel measures

151

Remarks. 1. For real-valued, non-negative functions f, g E pl (Ad) the function f * g need not be finite everywhere. It suffices to consider any real-valued, non-negative, even function f which lies in Y1 (A") but not in 22(Ad) and to take g = f. Then f * g(0) = +oo. In case d = 1, such a function is

f(x) :=

forlxI>Iorx=0

10 1

IXI-112

for 0 < IxI < 1.

2. In passing to Le(ad) - cf. Remark 1 in §15 - the difficulties high-lighted above with the definition of f * g disappear. Indeed, let f H f be the canonical mapping of .1 (Ad) onto Ll (Ad). One defines f * g for arbitrary f , § E Ll (Ad) as the image h of a function h E 21 (Ad) which coincides Ad-almost everywhere with f * g. This definition is independent of the special choice of representing functions f, g and h from 21 (Ad). The new operation * renders the vector space Ll (Ad) an algebra over R.

Exercises.

1. Show that for any it, v E dii (Rd) and any linear mapping T : Rd - Rd, T(µ * v) = T(p) * T(v). To this end, first observe that T o A2 = A2 o (T (& T), where T 0 T denotes the mapping (x, y) -+ (T (x), T (y)) of Rd x Rd into itself. 2. Compute the nlh convolution power of the function f defined on R by f (x)

ethat is, the convolution f * ... * f with n(E N) factors. Is it true that for every n E N, f has an "nth convolution root"? That is, is f the nth convolution power of some A'-integrable function g > 0? 3. If we set N1(f) f I f I dAd (this is (14.1) for it := Ad), then

N, (f *g)
III*9II1 5 II/II,

119111

holds for all elements f and g of the Banach space L1(Ad). The latter is therefore a Banach algebra.

Chapter N

Measures on Topological Spaces

In view of many applications in analysis, geometry and probability theory it turns out to be unavoidable to subject the Borel measures on Rd to more precise analysis. These measures possess a host of remarkable properties involving the topology of Rd. Up to this point topology has only entered the picture in the generation of the Borel sets. We will see that the completeness of the euclidean metric, respectively, the local compactness of Rd were responsible for the aforementioned properties. But there are more general topological spaces, important in their own right, which share these properties with Rd. Therefore from the start we will set our exposition in an essentially more general framework: Instead of Borel measures on Rd we will study Radon measures on Polish and locally compact spaces. In the process new facts, even in the Rd environment, about the nature of the integral and the measurability concept will emerge. A natural and useful convergence concept will play a role. In what follows some simple things from general (point-set) topology will be pre-supposed. The textbooks of KELLEY [1955] and WILLARD [1970] are good sources for these, and explicit references to them will be given at the appropriate points in the text.

§25. Borel sets, Borel and Radon measures Initially E will be an arbitrary topological space. The system of its open subsets which defines the topology will be denoted B. In the case of Rd we had determined (cf. 6.4) that the o-algebra of Borel sets is generated by the open sets. Consonant with this we now make the general

25.1 Definition. The a-algebra in E generated by 6 is denoted by V(E) and called the Borel or-algebra in E: (25.1)

.l

(E) := Q(6) .

The closed sets being the complements of the open ones, _V(E) is also generated by the system of all closed subsets of E. In this respect the analogy with 6.4 extends

a bit farther. The intersection of a sequence of open sets is called a G6-set, and the dual, the union of a sequence of closed sets is called an Fa-set. All such sets are clearly Borel.

§25. Borel sets, Borel and Radon measures

153

From now on E will be a Hausdorff space. Then every compact subset of E is closed, hence Borel. The second example below will show, however, that generally

the a-algebra generated by the class Xfof all compact subsets of E is strictly smaller that _4(E). So at this point the analogy with 6.4 falters. Examples. 1. From 6.4, as has already been mentioned, -4 (Rd) = .Vd,

(25.2)

E := Rd here carrying its euclidean topology. 2.

Let E be a discrete space, meaning that 6 = 9(E). Then the system

' of

compact sets consists just of the finite subsets of E. Consequently (cf. Examples 2 and 7 in §1) o(..iE') is the countable and co-countable a-algebra, comprised of all countable subsets of E and their complements, and so o ,(X) = -V(E) if and only if E is countable.

Let Q be a subspace of the Hausdorff space E. Then .V(Q) is the trace of R(E) on Q: -v(Q) = Q n 9(E). 3.

In fact, by definition the subspace topology of Q consists of the sets {Q n G : G E

6}, so this system generates ..(Q). Since Q n B(E) is a a-algebra in Q which contains this system, it follows that 9(Q) C Q n . f(E). On the other hand, the system {A C E : Q n A E .9(Q)} is obviously a a-algebra in E which contains all the open subsets of E, a generating system for .V(Q). Hence Q n M(E) C SR(Q). If Q itself is a Borel set in E, then ..(Q) just consists of all the Borel sets in E which are subsets of Q. 4.

The compactified number line i is a topological space which is homeomorphic

to the compact interval [-1,+1]. For it

.9(i) = R1 In fact, R n..(i) = ..(R) = V1 by Examples 1 and 3 above. The subsets {-oo} (25.3)

.

and {+oo} are closed in R and the subset R is open in R, hence all three are Borel sets in K. Equality (25.3) therefore follows from the definition of R1, given in §9.

In the sequel we will be studying measures on R(E) for two important classes of spaces E. In preparation for which we make

25.2 Definition. Let E be a Hausdorff space. A measure p on the a-algebra..(E) is called: (i)

a Borel measure on E if

µ(K) < +oo

for every compact K C E;

(ii) locally finite if every point of E has an open neighborhood of finite µ-measure;

(iii) inner regular if for every B E ..(E) (25.4)

µ(B) = sup{µ(K) : K compact C BI;

154

IV. Measures on Topological Spaces

(iv) outer regular if for every B EL(E) (25.5)

p(B) = inf {p(U) : B C U open} ;

(v) regular if it is both inner regular and outer regular.

Note that a Borel measure is more than just a measure defined on 69(E): in addition finiteness on the system of compact sets is demanded. The inner and outer regularity conditions say that the measure is determined on every Borel set by its values on the compact, resp., the open sets. The Borel measures on E = Rd are already familiar to us from §6. Every finite measure on M(E) is obviously a Borel measure; as in §24 where E = Rd, we naturally call it a finite Borel measure on E. The notation introduced there for the total mass of a finite Borel measure will be carried over to this more general setting: For every finite Borel measure i on a Hausdorff space E (25.6)

IIiII := p(E)

is called the total mass of it. Already at this point we can observe that every locally finite measure p on R(E) is a Borel measure, that is, that (25.7)

(ii)

(i) .

Indeed, each point x in the compact set K has an open neighborhood V,, with p(Vr) < +oo, and compactness means that finitely many of these, say those corresponding to x, , .... x,,, cover K. Then n

p(K) < p(VV, u ... U

p(Vxf) < + 0 C .

The converse of (25.7) is, however, not generally valid. Exercise 2 below furnishes eui example.

Because of the implication (25.7), instead of locally finite measures defined on 1(E), we will henceforth say simply locally finite Bore! measures.

For the moment we will be content to illustrate the regularity concept with some examples.

Examples. 5. Let E be an arbitrary Hausdorff space, a a point in E. The measure eq on .(E) defined by unit mass at a: (25.8)

e .(A) = 1A(a)

for A E R (E)

is both inner and outer regular on E. Henceforth it will be called the Dirac measure

on Eat a. As in Example 2, let E be a discrete space, so that t9 = .9p(E). The compact sets are just the finite ones. The measure defined on .9(E) by if A is countable J0 p(A) 1 +oo otherwise 6.

§25. Borel sets, Borel and Radon measures

155

is a locally finite Borel measure which is obviously outer regular. It is, however, inner regular if and only if the set E is countable. 7. On -41 = .a(R) consider the counting measure. It is not a Borel measure, is however inner regular, but not outer regular. In fact, equality (25.5) fails even for one-point sets B.

L-B measure Ad on ( d =M(Rd) is a (locally finite) Borel measure. In §26 we will see that it - and indeed every Borel measure on Rd - is regular. 8.

Developments stretching over decades attest to the fact that on a Hausdorff space those Borel measures which are locally finite and inner regular play a distinguished role. Such measures are nowadays named after J. RADON (1887-1956). A work of his from the year 1913 (cf. the bibliography), which has since become classical, set this development in motion. 25.3 Definition. A measure defined on the Borel a-algebra . (E) of a Hausdorff space E is called a Radon measure on E if it is both locally finite and inner regular.

More precisely the term used is "positive" Radon measure, but in this book we dispense with that adjective because non-negativity is built into our definition of measure, that is, we consider only measures with values in [0, +oo]. Example 5 says that the Dirac measure at any point a E E is always a Radon measure on E. We have already noted that Borel measures are not automatically locally finite. Nevertheless for many spaces Radon measures can be defined simply as the inner regular Borel measures. That is the import of

25.4 Lemma. On a Hausdorf space E in which every point has a countable neighborhood basis, every inner regular Borel measure p on E is also locally finite and hence a Radon measure. Prof. We argue by contradiction: Suppose that it is not locally finite, which means there is a point x E E such that u(V) = +oo for every open neighborhood V of x. By hypothesis x has a neighborhood basis consisting of a sequence (Vn) of open sets, and by replacing each V. with V1 fl ... fl V,,, we may suppose that V. 1. {x}. Since p(Vn) = +oo and p is inner regular, there exists a compact subset Kn C V. such that p(K,,) > n, and this is true of each n E N. Now the set

K := {x} U U Kn nEN

is compact. For if °1! is an open cover of K, then some U E P1 contains x and since (Vn) is a neighborhood basis at x, Vno C U for some no E N. It follows that C U for all n > no. Since Kl U ... U Kno is a compact subset of K, K, C Vn C it is covered by finitely many sets in 9l. These together with U then furnish the desired finite covering of K. On the one hand then p(K) < +oo, since p is a Borel

156

IV. Measures on Topological Spaces

measure, and on the other hand since K C K

µ(K) ? p(KK) > n This is the contradiction sought. O

for allnEN.

Exercises. 1. Let (Q, .W) be a measurable space, 8 a generator of &V and ! ' a subset of Q.

Consider the traces a' and d" of a' and 8, reap., on S2' and show that e' is a generator of the a-algebra .rah' in ff. Example 3 above is a special case. 2. Equip the set R with the so-called right-sided topology (which is also sometimes named after SORGENFREY [1947) whose system 0, of open sets is defined as follows: A subset U C R lies in ®r if and only if for each x E U there is an e > 0 such that [x, x + E[ C U. The topological space thus created will be denoted R,. Establish, one after another, the following claims: (a) Every right half-open interval [a, b[ is both open and closed in R,.. The rightsided topology on R is strictly finer than the usual topology. In particular, R, is a Hausdorff space.

(b) .W(R,) =0. (c) Suppose (x,e) is a strictly isotone sequence of real numbers possessing the supremum b E R. Then the set {z : n E N} U {b} is closed but not compact in R,. By contrast, if (y,,) is a strictly antitone sequence of real numbers possessing the infimum a E R, then {a} U {y : n E N} is compact in R,.. (d) Let K be compact in R,. Then there exists (from the first part of (c)) for every x E Kay E Q with y < x and [y, x[f1K = 0. If for each x E K, p(x) designates such a rational number y, then a mapping B : K -+ Q materializes which is strictly isotone, and hence injective. (e) Every compact subset of R, is countable. (But (c) shows that the converse is not true.) (f) Consider on .W(R,) = . 1 the measure p which assigns to every countable set

the value 0 and to every uncountable set the value +oo (cf. Example 6). Then p is a Borel measure on R, for which no point of R, has a neighborhood of finite measure. In particular, the measure p is not locally finite and is neither inner regular nor outer regular.

(g) Consider the measure v := IA' with density f(x) := x-'

llo,+ool(x)

(x E R)

and show that it too is a non-locally-finite Borel measure on R,.

(h) Investigate the L-B measure Al, thought of as a Borel measure on R in respect to its inner and outer regularity.

§26. Radon measures on Polish spaces

157

§26. Radon measures on Polish spaces For two extensive classes of Hausdorff spaces Borel measures come up very naturally. The first of these classes will be discussed in this section, beginning of course with its

26.1 Definition. A topological space E is called Polish when its topology has a countable base and can be defined by a complete metric. The terminology is due to N. BouRBAKI and commemorates the achievements of Polish topologists in the development of general topology. A metric is called complete when the associated metric space is complete: every Cauchy subsequence in it converges. A countable base or basis for the topology is a countable system of open sets such that every open set is the union of those from the system which are subsets of it. For a metrizable space E the existence of such a basis is equivalent to the existence of a countable dense subset.

Examples. 1. The euclidean spaces Rd of every dimension d > 1 are Polish, the ordinary euclidean metric being complete. The product E' x E" of two Polish spaces is another, when given the product topology. For if d, d" are complete metrics generating the topologies of E' and E", reap., then the product topology of E' x E" is generated by the metric 2.

d(x, y) = d'(x', y) + d"(:r", y"), x := (x', x"), y (y', y"). which moreover is complete. If 9',9" are countable bases for E', E", resp., then {G' x G" : G' E 91, G" E 9") is a countable basis for E' x E". Every closed subspace F of a Polish space E is Polish. Just restrict to F any complete metric that generates the topology of E. 3. 4.

Every open subspace G of a Polish space E is Polish.

Proof. We may suppose G # E. By 1. and 2. R x E is Polish. Let d be a complete

metric giving the topology of E, and consider the set F of all (A, x) E R x E E\G) = 1. Here, as usual, for 0 0 A C E. d(x., A) := inf{d(x, a) a E A} is the distance from the point x E E to A. The mapping x H d(x, A) is continuous on E, in fact., as the reader can easily check, ld(x, A) - d(y, A)l < satisfying

d(x, y) for all x, y r= E. Consequently, (A, x) Fa A d(x, E \ G) is a continuous real function on R x E, and F is a closed subset of R x E, hence itself a Polish space, by 3. Finally, (A, x) H .r. maps F homeomorphically onto G. To see surjectivity, we only have to notice that, because E \ G is closed, G coincides with the set {x E E : d(x, E \ G) > 0}. 5.

More generally it is true (cf. COHN [1980], Theorem 8.1.4 or WILLARI) [1970],

Theorem 24.12) that a subspace A of a Polish space E is Polish if A is a Ga-set in E, that is. A is the intersection of a sequence of open subsets of E. Thus, for

158

IV. Measures on Topological Spaces

example, the set J of all irrational numbers with its topology as a subspace of R is Polish, since

J= n (R \ {x}) . 2E'Q

Every compact space E with a countable basis is Polish. For a famous theorem of P.S. URYSOHN (1889-1924) (cf. KELLEY [1955], p. 125 or WILLARD [1970], 6.

Theorem 23.1) guarantees that E is metrizable, and in Remark 3 of §31 we shall even give a proof of this. The compactness of E easily entails that every metric defining its topology is complete.

The key to the further discussion is the following lemma, which is here just a preliminary to the big theorem that follows it, but nevertheless is significant in its own right. In it we encounter our first extensive class of Radon measures. 26.2 Lemma. Every finite Borel measure it on a Polish space E is regular. Proof. We consider the system .9 of all B E -W(E) which satisfy both

p(B) = sup{µ(K) : K compact C B}

(26.1)

and

µ(B) = inf {it(U) : B C U open). The goal of course is to show that .9 = M(E). We block off the work into five sections. Let d be a complete metric defining the topology of E. 1. E E 9: Only (26.1) needs proof when B = E. Let (X,,)-EN be a sequence which is dense in E, and for x E E, real r > 0 let Kr(x) denote the open ball of center x and d-radius r. For every r then E _ U K,.(xn), because in every ball Kr(x) lies (26.2)

nEN

some x,, so that x E Kr(xn). Sincep is continuous from below k

p(E) = kunµ(U Kr(xj)) . j=1

Therefore, for each e > 0 and n E N there exists kn E N such that

k

µ

K1/,, (xj)) > p(E)

-F2'°

j=1

kp

Each set Bn

U K 1 / (x j ), hence also their intersection K:= f Bn is closed, nEN

j=1

and we have

u(E)-µ(K)=µ(E\K)=p(U (E\B,)) 5 nEN

p(E\Bn)<_

n=1

Ee2-n=e. 00 n=1

This will prove (26.1) if we can confirm that the closed set K is actually compact. For every n E N

K C B. E l l , ,

§26. Radon measures on Polish spaces

159

and each set in this union has diameter no greater than 2/n. This shows that K is pre-compact (=totally bounded) and in a complete metric space that is equivalent to compactness, by very easy arguments (cf. WILLARD [1970], Theorem 39.9 or KELLEY [1955], p. 198).

2. Every closed set C lies in 9: Let F > 0 be given. We already know that there is a compact set K with µ(E) - IA(K) < e. According to 3.5 however

µ(C) - µ(C fl K) = p(C U K) - µ(K) < µ(E) - µ(K) < £ and this proves (26.1) for B :

C, because C fl K is compact. As a closed subset

of a metric space, C is a G6-set, that is, there are open sets G. J. C. To see this we may assume C 9& 0, so that G := E \ C is an open proper subset of E. Consequently, x H d(x, C) is a continuous mapping whose zero-set is C, as was

shown in treating Example 4. The sets Gn :_ {x E E : d(x,C) < 1/n} are therefore open and decrease to C. From the finiteness of µ and 3.2(c) we then have that µ(G.) 4. µ(C), showing that (26.2) is also satisfied by B := C. 3. Whenever B lies in 9 so does CB: First note that for every compact K C B

µ(CK) - p(CB) = µ(B) - µ(K) , and so CB satisfies (26.2) whenever B satisfies (26.1). Moreover, if G is an open superset of B, then CG is a closed subset of CB with µ(CB) -,u(CG) = µ(G) - µ(B) ,

showing, at least, that CB satisfies (26.1) weakened by replacing "compact" there

by "closed". But then application of step 2 to these closed sets gives us the full (26.1) for CB.

4. Whenever pairwise disjoint sets Dn lie in 9 (n E N), their union D also lies in 9: First of all

µ(D.)

µ(D) _ n=1

Letting e > 0 be given, we therefore have an nr E N such that n, (26.3)

µ(D) - E p(Dn) < c/2. n=1

Every Dj contains a compact K,j such that

µ(Di) - µ(Ka) <

(7 = 1, ... , ne)

2nE

since each D, E 9. Then K := K1 U...UKn, is a compact subset of D1 U...UD0, C D which satisfies

( n,

=

n,

µ(D1 U... U Dn.) - µ(K) S µ U (D, \ Ki )) j=1

j=1

µ(Di \ K,) < e/2

IV. Measures on Topological Spaces

160

from which, in view of (26.3),

IA(D) - µ(K) < e .

Again, D. E .9 means there exists open Un 7 D,, such that

e/2"

for each n E N.

Then the open set U := U Un contains D and satisfies nEN 00

l(U) - p(D) < µ( U (Un \ Dn)) < E li(Un \ D,,) < C. n=2

nEN

In summary, we have shown that (26.1) and (26.2) hold for B := D = U D,,. 5. The result of the first four steps is that 9 is a Dynkin system which contains the system .$ of all closed sets. The claim, namely that -9 = R(E), now follows

in the familiar way: Because 9 is n-stable, 6(.F) = o(Jr) = R (E). From Or C 9 c£(E) follows . (E) = J(9) c9 c . (E), and thus the equality sought. o We come now to the principal result of this section. It generalizes the foregoing lemma.

26.3 Theorem. On a Polish space E every locally finite Borel measure p is a ofenite Radon measure.

Proof. The hypothesis is that every point x E E has an open neighborhood U. of finite u-measure. The family (U:)XEE is an open cover of E. Because the topology of E has a countable basis, a theorem of E. LINDEt.oF (1879-1946) insures that this cover contains a countable subcover. That is, there is a sequence (xn)fEN in E already covers E. [It is easy enough to prove such that the sequence Lindeldf's result right here: Let V be any open cover of E, 0 a countable basis for the topology of E, and define d' to be the system of all A E d such that A C U for some U E 9l and let U(A) be one such member of 'Pl. The subset 0' of at, and therewith the system of all these U(A), is countable. This system covers E. For if x E E, then there is some U E Pl that contains x, and since d is a basis

there is some AEsi such that xEACU.Thus AEii'and xEAcU(A).j The system of sets Gn := U,z, U ... U US,,, n E N, satisfies

u(G,) < +oo

(26.4)

for every n E N, and G,, ? E.

Via

A E R(E)

1,6. (A) :_ p(AnG.),

a finite Borel measure µ,, is defined on E for every n E N. Each such measure is inner regular by the preceding lemma. It follows that for each A E SR(E)

µ(A) = sup p(A n Gn) = sup µ,(A) = sup sup µ (K) . nEN

nEN

nEN KEA

§26. Radon measures on Polish spaces

161

After commuting the two suprerna this reads

jt(A) = sup suptin(K) = sup p(K), KEr

KEr nEN

KCA

KCA

proving the inner regularity of tt. The a-finiteness of it is affirmed by (26.4), so the proof is complete.

The question now suggests itself whether - in analogy with 26.2 - the outer regularity of p can be proved. This is in fact the case. 26.4 Corollary. Every Radon measure on a Polish space is outer regular.

Proof. We have to show that every B E 4(E) satisfies (25.5). So let B E .4(E) and e > 0 be given. Consider the open sets G. and the finite measures tt created in the preceding proof. Lemma 26.2 furnishes open sets U. J B such that ti((U,, \ B) n

(26.5)

Let U

p. (U,. \ B) < e/2"

for each n E N.

U U n G,,, an open set. Since nEN

B = B n E = B n UG,, U BnC,,, nEN

nEN

it follows from B C U for every n, that B C U. Moreover, this representation of B shows that

U\B = U (UnnG,,)\ U (BnGn) C U (UnnGn)\(BnGn) = U (Un\B)nGn nEN

nEN

nEN

nEN

and consequently x,

x

n=1

n=1

e/2" =E.

tt(U\B) < by (26.5). It follows finally that

µ(U) = u(B) + tt(U \ B) < µ(B) + c, which confirms (25.5).

The regularity conditions (25.4), (25.5) make sense for outer measures px and together with one other minimal demand on p* they assure that all Borel sets are ,W-measurable. In fact, these conditions on an outer measure come up naturally in the course of proving the famous Riesz representation theorem in §29; cf. also 28.3.

26.5 Lemma. Let E be a Hausdorf space and tt' an outer measure on E with the following three properties: (i) for every set A C E

tt'(A) = inf{tt'(U) : A C U open 1;

IV. Measures on Topological Spaces

162

(ii) for every open set U C E

p* (U) = sup{Ec*(K) : K compact C U}; (iii) for any two disjoint compact sets K1, K2 C E JL*(Kl UK2) = p*(Kl) +{l*(K2) Then the restriction of µ* to R (E) is a measure.

Proof. We consider the a-algebra d* of all µ*-measurable sets, that is, according to (5.6) the set of all A E .9(E) which satisfy (26.6)

k*(Q) > µ*(Q n A) + p*(Q \ A)

for all Q E .9(E).

First note that it suffices that this hold for all open sets Q in order that it hold for all Q whatsoever. In other words, what we need to check for an A to be in d* is that (26.6')

p*(U) > p*(U n A) +,t.*(U \ A)

for all U E 0.

Indeed from (26.6') it follows for any Q C E that p*(U) > p*(Q f1 A) + p*(Q \ A) whenever U is an open set containing Q; then (26.6) itself follows by taking the infimum over such U and invoking (i). So now let A = G be an open set; we will

use criterion (26.6) to show that G lies in W*. To this end consider any open U C E; further, consider any compact Kl C U n G and any compact K2 C U \ K1. Since then K1 n K2 = 0 and Kl U K2 C U, it follows from (iii) that y* (U) > {b' (K1 UK2) =A* (KI) +Ft*(K2) The set U\Kl is open, so if we take the supremum over all such K2 in the preceding inequality and appeal to (ii), we get

it* (U) > IA*(Kl) + u* (U \ K1) > u'(Ki) + t,* (U \ G), the last inequality because U\Kl D U\G. This holds for all compact Kl C UnG, and so after a second appeal to (ii) it yields

p*(U) > p*(UnG)+µ'(U\G), holding for all U E 0. That is, (26.6') holds for A = G, and consequently G E d9*.

the latter This all proves that B C W*. But then .9(E) = a(®) C j W* is a a-algebra, by Theorem 5.3. That theorem further affirms that the restriction of u* to W* is a measure.

The foregoing Theorem 26.3 and its corollary show in particular that the L-B measure Ad is a regular Bored measure on Re in e a c h dimension d = 1, 2, ... . In fact every Bore] measure on Rd is regular (cf. also Theorem 29.12). Following STROMBERG [19721 we derive from the regularity of Ad a purely topological result of H. STEINHAUS (1887-1972). It shows, incidentally, that every set of positive L-B measure has the cardinality of R.

§26. Radon measures on Polish spaces

163

26.6 Theorem (of Steinhaus). Let A E Rd be a Borel set in Rd of positive ddimensional Lebesgue measure. Then 0 is an interior point of the set A - A of differences of elements of A.

Proof. The inner regularity of Ad means that A contains a compact subset K with Ad(K) positive. It suffices to prove the claim with K in place of A. Outer regularity furnishes an open set U D K with Ad(U) < 2Ad(K). There is an open ball V centered at 0 of positive radius such that the sum set satisfies K + V C U. One only has to choose the radius less than the (positive) distance between the compact set K and the closed set CU from which it is disjoint. We will show that V C K - K, which makes 0 an interior point of this difference set. Consider any v E V. The translated set v + K cannot be disjoint from K, for otherwise from K U (v + K) C K + V C U and translation-invariance of Ad would follow that 2Ad(K) = Ad(K) + Ad(v + K) = \d (K U (v + K)) < Ad(U),

contrary to the choice of U. But K fl (v + K) 0 0 means that for some x, y E K, x = v + y; which says that the given point v = x - y lies in K - K. 0 In closing we turn to a remarkable consequence of Theorem 26.3 and its Corollary 26.4. It concerns the analogy, pointed out in §7 as measurable mappings were being introduced, between the notions of measurability and continuity. Initially this analogy is merely an analogy. Namely, if f : E -+ E' is a mapping of one topo-

logical space into another, then f is Borel measurable (i.e., .(E)-.(E')-measurable) just if the pre-image f - i (G') of every open set G' C E' is a Borel set in E. This follows from Theorem 7.2 and the fact that the Borel o-algebra M (E") is generated by the open subsets of E'. By contrast, f is continuous just if f-1(G') is open in E for every open set G' C E. What is quite remarkable is that for Polish spaces E a much closer connection between those two concepts exists. This is brought out by the following theorem, discovered in its definitive form by N. LUSIN (1883-1950).

26.7 Theorem (of Lusin). Let ,a be a locally finite Borel measure, thus a Radon measure, on a Polish space E, and E' be a topological space with a countable basis. Then for every mapping f : E -+ E' the following are equivalent: (a) f coincides p-almost everywhere with a Borel measurable mapping of E into E'. (b) There is a decomposition of E into a p-nullset N E R(E) and a sequence (K,.)nEN of compact sets, such that the restriction off to each K is continuous.

If the measure µ is finite, (a) and (b) are further equivalent to: (c) For every e > 0 there is a compact subset KK C E such that p(CKE) < e and the restriction off to K, is continuous. Proof. Let us first suppose that p is finite. Let 9' be a countable base for the topology of E' and (Gn)nEN a sequential arrangement of its elements. Notice that 9' is a generator of the Borel o-algebra because every open subset of E' is a (countable) union of sets from s'.

IV. Measures on Topological Spaces

164

(a)=(c): By hypothesis there is a Borel measurable mapping g : E -* E' and p-nullset N E .£(E) with f (x) = g(x)

(26.7)

for all x E CN.

For every set Gn, g-1(Gn) E . (E). Because every Radon measure on E is regular, given E > 0, there exist compact sets Kn and open sets Un such that (26.8)

K C g-1(G'n) C Un and p(Un \ Kn) < 2-ne

The set A

for each n E N.

U (Un \ Kn) is open, being a union of open sets. For its measure nEN

we have the obvious inequality 00

p(A) s E p(Un \ Kn) < C. n=1

Using once more the (inner) regularity of 1S, we find a compact K C C(A U N) _ CA n CN such that

p(CAnCNnCK) <e-p(A), thus (since A U N C CK and A U N U (CA n CN = E) such that p(CK) = p(A U N U [CA n CN n CKI) < p(A) + p(N) + E - p(A) = E .

This set K does what is wanted in (c), because by (26.7) f and g coincide in K and because the restriction go of g to CA is continuous, as we now confirm. For each set Gn, go 1(Gn) = g-1(Gn) n CA;

from (26.8) and the fact Un \ Kn C A follows therefore

UnnCA =KnnCA cg'(G')cUnnCA, which means that

goI(Gn)=UnnCA =KnnCA, showing that the go-pre-image of G;, is open (as well as closed) in CA. Since (Gn)nEN is a base for the topology of E', this is enough to guarantee the continuity

of go=gICA. (c)=(b): It suffices to find pairwise disjoint compact subsets Kn of E such that f I Kn is continuous and K3) < p(C ?=1 U J n =

for each n E N. For then

N:=CUKn= nCKn nEN

nEN

is a Borel set disjoint from each Kn and satisfying p(N) < 1/n for every n E N, i.e., p(N) = 0. The sequence (Kn) is gotten inductively from (c) as follows: To start off, there is a compact K1 C E such that u(CKI) < 1 and f I K1 is continuous.

§26. Radon measures on Polish spaces

165

If Ks,. .. , Kn have been defined having the desired properties, we will get K"+1 from (c) and the inner regularity of p. By (c) there is a compact K' C E such that

p(CK') < (2n + 2)-' and f I K' is continuous. With L := K, U... UKn the inner regularity of p supplies a compact Kn+1 C K' \ L such that

µ(K' \ L) - p(Kn+1) = µ(K' n CL n CKn+,) < (2n + 2)' 1

.

Because

p(C(L U Kn+,)) = p(CK' n CL n CKn+1) + µ(K' n CL n CKn+, )

< p(CK')+p(K'nCL nCK,,+,) < (n + 1)-', with this set Kn+, the inductive construction is complete. (b)=(a): If E = N U K, U K2 U ... is the given decomposition, one defines a mapping g : E -* E' as follows. In case N = 0, let g := f. In case N 96 0, choose yo E f (N) arbitrarily and set

g(x) := f (x) for x E E \ N,

g(x) := yo for x E N.

What has to be shown is that g is Borel measurable, which is done as follows: For every open G' C E' 9_1(G')

= (g-1 (G') n N) U U (g-1(G') n Kn) = No U U g; 1(G') nEN

nEN

where No := g-1(G') n N and gn := g I Kn. Now No is either N or 0, according as yo E G' or yo V G'. Moreover, gn coincides with the restriction of f to Kn, so that by hypothesis gn 1(G') is open in Kn, that is, of the form Kn n Un for some open subset U,, of E. Therefore only Borel sets occur in the above decomposition of g-1(G') and we conclude that g-1(G') is a Borel set. This being true of every open G' C E', the Borel measurability of g follows from 7.2. Now consider an arbitrary locally finite measure p on R(E). According to 26.3, p is a-finite. Lemma 17.6 therefore furnishes a strictly positive p-integrable real function h on E. The measure v := hp is then a finite Borel measure on E which has exactly the same nullsets as p. The proven equivalence of (a) and (b) for the measure v therefore entails the validity of this equivalence for the measure it. Thus the whole theorem is proved.

Remarks. 1. The equivalence of (a) and (b) in Lusin's theorem may be lost if (a) is

strengthened to the 9(E)-9(E')-measurability of f. It suffices to take for E the compact set [0,1] x [0,1] and for p the L-B measure .X E. As was noted in the second part of Remark 4, §8, E contains a p-nullset N which contains a non-Borel subset. If M is such a set, its indicator function f = l,w is not Borel measurable, although f is p-almost everywhere equal to the Borel measurable function 1N On the other hand, if f is . (E)-. (E')-measurable, there is a Polish topology r on E, stronger than the original but generating exactly the same Borel sets, such that f is r-continuous. See 3.2.6 of SRIVASTAVA [1998] for the proof, which is not difficult.

166

IV. Measures on Topological Spaces

2. The Dirichlet jump function (cf. Remark 1 of §16) is continuous at no point of its domain of definition 10, 1], yet it is Borel measurable. This shows that in assertion (c) of Lusin's theorem one cannot hope to be able to replace the continuity

of the function f I K by the continuity of f at each point of K.

Exercises. 1. Show that every inner regular finite Borel measure on a Hausdorff space is outer regular.

2. Show that in a Polish space E the Dirac measures are the only non-zero Borel measures it which take only the values 0 and 1. [Hint: Show that the system of all compact K C E such that tt(K) = I is fl-stable and investigate the intersection of all itssets.]

3. Show that AE x E') _ i(E) ®M(E') for any Polish spaces E,E'. 4. Consider K compact C U open C Rd, and for each n E N let V denote the open ball of radius 1/n and center 0. Show that K + V C U for some n. [Hint: n CU # 0 for every it E N, find xn E K, vn E V,,, zn E CU such that If (K + x + v = z,,, for every n E N. Some subsequence of (xn) converges to a point xo E K and because CU is closed we even have x0 E K fl CU, which contradicts the fact that K C U.]

5. Let p be a locally finite Borel measure on a Polish space E and f : E - E' a mapping into a topological space E' with a countable base. Show that assertions (a) and (b) in Lusin's theorem are equivalent to (c'): For every e > 0 and every compact K C E there is a further compact Kf C K such that p(K\Kf) < c and f I KE is continuous.

§27. Properties of locally compact spaces A topological space is called locally compact if it is Hausdorff and if each of its points has at least one compact neighborhood. Examples of such spaces are the euclidean space Rd, every manifold (i.e., every locally euclidean Hausdorff space), every discrete space, and every compact space. When an arbitrary point is removed from a compact space the remainder is a locally compact space. Actually every locally compact space is of this form. For if © is the system of all open subsets of the locally compact space E and wo is any (so-called ideal) point not in E, then a topology can be defined on E' := EU {WO} as follows: The system d' of open sets in E' shall consist of ® together with the sets E' \ K for all the compact subsets K of E. This defines a compact topology on E', E is an open subset of E' and the topology that E inherits from t9' is its original topology. E was compact to start with if and only if wo is an isolated point in E'. If E is not compact, then it is dense in E'. These claims are easily confirmed, or the reader can consult KELLEY [1955], p. 150, or WILLARD [1970], 19.2. The space E'

§27. Properties of locally compact spaces

167

is called, after its creator P.S. ALEXANDROFF (1896-1982), the (Alexandroff) one-point compactification of E and wo its infinitely remote point. We will pursue the further theory of locally compact spaces via this compactification. First we study some distinguished continuous functions in this environment. For an arbitrary topological space E we denote by C(E) and

Ct(E)

the vector space of all, respectively all bounded, continuous real functions on E.

27.1 Definition. Let f : E -> JR be a real function on a topological space E. The set (27.1) supp(f) := If 34 0} is called the support of f.

The complement of supp(f) is thus the largest open set at every point of which f takes the value zero. If E is locally compact. we will designate by CA(E)

the set of all f E C(E) with compact support supp(f). A function f E C(E) lies in CA(E) just if there is some compact subset of E in the complement of which f is identically zero. Clearly (27.2)

C (E) C Cb(E) C C(E),

since an f E CA(E) is bounded on its compact support, hence throughout E. C,.(E) is a vector subspace of Cb(E). More generally for any n E N, E C(1R") with V(O) = 0 and fl,.. . E C,.(E), the composition f,,) lies in CA(E), rr

and indeed its support is a subset of f supp(fj). In particular, whenever u, v E j=1

C,.(E) the functions Jul, u V zv. u A v, and therewith u+ and u.-, all lie in C'(E). The needed continuity of y,(x, y) := r V y on 1R2 follows from the identity r V y =

(.x+y+I.e-yI) In the special case of a compact space E, all three function spaces in (27.2) coincide.

A fundamental property of the space C,.(E) is the following:

27.2 Theorem (on partitions of unity). Suppose that the compact subset K of the locally compact space E is covered by the n open sets U1, ... , U,,. n E N. Then

there are functions fl.... , f E C,.(E) with the following properties (27.3)

fj>0

(27.4)

supp(fj) C Uj

for j = 1.....n; for j = 1,....n:

r4

f(x) < 1

(27.5) j=1

for all r E E;

168

IV. Measures on Topological Spaces n

rfj(x)

(27.6)

forallXEK.

j=1

Proof. We work in the one-point compactification E' := E U {wo} of E. The given open sets together with Uo := E' \ K constitute an open cover of E'. Because compact spaces are normal topological spaces (cf. KELLEY [1955], p. 141 Or WILLARD [1970], Theorem 17.10), this covering can be "shrunk" to an open covering Ui, ... , Un of E' satisfying UUCUj for each j =0,...,n, where of course the bar denotes closure in E'. The theorem on partitions of unity in normal spaces (KELLEY [1955], p. 171 Or WILLARD [1970], 20 C) provides functions

fo..... fn E C(E') such that fj' > 0,

(i)

supp(f f) C Uj,

for j = 0,..., n;

n

Ef,(x)=1

(ii)

for all xE E'.

j=o

The restrictions f I , ... , fn to E of f f,i lie in C(E) and it will be easy to show that they have all the properties wanted. From (i) and (ii) properties (27.3)-(27.5) follow almost immediately. One only has to notice that for each j = 1,.. . , n

supp(fj)=supp(ff)flECUUflE=UUCUj since UU C Uj C E. In particular, Uf being a closed subset of the compact space E',

is a compact subset of E. From supp(fj) C W therefore follows the compactness of this support. Thus f I, ... , f,, all lie in CA(E). The remaining property (27.6) likewise follows from (ii) because supp(fo) C Uo = E \ K entails that fo(x) = 0

for all x E K. 0 Two consequences of the foregoing will turn out to be especially useful. The first - known as Urysohn's lemma - often serves as the starting point for inductive constructions of partitions of unity (see, e.g., RUDIx[1987J, p. 39). The second can also be proven directly, as indicated in Exercise 1 below.

27.3 Corollary 1. In the locally compact space E, U is an open neighborhood of the compact subset K. Then CA(E) contains a function f which satisfies (27.7)

0:5f:51, f(K)=fl),

and

supp(f) C U .

In particular, supp(f) is a compact neighborhood of K.

Proof. We have only to apply 27.2 for n = 1. Since K C (f, > 0} C supp(f3), the fact that (f, > 0) is open means that supp(f 1) is indeed a neighborhood of K. 0 27.4 Corollary 2. In the locally compact space E the compact subset K is covered

by then open sets UI,... , Un, n E N. Then K can be decomposed as K = KI U ... U Kn with Kj a compact subset of Uj for each j = 1, ... , n.

§27. Properties of locally compact spaces

Proof. Let fl,

169

, fn E Cc,(E) be as provided by 27.2. The compact sets

K; := K n supp(f3 ),

j = 1, ... , n

do what is wanted; for if x E K, then 1 = f i (x) +... + f n (x) means that f, (x) j4 0 for some j, and therefore x E K3.

For a locally compact space E there is another function space besides CC(E) that is of importance. To define it we assign to every bounded real function f on an arbitrary space E its supremum norm, also called its uniform norm, via Ilf11

sup If W1 sEE

The mapping (f, g) -+ If -gIi makes Cb(E) - more generally even the vector space of all bounded real functions on E - into a metric space. One speaks of the metric of uniform convergence (on E). A sequence (fn) of bounded real functions on E converges uniformly on E to a bounded function f just means that lim Ilfn - f 1l = 0 . nloo

27.5 Definition. A continuous real function f on a locally compact space E is said to vanish at infinity if it lies in the closure Co(E) of CC(E) in Cb(E) with respect to the metric of uniform convergence. Denoting closure in this metric by bar, we thus have Co(E) := CC(E) C Cb(E). The terminology "vanishing at infinity" is both clarified and justified by

27.6 Theorem. For a real function f on a locally compact space E the following statements are equivalent:

(a) f E Co(E); (b) f E C(E) and {If I > e} is compact for each e > 0; (c) the function

f'(x) :_ { f (x), for all x E E for x = wo 0, is continuous on the one-point compactification E' of E.

Proof. (a)=(b): Given e > 0, there is by definition off E Co(E) a g E Cc(E) with Ilf - gfl S e/2. Every x E E satisfies If (x)I - Ig(x)I <- If (x) - g(x)I S Ilf - gAI, so we see that (If 1> e} C {IgI > E/2} C supp(g). This shows that (If 12: c} is a relatively compact set. But, due to the continuity of f, it is also closed. Hence it is compact. (b)*(c): Since the subspace topology of E in E' is its original topology and E is an open subset of E', continuity of f' at each point of E is assured by f E C(E). As to continuity at the ideal point wo, given e > 0, we have I f'(x) - f'(wo) I = l f'(x) I <

170

IV. Measures on Topological Spaces

e for all x in the set E' \ {If I > E}, which by definition of E' is a neighborhood of wo, since (If I > e} is a compact subset of E. (c)=:>(a): Continuity of f' at wo and the definition of the topology in E' mean that for each e > 0 there is a compact K C E such that If (x)I = If'(x) - f'(wo)I < E for all x E E \ K. 27.3 supplies a g E CA(E) with 0 < 9< I and g(K) = {1}. Then fg E CA(E) and satisfies

If

- f(x)I = If(x)I (1-g(x)) < E

for all x E E, so Ilfg - f II < E. As e > 0 is arbitrary, this proves that f E CA(E).

Exercises. 1. Without resort to partitions of unity, prove Corollary 27.4 directly. [Hint for the case n = 2: Separate the disjoint compacta K \ U1, K \ U2 with disjoint open neighborhoods V1, V2 and set Kl := K \ V1, K2 := K \ V2.] 2. Let E' = E U {wo } be the one-point compactification of a locally compact space E. Describe the Borel sets in E' by means of the Borel sets in E. In particular, see how your description fits into the following general picture: For a measure space (E,.o), a point wo it E and the set EWO := E U {wo}, the a-algebra d"'O in E"'° generated by d and {wo} consists of all A' C El- such that All fl E E St.

§28. Construction of Radon measures on locally compact spaces In what follows E will be a locally compact space. We consider a Borel measure p

(defined on R(E)). Here the requirement µ(K) < +oo for every compact set K is the same as the local finiteness requirement, because every point of E has a compact neighborhood and the implication (25.7) holds in general. So in the present context the concepts of Borel measure and locally finite measure on .W(E) coincide. The Radon measures on E are thus (cf. 25.3) those Borel measures which are inner regular. For a Borel measure it every u E CA(E) turns out to be p-integrable. For, being continuous, u is Borel measurable. Denoting by K the compact support of u, we have 1111 5 IIuII 1K. Since It is a Borel measure, 1K is p-integrable, and the pintegrability of u follows. Therefore corresponding to the Borel measure is a linear form 1,, on C,;(E) defined by (28.1)

lu(u) := Judy.

This is an isotope linear form in the sense of (12.3): From u < v follows I,,(u) < I,,(v). Because of the linearity of I,, this is equivalent to

0
1,,(u)>0,

§28. Construction of Radon measures on locally compact spaces

171

which is why I,, is usually called a positive linear form. This brings us to a key question for our further work: Is every positive linear form on C,.(E) an I,, for some Borel measure p on E, or are there possibly positive linear forms of a completely different kind? Even for compact intervals J := [a, b] on the number line, answering this question is by no means a trivial task. In this case however, as early as 1909 F. Riesz showed (cf. RIEsz (1911]) that besides the

linear forms I,, arising from Borel measures it on J, there are no other positive linear forms on Q,,(J) = C(J). One of our goals is to show that every locally compact space E shares this property with J. The result in question will, in view of this pioneering work, be called the Riesz representation theorem. En route to it we will naturally be led to the construction of Radon measures on E. Besides the locally compact space E. let now a positive linear form

I : Cr(E) -+ R be given. What follows will prepare the way for the proof of the Riesz representation theorem. For every compact K C E we set (28.2)

p.(K) := inf{I(u) : 1K < it E C.,,(E)}.

Such functions u exist thanks to Corollary 27.3. Consequently, (28.3)

0 < p. (K) < +oc.

Moreover, the mapping K ' p.(K) is obviously isotone on the system ..l' of all compact, sets. For an arbitrary A E -1P(E) we set (28.4)

p.(A) := sup{p.(K) : K compact C Al.

Because of the above noted isotoneity of it. on ..it', this new definition is consistent with (28.2). Finally, for A E .9(E) we define (28.5)

p'(A) := inf{p.(U) : A C U open}.

Then it. and p` are isotone functions on . (E). Moreover (28.6)

p. (A) < y* (A)

for all A E .0(E),

as follows from the obvious fact that it.(A) < p.(U) for every open U D A; and (28.7)

p.(U) = /I* (U)

for all open U E Y(E),

which follows from (28.5) and the isotoneit.v of it.. Somewhat more effort is required

to check that (28.8)

p.(K) = p`(K)

for all K E X.

For every e > 0 definition (28.2) supplies a u E C,.(E) with to > 1K and

I(u) - p.(K) < E.

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IV. Measures on Topological Spaces

For0a} is an open superset of K and 1Ue <

U.

If therefore Lisa compact subset of Ua, then 1y < u and so from (28.2) P. (L) < a 1(u). From definition (28.4) therefore

ps(Ua) < I(u) and so, since K C Ua,

0<1 s(Ua)-ps(K)

=(a-l)p.(K)+a. As a 1 1 this majorant converges to e, which shows that

inf{ps(U) : K C U open} < IA. (K) +e holds for every e > 0; that is,

p`(K) = inf{µ.(U) : K C U open} < µ.(K). This confirms (28.8), the reverse inequality being part of (28.6). Of critical importance is the following result:

28.1 Lemma. W is an outer measure on E. Proof. Obviously p*(0) = 0, so what we have to prove is that 00

(28.9)

ias (U Q-):5 E /bs (Qn) nEN

n=1

holds for every sequence (Qn) in .9(E). We proceed in three steps. First step: For any two compact sets K1, K2

p`(K1 UK2) 1K, for j = 1, 2. Then IK,UK2 so (28.2) says that /L.(K1 U K2) < I(u1 + U2) = I(u1) + I(u2) . The claimed inequality now follows from (28.2) and (28.8). Second step: For any finitely many open sets U1,.. . , U.

A*(U1U...UUn)

ps(U1)+...+AV.).

U1 + U2,

§28. Construction of Radon measures on locally compact spaces

173

It suffices to settle the case n = 2, as induction then takes care of the rest. If K is a compact subset of Ul U U2, then 27.4 provides compact Kj C Uj, j = I, 2, such that K = Kl U K2. Then by the result of our first step

,u*(K) < lj*(KI) + p*(K2) <;t'(U,) +p`(U2) The claimed inequality (with n = 2) then follows from (28.8), (28.4) and (28.7). Third step: Now we will prove (28.9). In doing so we may obviously assume that p'(Q,,) < +oo for every n. E N. Given e > 0, there then exist open U. J Q,, such

that for every n E N.

2-11e

The open set U := U U contains Q :_ U Q. If now K is a compact subset nEN

"EN

of U, then K C U1 U ... U U for sufficiently large n.. From this it follows that :,

x

p.(K)_p*(K)
j=1

where we used the second step. As this last inequality is satisfied by every compact subset K of U, definition (28.4) and equation (28.7) give

a it. (U) = Et'(U) <- E; (Qj) +e, j=t and since Q C U we will then have as well 00

(Q):5 EW (Qj) +e.

j=

Finally, e > 0 being arbitrary here, (28.9) is proven. 0 The next corollary sharpens the inequality proved in the first step above.

28.2 Corollary. For any two disjoint compact subsets K1, K2 of E

p"(Ki U K2) = p'(K1) + p'(K2) Proof. Consider any u E C,(E) satisfying

u.>1K,uK2=1K,+1Ks. I}, and According to 27.3 there is a v E C,(E) with 0 < v < 1, v(K1) supp(v) C CK2, hence with v(K2) = {0}. The functions vu and (1 - v)u lie in CA(E) and satisfy vu > 1K,

and

(1 - v)u > 1K2.

Therefore

p.(Ki) +p.(K2) < I(vu) + I((1 -v)u) =1(u) ,

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IV. Measures on Topological Spaces

which, because of (28.2), has the consequence that

p.(Ki) + µ.(K2) < u.(K1 U K2). In view of (28.8) this inequality is half of the equality being claimed. The other half is simply the subadditivity of the outer measure µ'. The first important consequence of all this is:

28.3 Theorem. The restriction of µ' to M(E) is a Borel measure. The proof is immediate from Lemma 26.5 and the facts accumulated to this point. Notice that (28.7) and (28.5) say that hypothesis (1) of 26.5 is fulfilled, while (28.7), (28.8) and (28.4) insure that hypothesis (ii) of 26.5 is fulfilled.

The Borel measure µ' I ..(E) has a series of further remarkable properties:

28.4 Theorem. Every Borel subset A C E with µ'(A) < +oo satisfies

µ.(A) = µ`(A) Proof. Given e > 0, there is an open U D A such that

It* (U) - µ'(A) < e/2, which, due to µ' (A) < +oo and µ' being a measure on 9(E), can be written as

µ'(U\A) =µ'(U) -µ'(A) <e/2. From (28.4) we get compact L C U such that

µ'(U\L)=µ'(U)-li (L) <e/2. The set

Q:=(U\A)U(U\L) then satisfies p* (Q) < e. Hence there is an open G Q such that µ'(G) < C.

Now K := L \ G is a (closed, hence) compact subset of L with the properties

K C A and A\ K C G.

(28.10)

In fact, on the one hand

K = L \ G C L \ Q C L \ (U \ A) = L n A, since L C U, and on the other hand

A\K=A\(L\G)=(AnG)U(A\L)CGu(U\L)=G, since U \ L C Q C G. From (28.10) we get

µ'(A) - µ'(K) = µ'(A \ K) 5 µ'(G) < e,

§28. Construction of Radon measures on locally compact spaces

175

and so u* (A) < µ'(K) + e <- µ.(A) + e. As e > 0 was arbitrary, this says that µ'(A) < µ.(A), which with (28.6) finishes the proof. The finiteness hypothesis in the preceding theorem can be weakened. In doing so we make use of the terminology introduced just before the proof of Theorem 13.6.

28.5 Corollary. The equality p. (A) = u* (A) also holds for every A E -V(E) which has o'-finite µ'-measure.

Proof. The terminology means that there exist An E R (E) (n E N), each of finite µ'-measure, such that An T A. The preceding theorem and the isotoneity yield

µ'(An) = p.(An) < µ.(A) , from which and the continuity of µ' from below on R (E) follows µ'(A) = sup p* (An) <_ p. (A). n

Together with (28.6) this proves the claimed equality. Another central result, analogous to 28.3, emerges:

28.6 Theorem. The restriction of µ. to ..(E) is also a Borel measure.

Proof. Since all compact K satisfy µ.(K) = p'(K) < +oo, all that has to be proved is that p. I M(E) is a measure, i.e., that p. is countably additive on M (E). To that end, let (An) be a sequence of pairwise disjoint sets from R(E), whose union is A. For every compact K C A, K = U (K n An), so from 28.3 and 28.4 nEN

we get 00

00

00

µ.(K)=ii (K)=1: µ'(KnAn)=1: 1.(KnAn)<Eµ.(An). n=1

n=1

n=1

Taking the supremum over such K on the left, (28.4) gives 00

,u. (A) S !L=(An) n=1

In proving the reverse inequality we may assume that µ. (A) < +oo, and therefore

P. (An) < +oo for every n E N. There is then, given e > 0, a compact Kn C A. satisfying

p. (An) - µ.(KK) < 2-ne

for each n E N.

Since the sets Kj are pairwise disjoint, UKj)=µ*\UKj/IL_(Kj)A.(Kj)

j=1

j=1

j=1 n

> Ep.(Aj) - E j=1

j=1

j=1

n

j=1

for every n E N.

176

IV. Measures on Topological Spaces

Letting n -+ oo we infer that 00

(A) ? Eµ.(A.i) -e, 00

holding for every c > 0. That is, µ. (A) > E µ. (A,,), the complementary inequality we needed to finish the proof. We now set (28.11)

µo := µ. I .4(E) a n d µ° := µ* I R(E)

and, inspired by COURREGE [19621, call these the essential measure determined

by I and the principal measure determined by I, respectively. Each is a Borel measure (28.3 and 28.6).

Obviously the essential measure tb is inner regular, hence is a Radon measure on E. By contrast the principal measure µ° is outer regular. It turns out that µ° is the more important of the two. Thus to the given positive linear form I on CA(E) we have associated two Borel measures. The further relation of these measures to I and the questions of whether and when they coincide will be clarified in the next section. The closing lemma of this section recasts definition (28.4), when A is open, into a equivalent form. It has a preparatory character.

28.7 Lemma. Every open set U C E satisfies (28.12)

110(U) =11°(U) = sup{I(u) : u E C0(E), supp(u) C U, 0 < u < I}.

Proof. The first equality is just (28.7). Denote the right side of (28.12) by y, and consider any compact K C U. Corollary 27.3 provides a function u E CA(E) with

0 < u < 1, u(K) = {1} and supp(u) C U. In particular, 1K < u and so by (28.2) µ.(K) < I(u) < y, that is, µ.(K) < y for every such K. It follows that µ°(U) = µ`(U) = µ.(U) < y, by (28.4). The reverse inequality y < µ°(U) is derived as follows: Let u E CA(E) be a typical function involved in the definition of y. Set L := supp(u) and consider a typical v E C0(E) involved in the definition (28.2) of µ.(L). Evidently then u < v, so 1(u) < I(v); that is, I(u) < µ.(L) = µ0(L) = µ°(L) < µ°(U). Taking the supremum over eligible u gives finally the desired complementary inequality -y:5 µ°(U).

A sharpening of equality (28.12) will be presented in Exercise 2 of §29. The special case U = E of lemma 28.7 furnishes the following useful description of the total masses of it. and µ°: (28.13)

11µo11 = 11µ°II = sup{1(u) : u E CC(E),0 < u < 1).

§29. Riesz representation theorem

177

Exercises.

1. For a locally compact space E and a measure p defined on ..(E), show that it is a Borel measure if and only if Cc(E) C 21(p). 2. Let p be a Radon measure on a locally compact space E and (Gi)1EI a family of open sets which is upward filtering, that is, for any i, j E I there is a k E I such that Gi U G; C Gk. Show that C := U Gi satisfies iEI

p(G) = sup{p(Gi) : i E I} . 3. Using the preceding exercise, show that for any Radon measure p on a locally compact space E:

(a) There exists a largest open set G with p(G) = 0. The set CG is called the support of the measure p and is denoted supp(p). (b) A point x E E lies in supp(p) if and only if every open neighborhood of x has positive p-measure.

(c) For a non-negative f E C(E), f f dµ = 0 if and only if f = 0 throughout supp(p). Determine supp(Ad) for L-B measure Ad on Rd, and supp(E°) for every Dirac measure ea on E. 4. Let p be a Borel measure on a locally compact space E. Show that every set A from the a-ring p0(X) generated by the system ..iE' of compact subsets of E is a Borel set which satisfies p.(A) = p°(A). Here a ring .4 in a set 0 is called a aring if the union of every sequence of sets in .9 is itself a set in R. In complete analogy with a-algebras, every subset of .9(0) is contained in a smallest a-ring. Sometimes it is only the sets in pe(a') which get called "Borel sets"; this is the case, e.g., in the classic exposition of HALMOS [1974]. Why is it generally the case that po(..1E') 3 .9(E)?

§29. Riesz representation theorem Again let E be a locally compact space. Every Borel measure p on E defines a positive linear form

I,,(u) := fudp on CA(E). The question posed in §28 was: Is it true that for every positive linear form I on CA(E) there is a Borel measure p on E such that Iµ = I, that is, such

that

I(u) = Judp

foralluECC(E)?

Any such Borel measure p will be called a representing measure for I. The answer, leaked earlier, to this question reads:

178

W. Measures on lbpological Spaces

29.1 Riesz representation theorem. If E is a locally compact space, every positive linear form I on CA(E) has at least one representing measure. In fact, both the essential measure Po determined by I and the principal measure p° determined by I are representing measures for I.

Proof. po and p° are Borel measures. It must be shown that (29.1)

I(u)= fud = Judpo

for all uECC(E),

and because of linearity and the fact that the positive and negative parts of each u E CA(E) also lie in C°(E), it suffices to show this for non-negative u. So let such be given and let the real number b > 0 be an upper bound for u. Fbr auE a given e > 0 choose real numbers yp,... , y,, with

0=yo
for each j = I,-, n.

yj-yj-1< C We set

(j = 1,...,n)

uj :_ (u - yj-1)+ A (yj '- yj-1)

and get non-negative continuous functions, each having its support in supp(u), which satisfy n

(29.2)

u=Euj, j=1

as the following deliberations will confirm. If x E E and u(x) = 0, then uj(x) = 0 for each j = 1, ... , n. If x E E and u(x) > 0, then there is a unique j E {1,...,n}

such that yj-1 < u(x) < yj. In that case uj(x) = u(x)-yj-1 and uk(x) = yk-yk-1 for k < j and uk(x) = 0 for k > j. Equality (29.2) follows. Next we set Ko := supp(u) and Kj := {u > yj }

for j = 1, ... , n

and have (29.3)

(yj -yj-1)lx, < uj < (yj

-yj-1)1K1_,,

for j = 1,...,n,

which becomes clear from considering the three properties (29.4)

O!5 uj :5 yj-yj-1,

(29.5)

CKj_1 c {uj = 0},

(29.6)

Kj c {uj = yj - yj_1},

valid for j = 1, ... , n. Integrating in (29.3) with respect to p° gives (29.7')

(yj - yj-1)p°(Kj) <_ 1 uj dp° _< (yj - yj-1)p°(Kj-1),

§29. Riesz representation theorem

179

and from (29.3) we will - momentarily - infer the analogous inequalities (29.7")

(yj -Eli-1)lL°(Kj) 5 1(uj) 5 (yj -

valid for all j E {1, ... , n}. The left half of (29.7") follows from the left half of (29.3)

when account is taken of (28.2) and the fact that u.(Kj) = U*(Kj) = µ°(Kj). From (29.5) we have supp(uj) C Kj_1. For every open U i Kj_1, the function v :_ (yj - yyj_1)-luj is therefore an element of Cc(E) with supp(v) C U and satisfying, by (29.4), 0 < v < 1. From Lemma 28.7 then 1(v) < p°(U) and hence

1(uj) 5 (yj -yj-1)/P(U). According to (28.7) p°(U) = p.(U) and therefore from (28.5) and the arbitrariness of U we have confirmation of the right-hand side of (29.7"). Upon adding up the inequalities in (29.7') and those in (29.7") and recalling (29.2), we find that both of the numbers f u dµ° and I (u) lie between n

n

E(yj - yi-1)µ°(Kj)

E(yj - yj-1)1°(Kj-1)

and

j=1

j=1

and consequently 5 n

E( yj - yj -1)Fz°(Kj-1 \ Kj),

if

j=1

since Kn C Kn_1 C ... C Ko. Due to the choice of the yj it follows that

Jud1L0-

Eu°(KK-1\K3)-Fµ°(Ko\K.)<EIp°(Ko)

I(u)I <_F, j=1

The extreme inequality being valid for every e > 0 and p°(Ko) being finite, the desired equality (29.8)

I(u) =

f

J

udµ°

emerges.

The measures of the compact sets Kj, j = 0, ... , n do not change, thanks to (28.8), when µ° is replaced by p ,. Another pass through the preceding derivation therefore leads to the conclusion that µO is also a representing measure for 1. O

These two representing measures can be characterized by extremality properties:

29.2 Lemma. Every representing measure p for I satisfies

p(K) 5 p.(K) and p°(U) < µ(U) for all compact subsets K and all open subsets U of E.

IV. Measures on Topological Spaces

180

Proof. Given K and U, consider functions u,v E CA(E) with iK < v, 0 < u < 1, and supp(u) C U. Integrating these inequalities,

µ(K) < Jvd de = I(v) and I(u) =

r

J

udp < p(U).

From (28.2) and Lemma 28.7 therefore the claimed inequalities follow. 0 After this preparation we can enhance the statement of the Riesz representation theorem by characterizing the measures p and µ°, thereby putting into relief the role of Radon measures.

29.3 Theorem. For every positive linear form I on CA(E) the associated essential measure F4° is the unique Radon measure among the representing measures of 1. Proof. Let p he a representing measure for I which is inner regular, thus a Radon measure. Since 1I° is also inner regular, it follows from the first part of the preceding

lemma that

p(A) < p,(A)

for every A E .R(E).

In particular then all open U C E satisfy µ(U) < p0(U) < p°(U) and when this is combined with the second part of 29.2 we have p(U) = {I°(U)

(29.9)

for every open U C E.

If compact K C E is given and U is an open, relatively compact neighborhood of K, then U \ K is open, so that (29.9) is applicable and

p(U) - p(K) = p(U \ K) = po(U \ K) = p, (U) - p0(K) Another appeal to (29.9), remembering that p0(U) < +oo, gives the equality

p(K) = po(K) , valid for every compact K C E. This fact and the inner regularity of both measures

results in their equality. 0 29.4 Theorem. Among all representing measures for a positive linear form I on CA(E) the principal representing measure 1° is characterized by each of the following two properties: (i) p° is the smallest among all outer regular representing measures. (ii) p° is the unique outer regular representing measure p which is inner regular

on open sets, that is, satisfies (29.10)

p(U) = sup{µ(K) : K compact C U}

for every open U.

Proof. Let p be an outer regular representing measure. By Lemma 29.2, p°(U) <

p(U) holds for all open sets U. Since, however, µ° is also outer regular, that inequality passes over to Borel sets generally:

µ°(B) < p(B)

for all B E M(E),

§29. Riesz representation theorem

181

which confirms (i). If K is a compact set

u(K) 5 A.(K) = A(K) by Lemma 29.2 and (28.8), so by what has already been proven equality prevails here. That is, k and µ° coincide on the system .X' of all compact sets. Now p° satisfies the inner regularity condition for open sets in (29.10), as we know from (28.4), (28.7) and (28.8). If p also satisfies these conditions, then for every open set U

µ(U) = sup{µ(K) : U D K E ..'} = sup{p°(K) : U D K E JL'} = µ°(U), an equality which passes over to all Borel sets via the outer regularity of both measures; i.e., p = µ° on M(E). Remark. 1. Some authors (cf. HEWITT and STROMBERG [1965] and COHN [1980])

employ the adjective "regular" for just those outer regular Borel measures p that have property (29.10), in contrast to our usage. The following example shows that in general uO is not the only outer regular representing measure. Example. 1. Let E be an uncountable set and equip it with the discrete topology. For I take the identically 0 form. Then from the last two theorems it follows that µ° = µ° = 0. However the measure it from Example 6 of §25 is an outer regular representing measure which is not identically 0.

Example 1 - there u. and p° are identical - leads to the important question whether the essential and the principal measures coincide in general, or under appropriate supplemental conditions. Although according to 28.5 µ°(A) = µ°(A) for all A E M (E) having a-finite p°-measure, generally A. 96 A. An example due to C.H. DOWKER (cf. the reference in EDWARDS [1953], p. 160) will be presented in Exercise 7 below. Nevertheless in many important situations these measures do coincide and we are going to look into this now. We will encounter two types of supplemental hypotheses which will entail the

equality p° = p° on M(E). The first imposes conditions on the space E, but none on the linear form I. We already know, for example, that for a compact space E the representing measures p. and p° determined by a given positive linear form I on CC(E) coincide. This follows immediately from Theorem 28.4. The reasons that underlie this need to be examined more closely.

29.5 Definition. A locally compact space is called countable at infinity (also sometimes o-compact) when it can be covered by a sequence of compact subsets.

Examples. 2. The following spaces are countable at infinity: (i) every compact space; (ii) the euclidean spaces Rd, d E N: The closed balls with any fixed center and integer radii provide a countable covering by compact sets.

182

N. Measures on Topological Spaces

(iii) every locally compact space with a countable basis W. For 90 := {G : G E 9, G relatively compact} is a countable system of compact sets which covers E. Indeed, each x E E possesses by definition a compact neighborhood V, and since 9 is

a basis, x E G C V for some GE 9. Of course then GE40. 3.

A discrete space is countable at infinity just if it is a countable set.

Every subset A of a space E which is countable at infinity is of course covered by a sequence of compact subsets of E, so from 28.5 we immediately get:

29.6 Theorem. If the locally compact space E is countable at infinity, then the representing measures ii° and p° determined by any positive linear form I on CA(E) coincide. A simple consequence is:

29.7 Corollary. On a locally compact space E which is countable at infinity every Radon measure (inner regular by definition) is also outer regular.

Proof. Every Radon measure it on E defines a positive linear form I. on CA(E) of which it is a representing measure. According to 29.3 p must coincide with the essential measure pO determined by Iµ. Since µO = p° and the latter is outer regular, so must be A. 0

To justify the terminology "countable at infinity" we sharpen the covering condition featuring in Definition 29.5.

29.8 Lemma. Let E be a locally compact space which is countable at infinity. Then E can be covered by a sequence (Ln)nEN of compact subsets each contained in the interior of its successor. Every compact subset of E is therefore a subset of some (hence of all but finitely many) L.

Proof. First of all there is a sequence (Kn) of compact sets K such that Kn t E. Using Corollary 27.3 we find 0:5 u,, E CA(E) with u, t 1E. But then the sets

Ln:={un>1/n},

nEN,

do what is wanted: Each is closed and, since Ln C supp(u,,), it is compact. Because (zun) is isotone

L C {Yin+i > 1/n} C

1/(n + 1)} open C Ln+l,

whence L C I n+t, where A denotes the interior of a set A. As a result, (t )nEN is an open covering of E, so finitely many of its sets suffice to cover any given compact subset of E. 0 A simple interpretation of countability at infinity now emerges: A locally compact space E is countable at infinity if and only if the infinitely remote point wo

§29. Riesz representation theorem

183

in the one-point compactification E' has a countable base of neighborhoods. Such a countable neighborhood basis is furnished by the complements E' \ Ln of any sequence (L,,) with the properties described in 29.8.

We come now to the second type of supplemental hypotheses. Here E is an arbitrary locally compact space and conditions will be imposed on the positive linear form I on Cc(E).

29.9 Definition. A positive linear form I on Cc(E) is called bounded if there is a real number M such that II(u)1 < M IIuII

(29.11)

for all u E CA(E)-

Here IIf II denotes the supremum norm of any bounded real function f on E. The requirement (29.11) means that I is continuous with respect to the metric (of uniform convergence) in CA(E) derived from this norm. Remark. 2. If the space E is compact, then every positive linear form I on Cc(E)

is bounded, because CA(E) = C(E) so the constant function 1 lies in Cc(E). Therefore from - Dull 1 < u < IIuII . 1 and the positivity of I we infer that - Hull 1(1) < I(u) <_ Hull 10),

so that (29.11) holds with M := 1(1). The next theorem - like its predecessor - covers compact spaces as a special case.

29.10 Theorem. If I is a bounded positive linear form on a locally compact space E, then its principal representing measure µ° is finite and coincides with the essential measure µO. Proof. According to (28.13)

Il,0Il=sup{I(u):0
Thus µ° is a finite measure and the rest follows from 28.4. 0 Proceeding via Iµ as before (cf. 29.7) yields

29.11 Corollary. Every finite Radon measure µ on a locally compact space E is also outer regular.

IV. Measures on Topological Spaces

184

Indeed, the positive linear form I4 on C°(E) defined by It is bounded, by M := Ilicll < +00:

I,,(-)I = if

<

r Jul du <_ Dull M

for every u E C°(E),

and we can conclude as in the proof of 29.7. 0

Remarks. 3. From the proof of Theorem 29.10 it also follows that the total maw [l;t°II of u° is the smallest real number M > 0 that can serve in Definition 29.9.

4. It is not to be expected that in every locally compact space E which is countable at infinity every positive linear form on CA(E) will have exactly one representing measure with no further qualification. Still less is unqualified uniqueness of representing measures for bounded positive linear forms on C°(E), when E is only a locally compact space, to be expected. There is a counterexample to both in HALMOS [1974), p. 231 - DIEUDONNIi [1939) is also cited there - in which

the space E is even compact: It is the interval [1, Q] of all ordinal numbers not greater than the first uncountable ordinal f2, equipped with the order topology. The positive linear form IEn on C([1,52]) defined by the Dirac measure en has a representing measure it which is neither inner regular nor outer regular. Thus f f den = f f dp for all f E C([1,1z]) although It 96 eS2. Details can be found in PFEFFER [1977], p. 116.

In view of the last remark the following theorem is especially noteworthy, as well as useful:

29.12 Theorem. If the locally compact space E has a countable base for its topology, then every Borel measure on E is regular, hence in particular a Radon measure.

Proof. Let It be a Borel measure, I, the associated positive linear form on CA(E) and p° the principal representing measure for I. Along with E each of its open subspaces U also has a countable base. From Example 2 therefore U is countable at infinity; there exists a sequence of compact sets such that K 1' U. Since the measures It, p° are continuous from below, it follows that

u(U) _ rn p(K,,) and p°(U) _ im° p°(K,,). But u(K,,) <

u°(K,,) for every n E N, by Lemma 29.2. So we get u(U) < u°(U), from which and a second appeal to 29.2 (29.12)

u(U) = u°(U),

for every open U C E.

For an arbitrary Borel set A and open U D A we then have u(A) < u(U) = u°(U) and so, on account of the outer regularity of u°, (29.13)

u(A) < u°(A),

for every A E ..(E).

§29. Riesz representation theorem

185

If A E .4(E) is relatively compact, we can choose an open relatively compact neighborhood U of A and apply the last inequality to U \ A, getting

u(U) - u(A) = u(U \ A) < u°(U \ A) = u°(U) - u°(A). Subtracting (29.12) from this gives us the reverse inequality to (29.13). In summary,

tt(A) = p°(A),

(29.14)

for every relatively compact A E .V(E).

Now, E is, as already noted, countable at infinity. So we have a sequence of compact sets which increase to E. (29.14) is applicable to B n L for any Borel set B and any n. E N. We therefore get

u(B) = 'x-+x lim u(B n Lg) = n-x lim u°(B n L,) = u°(B). That is, u and u° coincide throughout .W(E). Since the essential measure u° is a representing measure for I,,, this fact insures (as does Theorem 29.6, for that matter) that u = u°. From the double equality it = u° = p° follows finally the

regularity of u. 0 In this situation the Riesz representation theorem can therefore be expressed thus:

29.13 Corollary. For a locally compact space E whose topology has a countable base, every positive linear form I on CA(E) can be represented as

1(u) = Judprt E by exactly one Borel rrteasur p on E. Example. 4. For cacti u E CS(R) choose real numbers a < 13 such that supp(u) C (a,131 and define

L(u) :=

j a u(x) dx, a

the integral being the usual Riemann integral: it is independent of the specific numbers a and,3 used. Evidently L is a positive linear form on CS(R). According to 16.4 L-B measure A' represents L, and by 29.13 it is the only representing measure.

Remark. 5. It is also possible to deduce Theorem 29.12 from Theorem 26.3 and its Corollary 26.4 because every locally compact space E whose topology has a countable basis is Polish. In fact along with E, its one-point compactification E' also has a countable base, as follows from Lenima 29.8 and the commentary after it. It will be shown in Remark 3 of §31 that E' is consequently ntetrizable, and completeness

of the metric follows easily from compactness (cf. Example 6, §26). Thus E' is Polish and E is an open subset of it. Therefore according to Example 4, §26 E itself is Polish.

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IV. Measures on Topological Spaces

Summarizing, we can say that for every locally compact space E, the mapping that associates to each Radon measure p on E the positive linear form 1. on Cc(E) is a bijection between the set of Radon measures on E and the set of positive linear forms on CA(E). That is the reason why in BOURBAKI [1965) the positive linear forms on CA(E) are themselves designated as (positive) Radon measures.

If the space E is countable at infinity as well, the Radon measures on E are all outer regular. If moreover the topology of E has a countable base, the Radon measures and the Borel measures on E coincide. We give now an application to integration that is of fundamental importance.

29.14 Theorem. For any regular Borel measure p on a locally compact space E and any p E [1, +oo[, the vector space CA(E) is dense in 2P(p) with respect to convergence in pen mean.

Proof. First of all, CA(E) C .`(p), because CA(E) C .2"(p) by (28.1) and Iulp E CA(E) whenever u E CA(E). The denseness claim requires that for each f E gy(p) and each number e > 0, a function u E CA(E) be produced with

Np(f -u):=

(f If - uIp du) "P <e.

We accomplish this by a stepwise simplification of the function f to be approximated. Since along with f , both f+ and f - are in .`gy(p), and Np is a semi-norm, we can assume that f > 0. By 11.3 and 11.6 there is an isotone sequence (fn) of SR(E)-elementary functions such that f,, t f. All these functions also lie in £"(p), due to 0 < fn < f, Therefore from the dominated convergence theorem lim

nioo

Np(f - f,,) = 0.

This makes it clear that only . (E)-elementary functions need be approximated by CA(E), and because of the semi-norm properties of Np the matter even comes down to approximating the indicator functions 1A of Borel sets A having p(A) = (Np(lA)Jp < +oo. For such an A the outer regularity of p supplies an open U J A such that [p(U) - p(A)J1/p = Np(lu\A) = Np(lU - 1A) < e/2.

In particular, p(U) < +oo. Therefore the inner regularity of p insures that for some compact K C U

that is,

Np(lu - 1K) < e/2. Finally, we use 27.3 to select u E CA(E) satisfying 1K < u:5 1U, whence

0<1u-u<1u-1K

§29. Riesz representation theorem

187

and so

Np(lu - u) < e/2. For the function f = 1 A to be approximated we now have

Np(f - u) < Np(lA - lu) + N,(lu - u) < e, completing the proof. 0 The proof actually uses the inner regularity of µ only on open sets. So what is involved here are conditions which according to 29.4(ii) characterize the principal representing measure. We will not pursue this any further but interested readers can in BOURBAKI [1965) and BAUER [1984), where this remark is placed in a more general framework.

Exercises. 1. Let E be an uncountable discrete space. Using the Borel measure from Example 6 in §25, show that every positive linear form on CA(E) has at least two different representing measures. This sharpens Example 1 of this section. 2. Let E be a locally compact space and I a positive linear form on CA(E). With the help of the R.iesz representation theorem prove the following refinement of equality (28.12): For every open U C E

µ°(U) = sup{I(u) :0:5 u:5 lu, u E CA(E)}

.

3. A Ko-set is a union of countably many compacta. Prove that in a locally compact space in which every open set is a Ks-set, every Borel measure is regular. (Hint: Re-examine the proof of Theorem 29.12.) 4. Show that a locally compact space E is countable at infinity if and only if there exists a strictly positive function in Co(E). 5. Prove that for an arbitrary Borel measure µ on a locally compact space E the following two assertions are equivalent: (a) it is finite. (b) Cb(E) C 2l(µ). Show that if µ is a Radon measure, the assertion C0(E) C 2l(µ) is equivalent to each of (a) and (b). 6. Let E be a locally compact space, I a positive linear form on Co(E). Show

that there is exactly one finite Radon measure µ on E such that 1(f) = f f dµ for every f E Co(E). (Hints: Indirect proof. Or: For every e > 0 and non-negative f E Co(E) there is a u E CA(E) with If - uI 5 7. Let El, E2 be the interval [0, 1) equipped with the discrete topology, respectively, the usual euclidean topology, and consider the product space E = El x E2. Show

that (a) E is locally compact. (b) Every product

xE:_{x}x[0,1), is a compact subspace of E, which is also open in E.

0<x<1,

188

IV. Measures on Topological Spaces

(c) A set U C E is open if and only if U fl xE is open for each x E 10111(d) Every compact subset of E is covered by finitely many of the sets xE. Now consider u E CA(E). By (d) u vanishes in the complement of the union of finitely many xE sets, and for each fixed x, y u-+ u(x, y) is a continuous function on the compact interval Ea = [0, 1]. Therefore I (u)

II

u(x, y) dy

O<x
is a well defined finite sum, evidently a positive linear form on Cc(E). Show that (e) The essential and the principal representing measures for I do not coincide.

[Hint: Show that the set A := El x {0} is closed and that s°(A) = 0, while u°(A) = +oo.] (f) In passing from u° to the Borel measure 1Bµ° for B E M(E) outer regularity may be lost. [It suffices to consider B := E \ A, for the set A in the preceding hint.]

§30. Convergence of Radon measures For locally compact spaces E we will henceforth use the notation .4'.. (E) for the set of all (positive) Radon measures on E. The Riesz representation theorem furnishes a canonical bijection of fl+(E) onto the set of all positive linear forms on CA(E).

With p, v E 4+ (E) and real numbers a > 0, i3 > 0 the measure aµ +)3v also lies in .ill+(E), as is easily checked. That is, .0+(E) is what is called a convex cone. Besides . W+ (E) we often consider the following subsets

.'+(E) = (1A E 4'(E) : p(E) < +oo}

-#+'(E) =fu E-0+(E):µ(E)=1}, the set of all finite (or bounded) Radon measures and the set of all Radon pmeasures on E, respectively. Evidently

-&+' (E) C.-W+(E) C .4+(E) .

In .f+1 (E) are to found all the Dirac measures on E. And 4 (E) is a convex subcone of 4f+ (E). In the special case E = Rd the set ..W+b (W') is the set of all finite Borel measures

on Rd, already familiar to us from §24. That the definition there is equivalent to the present one is due to Theorem 29.12, according to which every Borel measure on Rd is a Radon measure. Depending on whether one thinks of the elements of . W+(E) as measures on -V(E) or as positive linear forms on CA(E), two notions of convergence suggest themselves: One can define the convergence of a sequence (ta,,) in 4'+(E) to

§30. Convergence of Radon measures

pE

189

by requiring either that lim An (A) = p(A)

n-+oo

for all A E R(E)

or

lim

n-+oo

J

f dp = J f dp J

for all f E CC(E).

We will forthwith show that the first of these is of limited interest, while the second is of considerable significance.

30.1 Definition. A sequence (pn)nEN of Radon measures on E is said to be vaguely convergent to a Radon measure y if (30.1)

lim

-oo

for all f E CA(E).

A sequence (pn) in 4'+(E) is vaguely convergent just when the sequence of real numbers (f f dpn) converges in R for every f E CA(E). For in this case f H lim f f dpn evidently defines a positive linear form on CA(E), so by the Riesz n representation theorem together with Theorem 29.3 there is a unique Radon measure p to which (An) vaguely converges. At the same time we see that a sequence in . K+(E) can have at most one vague limit.

Examples. 1. Let (xn) be a sequence in E, x E E. If (xn) converges to x, then (e2 ) converges vaguely to eZ, for the latter just amounts to lim f (xn) = f(X)In general however lime= (A) = ex(A) does not hold for all A E -V(E); in fact, if all xn are distinct from x, A := {x} is such a set. Conversely, if (es,) vaguely converges to ey, then (xn) converges to x. For if this were not so, there would be a subsequence of (xn) which remains outside of some neighborhood U of x. 27.3 furnishes an f E CA(E) with f (x) = 1 and supp(f) C U. Evidently the (f (xn)) does not converge to f f de,. sequence (f f Let (an) be an arbitrary sequence of non-negative real numbers and (xn) a sequence in E with the property that {n E N : xn E K} is finite for every compact K C E. (In other words, E is not compact and limxn = wo E E'.) Then the sequence of measures An := ane: (n E N) is vaguely convergent to the zero measure p := 0. For f f dpn = an f (xn) = 0 for all n except the finitely many for which xn E supp(f), whenever f E Cc(E). 2.

The fact, illustrated by Example 1, that the vague convergence of (An) to A does not generally entail the convergence of (pn(A)) to p(A) for each A E . (E), while, as 30.2 will show, the converse is true, seems to indicate that the first mode of convergence mentioned above is too restrictive to be of much use. Actually, vague convergence of (An) to p follows just from knowing that (An (A)) converges to p(A) for certain special sets A E R(E). Even more:

190

IV. Measures on Topological Spaces

30.2 Theorem. A sequence (pn) of Radon measures on a locally compact space E converges vaguely to a Radon measure p if and only if the following condition is fulfilled: (30.2)

lim pp 1zn (K) < p(K)

and

lim oinµn (G) > jz(G)

for every compact K C E and every relatively compact, open G C E. converges vaguely top and that K and G are any compact and open sets, respectively. Consider functions u,v E CC(E) with u > 1K, 0 < v < 1 and supp(v) C G. Then for all n E N Proof. Suppose

µn(K) < J udjcn and JVdPn
limss op jln(K) <

J

udp and

J

vdµ < liimianfµn(G).

From these inequalities (30.2) follows via (28.2) and (28.12). One only has to recall

that the Radon measure p coincides, thanks to Theorem 29.3, with the essential measure po determined by the linear form Iµ. Now suppose conversely that condition (30.2) is fulfilled and that an f E CA(E) has been given. Since our goal is to confirm (30.1), we lose no generality by assuming that f > 0. For a pre-assigned e > 0 we choose finitely many numbers

0=yo IIfII and yj - yj_1 = e for each j = 1,...,k. Set

K:= supp(f) and Aj :_ {yj_1 < f < yj} f1 K,

j = 1,...,k.

Denoting the compact set { f > yj } fl K by Kj for j = 0,..., k (so Kk = 0 and Ko = K), we have K, -I D Kj and

(j=1,...,k).

A =Kj-1\Kj Because of the obvious inequalities k

k

Eyj-11A; <.f <_ Eyj1A,, j=1

j=1

every Radon measure v on E satisfies k

1: yj_1v(Aj) < Jfthi < j=1

k

j=1

yjv(A,),

§30. Convergence of Radon measures

191

from which and a simple calculation using the facts v(A,) = v(Ki_1) -v(K,) and yi - yi _ 1 = e, we get k

k

k

e E v(KK) - ev(K) = e E v(KK) < Jfdziv(Ki).

i=

i=o

i=o

For v := it,, the right-hand inequality gives us k

Jfd/<EJL(Kj)

for all n E M,

i=o

and therefore from the first half of hypothesis (30.2)

r

k

limsopJ fdµn<eEµ(K1) i=o

But this right-hand side can be estimated by using the left end of the earlier chain of inequalities, with v:= µ. We thereby get lim sup

f f dµn < r f dµ + eµ(K),

valid for every e > 0. Consequently, lira sup

Jfd µn <

ffd.

The complementary inequality that we need is

f fdµ
One sets Gi := If > yi }, j = 0, ... , k, which are open, relatively compact subsets of K with

Gi-i \ Gi = {yi-i < f < yi} _ {yi_1 < f < yi} fl K. These sets take over the role of the Ki. 0 The second example above (for the case in which, say, all the an equal 1) shows that a vaguely convergent sequence of measures from .41+(E) need not converge to a measure in .,W+l (E): mass can be lost. This illustrates the following general phenomenon:

30.3 Lemma. If the sequence (µn)nEN of Radon measures on the locally compact then the associated total space E converges vaguely to the measure µ E masses satisfy (30.3)

IIiII < Inm onf IIIinII

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IV. Measures on Topological Spaces


holds for n E N, so from (30.1) follows that

f udp < liminf IIpnII J n-00 Take the supremum of these integrals over all such u and you get, according to (28.13), the total mass p(E) = IIpII of p. The inequality persists after this operation

Vague convergence of sequences in .4'+(E) is convergence in a certain topology on ..ff+(E), called, naturally, the vague topology. It is defined as the coarsest topology on .4f+ (E) with respect to which all the mappings

p y J f dp

(30.4)

(f E CA(E))

are continuous. A fundamental system of neighborhoods of a typical po E 4' (E) consists of all sets of the form (30.5) Vi...... t..:E(WJ)

1/1 E 4+(E) : if fi d1a -

J fi

dPol < s,.1 = 1, ... , n}

in which n E N, 0 < E E R and fl,..fn E CA(E) are all arbitrary. The vague topology is Hausdorff because the uniqueness aspect of Riesz's theorem says that if p, v are different Radon measures, then I, 36 It,., which just means that f f du 34

f f dv for some f E C (E). In this context it is now clear too what should be understood by the vague convergence of a mapping t i-+ p of a subset A of a topological space T into W+ (E)

when t converges to a point to E A. With respect to the vague topology the convergence

lim µt = µ 10

t

tE A

for some U E 4'+(E) just means that (30.6)

lim ffdt = ffd

forevery f E C(E).

tEA

Example. 3. Let K be a non-negative Ad-integrable, real function on E := Rd with f K dAd = I (for example, the indicator function of the unit cube [0,1] ). For every real r > 0 set K,.(x) := rdK(rx) (x E Rd). Then K, is also non-negative and Ad-integrable, and f K, dAd = 1 as well. To see this we only have to recall (7.10), according to which the homothety H,(x) := rx on Rd transforms L-B measure thus: Hr(Ad) = r-dAd. For from that it follows

§30. Convergence of Radon measures

193

that

J KrdAd=rd I K0HrdAd=rdJ Kd(Hr(Ad))= I KdAd = 1. Now r -+ Kr)1d is a mapping of JO,+oo[ into dl. (Rd), and in the sense of the vague topology it satisfies

lim KrAd = e0

(30.7)

r-a+oC

To confirm this, first notice that for every f E

.F

f f Kr dad = rd J f (K o Hr) dad = rd f (f o Hr-') K dHr(Ad) = f(f oHH')KdAd= ff(f_1x)K(x)Ad((fr)

this and the Lebesgue dominated convergence theorem the claim (30.7) follows upon checking that, on the one hand

lint f (r-'x)K(x) = f (0)K(x)

r-++oo

for every x E Rd,

and on the other hand for all real r > 0 and all x E Rd

If (r-'x)K(x) I <_ Ilf11. K(x), so that 11111 K is an integrable majoraut for all functions. The "approximation of the identity" co expressed by (30.7) plays an important role in Fourier analysis (cf. the exercises in §23 of BAUER [1996] ). For the algebra L' (ad) (cf. Remark 2, §24)

has no identity element with respect to convolution, but it is not hard to show that II Kr * f - f 11 -+ 0 as r -+ +oo for each f E L' (Ad), and in many situations this is almost as useful as having an identity. To .,W+b (E) belong in particular all discrete Radon measures on E. These are the measures 6 which can be represented in the form k

5 = E aic", 7=1

f o r some finite number of points x1, ... , xk E E and non-negative real numbers at, ... , ak. Every 5 admits many such representations. Every Radon measure can be approximated, in the sense of the vague topology, by such 5, as we next show. 30.4 Theorem. For every locally compact space E the set of discrete Radon measures on E is dense in .4f+ (E) in the vague topology. Proof. Let a measure tso E .W+(E) and a vague neighborhood V of be given. As noted after (30.5), we can suppose V is Vj, ,....I,, :1(0) for some non-zero Ii..... f E ,,(E). We have to find a discrete measure 6 in V. To that end, consider the com-

194

IV. Measures on Topological Spaces

pact set

n

K := U supp(fi) i=1

and g > 0 such that npo(K) < 1. Every y E K has an open neighborhood U. in E such that 1 fi (y') - fi (y") I < q for all y', y" E U. and all j E {1, ... , n}. Finitely many Us,, say Uy...... Uy,, suffice to cover K. Set

Al :=KnU,,, A2:=(KnU,,)\Al,...,Ak:=(KnUYk)\(ALU...UAk_1). These are pairwise disjoint, relatively compact Borel sets whose union is K, and for all j E { 1, ... , n}, i E { 1, ... , k} and y', y" E A. the inequality I f i (y') - fi (y") 15 rl holds. Since only these properties of the A; are used in the sequel, we can discard those that are empty (not all are because 0 0 K = Al U ... U Ak), and re-index the

others. That is, we can suppose all the A; are non-empty and then select a point xi E A, for each i. The discrete measure k

i=1

(notice that po(A;) is finite because A; is relatively compact) will be shown to lie in V and that will complete the proof-

i=1

f fi dpo +

po(A:)fi(xi) i=1

I:k fA.' -f(x))dpo fA,

Ifi - fi(xi)I dpo<Eipo(A.)=rlpo(K), iel

using the fact that Ifi(x) - fi(xi)I < 17 for all x E A;, all i E {1,...,k}. This holds for each j E { 1, ... , n}, and gpo(K) < 1 by choice of q. Therefore b E V1,,..., f,,;1(po) = V, as was to be shown. 0

30.5 Corollary. The discrete p-measures on E are dense in di. (E) in the vague topology.

Proof. We take over the notation of the preceding proof. Now po is a measure in 4+' (E), but the discrete measure 6 = F, po(A;)ez, may not be a p-measure, so more work is required. Set a; := po(A1), i = 1, ... , k. If K = E (in which case E had to be a compact space), then a1 +... + ak = po(K) = 1 and b actually is a p-measure. In general what we have is

a1 + ... + ak = po(K) < uo(E) = 1

§30. Convergence of Radon measures

195

and if K 0 E we can choose another point, xk+l E E \ K, and set

(al +... + ak),

ak+l which is non-negative. Then

is a discrete p-measure with f fj dd = f fj db' for each j = 1, ... , n, since xk+l lies outside the supports of all these functions. Consequently, 6 E V = Vf...... f,,;I(P0)

yields that also 6' E V. 0 Next we will investigate whether the equality (30.1) and the continuity assertion (30.4) remain valid for classes of continuous functions more general than C..(E).

Recall in this connection that for a measure µ E .,&+' (E), every f E Cb(E) is uintegrable: it is g(E)-measurable and its modulus is majorized by a real constant, hence p-integrable, function. We will formulate the relevant results for sequences only; their extensions to mappings t u-+ pt are routine.

30.6 Theorem. If a sequence (µn)nEN in .14(E) is vaguely convergent to µ E .1/+(E) and if the sequence (IIµnII)fEN of total masses is bounded, then along with all the pn the measure µ is also finite, and for every f E Co(E) lim

f

f dµn =

Jfd.µ

Proof. If we set a := sup{11µn11: n E N}, which is finite, then 111411:5 a, by (30.3),

so µ is a finite measure. Definition 27.5 says that for each e > 0 there is a g = gf E CA(E) such that 11f - g11 5 e. Therefore for each n E N

if and

if f du

< ae,

so that via the triangle inequality

if f dµn - f f dµ

I< 2ae + 119dJun

- jgd.Uj

for all n E N.

Since the hypothesis of vague convergence means that f g dµn -1 f g dµ, we get

lim sup if f dAn - f f dµ < 2ae, 1

valid for every e > 0. That is, the limit exists and is 0. 0 Remarks. 1. If one considers measures pn and µ E .-W+6 (E) without the hypothesis sup 11µn 11 < +oo, the above conclusion can fail. The special case of Example 2 in

196

IV. Measures on Topological Spaces

which E := R, x := it and a := it for all n E N illustrates this. For the function f defined by

f (x) := min (1, Ix[-1}

for x # 0, f(0) := 1

lies in C0(R). But f f dpn = 1 for every n E N, while f f dµ = 0, because here the vague limit p is the 0-measure. 2. Example 2, again with E := R and xn := n for all n, considered earlier, but

this time with the constant sequence a := 1, shows that indeed lim f f dey = f f dµ for the measure p := 0 and all f E Co(R), but this equality is already false for the constant function f := 1E in Cb(R).

The passage from Co(E) to Cb(E) therefore calls for a special investigation, which we stress by introducing a new definition:

30.7 Definition. Let p, p1, p2.... be measures in 4(E). The sequence is said to be weakly convergent to p if lim

(30.8)

n-+00

JfdP=Jfdp

for all f E Cb(E).

30.8 Theorem. Suppose the sequence (An)nEN in ..4+(E) converges vaguely to the measure it E .W+ (E). Then the following statements are equivalent: (i) The sequence converges weakly to it. 11m IIpnll = IIEiII (ii)

(iii) For every e > 0 there exists a compact subset K = K, of E such that

(E\K)<e

forallnEN.

Proof. (i)*(ii) is obvious because 1 E Cb(E). Let c > 0 be given. The inner regularity and finiteness of p yield that there is a compact subset L of E such that p(E \ L) < e. According to 27.3, L has a compact neighborhood KO, so there is an open set G with L C G C Ko. By (30.2)

lim inf µn(G) > p(G) > p(L) > IIp1I - e, ,l-+00

so if we choose a E I 11p 11 - e,p(L)[ there will be an no E N such that pn(G) > a for all it > no. Moreover, in view of (ii) this no may be supposed large enough

that IIpnII < a+e for all n > no. Consequently, pn(Ko) > pn(G) > a > IIII -e, so that p.n (E \ KO) < e, for all n > no. For each n E { 1, ... , no) inner regularity

and finiteness of pn give us a compact K C E such that pn(E \ Kn) < e. The compact set K := Ko U K1 U ... U Kn0 then satisfies (iii). Given e > 0, let K = K, be as described. Again from (30.2) we have

p(E \ K) < lim inf K) < e. There is a function u E C,:(E) with 0 < u < 1 and u(K) _ { I). It satisfies 0 < 1 - it < 1CK and so for each f E Cb(E)

ifl 5 I[f11 f(i-u)dp,,<11fllM(CK)<-IIIIIk

forallnEN

§30. Convergence of Radon measures

197

and by the same argument

J(i- u)fd/)

<- If 11£.

As in the preceding proof, the triangle inequality then gives

if

Jfdµl

<2IIfIIE+11ufd/Ln- Juid µl

for all n EN.

Since of E CA(E), the hypothesis of vague convergence insures that (f of dµ,,) converges to f u f dµ, so the preceding inequality yields

limsup if f dILf

-1 f dµl s 2IIf1I e,

valid for every e > 0. That is, this limit exists and equals 0, for every f E Cb(E). Which proves (i). 0

30.9 Corollary. A sequence (µn)fEN in ..,f+ (E) is vaguely convergent to µ E 4' (E) if and only if it is weakly convergent to p.

Remark. 3. A sequence (µn) in .f+(E) which satisfies condition (iii) is called tight, whether or not any convergence is going on. If a tight sequence from _f+1 (E) vaguely converges to a measure µ E .ill+(E), then first of all, IIµII S 1 by (30.3), so that µ E _&+6 (E). The preceding theorem then guarantees the weak convergence of (µn) to p and therewith µ E _W+' (E). In particular, with vaguely convergent tight sequences in ..f+ (E) no mass is lost (cf. the remark preliminary to Lemma 30.3). Consequences like these constitute the real significance of the tightness concept.

At this point it is worth returning once more to Theorem 30.2. If the measures µ,µn there are all finite and of the same total mass, e.g., if they are all p-measures, then the two components of the compound condition (30.2) become equivalent. The result is the following portmanteau-theorem:

30.10 Theorem. Let µ,µl,µ2, ... be measures in &+' (E). Then the following three assertions are equivalent: (i) The sequence (µn)nEN converges vaguely (and therefore also weakly) to p. (ii) For every closed F C E

(30.9)

lim so p µn (F) < µ(F) .

(iii) For every open G C E (30.9')

lim of µn (G) >- IL(G)

Proof. The first paragraph of the proof of 30.2 actually established that (i)=(iii), under the less restrictive hypotheses prevailing there. Since that theorem further shows that the conjunction of (ii) and (iii) implies (i), it only remains to establish

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IV. Measures on Topological Spaces

the equivalence of (ii) and (iii). That follows from the trivial observation that

v(CA) = v(E) - v(A) = 1 - v(A) holds for all A E -4(E) and all v E _W+1(E).

Example 1 in this section shows that the weak convergence of a sequence (µn) in .4/+(E) to a It E 4' (E) does not imply the convergence of (f f dµ,+) to f f dµ for every bounded Borel measurable function f . Nevertheless the continuity of the functions f which define weak convergence can be relaxed somewhat. To this end, we consider bounded, real-valued, Borel measurable functions f on E which are p-almost everywhere continuous for a p E .A"+(E): After excision of a p-nullset N E .3(E), f is continuous at each point of E \ N. Important examples of such are the indicator functions of boundaryless Borel sets. The latter are defined as follows:

30.11 Definition. A Borel subset Q of a locally compact space E is called boundaryless with respect to a measure p E .AY+(E), p-boundaryless (or p-quadrable) for short, if the boundary Q' \ $ of Q is Eo-mill:

µ(Q') = 0.

(30.10)

Examples. 4. Every interval of the number line R is A'-boundaryless. 5.

A set Q E V(E) is boundaryless with respect to a Dirac measure ea if and

only if a E E \ Q*. Look back at Example 1 with this observation and the following theorem in mind.

30.12 Theorem. Suppose the sequence (µn),+EN in .Al!+(E) converges weakly to

it E J4 (E). Then (30.11)

lim

n-,00

JfdPn=JfdP

holds for every bounded Borel measurable function f that is p-almost everywhere continuous on E. In particular, (30.12)

lim p,,(Q) = µ(Q)

n-,OC

holds for every p-boundaryless set Q E .O(E).

Proof. By hypothesis there is a Borel set Eo C E with µ(E \ Eo) = 0 such that f is continuous at the each point of E0. Let e > 0 be given. Since p is a Radon measure, there is a compact K C Eo with

p(Eo\K)<e. Every x E K has an open neighborhood Ux on which the oscillation of f is at most e, meaning that for all y1, y2EUx. If(yi)-f(Y2)I <_e

§30. Convergence of Radon measures

199

Choose a compact neighborhood V= of x with VV C Ux and then use the compactness of K to find finitely many points x 1, ... , x, E K such that V=, , ... , V=,, cover K. If we now set

a := inf f (E),

aj:= inf f (U=; ),

13 := sup f (E),

Q3 := sup f (U , )

for j = 1, . . . , n, then for each such j there exist functions gj, h3 E Cb(E) satisfying

9i( x) _

(aj a

as well as

if x E Vx ifxECUU,

and h (x) =

{ ,Qi [3

if x E Vj ifxECUU,

a
This follows at the once from 27.3 and the application of an appropriate affine transformation in the range space R. From these properties and definitions it follows in particular that gi S f < hj for all j. Therefore if we set

g:= g1 V... Vg,, and h:=h1 A...Ahn, then both these functions lie in Cb(E) and they satisfy a < g < f < h < ,0. Moreover,

0
forallxEK.

For each x E K lies in some V1, C Us, and because of the way Ux; was chosen with respect to the oscillation of f, it follows that h(x) - g(x) < h,(x) - gj(x) _ /31 - aj < E. We are now in a position to finish the proof, as follows:

dµ+JE\K-g)dit J(h-g)di=IK-g) <

eµ(K) + ((3 - a)µ(E \ K) < e(µ(E) + 3 - a) ;

and, because g < f < h and g, h E Cb(E), the weak convergence hypothesis gives

J

g dp = n-too lim

< lim

J

-n +00 J

g dµn < lim inf f f dttn < lim sup if f dµn

nloo J

h dµn =

J

n-+00

h du.

Of course we also have f g dµ < f f dµ < f h dµ. Putting all this together shows that any pair of the numbers f f dµ, lim inf f f dµn and lim sup f f dµn differ by at most e(µ(E) +,3-a). Since e > 0 is arbitrary, (30.11) holds. 0

Let us now look at an application of this theorem which relates the vague convergence of p-measures on the number line to their Theorem 6.6 description in terms of distribution functions. This is the way that weak (and hence vague) convergence made its original historical appearance. 30.13 Theorem. Let µ, Al, A2.... be measures in 4+1(R), that is, probability measures on .41, and F, F1, F2 ... their distribution functions. If the sequence (µn)nEN

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IV. Measures on Topological Spaces

converges weakly to p., then (30.13)

limo F,,(x) = F(x)

n +0

holds for every x E R at which F is continuous. If F is continuous throughout R, then this convergence is uniform on R. Proof. According to Theorem 30.12, 1im p.,, (Q) = p(Q) for every p-boundaryless set Q E .£1 and thus, after (6.11), lim F,,(x) = F(x) for every x E R such that the oo, x( is p-bounda.ryless. We have interval Qx

] - oo, x] = Qx = n

Q=+1 /k

kEN and therefore

t (Qx) = klim u(Q.+1/k) =kin F(x + Ilk) . Consequently, Q, is L-boundaryless just if the (isotone) function F is right-continuous at x, that is (since distribution functions are everywhere left-continuous), just if x is a point of continuity of F. This proves the first assertion. Let us now hypothesize that F is continuous on the whole line, and let e > 0

be given. First of all, (6.13) supplies numbers a < b such that F(a) < e and 1 - F(b) < c. The uniform continuity of F on the compact interval [a,b] insures that points a = xo < x1 < ... < xk = b exist such that

F(xj)-F(xj_1)<e

forj=1,...,k.

From what has already been proven we know that there exists nE E N such that

IFn(xj) - F(xj)I <,E

for each j E 10,..., k} and all n > nE.

But then, as we will show, the inequality (Fo(x) - F(x)] < 2e prevails for every x E R and all n > ne1 which proves the uniform convergence of (Fn) to F. For if x < x0, then

0 < F(x) < F(xo) < e and 0 < Fn(x) < Fn(xo) < F(xo) +e < 2e, that is, I F,,(x) - F(x)j < 2e. And a similar argument works if x > xk. The remaining x fall into [x j _ 1, x j [ for an appropriate j E {1,...,k}, so

F(xj_1) < F(x) < F(xj) < F(xj_1) +e and

F(xj_1) - c < Fn.(xj_1) < Fn(x) < F,,(xj) < F(xi) +e < F(xj_1) +2e, confirming that in this case too IFn(x) - F(x)I < 2E. Remarks. 4. At a point x E R of discontinuity of F limit relation (30.13) generally fails, as the example Ee := n E N, confirms.

§30. Convergence of Radon measures

201

5. Condition (30.12) for every p-boundaryless set Q E R (E) is also sufficient for the weak convergence of the sequence to p (cf. Exercise 6 below). The same is true of condition (30.13) (cf. Exercise 7). The concept of weak convergence (with the same definition) is also meaningful

if E is a Polish space (or even just a metric space) if the measures involved in Definition 30.7 are all finite Borel measure on E. Only the uniqueness of limits calls for discussion:

30.14 Lemma. Finite Borel measures p and v on a metric space E are equal if f f dp = f f dv for all f E Cb(E). Proof. Let d be a metric giving the topology of E and consider closed subsets F C E. Suppose we can always find a sequence (fn) in Cb(E) with fn .1. 1F. Then it would follow from the hypothesis and from Lebesgue's dominated convergence

theorem that u(F) = v(F). The system of closed subsets F of E is an fl-stable generator of the Borel a-algebra R(E) and it contains the whole space E. The equality µ = v would thus follow from the uniqueness theorem 5.4.

It remains therefore to prove the existence of such sequences (fn) and we can suppose F 0 0. For this purpose we use the (uniformly) continuous antitone function h : R -+ R which is constantly 1 on ] - oo, 01, constantly 0 on [1, +oo[ and defined by h(t) := 1 - t on [0, 1], together with the function x H d(x, F) := inf{d(x, y) : y E F}. The latter is a (uniformly) continuous function on E, as we showed in the proof of Example 4, §26. Moreover, its zero-set is exactly F, because F is closed. Apparently then the sequence of (uniformly) continuous functions fn(x) := h(n d(x, fl), x E E, n E N

does what is wanted. 0 Remarks. 6. The concept of p-boundaryless sets is also meaningful for finite Borel measures p on Polish spaces. One easily convinces himself that Theorem 30.12

remains valid in this new situation. In the proof one merely has to secure the existence of the needed functions g3 and h2 somewhat differently: To this end one engages Urysohn's lemma (WILLARD [1970], p. 102 or KELLEY [1955], p. 115). 7. Weak convergence in the set of finite Radon measures on a Polish or a locally compact space E derives from a topology in the same way that vague convergence does. It is called, naturally, the weak topology and it is defined by letting Cb(E) take over the role of CC(E) in (30.4). Weak convergence in (non-locally compact) Polish spaces plays only a marginal role in this book, but is thoroughly investigated in BILLINGSLEY [1968] and PARTHASARATHY [1967].

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IV. Measures on Topological Spaces

Exercises. 1. Let E be a locally compact space, (µn)fEN a sequence in ..Wb(E) which is vaguely convergent to µ E . +(E). If 11µI.11 !5 1 for every n E N, then R o.D exists and equals 1. be a convergent sequence of real numbers, with slim an = a E 2. Let +00 be a sequence of non-negative real numbers such that al > 0 Further, let (a and the series E a,,, is divergent. Then lim

n-+no

alai +...'+'anon =a a,

the case in which all an = 1 being the best known instance. Here is an outline for a measure-theoretic proof: The equations /tn :=

x161 + ... + anEn

n E N,

al+...Ian

define a sequence of measures in -0 (N) which vaguely converges to 0. Therefore

according to 30.6, line f f dt. = 0 holds for every f E Ca(lm). The relevant f is the one defined by f (n) := a - a. 3. Let E be a locally compact space and T a subset of C0(E) with the following properties: Each compact K C E has a relatively compact neighborhood U such that every f E C0(E) with supp(f) C K is uniformly approximable on E by functions t E T whose supports He in U; and further, there exists a t E T with 0 < t < I and t(K) _ {1}. Show that: (a) A sequence (µn) in .1+(E) is vaguely convergent if and only if the sequence (f t dp) is convergent in R for every t E T. (b) For E := R, the set of all continuously differentiable real-valued functions with compact support is a T with the above properties.

4. With the help of Exercise 3 show that for the functions f, (x) := I - sin(nx) on R, the sequence (f .\'),,EN converges vaguely to A1, and deduce from this the Riemann-Lebesgue lemma: Elm

n -r00

f

f (x) sin(nx) dx = 0

for every f E

5. Let it be a finite Radon measure on a locally compact space E. Prove that: (a) The system . of all p-boundaryless sets is an algebra in E. (b) For every f E Cb(E) there is a countable set Al C R such that { f > a) E . for every a E R \ A f. [Hint: For every finite set {al , .... an ) of real numbers n

Eµ({f =aj)) <µ(E) < +oo.] i=1

6. µ,µ1,µz, ... are finite Radon measures on the locally compact space E. Show that condition (30.12) is also sufficient for weak convergence; that is, from limA.(Q) = µ(Q) for every p-boiundaryless set Q C E follows the weak convergence of to it. This is also true if E is a Polish space. [Hints: Imitate the proof

§30. Convergence of Radon measures

203

of Theorem 11.6 and show with the help of Exercise 5 that every 0 < f E Cb(E) is the uniform limit on E of an isotone sequence (un) in the vector space spanned by the indicator functions of the sets in -90-1 7. As an application of Exercise 6 show that in the context of Theorem 30.13 condition (30.13) there is also sufficient for the weak convergence of (µn) to p. 8. Let (an)nEN be a sequence of real numbers in J0, 1[. From [0,1] delete the open interval Ill centered at 1/2 having length al. There remain two disjoint closed intervals J11, J12. From J1j delete the open interval I2j of length a2A1(J13) whose midpoint is that of J13 (j = 1,2). Then there remain four pairwise disjoint closed intervals J21, J22, J23, J24. From J2, delete the open interval I3j of length a3.' (J23)

whose midpoint is that of J23 (j = 1,2,3,4). Then there remain 8 = 23 pairwise disjoint closed intervals J3j, j = 1, ... , 8. Continuing in this way one gets for each n E N pairwise disjoint closed intervals Jnj, j = 1, ... , 2n. The set

C:= n(Jn1U...UJn2n) nEN

is called a generalized Cantor discontinuum, and if all an = 1/3 it is simply called the Cantor discontinuum. Prove that: (a) C is compact and non-void, but C has void interior.

iim fln

I(,_ a,).

cc

(c) A' (C) = 0 4* E an = +00 n=1

[Hint: Recall the inequalities 1 + a < (1 - a)-1 and 1 - a < e_a for 0 < a < 1.] 00 an < +00, U :=]0,1[ \C is an open subset of R whose boundary (d) In case

_

n=1

U' := U \ U is not a A'-nullset. 9. Construct an open subset of ]0,1[x]0, 1 [ whose boundary has positive

\2-measure.

10. Let E be a metric space, with metric d, and let µ, µ1,p2, ... be p-measures on .R(E). Show that each of the following is necessary and sufficient for the weak convergence of the sequence (µn) to p: (a) lim f f dµn = f f dµ for all bounded functions f which are uniformly continuous on E.

(b) lim sup µn(F) < µ(F) for all closed F C E. (c) lim inf µn (G) > µ(G) for all open G C E. [Hints for (a) .(b): Re-examine the proof of 30.14. There it was shown how, for a closed non-empty F C E, to construct uniformly continuous functions fn satisfying fn 1F.1

204

IV. Measures on Topological Spaces

§31. Vague compactness and metrizability questions We again consider a locally compact space E along with its space &+ = .,a'+(E) of Radon measures, equipped with the vague topology. Our interest here is in the subsets of ..41+ which are compact or relatively compact in this topology. They are naturally called vaguely compact, resp., vaguely relatively compact. A necessary condition for the vague relative compactness of a set H C -W+ can be inferred at once from the very definition of the vague topology. According

to it, for each f E Cc(E) the real function p H f f dµ is continuous on W+. Therefore the image of any relatively compact H under each such mapping must be a relatively compact subset of R, that is, a bounded set. This observation leads to the following definition: 31.1 Definition. A set H C ..&+(E) is called vaguely bounded (sometimes simply bounded) if (31.1)

sup

ffd µl

< +00

for every f E CA(E).

Thus vague boundedness of a set H C -4'+ is a necessary condition for its vague relative compactness. We want to show that it is also sufficient:

31.2 Theorem. A set H C 4f+(E) is vaguely relatively compact if and only if it is vaguely bounded.

Proof. In view of the preceding, all that has to be shown in that vague relative compactness follows from the vague boundedness of H. To this end, let of denote the real number in (31.1), for each f E Cc(E), and Jf the compact interval (-a f, a fJ in R. Also denote the (vague) closure of H in W+ by H. First observe that

fid AEJf for all f E CA(E) and all p E H. In fact, if f E CA(E) and e > 0 are given

Vf;e6a):={vE.A"+:I ffdv_ffd ul <e} is a vague neighborhood of p, so if p E H then H fl Vf;e(p) 34 0. For any v in this intersection, f f dv E Jf and therefore

ifll

r


As the extreme inequality holds for every e > 0, we see that If f dpi < a f, that is, f f dµ. E Jf.

§31. Vague compactness and metrizability questions

205

Now consider the product space

P:= RC = X Rl IEC,

in which for each f E C, = CA(E) a copy RI := R of the number line appears as a factor. The product

J:= X JI I EC

is a subspace of P which, as a product of compact spaces, is compact, by the famous Tychonoff theorem (KELLEY [1955], p. 139 or WRIGHT [1994]). To each

Radon measure p E .A/+ we assign the mapping f -r f f dµ of C,,(E) into R. This is a point in P. In this way a mapping

4':.l+-4P is defined which is injective by the Riesz representation theorem. On the basis of what was shown in the opening campaign

4;(H) C J. Our goal will be realized if we can show that (a) 4' maps .4f+ homeomorphically onto

and

(b) 4'(4'+) is closed in P. is also closed in P. From 4)(H) lyFor then 4)(H), as a closed subset of ing in the compact set J it therefore follows that 4'(H) is compact, hence too its homeomorphic image H. As to (a): Continuity of a mapping 4> into a product means continuity of every "component" of 4), that is, of each mapping It P- f f dp (f E CA(E)). But this is true right from the definition of the vague topology. Continuity of the mapping 4' inverse to 4' means continuity of each mapping

4'(u)'-Jfd(4i(4'()))

Jfdt

of 4>(.q'!.+) into R (f E C'(E)). But this mapping is just the restriction to 4)(..C/+) of the projection of P = RC, onto its coordinate specified by f.

As to (b): Let I E P be a point in the closure of 4'(..E'+) in P. Then I is a positive linear form on CA(E). To see its additivity, for example, let f, g E CA(E)

and E > 0 be given. The set of all I' E P which satisfy

II'(u) - I(u)I < E

for u E (f, g, f + g}

is a neighborhood of I in P, and therefore contains a point I' = 4>(p) from I' is thus the positive linear form

u H I' (u) = Judu

206

IV. Measures on Topological Spaces

on CA(E). That means that we have

II(f +g) - I(f) - I(g)I

II(f +g) - I'(f +g)I + II'(f +g) - I(f) - I(g)I

=II(f+g)-I'(f+g)I+II'(f)-I(f)+I'(g)-I(g)I <e+II'(f)-I(f)I+II'(g)-I(g)I <3c,

and because e > 0 is arbitrary, the extreme inequality means that its left-hand side must be 0. In a completely analogous way one proves that I (a f) = aI (f) for every a E R, f E CA(E), and I(g) > 0 for every non-negative g E CA(E). With the linearity of I confirmed, the Riesz representation theorem supplies a Radon I. That is, I lies in confirming that measure v E + such that the latter is closed in P. lJ 31.3 Corollary. For every real number a > 0 the set

9a:={pE..t+(E):IItzII
Proof. For every f E CC(E) and p E 4, if f dpi < f If I du <_ a IIf 11. Consequently, tf,, is vaguely bounded, hence vaguely relatively compact. What therefore remains to be confirmed is the closedness of via in .4W+. According to (28.13)

6 is just the set of all Is E W+ such that f u dµ < a holds for all [0,1]-valued u c- CA(E). Because the mapping p '-+ f u dtp of .'+ into R is continuous, the set {µ E - ' + : f u du < a} is closed, for each u E CA(E), and by the preceding observation 4 is an intersection of such sets, those for which u(E) C [0, 11. Thus .9a is indeed (vaguely) closed. 0 Remark. 1. The set of all measures u E 4' (E) with IIpQ equal to a fixed positive number a is vaguely closed if E is compact (because in that case 1E E CA(E)). Example 2 of §30, with all the a there equal to a, illustrates this.

For a variety of applications it is important to know when, in terms of E, the vague topology of 4+(E) is metrizable. One reason is that sequences suffice for dealing with metric topologies, but generally not for non-metric ones. The following remark will prove useful in answering this question. Remark. 2. For every locally compact space E the, obviously injective, mapping (31.2)

V : E -+ .4f+ (E)

defined by V(x) := ex is a homeomorphism of E with cp(E) _ {ey : x E E}. For every point x E E the (open) sets Mf...... f..:n(x) = {y E E : If,(x) - f;(y)I < 17,,7 = 1,...,n}

form a neighborhood basis at x as the fj run through all finite subsets of CA(E) and 17 through all positive real numbers. In fact, if U is a neighborhood of some

§31. Vague compactness and metrizability questions

207

x E E, 27.3 furnishes a u E CA(E) with 0 < u < 1, u(x) = 1 and supp(u) C U, which implies that C U. Using the notation (30.5) it is obvious that = V(E) n Vf...... J..;,&.)

for all relevant functions, q E R+ and x E E. Together with the injectivity this clearly shows that cp is a homeomorphism.

As a result of the foregoing, the metrizability of the locally compact space E is clearly a necessary condition for the metrizability of the vague topology on .41+(E).

For the former the existence of a countable basis in E is sufficient, as was noted in Remark 5 of §29. It is useful to formulate this in terms of CA(E): 31.4 Lemma. For any locally compact space E the following assertions are equivalent:

(a) E has a countable basis. (b) There is a countable subset of CA(E) which is dense with respect to uniform convergence.

Proof. (a)=::-(b): Let 9 be a countable base for (the topology of) E,.? the set of all open intervals in R with rational endpoints. For every natural number n let us say that an n-tuple (C1,... , Gn) E 1n and an n-tuple (II, ... , In) E Mn are compatible with each other if a function f E CA(E) exists such that f(G,) C II

for each j = 1,...,n and supp(f) C Gl U ... U Gn. Any such f will be called a compatibility function for the pair of n-tuples. Obviously, the set

U(9" x,1n) nEN

is countable; there are therefore only countably many such pairs of n-tuples (n E N) that are compatible with each other. We choose a compatibility function for each

such pair and designate by F the set of functions chosen. It suffices to prove that F is a countable dense subset of CA(E). To prove its denseness, let u E CA(E) and e > 0 be given. Denote the support of u by K. Every x E K lies in an open neighborhood from 9 each point y of which satisfies Iu(x) - u(y) I < E. The compact set K is covered by finitely many such neighborhoods, say by C1,.. . , Gn. The diameter of each image set u(G,) is at most 2E. Consequently there are intervals I j E 9 of length less that & such that u(G3) C II, f o r j = 1, ... , n. Thus u is a compatibility function f o r the pair of n-tuples (G 1 i ... , G"), (I1, ... , In ). Hence there must also be such a compatibility function f in the representative

set F. Every X E Gj therefore satisfies Iu(x) - f(x)I < .A'(Ij) < 3e; that is, Iu(x) - f (x)I < 3e for all x E G1 U ... U Gn. But this latter inequality prevails as well for all x E E \ (G1 U ... U Gn) for the simple reason that both f and u vanish identically in this complement. In summary, llu - f II < 3F. This proves that F is dense in CA(E). (b)=*(a): Let D be a dense subset of Cc(E). We will show that the system 9 of all sets {u > 1/2} with u E D is a base for the topology of E. For every open U C E and every point x E U Corollary 27.3 furnishes an f E CA(E) with f (x) = 1

208

IV. Measures on Topological Spaces

and supp(f) C U. Since D is dense, there is a u E D with 1$u - f O < 1/2. Then

xE{u>1/2}C{f> 0) C supp(f) C U. If D is countable, so is If.

O

Remark. 3. It is easy to show directly that (b) implies the metrizability of E. To this end, let D be a countable dense subset of Ce(E). Now (cf. Corollary 27.3)

CA(E) separates the points of E, so D must also; that is, for any two distinct points x, y E E there is a u E D with u(x) 96 u(y). The functions in D \ {0} may be organized into a sequence ul, u2.... and we may then define (31.3)

1un(x) -'uw(y)1

d(x, y) :_ n=1

X, Y E E.

2" 11un11

Point-separation by D means that d(x, y) > 0 whenever x # y. All the other properties of a metric on E are obvious for d. This function d on E x E is a uniform limit of continuous functions and is consequently continuous. Therefore the topology generated by d, which we will call the d-topology, is coarser than the original topology of E. For any given point x E E and neighborhood U of x in the original topology of E there is, as was shown in the "(b)=(a)" part of the preceding proof, a u E D with

zEV:={u>1/2}CU.

This function u is however a u,,, so that by (31.3) u is d-continuous and V is d -open.

Therefore the d-topology is finer than the original topology of E. Consequently the two topologies in fact coincide. Now we can provide the final answer to the question posed after Remark 1.

31.5 Theorem. The following assertions about a locally compact space E are equivalent:

(a) .A+(E) is a Polish space in its vague topology. (b) The vague topology of 4+(E) is metrizable and has a countable base. (c) The topology of E has a countable base. (d) E is a Polish space. Proof. (a)=>(b): This follows from Definition 26.1 of a Polish space. In Remark 2 we learned that x o-4 ey is a homeomorphic mapping of E onto the subspace {e. : x E E} of all Dirac measures in .4'+(E). Since the property of having a countable basis clearly passes to subspaces, (c) follows. (c) .(d): This was shown in Remark 5 of §29. (d)*(a): Lemma 31.4 provides a countable Do C CC(E) which is dense in CA(E) with respect to uniform convergence. Furthermore, according to Example 2 of §29, E is countable at infinity, so that by 29.8 there is a sequence of compact

sets such that L. 1 E and every compact subset K of E satisfies K C L. for all but finitely many n. For each n E N choose an e,, E CA(E) satisfying 0 < e,, < 1,

§31. Vague compactness and metrizability questions

209

en(Ln) _ {1}. The subset

D:=Do

EDo,nEN}Ufen: nE N}

of CA(E) is still only countable and, of course, is dense in CA(E). Let d1, d2,... be an enumeration of its elements:

D={d,,:nEN}. Using this enumeration we define a mapping

e:

+x-&+-+ R+

by

(31.4)

e(µ, v) :_ E002-n min{1, I f do du - f do dvl },

µ, v E

n=1

All the properties of a metric save perhaps one are obvious for p. What needs checking is that µ = v follows from g(µ, v) = 0. In view of the uniqueness part of the Riesz representation theoremr this amounts to showing that from

J dodp=J dodo

for all nEN

follows the equality

f f dp =

J

fdv

for every f E C,(E).

So let us show this. Given f E CA(E) there is k E N such that

supp(f)CLkC{ek=1}. Further, given e > 0 there is u E Do with Ilf - ull < c, whence, since f = fek,

If - uekl < Eek.

(31.5)

Integration yields (31.6)

if

(31.61)

I ffdv_Juekdv l < F

J ek dv.

As the functions ek and uek are in D, the assumption that p(p, v) = 0 entails that their p- and the v-integrals coincide, and it follows that

Jfdi_Jfdu

2e l

<

J ek d,",

holding for every e > 0. That is, the desired equality f f dp = f f dv must hold. The next step is to show that the topology determined by P is none other than the vague topology. We will, to that end, make use of the fact that the sets defined in (30.5) are a neighborhood base at v E ..&+ in the vague

210

IV. Measures on Topological Spaces

topology, when all possible finite subsets {fl,..., fn} of C0(E) and all numbers e > 0 are considered. We will denote by Ue (v) the open ball of center v and radius e

with respect to the metric p. 1. Given e > 0 there exists m E N such that Vd,..... dm;e/2(V) C UU(V)

(31.7)

for every v E .4'+.

Indeed, one may take any m E N such that 00

E 2-n < e/2 n=m+1

and every le E Vd,..... d,,,;e/2(V) will then satisfy in E2-n

p(µ, V) <

+<e

n=1

and consequently lie in UE(v).

2. For finitely many f1,..., fn E CC(E), for every number e > 0 and every v E 4'+, there is a number i > 0 such that (31.8)

Un(v) C V11,---.fn;-(V)

First of all, choose k E N so that n U supp(fj) C Lk C {ek = 1}. j=1

We can find a number 8, dependent on v, so that

0<8<1 and b2+(1+2fekdv)8<e. For each j there is a function uj E Do with II fj - uj II < 6, hence with Ifj - ujekl !5 bet,

(j=1,...,n).

Integration with respect to v and any u E _W+ gives (31.9)

Jf)dL_Juiekd1zl<SJekdµ,

fjdv-

(31.9')

ujekdvl
if

show for j = 1, ... , n. Choosem so large) that all the functions ek, u1ek.... up among the first m functions dl,..., d,,, in the enumeration of D, to which they

all belong. Finally, set

,7] ._ d2-m

and consider any li E .,,v). It satisfies

2-'min{1,l fd;dle - fd;dvl}
§31. Vague compactness and metrizability questions

211

whence, since b < 1

if

p

for i = 1, ... , m.

Because of the way m was chosen (31.10)

for j=1,...,n

if

and

Jekd/L_fekdP
(31.10')

From (31.9) and (31.9'), as well as from (31.10) it follows, via the triangle inequality that

ffjd_Jfjdv l < (1+J ekdµ+J ekdv)b; while from (31.10')

J

ekdp <6+ / ekdv, J

so the preceding implies that

Jfid_ffidv<82+(1+2 J edv)S<eAs this holds for every j E{ 1, ... , n}, it asserts that p E V11 ,... j,,, (v) and confirms (31.8). Together (31.7) and (31.8) assert the equality of the vague and the p-topologies.

The next step will be to prove the completeness of the metric p, and we can do that via slight modifications in the foregoing arguments. Let (pn)nEN be a pCauchy sequence in W+. Instead of the functions fl,..., fn and the number e > 0 in 2. above, let an f E CA(E) and a number b E ]0, 1[ be given. We aim first to prove that the numerical sequence (f f dpn)nEN converges in R. Choose k E N with supp(f) C {ek = 1) and u E Do with Ilf - It < b. Then choose m E N large enough that the two functions ek and uek are among dl, ... , d,n and set 17:= 62-1. Since (µn) is a p-Cauchy sequence, there is a natural number N, dependent on 'q, thus on f and S, such that for all r, s > N. p(pr, ps) < 77 Just as in the earlier deduction scheme, we get that for such r, 8

for all i E {1,...,m}, which contains in particular the inequalities (31.11)

if

I

< 6 and

JekdPr_JekdPa < 6.

if

I

212

IV. Measures on Topological Spaces

Of course we also have the f-analogs of (31.9) and (31.9'), so that reasoning similar

to that used earlier deliversthe inequality

for all r, s > N.

if

The second of the (valid for all r,s > N) inequalities in (31.11) shows that the numerical sequence (f ek d

EN is bounded, say by M E R+:

forallnEN. The earlier inequality therefore yields

Jfdpr_JfdP8<62+(1+2M)

for all r,s>N.

Notice that M depends only on k, hence only on f. Furthermore N depends only on b and f. Therefore this last inequality affirms that (f f dpn)nEN is a Cauchy sequence in R. According to the remark following Definition 30.1 the sequence (tin)

is therefore vaguely convergent to some p0 E .4'... Since the vague topology coincides with the p-topology, as we have already confirmed, this means that the sequence (pn) converges to po in the p-metric. We finally need to prove that, like the topology of E, the vague topology of ..k+ has a countable base. Since the vague topology is generated by the metric p, it is enough to find a countable set 9o which is dense in . W+; because it is obvious that the set of all open balls with respect to the metric p centered at points of 9o and having rational radii is then a countable base for the p-topology of . '... Our candidate for 9o is the set of all discrete measures k

b :_

aifx,

with positive rational ai and points ai drawn from a countable set Eo which is dense in E. We get such a set Eo simply by taking a point from each set in a countable base for the topology of E. Evidently, this 90 is countable. We have to show that for every p E . fl+, every real e > 0, and every finite set F :_ {fl,..., fn} C CA(E), the basic vague neighborhood Vj,,... contains a measure from 90. At least, according to 30.4, this neighborhood contains a

with positive real Ui and Ti E E. Thus (31.12)

ip- Jfdbl-l Jfd

k

<e i=1

for all f EF.

§31. Vague compactness and metrizability questions

213

Now for such f and d as above

if fdIt-Jfd.6l<

J

f

fdlt -

J

f &l +IJ fa-J fd6l k

k

fd,,- ffdal+Ea;If(=i)-f(xj)I+FI-;-ailIlfII

Inequality (31.12) says that the number EHJfd,1

-ffdbl

is positive. If we choose a; from Q+ sufficiently close to i and x; from the dense set E o sufficiently close to T, (i = 1, ... , k ), then because of the continuity of the (finitely many) functions f, we can obviously see to it that the two sums in (31.13) together are less than this, so that the right side of inequality (31.13) is less than e, for each f E F. But that means that b E 9o n V1..... f,,;, (It).

Remarks. 4. The reader should recall the rather elementary fact that for a metric space compactness and sequential compactness are equivalent (see (6.37) in HEwrrr and STROMBERG [19651). In view of this, a very useful consequence of Theorems 31.2 and 31.5 for a locally compact space E with a countable base is that every vaguely bounded sequence in _J!+(E) contains a vaguely convergent subsequence..

In particular, for such E every sequence (p,,) in ..#+(E), that is, every sequence of p-measures, contains a vaguely convergent subsequence. Moreover, in case all convergent subsequences have the same limit e, the original sequence (p,,) itself converges vaguely to /t: Otherwise there would be an f E CA(E) for which (f f dlt )

sloes not converge to f f dlt, and so an e > 0 and integers I < n1 < n2 < ... such

that If f dlt,,; - f f ditl > e for all j E N. The sequence

)jEN would have a vaguely convergent subsequence and its vague limit could not be iz. If we further that it is tight, then with the aid of Remark 3 in §30 we can hypothesize of even converges weakly to it. conclude that it E .W+(E) as well, and that

5. The foregoing deliberations show (for locally compact. E with a countable base) that tight sequences in &+'(E) always contain weakly convergent subsequences. Explicitly formulated this says: A set H C .,i.+ (E) is relatively compact (= relatively sequentially compact) in the weak topology if it is tight, meaning that for every e > 0 a compact Kf C E exists such that p(E \ KE) < e for every it E H. A theorem of Yu.V. PROHOROV asserts that the lightness of H is even equivalent to its weak relative compactness. More is true: This equivalence prevails as well whenever E is any Polish space. For details the reader can consult BILLINGSLEY [1968[.

214

IV. Measures on Topological Spaces

The ideas employed in the proofs of Theorems 31.4 and 31.5, slightly modified, lead to a further interesting result. It concerns the space

C := C(R+, E)

of all continuous mappings f of R+ := [0, +oo into a Polish space E, for example, Rd. We endow C with the topology of uniform convergence on compact subsets of R+.

31.6 Theorem. Along with E, the space C(R+, E) is also Polish. Proof. Consider any complete metric B which generates the topology of E. Another

such metric is given by (x,y) H min{1, p(x,y)}, and using it if need be, we can simply assume that L< 1. This lets us define do in C for each n E N by dn(f,g) := sup{p(f(x),g(x)) : x E [0, n]),

f,g E C;

and

(31.14)

d(f,g) :_

00

E2-ndn(f,g),

f,g E C.

n=1

Just as earlier (cf. (31.3) and (31.4)), one easily confirms that d is a metric on C (with all its values in [0,1]) which satisfies (31.15)

2-nd(f,g)
for allnEIN,

the right-most inequality following from the fact that d< < d,+1 for all i E N, resulting in n

00

d(f, g) 5 E 2-`dt(f, g) + E 2-'. i=1

i=n+1

It follows from (31.15) via by-now-familiar reasoning that the d-topology coincides with the original topology of C, and moreover that d is a complete metric.

So it only remains to prove that the topology of C has a countable base. As we showed in the very last phase of the proof of Theorem 31.5, the Polish space E contains a countable dense subset E0. The system 9 of all open balls with respect to the metric o with centers in Eo and with positive rational radii is then a countable base for E. Together with it we consider a countable base 0 for R+. Thus n-tuples (01, ... , On) E 0n and (G,,.. -, E [9n are called compatible if there is a function f E C such that f (O,) C G,, for each j = 1, ... , n. And, as before, any such f will be called a compatibility function. Because

U(®n

nEN

is countable, there is a countable set F C C which contains a compatibility function

for each pair of compatible n-tuples, for each n E N. The open d-balls having centers in F and rational radii are a countable set, and it is easy to see that they constitute a base for the d-topology of C once we confirm that F is dense in C.

§31. Vague compactness and metrizability questions

215

So that is now our goal. Consider then an arbitrary fo E C and N E N. Set

c:= 2-N-2. Since f is continuous, every x E [0, NJ lies in a set 0 E 6' such that for all y E 0.

p(fo(y), fo(x)) <,E/2

Finitely many such sets 0 suffice to cover [0, NJ, say 01,..., 0,,. By the triangle inequality

Q(fo(y),.fo(x)) < e

for all x, y E Oj, j E { 1, ... , n}.

Choose a point xj from each Oj. Then

p(fo(x),fo(.., j))<e

for allxEOj.jE

The open Lo-ball of radius c centered at fo(xj) meets the dense set E0, say in the point zj. As E is rational, the open p-ball of center zj and radius 2e is a set G j E I. Then every x E Oj satisfies

P(fo(x),zj) <_ P(fo(x),fo(xj))+P(fo(xj),zj) <2e, which means that fo(Oj) C G j, all this for each j E { 1, ... , n}. This shows that

fo is a compatibility function for (01,._O.) and

Consequently,

this pair of n-tuples has a compatibility function f E F, that is, f E F satisfies

f(0j)CGj

forj=1,....n.

It follows that f(x), fo(x) both lie in Gj whenever x E Oj and so e(f (x), fo(x)) < 4c.

As the Oj cover [0, NJ, this inequality holds for every x E [0, NJ. It affirms that dN(f, fo) < 4E, and so thanks to (31.15) and the definition of e, d(f, fo) < 4E + 2-N = 2-N+'. As N E N is arbitrary, this shows that F is d-dense in C, which, as noted earlier, completes the proof. The significance of Theorem 31.5 lies partly in the fact that for a locally compact space E whose topology has a countable base the space .41+(E) of all (positive) Radon measures - which according to 29.12 is the set of all Borel measures on E - being also a Polish space, is itself an environment in which measure theory can be pursued. And this happens in convex analysis, in integral geometry, and in stochastic geometry, a meeting point between geometry and probability theory.

The path-space C(R+, E) of all continuous paths or curves t H f (t) 1 t E R+, in a Polish space E (Theorem 31.6) plays a fundamental role in the theory of stochastic processes. For example, the Polish space C(R+, Rd) carries the famous Wiener measure; it is the steering mechanism of the Brownian motion in Rd (cf. BAUER [1996]).

Exercises. 1. Let E be a locally compact space, v E ..#+(E). Show that the set of all p E ..#+(E) which satisfy 0 <_ f u. d,u < f udv for every non-negative u E CA(E) is vaguely compact.

216

IV. Measures on Topological Spaces

2. Let E be a locally compact space with a countable base. Prove that there is a countable subset of C0(E) that has the properties of the set T in Exercise 3, §30. [Hint: Try the set D that featured in the proof of Theorem 31.5.] 3. (Selection theorem of E. HELIX (1884-1943)). Prove the original form of Corollary 31.3: To every sequence (Fn)nEN of distribution functions on R corresponds a measure-generating function F : R -+ R and a subsequence (Fn,, )kEN of the original sequence such that lim Fnk (x) = F(x) for every continuity point x of F. k-roo

Why is F generally not a distribution function? How does one recover 31.3 (for the case E := R) from Helly's theorem? 4. For a Polish space E consider the topology (introduced in Remark 7 of §30) of weak convergence on the set of finite Borel measures (the finite Radon measures - cf. 26.2) on E. By adapting the ideas in the proof of Theorem 31.5, show that this topology is metrizable. 5. For what more general spaces taking over the role of R+ in the definition of C(R+, E) does Theorem 31.6 remain valid?

Bibliography

e_ls dr", Bull. Sci. Math. (2)13, 84. U. ANONYME [1889]: "Sur l'integrale JIx G. AUMANN [1969]: Reelle Funktionen. Grundlehren Math. Wiss. 68 (2nd edition), Springer-Verlag, Berlin-Heidelberg-New York. S. BANACH [1923]: "Stir le problenne de la mesure", Fund. Math. 4, 7-33.

R.G. BARTLE and J.T..JoicHI [1961]: "The preservation of convergence of measurable functions", Proc. Amer. Math. Soc. 12, 122-126. H. BAUER [1984]: Mafle auf topologischen Raumen, Kurs der FernuniversitatGesamthochschule-Hagen. - 11996]: Probability Theory, de Gruyter Stud. Math. 23. Walter de Gruyter. Berlin-New York. S.K. BERBERIAN [1962]: "The product of two measures", Amer. Math. Monthly 69, 961-968. P. BILLINGSLEY [1968]: Convergence of Probability Measures. John Wiley & Soils,

Inc., New York-London-Sydney-Toronto. G. BIRKHOFF and S. MACLANE [1965]: A Survey of Modern Algebra (3rd edition). The Macmillan Co., New York.

N. BOURBAKI [1965]: Integration, Chap. 1-4. Hermann, Paris. A. BROUGHTON and B.W. HUFF [1977]: "A comment on unions of a-fields", Amer. Math. Monthly 84, 553-554. S.D. CHATTERJI [1985-86]: "Elementary counter-examples in the theory of double integrals", Atti Sem. Mat.. Fis. Univ. Modena 34, 363-384. G. CHOQUET [1969]: Lectures on Analysis. Vol. 1. W.A. Benjamin, New YorkAmsterdam. .I.P.R. CHRISTENSEN [1974]: Topology and Borel Structure. Mathematical Studies 10. North-Holland Publ. Co., Amsterdam-London. D.L. COHN [1980]: Measure Theory. Birkhauser Verlag, Basel-Boston-Stuttgart. P. COURREGE [1962]: Theorie dc la mesue. Les cours de Sorboune. Centre de Documentation Universitaire, Paris 5'. C. DELLACHERIE et P.-A. MEYER [1975]: Prnbabilites et potentiel, Chap. I a IV. Hermann, Paris. P. DIEROLF and V. SCHMIDT [1998]: "A proof of the change of variable formula for d-dimensional integrals", Amer. Math. Monthly 105, 654-656. J. DIEUDONNE [1939]: "Un exemple d'espacc normal non susceptible dune struc-

ture uniforme d'espace complet", C. R. Acad. Sci. Paris Ser. I Math. 209, 145-147.

218

Bibliography

- [1978]: Abreg6 d'Histoire des MathEmatiques, 1700-1900, tome II. Hermann, Paris. E.B. DYNKIN [1965]: Markov Processes, I, II. Grundlehren Math. Wiss. 121, 122. Springer-Verlag, Berlin-Heidelberg-New York. R.E. EDWARDS [1953]: "A theory of Radon measures on locally compact spaces", Acta Math. 89, 133-164. B.W. GNEDENKO [1988]: The Theory of Probability (translated from Russian by G. Yankovsky) 6th printing. Mir Publishers, Moscow. C. GOFFMAN and G. PEDRICK [1975]: "A proof of the homeomorphism of Lebes-

gue-Stieltjes measure with Lebesgue measure", Proc. Amer. Math. Soc. 52, 196-198.

H. HAHN and A. ROSENTHAL [1948]: Set Functions. The University of New Mexico Press, Albuquerque. P.R. HALMOS [1974]: Naive Set Theory. Undergrad. Texts Math., SpringerVerlag, New York-Heidelberg. - [1974]: Measure Theory. Grad. Texts in Math. 18, Springer-Verlag, New YorkHeidelberg-Berlin. F. HAUSDORFF [1914]: Grundziige der Mengenlehre. Verlag von Veit and Comp., Leipzig; reprinted (1949), Chelsea Publishing Comp., New York. T. HAWKINS (1970]: Lebesgue's Theory of Integration. University of Wisconsin Press, Madison-Milwaukee-London. J. HENLE and S. WAGON [1983]: "A translation-invariant measure", Amer. Math. Monthly 90, 62-63. E. HEwITT and K.A. Ross [1979]: Abstract Harmonic Analysis I. Grundlehren Math. Wiss. 115 (2nd edition). Springer-Verlag, Berlin-Heidelberg-New York. E. HEwITT and K. STROMBERG [1965]: Real and Abstract Analysis. Grad. Texts in Math. 25. Springer-Verlag, New York-Heidelberg-Berlin. J.L. KELLEY [1955]: General Topology, Grad. Texts in Math. 27. D. Van Nostrand Co., Inc. Princeton; reprinted (1975), Springer-Verlag, New York-HeidelbergBerlin. L. MATTNER (1999]: "Product measurability, parameter integrals, and a Fubini counterexample", Enseign. Math. (2) 45, 271-279. P.-A. MEYER [1966]: Probability and Potentials. Blaisdell Publ. Comp., Waltham, Massachusetts-Toronto-London. L. NACHBIN [1965]: The Haar Integral. The University Series in Higher Mathematics. (Translated from Portugese by L. Bechtolsheim.) D. Van Nostrand Co., Inc. Princeton; reprinted (1976), R.E. Krieger Publ. Comp., Huntington, New York. J. VON NEUMANN (1929]: "Zur allgemeinen Theorie des MaBes", Fund. Math. 13,

73-116+333. W.P. NOVINGER [1972]: "Mean convergence in Lp-space", Proc. Amer. Math. Soc. 34, 627-628.

Bibliography

219

D.A. OVERDIJK, F.H. SIMONS and J.G.F. THIEMANN [1979]: "A comment on unions of rings", Indag. Math. 41, 439-441. J.C. OXTOBY and S. ULAM [1941]: "Measure-preserving homeomorphisms and metrical transitivity", Ann. of Math. (2) 42, 874-920. K.R. PARTHASARATHY [1967]: Probability Measures on Metric Spaces, Academic Press, New York-London. W.F. PFEFFER [1977]: Integrals and Measures. Marcel Dekker. New York-Basel. J. RADON [1913]: "Theorie and Anwendungen der absolut additives Mengenfunktioncn", Sitzungsber. Kaiserl. Akad. Wiss. Wien, Math.-NaturYaiss. K1. 122, 1295-1438.

H. RICHTER [1966[: Wahrscheinlichkeitstheorie. Grundlehren Math. 1Viss. 86 (2nd edition). Springer-Verlag, Berlin-Heidelberg-New York. F. R.IESZ [1911]: "Sur certaines systemes singuliers ('equations intrgrales", Ann. Sci. Ecole Norm. Sup. (3) 28, 33-62. J.B. ROBERTSON [1967]: "Uniqueness of measures", Amer. Math. Monthly 74, 50-53. W. RUDIN [1962]: Fourier Analysis on Groups. Interscience Tracts in Pure Appl. Math. 12. John Wiley & Sons, New York-London. - [1987]: Real and Complex Analysis (3rd edition). McGraw-Hill Book Comp., New York-Hamburg-Tokyo--Toronto. S. SAEKI [1996]: "A proof of the existence of infinite product probability measures", Amer. Math. Monthly 103, 682-683. W. SIERPINSKI [1928]: "Un thboreme general sur les families d'ensembles", Fund. Math. 1, 206-210.

R.M. SOLOVAY [1970]: "A model of set-theory in which every set of reals is Lebesgue measurable", Ann. of Math. (2) 92, 1-56. R.H. SORGENFREY [1947]: "On the topological product of paracornpact spaces",

Bull. Amer. Math. Soc. 53,631-632. S.M. SRIVASTAVA [1998]: A Course on Bore! Sets. Grad. Texts in Math. 180. Springer-Verlag, New York-Berlin.

K. STROMBERG [1972]: "An elementary proof of Steinhaus's theorem", Proc. Amer. Math. Soc. 36, 308. - [1979]: "The Banach-Tarski paradox", Amer. Math. Monthly 86, 151-161. - [1981]: An Introduction to Classical Real Analysis. Wadsworth International, Belmont, California.

H.G. TucKER [1967]: A Graduate Course in Probability. Academic Press, New York-San Francisco-London. J. VAN YZEREN [1979]: "Moivre's and Fresnel's integrals by simple integration", Amer. Math. Monthly 86, 691-693. D.E. VARBERG [1971]: "Change of variables in multiple integrals", Amer. Math. Monthly 18, 42-45.

220

Bibliography

S. WAGON [1985]: The Banach-Tarski Paradox. Encyclopedia Math. Appl. 24. Cambridge University Press, Cambridge. S. WILLARD (1970]: General Topology. Addison-Wesley Publishing Co., Reading, Massachusetts. J. YAM TING Woo [1971]: "An elementary proof of the Lebesgue decomposition

theorem", Amer. Math. Monthly 78, 783. D.G. WRIGHT [1994]: "Tychonoff's theorem", Proc. Amer. Math. Soc. 120, 985-987.

Symbol Index

The numbers beside the symbols refer to the pages where the symbol in question is defined.

C, u,n u, n, c, \, xii 0,33 -00, (+)oo, xi

f * v (convolution of a function and a measure), 149

If < g}, If < g}, If = g}, { f 76 g},

If >g}, If > 0, 50

IR, xi

N, Z, Q, R, xi, Z+, Q+, R+, R+, xi

f f du, f f (w)A(dw), f f (w) da(w), f u dµ, 55 58 64

R+, 141

f f dF, 65

R" (multiplicative group R \ {0}), 44

fA f dµ, fB f

R,., 156 Qd, 45.

T (unit circle [torus]), 38

(X)

dx, fa f dAd, f f d4

67 90

f If, 6D fnTf, 00 F fn, E fn, xiii

n=1

a
0,11 0,i, 0,1, 38 39 avb, aA xi a-, a+, xi

lim sup An, lim inf An, 61 n-+00

n-4o0

limsup fn, liminf fn, xiii lim fn, xiii n--,oo n = (n,.. . , n) E Rd (usually n E Z), 23,32

(an)nEN, (a.).=j.2...,, Xii an I a, an T a, xiii

91

(ai)iEJ, Xli d(x, A), 157, 201

sup fn, inf f, xiii supp(f) (support of a function), 167

det T, 43

supp(p) (support of a Borel measure),

f: A -. B, x H f (x) (mapping), xii f I A' (restriction of a mapping), xii f-1(B') (pre-image), xii fA, 62

8Q, 1,36

177

ix - yl (euclidean norm), 146 (x, y) (euclidean scalar product), 41

f";, 137 F,, 31

u 12

Ilf il (supremum norm), 169. 183

t (topological interior), 182 A* (topological boundary), 198

Ilflip, Ilflloo, 86, 8-2

f-, f+, [LL, 53 f, 86 fog (composition of mappings), xii f = g (ti-)almost everywhere, 70

f +g, fg, af, xii, 66 f * g (convolution of functions), 150

A (topological closure), 17

An, 147 1A (indicator function), 49

a+A, A+a, 36 A:= B, B =: A, xi AC B, xii A\ B, xii

222

Symbol Index

A - A (algebraic difference), 1.63

.2l'(µ), 1 < p < +00, 71

AD B,5

Y°°(µ), 78

C(E), Cb(E), Co(E), C°(E), 167,169 C(R+, E), 214

.4V+ (Rd), L47

Dye, 44

..41+(E), .4l+(E), _W+1 (E) (spaces of Radon measures), 188

E = E(1l, sd), 53

,,,Y (negligible sets), 86 87 100 .N,, 13,106, 107

E`(1,d), 58

® (open sets), 152

E, 10 E. T E. G:= f e-` )'(dx), 88. 93, 145. 146

..(St) (power set), xii

Dai), 31

GL(d,R), 43 H,. (homothety), 37 Ij,,171, K,.(xo) (closed ball), 146 K,.(xo) (open ball), 158 L'(Ad), 151, 123 L°°(µ), LP(µ), 86, 81 L3 (lower sum), 91

M(sd), 120 Mot(] d), 42

N,(f),Np(f),87,74 Q., Q.,, 135 S(i'k) (skewing transformation), 30 S,,(0) (euclidean sphere), 37 SL(d, R), 43 Ta (translation), 36. 149 T(µ) (image measure), 36

T-'(sd), 3

(x,27 o(cia), 7 Odd

Qa

...

5:,

F, 32

e,, eZ (unit point mass, Dirac-measure), 8 154 (counting measure), 12, 13 Ad (d-dimensional L-B measure), 18, 26, 27 Ad (d-dimensional L-B measure on C), 27

(total mass), 14L 1.54 µ., 171

9 171 µ° (principal measure), 176 µO (essential measure), 176 µA (restriction of µ), 68 µF, 30 µ- lim (stochastic limit), 113

U3 (upper sum), 91

µ-v,1.0Z IKv,P Lv,1455

Sv-, 170

µ1 ®µ2, 1M ®n 11+j, 143 v convolution of measures), 147

a(-s.P-measurable, 34

®.di = e®®... ® dd, (product of i=1

a-algebras), 132 .Vd (Borel a-algebra in Rd), 27

4'

= V(K), 49, 1553

.i(E) (Borel a-algebra in E), 152 (systems of closed, open, compact subsets of Rd), 27

`6'd, Cd,

.

9,,, 206

jd

14

.,1E', 13. 171

.2'(µ), 6fi

**µn,147

P(S), 4 P."W ), 171

e(x, y) (euclidean metric), 41

P+,r, M2 o'(8), Q(T), a(T1,... ,T,),

o(Ti:iEI),3 35 62

(Q, d) (measurable space), 34 (S1,.,, µ) (measure space), 34

0 7-119,*,µi), 1.44

S 1' 0 sd (trace of a-algebra), 2

Name Index

Alexandroff, Pavel Sergeevich (1896-1982),167 Anonyme, U., 25 Aumann, Georg (1906-1980), 46 Banach, Stefan (1892-1945), 46 Bartle, R.G., 119 Bauer, Heinz, 130, 144. 187, 193, 215 Berberian, S.K., 141 Billingsley, P., 201, 213 Birkhoff, Garrett (1911-1996), 44 Borel, Emile (1871-1956), ix, 18 Bourbaki, Nicholas, 87, 157, 186, 187 Broughton, A., 5

Caratheodory, Constantin (1873-1950), 20 Cauchy, Augustin Louis (1789-1857), ix

Chatterji, S.D., 141 Choquet, G., 47 Christensen, J.P.R., 47 Cohn, D.L., 92,157, 181 Courrisge, P., 116 Dellacherie, C., 130 Dieroif, P., 43 Dieudonn6, Jean (1906-1992), 18 1.84

Dirichlet, Peter Gustav Lejeune (1805-1859), ix Doob J.L., 62 Dowker, Clifford Hugh (1912-1982), 181

Dynkin, E.B., 5

Edwards, Robert Edmund (1926-2000), 181

Fatou, Pierre (1878-1929), 80 Fischer, Ernst (1875-1956), 84 Fubini, Guido (1879-1943), 138 Gnedenko, Boris W. (1912-1995), 33 Goffman, C., 4Z Hahn, Hans (1879-1934), 108, 141 Halmos, Paul R., 45 141. 177. 184 Hausdorff, Felix (1868-1942), 46 Hawkins, T., 18 Helly, Eduard (1884-1943), 216 Henle, J., 39 Hewitt, Edwin (1920-1999), 46, 92, 144, 147, 181, 213 Holder, Otto (1859-1939), 75, 78 Huff, B.W., 5

Joichi, J.T., 119 Jordan, Camille (1838-1922), ix Kelley, John Leroy (1916-1999), 1Z, 152, 158, 159, 166, 168, 201., 205

La Vallee Poussin, Charles de (1866-1962), 130 Lebesgue, Henri Loon (1875-1941), ix, 18.

Levi, Beppo (1875-1961), 59 Lindelof, Ernst Leonhard (1870-1946), 160

Lusin, NikolaT Nikolaevich

(1883-1950),163 MacLane, Saunders, 44 Mattner, L., L41 Meyer, P: A., 130 Minkowski, Hermann (1864-1909),75 83

Egorov, Dmitrii Fedorovii` (1869-1931), 120

Nachbin, Leopoldo (1922-1993), 39

224

Name Index

Neumann, John von (1903-1957), 46, 105

Nikodym, Otto Martin (1888-1974), 105

Novinger, W.P., 82

Overdijk, D.A., 5 Oxtoby, John Corning (1910-1991), 4Z

Parthasarathy, K.R., 201 Peano, Giuseppe (1858-1932), ix Pedrick, G., 47 Pfeffer, W.F., 184 Prohorov, Yurii Vasil'evich, 213

Solovay, R.M., 45

Sorgenfrey, Robert Henry (1915-1996), 156 Srivastava, S.M., 47. 165 Steinhaus, Hugo (1887-1972), 162 Stieltjes, Thomas Jan (1856-1894), 32 Stromberg, Karl R. (1931-1994), 46.92 144162, 181, 213

Thiemann, J.G.F., 5 Tonelli, Leonida (1885-1946), 95 138, 144

Tucker, H.G., 33 Ulam, Stanislaw Marcin (1909-1984),

Radon, Johann (1887--1956), 105, 155 Richter, Hv 33 Riemann, Georg Friedrich Bernhard (1826-1866), ix Riesz, Frigyes (1880-1956), 82. 81171 Robertson, J.B., 23 Rosenthal, Arthur (1887-1959), 141 Ross, Kenneth A., 46, 147 Rudin, Walter, 105, 147, 168 Sacki, S., L44 Schmidt, V., 43 Sierpinski, Waclaw (1882-1969), 5 Simons, F.H., 5

47

Urysohn, Pavel Samuilowich (1898-1924),158 Varberg, D.E., 45 Wagon, S., 39, 46 Willard, S., 17 152, 157-159, 166, 168, 201

Woo, J. Yam Ting, 105 Wright, D.G., 205

Yzeren, J. van, 95

Subject Index

fl-stable system, 7 U-stable system, 2 p-fold (p-)integrable, 7f p-measure, 31 p-space, 34

pth-power integrable, 76

p -measurable, 20 a-additivity, 8 a-algebra, 2 a-algebra of Borel sets in Rd, 27

-R 42

- topological space, 152 a-algebra generated by mappings, 35

Cl-diffeomorphism, 44 111

F,-set, 152 Ga-set, 47 152, 157, 1.59

K,-set, 187

29-convergence, 72 ."P-functions, 77 .`gyp-pseudometric, 79

2-semi-norm, 79 e-bound, 121

p-almost all points, 7Q p-almost everywhere, 70

- continuous, 128 - defined measurable function, 73 p-boundaryless, 19-8 p-completion, 26 p-continuous measure, 99 p-essentially bounded, 78

p-integrable over a set, 62 p-integrable function, 64 p-integrable set, 68 p-integral of function, 55, 5$, 64

-- by a set, 3 a-compact, 181 a-finite content, 23

a-finite measure, 23 72 28 a-finite measure space, 34 a-ideal, 13, 100, 1117

a-ring, 177 -- generated, 177 absolutely continuous (see p-continuous)

absolute value of function, 52 additivity, finite, 8

-,a-,8

- , sub-, 9 Alexandroff compaetification (see onepoint compactification) algebra, 1512 193

-,a-,2

- of sets, 4 almost everywhere, 70

- bounded,71

- over a set, 67, 62

- defined function, 73

p-negligible, 13 p-nullset, 13, 70 p-quadrable, 198 p-singular, 105 p-stochastically convergent, 113

- equal, 70 - finite, 74 analytic set, 47 antitone, xiii strictly, xiii

226

Subject Index

approximation by discrete measures,

continuous with respect to a measure,

193, 194

of the identity, 193 property, 24

99

continuous part with respect to u, 105 convergence almost everywhere, 70 almost uniform, 120

Banach algebra, L51 Banach space, 87 basis, base (topological), 157 Bernoulli inequality, 75 Borel Q-algebra, 27, 152 Borel (measurable) function, 50 1523 Borel measure, 311, 153

- , mean square, 80

- , regular, 154, 158, 184

convex cone, 188 convolution of functions, 151 functions and measures, 150

locally-finite, 154 bounded, 147 Borel set in Rd, 26

in mean, 80

in eh mean, 71, 114 124 in measure, 113 - , stochastic, 113 - , vague, 189

-

, weak, 196, 2011

- measures, 147

- - topological space, 152

convolution power, 151

boundaryless, 198 bounded Borel measure, 147

- root, 151

(z)-essentially, 78

Cantor discontinuum, 203 - , generalized 2113 carried by a set, 105 Cauchy criterion for stochastic convergence, 114 Cauchy sequence in Y P, 84 85 Cauchy-Schwarz inequality, 78 characteristic function, 44 charge distribution, 108 Chebyshev-Markov inequality, 112 Chebyshev inequality, 112 compatibility function, 207 complete measure, 26, 46 completeness of LP, 82 completion of a measure, 24 56 composition of functions, xii

- measurable mappings, 35 content, 8 content-problem, 411

continuity at 111 continuity from above, 10 - from below, 10 continuity lemma, 88

unit, 142 countable additivity (see Q-additivity) countable at infinity, 181 countable and co-countable o'-algebra, 2

countable (neighborhood) base, 157 countable set, xii counting measure, 12

density of a measure, 96

denseness of C, in 2, 186 denseness of discrete measures, 1194 diffeomorphism, 44, 111 difference set-theoretic, xii

- , symmetric, 5, 14 24, 87 differentiation lemma, 82 Dirac function, 146 Dirac measure, 12. 154 Dirichlet jump function, 57, 92, 166 disjoint sets, xii distribution function, 31, 201 dominated convergence theorem, 83 --- , sharpened version, 124 Doob's factorization lemma, 62 Dynkin system, 6 --- generated by ', 7

Subject Index

elementary content, d-dimensional, U. 16,27 elementary function, 53 envelope, lower, xiii

- , upper, xiii equi-(hµ)-continuity, 128 equi-p-continuous, 131 equi-continuous at 0, 131 equi-integrable, 121 if. essential measure, 176 extension theorem, 19 Factorization lemma, 62 family, xii

Fatou's lemma, 81

- , dual version, 130 figure, d-dimensional, 14

finite (or bounded) Radon measure, 188

finite additivity, 8 finite Borel measure, 3 147, 154 finite signed measure, 101

finite-co-finite algebra, 4 8 U Fubini's theorem, 13(1 function, additive, 59

- , antitone, xiii - , integrable of order p, 76 - , isotone, xiii - , Lebesgue integrable, 65 - , Lebesgue-Stieltjes integrable, 65 - , measurable, 34, 49 - , measure-generating, Q. 32 - , numerical, 49 - , positively homogeneous, 59

- , real, xii - , Riemann integrable, 91, 92

- , step, 53 - , with compact support, 167 Gaussian integral, 88, 93 general linear group, 43 generator, of a a-algebra, 3 -- , of a product a-algebra, 132 Haar measure, 39, 107

227

Hahn decomposition, 108, 109 Hilbert space, 87 Holder inequality, 75

- - , generalized, 78 - , reversed, 72 homothety, 37

hull, measurable, 25 ideal, 5 -- of 1A-null sets (see a-ideal) image measure, 3366 110

indicator function, 49 input-output formula, 13 integrable, 64 , equi-, 121

-

- , Lebesgue, 65 - , Lebesgue-Stieltjes, 65 -- quasi-, 64 - of order p, 76 - over a set, 69 integral of f exists, 65 integral over a set, 67 intervals in Rd, 14 isotone, xiii, 59 170 -- , strictly, xiii isotoneity, 9 Jordan decomposition, 109 L-B measure, 27 L-B measure space, 34 L-B-nullset, 28, 29, 33 Lebesgue decomposition, 105 Lebesgue integrable function, 65 Lebesgue integral, 65 Lebesgue measure, 46 Lebesgue-Borel measure (see L-B measure) Lebesgue-Stieltjes integral, 65 Lebesgue's convergence theorem, 83

- , sharpened version, 124 Lebesgue's decomposition theorem, 105,143 left half-open interval, 29 left-continuous, 30 lemma of Doob, 62

228

Subject Index

- Fatou's, 81 - Urysohn's, 168

- - , reversed, 79

--- Riecnann-Lebesgue, 202 --- on differentiation of integrals, 89 linear form, 66, 68

- , positive (isotone), 66, 171 Lusin's theorem, 10 mapping, xii mass distribution, 12, 108 measurable mapping, 34 measurable numerical function, 49 measurable sets, 34 measurable space, 34 measurable, Borel, 34, 103

- , Lebesgue, 46 with respect to an outer measure, 21)

negative part of a function, 53 - of a signed measure, 109 non-Borel set, 45 47 non-denumerable, xii norm of uniform convergence, 169 normal representation, 54 nullset, 13

- , L-B, 28 20 33, 43 - , Lebesgue, 46 totally, 1119

number line, xi

- , compactified, xi - , extended, xi

measure, 11 Borel, 31L 153

- , carried by a set, 105

one-point compactification, 167 outer measure, 20

finite, U finite signed, 11)2 inner regular, 1.54

point, ideal, 106 point, infinitely remote, 167 point mass (see Dirac measure) Polish space, 157, 208, 214 portmanteau-theorem, 197 positive part of a function, 53

L-B,27

-

motion, 41 motion group, 42 motion-invariance of ad, 42 motion-invariant content, 46 mutually singular (measures), 1.05

, Lebesgue, 46

- , locally finite, 153

-, of a set, 11 outer regular, 153 positive, 1519

- of a signed measure, 109

- , regular, 15.4 - , u-continuous, 99 - , a-finite, 23, 72, 98 - , signed, 102

positively-homogeneous function, 59 power set, xii, 2 pre-image, xii premeasure, 8

with density, 96 measuue-defining function, 311

measure-extension theorem, Q. 21 measure-generating function, Q. 32 measure space, 26. 34

- , a-finite, 34 metric of uniform convergence, 169 metrizability of locally compact spaces,

- , Lebesgue, 18 principal measure, 176 probability measure, 31 probability space, 34 product measure, 137, 143 product of measure spaces, 144

- of a-algebras, 132, 142 pseudometric, 79

208

- of vague topology, 208 Minkowski inequality,

70 83

Radon measure, 155 --- , bounded, 188

Subject Index

- , finite, 188 - , p-measure, 188 - , regularity of, 156, 161, 183 Radon-Nikodym density, 105 - integrand, 105 - theorem, 101

signed measure, 107 singular part of a measure, 105 singular, mutually, 103

- , to each other, 105 Sorgenfrey topology, 156

- integral, 91

Souslin (analytic) subset, 4Z' space, locally compact, 186 - , Polish, 157, 208, 214 special linear group, 43 square-integrability, 76 Steinhaus' theorem, 163 step function, 53 Stieltjes integrable, 65 Stieltjes measure function, 32 stochastic convergence, 113 stochastic limit, 112 subadditivity, 9 subsequence principle, 118, 120 subtractivity, 9 support of a function, 167

- - , improper, 92

- of a measure, 173

Riemann-Lebesgue lemma, 202 Riesz representation theorem, 171,

supremum norm, 189 symmetric difference, , 14, 24 87

reflection-invariance, 37 regular, inner, 183

- , outer, 154, 181 regularity of Borel measures, 184

- of L-B measure, 162 relatively compact, vaguely, 204

- , weakly, 213 representing measure, 173

- , essential 176 - , principal, L76 restriction of p, 19, 68 restriction of f, xii Riemann integrable, 91, 92

229

178, 185

right half-open interval, 14 ring (of sets), 4

- generated by intervals, 14

tensor product, 145 theorem of Caratheodory on outer measures 21

section of a function, 138

- Egorov, 120 - Fubini, 139

- of a set, 135

-Helly,218

semi-norm, 79

- Lebesgue, 83, 103 - Lebesgue-Radon-Nikodym, 1125

- of convergence in pth mean, 79

- Levi, 59 - Lusin, 183

-,2'-,79

sequence, xii sequentially compact, 213

- , relatively, 213 set, analytic, 47 - , Borel, 26, 49, 152, 172 - , difference, 183 - , Lebesgue measurable, 47 - , non-Borel, 45, 47 - of a-finite measure, 72, 175 - , (partially) ordered, xiii - , quadrable, 198 Souslin, 47

- Prohorov, 21.3 Radon-Nikodym, 194 F. Riesz, 82

- Riesz-Fischer, 84 - Steinhaus, 163 Tonelli, 13$ theorem on dominated convergence 83

- monotone convergence, 59 - partitions of unity, 167 tight, 197, 213 topological basis (base), 157

230

Subject Index

-- , countable, 157, 2174 208 topology, right sided, 157 vague, 192 weak, 211 total mass, 147. 154, L41 in vague convergence, 191, 195,19%

in weak convergence, 19& trace, 3

transformation theorem for general integrals, 111 for Lebesguc integrals, 111 transitivity of image measures, 36 translation-invariance of Ad, 36

translation-invariant measure, 4 39 ultimately all, xii uncountable, xii uniformly integrable, 122 uniqueness theorem, 22 unit mass at w, 8. 12 (see also Dirac measure)

Urysohn's lemma, 168

vague density of discrete measures, 193, 194

vague limit, 189 vague topology, 192 vaguely bounded, 204 vaguely compact, 204 vaguely convergent, 189

vanish at infinity, 10 vector space, 66,7,778 Z8 weak convergence, 196, 211 --- and distribution functions, 2(111 weak relative compactness, 213 weak topology, 201 Wiener measure, 216

zero-measure, 11

This book gives a straightforward introduction to the field as it is nowadays required in many branches of amtlysis and especially in probability theory. The flrst three chapters Measure Theory. Integration Theory. Product Measures) basically Follow the clear and approved exposition given in the authors earlier book on Probability Theory and Measure Theory'. Special emphasis is laid on a complete discussion of the transformation of measures anti integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon— Nikodym theorem. The final chapter. essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem. the

representation theorem, the Portimtiitcau theorcm. and a characterization ol locally compact spaces which are Polish, this Riesz

chapter is a true invitation to study topological measure theory.

'lie (ext addresses graduate students, who wish to earn the Fundamentals in measure and integration theory as netded in modern analysis and probability theory. It will also bc an important source for anyone teaching such a course.