VECTOR MEASURES A N D CONTROL SYSTEMS
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VECTOR MEASURES A N D CONTROL SYSTEMS
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NORTH-HOLLAND MATHEMATICS STUDIES
20
Notas de Matemstica (58) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Vector Measures and Control Systems
IGOR KLUVANEK and GREG KNOWLES The Flinders University of South Australia
1976
.
NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
0 NORTH-HOLLAND
PUBLISHING COMPANY
- 1975
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
North-Holland ISBN for this Volume: 0 7264 0362 6 American Elsevier ISBN: 0 444 1 1 040 2
Publishers : NORTH-HOLLAND PUBLISHING COMPANY
- AMSTERDAM
NORTH-HOLLAND PUBLISHING COMPANY, LTD. - OXFORD Sole distributors for the U.S.A. and Canada:
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
Printed in The Netherlands
Y
PREFACE These notes are the result of our effort to present in a systematic way the theory needed for investigating the range of a vector-valued measure. The inclusion of the term control systems in the title has two reasons. We are convinced that we are dealing with those parts of the theory of vector measures which will allow the extension into infinite dimensional spaces of the results obtained f o r finite dimensional linear control systems using finite dimensional vector measures. This extension is motivated by the desire to have the techniques described in the monograph of Hermes and LaSalle o r the article of C. Olech (both quoted in the Bibliography), available for control sytems governed by linear partial differential equations. The second reason for mentioning control systems is that we have included results about slightly more general objects than vector measures, We call these objects control systems as they serve as a suitable model for many control problems. ITe believe that these notes could also serve as an introduction to the general theory of vector-valued measures. Several aspects of the theory are missing, however, including chapters on construction of vector measures, Radon-Nikodym theory, representation of linear maps, etc. These are or will be covered by the works o f other authors who have the necessary expertise. In particular, we have learned that J. Diestel and J . J . IJhl are preparing a text where many subjects not treated here will be presented, From the many colleagues who have assisted us directly, o r indirectly, we would like to mention Peter Dodds. He discussed with us many aspects of the work, especially those involving order. While engaged in this work one of us (Knowles) was supported by a Commonwealth Post-Graduate Studentship, and later by a Flinders University Research Grant.
Igor Kluvinek Greg Knowles
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TABLE
I.
OF
CONTENTS
PRELIMINARIES
1. Locally convex spaces
1
2 . Extreme and exposed p o i n t s
4
3. Measure spaces
8
4. Conical measures Remarks 11.
14 16
1. Vector measures; v a r i a t i o n and semi-variation 2. Integration
16 21
3 . I n t e g r a b i l i t y of bounded f u n c t i o n s
26
4. L i m i t theorems
27
5. A s u f f i c i e n t condition f o r i n t e g r a b i l i t y
30
6. An isomorphism theorem
32
Remarks FUNCTION SPACES I
35 36 38
1. Topologies
38
2 . Some r e l a t i o n s between topologies
41
3 . Completeness
45
4. L a t t i c e completeness
49
5. Weak compactness
54
6. Completion
57
7. Extreme and exposed p o i n t s
59
8. Vector-valued functions
61
Remarks IV.
10
VECTOR MEASURES AND INTEGRATION
7. Direct sum of v e c t o r measures
111.
1
66
CLOSED VECTOR MEASURES
67
1. P r o p e r t i e s of t h e i n t e g r a t i o n mapping
67
2 . Closed v e c t o r measures
70
3. Closure of a v e c t o r measure
72
viii,
CONTENTS
1
V.
VI.
VII
VIII
.
.
4. Completeness of L (rn) 5. Lattice completeness 6. Weak compactness of t h e range 7. Sufficient conditions for closedness Remarks
73 74 75 78 80
LIAPUNOV VECTOR MEASURES
82
1. Liapunov vector measures 2. Consequences of the test 3. Liapunov decomposition 4. Moment sequences 5. Liapunov extension 6. Non-atomic vector measures 7. Examples of bang-bang control Remarks
82 85 88 89 93 94 98 110
EXTREME AND EXPOSED POINTS OF THE RANGE
112
1 . Extreme points 2. Properties of the set of extreme points
112 115
3 . Rybakov's theorem
120
4. Exposed points of the range Remarks
122
THE RANGE OF A VECTOR MEASURE
128
1. The problem 2. The conical measure associated with a vector measure 3 . The relation between rn and A h ) 4. Consequences of the test Remarks
128 130 134 137 139
FUNCTION SPACES I1
142
1. Set-valued functions 2. Measurable selections 3 . Sequences of measures 4 . Extreme points Remarks
142 145 148 152 153
127
CONTENTS
IX.
ix
CONTROL SYSTEMS
154
1. A t t a i n a b l e s e t
154
2 . Extreme p o i n t s of t h e a t t a i n a b l e s e t
156
3 . Liapunov c o n t r o l systems
158
4 . Non-atomic control systems
160
5. Time-optimal c o n t r o l
162
Remarks
165
BIBLIOGRAPHY
169
NOTATION INDEX
177
INDEX
179
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I , PRELIMINARIES
There is no pretence of a systematic o r complete presentation in this Chapter; it is meant to serve two purposes, The conventions and notations used throughout are fixed here. Also there are collected some results of direct relevance to problems treated in subsequent Chapters to facilitate reference. Hence the Chapter is meant to be use& as an Appendix, to be consulted only when needed.
1.
Locally convex spaces
All vector spaces used will be real vector space5,i.e. the field of scalars will be Ip, the real-number field. If X is a linear space, X* will stand for the space of all linear forms on X .
Given x*
E
X* and z
E
X , ( z * , x ) is the value of x* at x.
If X is a topological vector space then XI is the subspace o f X * consisting of all continuous linear forms on X . again, for the value of x'
E
X' at
We use, of course, the symbol
3: E
( X I ,
x)
X.
Only locally convex topologies wilI be used, The term "locally convex topological vector space" is abbreviated to 1.c.t.v.s. It is well known that the topology of such a space is given by a family P of semi-norms on X , in the sense that the family {x : p(xt.r
<
€ 1 , for every
E
>
0 and every p
E
P, is a
sub-base o f neighbourhoods of zero in P. I f P i s fundamental then this family forms a base of neighbourhoods. The family of all continuous semi-norms can be taken for P . A normed space is a 1.c.t.v.s. whose topology is given by
a
single semi-
norm which is separating, i.e. which is a norm. As usual, the norm of x denoted by llzll rather than p!x).
E
X is
A Banach space is a complete normed space. 1
2
LOCALLY CONVFX SPACES
If p is a semi-norm on a vector space X, then we put U = {x : x E Y , P p(x) i 11, and Uo = h* : x* E X * , /(x*,Z ) / S 1 , f o r every x E C } , If p is
P
P
a continuous semi-norm (in a given locally convex topology on X), then every element x*
E
~0
P
belongs to X'
,
In fact, a set F."
c
is equi-continuous if
X'
and only if there is a continuous semi-norm p on X such that W' A
c
Uo.
P topology on a 1.c.t.v.s. X is said to be consistent with the duality
between X andX' if an element x* only if x* belongs to X I .
E
X* is continuous in the topology if and
The weakest o f such topologies is called the weak
topology on X and is denoted by o f X , X ' I .
The strongest of them is called the
Mackey topology. I f X is a 1.c.t.v.s. and p a continuous semi-norm on Y , then p
Ix
:
x
X/p-'(O)
E
X , p(x) =
01
-1
(0)
is a closed vector subspace o f X. lrie denote by Y
the quotient space of X modulo p
-1
(0).
-1
Let
71
P
P
2
=
he the natural mapping
associating x k p ( 0 ) with any 3: E X . The semi-norm p induces P the norm x * p-l (O) * p ( x ) , x E Y , on X Then X becomes a normed space and P' P hence one can consider, say, the dual o f X the weak topology on X etc. P' P' of Y onto X
Y whose
THEOREM 1. Let W he a complete convex s e t in a l . c . t . v . s . topology is given by a famil-y P of sai-norms. TI
P
W ) is a weakly compact subset of X
If, f o r every p
E
P, t h e s e t
then I.I is weakly compact.
P'
This Theorem is stated only f o r the purpose of reference. Its proof is immediate from James' Theorem. However, such a deep theorem is not needed f o r the proof. Theorem 1 is an easy consequence of any weak compactness condition involving equi-coninuous families of linear functionals (e.g. r32! Theorem 17.12. (ii)). Let xn be an element of a 1.c.t.v.s. Y , for n = 1,2,. , the series l;=lxn is convergent and x
E
..
We say that
~ ~ , , z=n x, o r
X is its sum if limN+
if f o r every neighbourhood U of 0 there is a 6 such that
N
1,
xn
-x
E
U , for
PRELIMINARIES
1.1
3
every N > 6 . More g e n e r a l l y i f Wn
c
..
X , f o r n = 1,2,.
, we s a y t h a t t h e s e r i e s 1n=lWn i s
convergent i f l i = l x n is convergent f o r e v e r y c h o i c e of x
The s e r i e s of s e t s
I,,
m
1”n=13:n
n
E
Wn, n
1,2,.
.. We p u t
i s s a i d t o be u n c o n d i t i o n a l l y convergent i f t h e s e r i e s
{O,xnj i s c o n v e r g e n t .
This i s e q u i v a l e n t t o t h e e x i s t e n c e of an
element x such t h a t , f o r every neighbourhood U o f 0 , t h e r e i s a f i n i t e s e t of n a t u r a l numbers such t h a t ln,,xn numbers such t h a t
I
3
a f i n i t e set of i n d i c i e s c I c
I f Wi
c
K
c
E
X , i f , f o r e v er y neighbourhood U of
0 there is
E
X , for
-
I w i t h liE,xi
i
x
E
E
I.
The series
U , f o r ev er y f i n i t e s e t
I
such
I. X, i
THEOREM 2 .
p(&I-KWi)
of n a t u r a l
is said
E
I , we s a y t h e s e r i e s
convergent f o r ev e r y c h o i c e of x i
2.c.t.v.s.
I
1.Z E
Let x .
t o be convergent t o t h e sum x
K
U , f o r any f i n i t e s e t
E
K.
Let I be an i n d e x s e t .
that
-x
K
I f a series
E
liEIWi
Wi, i
1id W i E
I.
i s convergent if
1.2.6 f c i
is
We write
o f non-empty subsets Wi, i
E
I , of a
X converges then, f o r any continuous semi-norm p on X , l i m =
0,
where the l i m i t i s taken over the net of a l l f i n i t e subsets of
I ordered by inclusion. The proof of t h i s Theorem i s o m it te d as i t is o b t ai n ed by an easy ( al t h o u g h perhaps t e d i o u s ) argument of t h e 3~ t y p e .
( I f o r d i n a r y sequences a r e i n v o l v ed
see e.g. 1361.) LEMMA 1. L e t I and K be s e t s , X a 2.c.t.v.s.
k
E
K.
Let Wi = 1keKWik and l e t W =
liEIwi.
and Wik
Then W =
c
X, for i
lkEK& E I w i k .
E
I,
EXTREME AND EXPOSED POINTS
4
LEMMA 2 .
If
wi
c
X, i
E
I , are convex and
LEMMA 3 .
If
wi
c
X, i
E
I , are compact and
1.2
CiE1wi,
FI =
w
=
then
CiE1wi,
w
then
i s also convex.
w
is also
compact.
THEOREM 3 (Orlicz-Pettis). Let
x
A s e r i e s l;=lxn i s
be a 1.c.t.v.s.
weakly unconditionally convergent i f and only i f it i s unconditionally convergent i n any topology consistent with the duality between X and X A series IiEIWi,
where rJi
c
X, i
E
I .
I , i s weakly convergent i f and only i f
it i s convergent i n any topology consistent with the duality between Y and X I . The sequential part of this Theorem is classical, for Ranach spaces at least. The generalization represents no substantial problems.
2. Extreme and exposed points
If X is a l.c.t.v.s., A
c
X, we denote
the weak closure of A , coA
(resp. bcoA) the convex (resp. balanced) convex Iiull of A, =A
the closed
convex hull of A , and by exA the set of extreme points of A . A
point zo
E
A is called a strongly extreme point of A if xo is not in the
closed convex hull of A - U for any neighbourhood U of
TO.
The set of strongly
extreme points of A is denoted by st.exA. A point
such that
xo
( X I ,
E
A is called an exposed point of A if there exists an x '
xo )
<
(x', 20 ) whenever x
E
A and
x #
"0,
E
X'
The functional x' is
said to expose A at xo. The set of exposed points of A is denoted by expA. A point xo
E
A is called a strongly exposed point of A if x o is exposed
by a functional x' and if, ( 2 ' , x. )
z
implies that x
i
+
-+
xo ) f o r any net
(I!,
x o in the topology of Y .
in A ,
The functional x' is said to
strongly exposeX at z o . The set of strongly exposed points of A is denoted by
5
PRELIMINARIES
1.2
s t , expA. If A and B a r e non-empty compact, convex s u b s e t s of a 1 . c . t . v . s . X, we
t o be t h e s e t o f , a l l extreme p o i n t s x of A f o r which t h e r e e x i s t s
d e f i n e ex#
some extreme p o i n t y of B such t h a t x t y If a l i n e a r functional
point x o
A, i , e , ,
E
hyperplane Ix :
(XI,
( X I ,
“0)
XI
XO) =
=
( X I ,
ex(A
B).
t
achieves i t s maximal value i n a s e t 4 , a t a
sup(xfr A ) = s u p I ( x f , x ) : z
E
A ] , then t h e
xo ) > i s c a l l e d a supporting hyperplane of A .
xo i s c a l l e d a support p o i n t of
The p o i n t
E
A (and t h e hyperplane).
X, A i s a
LEMMA 1. Suppose B i s a closed, convex subset of a 1,c.t.v.s.
subset o f B, and A,B have the same supporting hyperplanes. and st.expB
c
Then expB
c
expA
st.expA.
Moreover, i f B i s weakly compact and exB
c
then A and B have the same
A,
exposed points. P r o o f.
I t i s c l e a r t h a t expB c expA a s A and I? have t h e same support-
ing hyperplanes. If b
E
st.expB then b
E
exp4 and a r o u t i n e argument shows t h a t b
E
st.expA.
Let a
E
expA
For t h e second p a r t i t s u f f i c e s t o show t h a t expA c expB.
and H be a supporting hyperplane t o A such t h a t H n A = { a ] . Then B is supported by H , s o B n H i s a non-void weakly compact, convex s e t and s o has From t h e assumptions, ex(B n IT)
extreme p o i n t s ,
A.
c
Thus ex!B n Hj
c
A n H =
{ a > , and so t h e Krein-Wilman Theorem gives t h a t B n H = { a ] . In o t h e r words a
E
expB.
Let A and B be non-empty compact convex subsets of a ~ . c . t . v . s .
THEOREM 1.
X.
If x
E
ex(A
that x = a + b.
t
B ) then there e x i s t s a unique a
Further a
E
exA and b
E
exB.
E
A and a unique b E B such
ConverseZy, i f an element x of
A + B has a unique representation i n the form x = a + b, a E A,
b
E
B , and i f
EXTREME AND EXPOSED POINTS
6
a
L
exA and b
exB, then x
E
P r o o f. a
f
al, b
f
ex(A t B )
E
Suppose x = a t b = al t b Then a t b
bl.
s i n c e a t b = al
+
b
1
1
1.2
a1
f
t
1
where a,ul
b , f o r , o t h er w i se, a
%(alt b ) , x cannot be an extreme p o i n t of A t B . E
A and a u n iq u e b
A , and b,b,
B and
b = al - b,,
-
Consequently i f z B w i t h x = a t b.
E
E
and
S i n c e x = % ( at b l ) t
we must have a = al and b = b,.
t h e r e must e x i s t a unique a
E
ex(A t B )
E
As
C451 § 2 5 . 1 ( 9 ) ) , t h e f i r s t p a r t of t h e Theorem
ex(A t B ) c ex4 t exB(e.g. f o l lows.
Conversely, suppose t h a t x = a t b , where a determined, and a
bl,b2
E
E
exA, b
B wi t h al t b
1
t
a
exB.
E
2
t
b
2
If z
a
E
A, b E B , are u n i q u el y
ex(A t B ) t h e r e e x i s t alJa2
E
A , and
and x = +(a, t b,) t +(a2 t b 2 ) . On r e a r r a n g -
i ng x = %(al t a,) t %(bl t b ), which i m p l i e s t h a t a = % ( a l t u 2 ) and 2
b = +(bl b
E
t
b 2 ) , s i n c e t h e r e p r e s e n t a t i o n of x i s u n i q u e.
exB we have a
Since a
E
exA and
= a2 = a and bl = b , = b , which g i v e s a c o n t r a d i c t i o n .
The c a n c e l l a t i o n law c o n t a i n e d i n t h e f o l l o w i n g Lemma was proved i n 1651, Lemma 2 , f o r t h e c a s e when X i s a Banach s p a c e .
The st at em en t h o l d s i n g e n e r a l
(with similar p r o o f ) a s p o i n t e d o u t i n 1281 Lemma 1. LEMMA 2 .
Let A and B be d o s e d , convex subsets of a 2 . c . t . v . s .
suppose there is a bounded subset C of X such that A THEOREM 2 .
I.c.t.v.s.
t
C = B
t
X, and
C. Then A
B.
If A and B are non-empty compact, convex subsets of a
X , then ex& i s dense i n exA.
P r o o f.
Let
c=
=(ex
8 ) . Then
ex(A
t h e Krein-Milman Theorem (C45]), A t B c C t B Hence, by Lemma 2 , A = C. (6)), i . e . ex#
t
B) c
A
c
C
t
B c A
+ B ; or
t
B , s o t h a t by
C t B = A t B.
I t now f o ll o w s t h a t exA = exC c cl ( ex $ l )
i s a dense s u b s e t o f ewl.
(1451 5 2 5 . 1
PRELIMINARIES
1.2
COROLLARY.
7
I f A,R are compact,convex subsets o f a 2.c.t.v.s.
ex(A t B ) i s closed, then exA and exB are c2osed and ex#
P r o o f. a
t b.
Consider the mapping t : A
x
B
+
A
t
= exA.
B d e f i n e d by t ( a , b ) =
S i n c e t i s c o n t in u o u s , t - l ( e x ( A t R ) ) i s cl o sed i n A Let P
compact.
A denote t h e p ro j ecti o n of A
PA(t-'(ex(A
t
B onto A .
x
B , and s o
By Theorem 1, ex@
B ) ) , and hence t h i s s e t i s compact, and s o c l o s e d .
From t h e
= exA.
i s d e n s e i n exA, and so ex$
Theorem, ex$
x
X, and
I f X i s a Banach s p a c e and K i s weakly compact, convex set i n X , we l e t
and we d e f i n e dK : X'
-+
IR t o be t h e map
K (2') = diamK,,
d L E l W 3.
x'
,
E
X' .
For any weakly compact convex subset K of a Banach space X,
the ;nap dK i s continuous at every x' i n X' which strong22 exposes K .
P r o o f.
Let x'
E
X' be a s t r o n g l y exposing f u n c t i o n a l o f K .
K (x') =
i s a s i n g l e t o n and s o d Then t h e r e e x i s t s a n
dK(2') t n
E ,
E
-
Suppose t h a t dK i s n o t co n t i n u o u s a t z ' .
> 0 and a sequence
f o r e v e r y n = 1,2,
i n Kx, s u ch t h a t Han
0.
b,"
Then K z l
... .
{x;} converging t o
2'
f o r which
Thus, f o r each n, t h e r e e x i s t s an and bn
1 % ~ .Since
K i s weakly compact t h e r e e x i s t s a
?l
subsequence {a,} ( r e s p . { b j } ) of { a n } ( r e s p . {b,}) converging weakly t o some 3 2'11 0 , we have p o i n t a ( r e s p . b ) i n K . S i n c e K is bounded and llx'
j '
x ) - (x', x ) 1
i
-+
-
-+
0 , o r o t h e r words, t h e sequence { x ' . ( K ) } of compact
3 i n t e r v a l s converges t o t h e compact i n t e r v a l x'(K) i n t h e Hausdorff metric on
t h e cl o s ed s e t s of H1.
S e t B . = s u p ( z ' K ) , and 8 = s u p ( x ' , K), and s o B 3 j ' j
-+
B.
8
I.3
MEASURE SPACES
Further,
..
since a + a weakly. But, f o r every j = 1,2,. , uj j Consequently s t ( a ) = limB = 6, and since x ' ( a . ) 'j j 3 exposes K at a , we have IIa all + 0.
.
j
Similarly, r ' ( b ) so IIa
j
-
Ksl,
that x 3! ( a3. ) = x ' ( a ) , and x' strongly so
J
+
-
6 and Ilb.
3
-
bII
and since x' exposes K, a = b , and
+ 0.
b.11 + 0. This contradiction gives that 3
THEOREM 3 .
E
d
K
is continuous at
2'.
I f K i s a weakZy compact, convex su5set of a Banach space X
such that the s e t o f strongly exposing functionals of K i s dense i n X', then the s e t of exposing functionals of K i s residual i n X'. Further i f every exposing functional of K i s strongly exposing, then the exposing functionals from a G6 residual s e t i n X'.
P r
Let C be the set of points of continuity of dK, and let XL
o o f.
and Xi be the sets of exposing and strongly exposing functionals of K in X'. Since X'S is, by hypothesis, dense in X' and dI< vanishes at every point of Xs,
d
K
d
K
is zero at every point where it is continuous. But x' exposes K whenever (XI)
= 0.
Thus from Lemma 3, XI
c C c
Xi.
Since the points of continuity of any real-valued function form a G6 set,
XI contains the dense G 6 set C, and so X'e is residual in X'. For the second part of the Theorem XA
c
Xi, whence Xi
C.
3 . Measure spaces
Suppose T is a set and S is a a-algebra of subsets of T. For a set E put SE = { F
:
F
E
S, F
c
El.
If T is a Bore1 measurable subset of lRn, the
t
S
PRELIMINARIES
1.3
9
usual n-dimensional r e a l space, o r i f T i s a compact Hausdorff space, l e t B ( T ) be t h e a-algebra of Bore1 s e t s i n T .
W e denote t h e s e t of a l l S-measurable real-valued f u n c t i o n s on '2 by M(S), and by BM(S) t h e s e t of a l l bounded, S-measurable f u n c t i o n s on T. V
C R = IR',
For a s e t
M ( S ) ( r e s p . BMV(S))i s t h e s e t of a l l f u n c t i o n s f i n M(S) ( r e s p .
v
E M ( S ) ) with f ( t )
V, t
E
E
T. C l e a r l y , i f V i s a bounded s e t , then MV(S)
BMV(S). We consider t h e usual l a t t i c e operations on M ( S ) , namely, f o r f,g
f
v g
= 4Cf
+
g
+
If - g l )
*g
f
and
= 4(f
+
g
If
-
-
E
M(S),
gl).
By ca(S) we mean t h e Banach space of a l l f i n i t e countably a d d i t i v e ( r e a l valued) measures on S with norm IIpll = 1p1 ( T I , p
ca ( S ) , where IpI i s t h e
E
v a r i a t i o n of u. The n o t a t i o n p 4 A means t h a t p i s a b s o l u t e l y continuous with r e s p e c t t o A.
Two measures p,A a r e c a l l e d equivalent i f h
< 11
and p 4 A. A family
A c caCS) i s c a l l e d uniformly a b s o l u t e l y continuous with r e s p e c t t o h
denoted by A respect t o
* A,
L, E
if A(E)
+
0, E
E
S, implies t h a t u ( E )
E
ca(S),
+ 0 uniformly with
A.
When convenient, we a l s o consider ca(S) a s a l i n e a r l a t t i c e with r e s p e c t t o n a t u r a l (set-wise) order.
I t i s well-known ( e . g . C171. 111.7.6) t h a t t h i s
l i n e a r l a t t i c e i s r e l a t i v e l y complete (Dedekind complete). A measure space i s a t r i p l e (2',S,A) where S i s a o-algebra of s u b s e t s of
T and X i s a p o s i t i v e , p o s s i b l y i n f i n i t e , measure on S. i n t e g r a b l e f u n c t i o n with r e s p e c t t o A on a s e t E hE(f)
=
jGfdX; and h(f) = A T ( f )
=
E
The i n t e g r a l of an
S w i l l be denoted by
lfdh.
A measure space ( T , S , A ) i s c a l l e d l o c a l i z a b l e i f , f o r every continuous 1 on L ( T , S , A ) t h e r e i s a f u n c t i m g 1 ~ ( f )= j f g d ~ , f o r every f E L (T,S,A).
linear functional
For a s e t E
@
E
E
BM(S) such t h a t
S we denote CEIAthe c l a s s of a l l s e t s F
E
s
f o r which
CONICAL MEASURES
10
X(E
* F) =
0 and set S ( X ) = { r E I X : E
E
1.4
Then SfX) is a Boolean algebra
S}.
with respect to the operations induced by those of S .
The measure space
( T , S , h ! is localizable if and only if S(X) is a complete Roolean algebra.
Further ( T , S , X ) is a localizable if and only if the ring of elements of S(X) corresponding to sets of finite measure is relatively complete. This is, in fact, the original definition in f731 of localizability. If X f T )
c m
then ( T , S , X ) is called a finite measure space. A measure
space ( T , S , X ) is said to he a direct sum of finite measure spaces if there is a family
F
c
S of pair-wise disjoint sets such that h ( F ) <
-, for every F
the o-algebra S contains every set E such that E n F belongs to
F
E
F, and X ( E )
=
1F EF
X ( E n F), for every 2
E
S.
A
s,
E
F.
for every
measure space which is a
direct sum of finite measure spaces is localizable. Suppose X is a 1.c.t.v.s. and ( T , S , h ) a measure space. Integrahility of a vector-valued function f : T
-+
X is meant in the sense of Pettis. In other
words, the function f is said to he A-integrable if for every function E A(?,)
E
( X I ,
f), i.e. t *(z',f(t)),t
S there is a point zE in X such that
of f on a set E
E
E
Z'E
XI, the
T, is 1-integrable and for every
(XI,
zE )
S is defined to be xE; A(f)
=
XE ( ( X I ,
f)). The integral
=zT.
4. Conical measures
Let X be a 1.c.t.v.s. with dual XI.
The elements of XI are treated as
continuous linear functions on X, and the smallest linear lattice of functions, with respect to point-wise order and linear operations, containing X' is denoted by h ( X ) . Every element z '
E
h ( X ) can be written as k
v xi, i=j+l
11
PRELIMINARIES
I .4
where 1 < j
5
k a r e i n t e g e r s and x!
E
X', i = 1,.. . , k .
A non-negative l i n e a r f u n c t i o n a l u on h ( X ) i s termed a c o n i c a l measure on
X. The s e t of a l l c o n i c a l measures on X i s denoted by M+(X). complete l a t t i c e w i t h r e s p e c t t o t h e o r d e r v 5 u i f v ( z ' ) 2' E
h(X), If u
I'
E
u ( z ' ) , f o r ev er y
2 0
2'
P(X) and
E
5
I t is a relatively
X', we w r i t e
I
I
i s a p o i n t of X such t h a t u ( z ' )
(XI,
I),
f o r ev er y
If t h e sp ace X i s
~ ( u and ) c a l l z the resultant of u.
s e p a r a t e d ( Hau s d o r f f ) , t h e r e s u l t a n t o f u i s unique i f i t e x i s t s . For u
E
M+(X) we w r i t e K
= {r(v) : v
u, v
5
E
M+(X)).
For t h e s e f u r t h e r f a c t s about c o n i c a l measures we r e f e r t o 1117 and [ 1 2 J
(88
f u r t h e r r e f e r e n c e s can be found t h e r e .
30,38,40);
Denote by C = C ( X ) t h e minimal u - a l g e b r a o f s u b s e t s o f X such t h a t ev er y f u n c t i o n i n k ( X ) i s C-measurable.
I t i s of c o u r s e t h e minimal o - a l g e b r a such
t h a t every f u n c t i o n i n X' i s C-measurable.
C i s t h e u - al g eb r a g en er at ed by a l l
s e t s o f t h e form {x : x'(z)E B } f o r a l l
E
I'
X' and a l l Borel s e t s B cIR. 1
For any c o n i c a l measure u on t h e s p a c e X = R e x i s t s a compact s e t T
c
, where
c a r d I I: No, t h e r e
X and a f i n i t e r e g u l a r measure A on t h e Borel o - a l g e b r a
S i n T such t h a t u ( z ' ) = ITzt(t)dA(t),f o r e v er y z '
E
h ( X ) , C121 Theorem 3 8 . 3 .
The n e x t aim i s t o i n c r e a s e t h e c a r d i n a l i t y of t h e index s e t I and t o show t h a t t h e s e t T and t h e o - a l g e b r a S can be found independent on t h e c o n i c a l measure u.
The p r i c e f o r t h i s improvement i s t h a t T w i l l n o t be compact and S w i l l n o t
be t h e o - al g eb r a o f Borel s e t s anymore and a l s o t h e measure X w i l l n o t b e finite.
The measure s p a c e ( T , S , h ) w i l l be a d i r e c t sum of f i n i t e measure s p a c e s ,
however. The f o l l o wi n g Theorem 1 i s o n l y concerned w i t h complete weak sp aces. t h a t X i s a complete weak s p a c e i f and o n ly i f i t i s a p r o d u ct of c o p i e s real-line
x
=
IR' f o r some s e t I.
Recal l of t h e
12
CONICAL MEASURES
THEOREM 1.
I Let X = IR and card
I 5
I .4
HI, There e x i s t s a s e t T
c
X and a
o-algebra S of subsets of T suck t h a t , f o r every conical measure u on X , there i s an extended-real-valued, non-negative, a-additive measure
f o r every
Z I E
on S such t h a t
A
h ( X ) , and the measure space ( T , S , A ) i s a d i r e c t sum of f i n i t e
measure spaces. It will be clear what modifications
P r o o f. Assume that card I = HI.
We assume further that the set I is well-ordered
are to be made if card I < Nl.
and, indeed, we take for I simply the set of all countable ordinals. For ~ ( 1 = )
K E
I, let T be the set of all points x =
0, for every
I < K,
and
Ix(K)(
It is clear that, for every x number a and a point t
E
= 1. E
Put T =
(
~
(
1
)
in ) X ~ such ~ ~ that
U
KE?'K*
X, x f 0 , there exists a unique positive
T such that x = a t .
It follows that every
Z'E
h!X!
is uniquely determined by its restriction f = z'IT to T. Denote by LO = {f
:
f = z'IT,
k(X).
Z'E
k ( X ) } the set of restrictions to T of all functions in
It i s clear that Lo is a linear lattice. Let Mo be the minimal class of functions containing Lo and closed with
respect to taking point-wise limits of sequences of its elements. Let So be the class of subsets of T with characteristic functions belonging to h40. Clearly, So is a a-ring. We establish next that every set T K . to S and that every function in Let
K E
less than
K.
I. Let
3.c
I,belongs
is So-measurable.
be a sequence whose terms are all elements of I
(:in}n=l
Put
zh(x)
K E
=
(Ixc(K)I
-
n n
1 Ix(ii)l)
i=1
v
o
1.4
PRELIMINARIES
for every
3:
13
= ( Z ( I ) ) ~ ~ ~ in X, and fn = z A / T , n = 1,2,
n = 1,2,. . , , and the sequence lfn}i=l istic function of T K'
tience TK
E
... .
Then fn
E
Lo for
tends monotonically to the character-
S O , for every
I.
K E
To show that every function in Mo is So-measurable, it suffices to prove that every function in LO is So-measurable. As every continuous linear function on X is the linear combination of a finite number of evaluations at points of
I and functions in h ( X ) are expressible in the form (I), for every €unction
f
E
L , there exists
I such that f(t) = 0 for every t
K E
characteristic function hence
f ~ xbelongs
to Mo.
real constant then
fAa
E
x
of the union of all TI with Clearly, fAx =
fAl.
So,
~AQ)) A 1
The
I.
belongs to Mo and
I
<
K
if f
E
LO and a is a
then the functions
t 0
+ m.
Let S be the u-algebra consisting of all sets E K E
<
monotonically tend to the characteristic function of
the set It : f(t)>a3 as n
for every
K
M D . To finish the proof that every function in L o
is So-measurable it suffices to observe that if f
f, = (n(f -
TI with
E
T such that E n TK
c
E
SO,
I.
To show that so defined T and o-algehra S have the claimed properties, assume that u is a conical measure on X. where f = z ' I T .
For every f
E
L o , let uo(f) = u ( z ' ) ,
Then u o is an unambiguously defined possitive functional on'Lo.
The fact that u is a Daniell integral On h ( X ) , 1121 Theorem 3 8 . 1 3 , and that a sequence { z h } of elements of h ( X ) tends point-wise monotonically to zero on
X if and only if the sequence { z ' IT) of resetrictions tends monotonically to n zero on T, imply that uo is a Daniell integral on L o . The theory of Daniell integrals implies the existence of a non-negative o-finite measure 10 on S O such that u ( z ' ) = uo(2IIT) =
for every
Z'E
h ( X ) . Clearly, X0(TK) <
m,
T
z'(t)dho(t),
for every
3: E
I.
14
I
REMARKS
FOT every E
E
S, d e f i n e now
I t i s c l e a r t h a t X i s an o - a d d i t i v e measure on S and t h a t t h e measure sp ace
( T , S , A ) i s a d i r e c t sum o f f i n i t e measure s p a c e s .
I t i s easy t o s e e , f u r t h e r ,
t h a t i f f is a h g - i n t e g r a b l e f u n c t i o n t h e n it i s a l s o A - i n t eg r ab l e and
A(f)
=
Xo(f).
I n p a r t i c u l a r u ( z ' ) = u g ( . z ' I T ) = Ao(z'IT) = A(z'1T). T h i s i s ( 2 ) . Remarks
There i s o n l y one i n t e r e s t i n g s t a t e m e n t i n S e c t i o n 1, namely t h e c l a s s i c a l O r l i c z - P e t t i s Lemma.
I t d a t e s back t o t h e e a r l y t h i r t i e s and ap p ear s i n t h e
p u b l i c a t i o n s o f Banach's s c h o o l , e s p e c i a l l y i n O r l i c z ' s p ap er s on o r t h o g o n al series.
The f i r s t known complete proof f o r ( co u n t ab l e) series i n a Banach
space i s i n P e t t i s ' c l a s s i c a l work CG21. statement,
There a r e now s e v e r a l p r o o f s o f t h e
The e x te n s i o n t o l o c a l l y convex s paces does n o t r e p r e s e n t any
d i f f i c u l t i e s ; it a p p e a r s i n d i f f e r e n t works, e . g . i n r 3 5 1 .
The same a p p l i e s
t o t h e e x t e n s i o n t o more g e n e r a l s e t s o f i n d i c i e s , n o t n e c e s s a r i l y t h e n a t u r a l numbers, S t r o n g l y extreme p o i n t s a r e a l s o c a l l e d d e n t i n g p o i n t s i n t h e l i t e r a t u r e e . g . R i e f f e l C681 and C6Jl.
Lemma 2 . 1 i s due t o Anantharaman C21.
i s due t o Husain and Tweddle C281. from C281.
Theorem 2 . 1
Theorem 2 . 2 and i t s C o r o l l a r y a r e a l s o
Anantharaman proved i n r31 t h a t i f X i s a Banach space and A,B are
weakly compact convex s e t s i n i t , th e n s t . e x p $ Theorem 2 . 2 i s an e a s y c o r o l l a r y of t h i s r e s u l t .
i s weakly dense i n exA. Lemma 2 . 3 and Theorem 2 . 3
a r e from Anantharaman r31. The concept o f a l o c a l i z a b l e measure s p a ce was i n t r o d u ced i n S e g a l ' s paper C731.
There a r e proved s e v e r a l e q u i v a l e n t c h a r a c t e r i z a t i o n s o f such
I
PRELIMINARIES
15
spaces and a l s o t h a t a d i r e c t sum of f i n i t e measure spaces i s l o c a l i z a b l e . s t i l l seems t o be t h e b e s t r e f e r e n c e on t h e s u b j e c t .
It u s e s terminology
s l i g h t l y d i f f e r e n t from ours. Concerning t h e p r o p e r t i e s of c o n i c a l measures we r e f e r t o Choquet Clll and [121.
Further r e f e r e n c e s can be found t h e r e .
Theorem 4.1 i s from C401.
It
11.
VECTOR
MEASURES
AND
INTEGRATION
In t h i s chapter t h e b a s i c concepts and conventions about v e c t o r measures a r e introduced.
The f u r t h e r aim is t o g i v e a workable d e f i n i t i o n of t h e i n t e g r a l
of a scalar-valued function with r e s p e c t t o a l o c a l l y convex space valued
measure, and t o i n v e s t i g a t e t h e p r o p e r t i e s of t h i s i n t e g r a l .
We d e f i n e t h e
analogue of t h e Lebesgue space of i n t e g r a b l e f u n c t i o n s , and show t h a t convergence theorems of t h e type of Beppo Levi and Lebesgue hold f o r t h i s i n t e g r a l .
1. Vector measures; v a r i a t i o n and semi-variation
Let X be a l i n e a r space.
Let S be a a - a l g e b r a of s u b s e t s of a set T,
and suppose rn : S + X i s a s e t - f u n c t i o n . I f x* i s a l i n e a r f u n c t i o n a l on X, we d e f i n e t h e s e t - f u n c t i o n (x*,rn) : S +lR by ( x * , r n ) ( E ) = (z*, r n ( E ) ) . E
S.
E
The r e s t r i c t i o n of rn t o Si w i l l be denoted by rnE and rn(SE) = rn,(S,) Irn(F) : F
E
SEj i s t h e range of rnE
Let p be a semi-norm on X.
;
=
rn(S) = rn(ST).
A mapping rn : S
+
X i s c a l l e d a p-measure if
rn is f i n i t e l y a d d i t i v e , and i f f o r every sequence of p a i r w i s e d i s j o i n t s e t s
{Ei as n
:
i
+
1,2
,...
3 from S, with E
l!,lrn(Ei))
+
0
-.
If X i s a 1 . c . t . v . s . P, then a mapping rn : S +
every p
m
= Uiz1 Ei, we have p(rn(E) -
E
with topology determined by a family of semi-norms
X i s c a l l e d a v e c t o r measure i f rn i s a p-measure f o r
P.
For t h e remainder of t h i s Section we suppose t h a t X i s a l i n e a r space, p a semi-norm on X, and rn : S + X i s a p-measure. Define t h e p - v a r i a t i o n v (m) t o be t h e s m a l l e s t non-negative measure P 16
v. 1
VECTOR MEASURES AND INTEGRATION
17
v ( E l , f o r every E E S. In t h e case m is a s c a l a r measure P we r e v e r t t o t h e u s u a l n o t a t i o n Iml f o r v a r i a t i o n . For a general l o c a l l y convex such t h a t p ( m ( E ) )
S
space valued measure m. u (rn) need not be f i n i t e .
For t h i s reason it i s more
P
u s e f u l t o consider t h e p-semi-variation of m , defined by,
LEMMA 1. Suppose p i s a semi-non on a linear space X and m :
p-measure
.
-+
3
f.
I(x*,
m ) l ( E ) : x*
UO), E
E
P
By Abel's p a r t i a l summation p ( r n ) ( E )
E
S.
n sup{p(&=laim(Ei))
where t h e supremum is taken over a l l f i n i t e c o l l e c t i o n s of s c a l a r s /ail and a l l p a r t i t i o n s of E i n t o a f i n i t e number of d i s j o i n t s e t s i n S. l y for E
E
Suppose m
:
s
+X
i s a p-measure.
m 1 : S + R defined by m l ( E ) = sup{p(z) : 5
5
1.
Consequent-
S,
LEMMA 2 .
p(m)(E)
X is a
fien p ( m ) ( E ) = sup{
P r o
s
2ml(E), E
E
S.
t E
Then the set-function
m(SE)), E
S, s a t i s f i e s m l ( E )
Both ml and p ( m ) are bounded, positive and o-sub-
additive.
P r o o f.
E
Clearly m l ( E )
5
p(m)(E), E
E
S and
5
11.1
VECTOR MEASURES
18
To show that p ( m ) is a-sub-additive,suppose iEi pairwise disjoint sequence of sets in 3. m
partition of E = UiZ1 Ei, then {Ei n F1, of EiJ for each i = 1,2.
so
.. . .
:
i
= 1,2,... 1 is a
If IF1,
.... Fk 1 is a disjoint
..., E 2.
n F k l is a disjoint partition
Thus if Ia.1 < 1, j = 1,2.. 3
.. k ,
we have
that,
The semi-variation will only be a measure if
S. In other words, if
(rn) is infinite, p ( m ) cannot be additive. To P avoid problems resulting from this fact we are going to show that there exists each E
E
2,
a finite positive measure equivalent to p h ) .
The proof will follow from the
next Lemmas. LEMMA 3 . limn-En
=
P,
If {EnIiZl is a decreasing sequence of s e t s in S and if
then limnd ( m ) ( ~ = ~ 0.)
Suppose the conclusion is false. Then p(rn)(E,)
P r o o f.
some 6 >
0
> 6,
for
and all n.
Start an inductive process by putting n 1 = 1 and selecting a set A1 E S such that p(rn(Al n E l ) ) > $6. The existence of A1 follows from the inequality p(rn)(El)
> 6
Let r
and from Lemma 2 .
t 1
select a set Ap
be an integer and assume that nr is already determined. Then E
S such that p ( m ( A , n
again, from p ( r n ) ( E , ) m
> 6
En )) >
r
and from Lemma 2 .
46. Its existence follows,
Now, limnd(rn(A,
n En)) = 0, since
is a p-measure. Let nr+l be an integer such that nr+l > nr and
VECTOR MEASURES AND INTEGRATION
11.1
19
m
This d e f i n e s i n d u c t i v e l y sequences {nrlr=l and {Ar}z=l.
-
Let Fr =
... .
1 , for r = 1,2. Then t h e s e t s F a r e pair-wise d i s j o i n t r r+l and, by a d d i t i v i t y of m and t h e t r i a g l e i n e q u a l i t y f o r p , p c m ( F r ) ) > 6/Q,
Ar n ( E n
En
r
r = 1,2, ...
.
This i s a c o n t r a d i c t i o n with requirement t h a t m be a p-measure,
i . e . t h a t p ( m ( F ) - C:=lm(Fr)) COROLLARY 1.
-t
0 a s s -+
m
u,,lFr.
where F
m,
I f { E n ) i s a monotone sequence of s e t s i n S, then
l i m d ( m ) ( E ) = p(rn)(lim
n-pDEn ) .
n
P r o o f. E > 0
I f { E n } is decreasing and E
t h e r e i s 6 such t h a t p ( m ) ( E n - E ) <
and s u b a d d i t i v i t y of p(m) then p i m ) ( E )
implies p(rn)(E) +
A(E) = 0, E
E
0
B
n
=
Uj=yPj.
A ( B n ) <'
By monotonicity
for a l l n > 6.
p(m)(En)
* p(rn)(E) +
if and only if X(E) = o implies p ( m ) ( E )
The "oply i f " p a r t i s obvious.
E.
-+
0, E
E
s,
0.
Conversely, suppose t h a t i f
S , then p ( m ) ( E ) = 0 and t h a t t h e r e e x i s t s 6 > 0 and a sequence
{ A n } i n S such t h a t p ( m ) ( A , ) > 6 and A(A,) m
by Lemma 3 , f o r every
I f A i s a f i n i t e p o s i t i v e measure then X ( E )
COROLLARY 2 .
P r o o f.
S
E,
limn-En,
n =
1.2.
... .
( 4 ) n . Thus i f
< ($)ntl,
for n = 1.2,.
.. .
Let
By monotonicity, p ( m ) ( B n ) > 6 , f o r a l l n, and then A ( B ) = 0 b u t , by Corollary 1,
B = limwBn
p(m)(B) 2 6. L E M 4.
If m : S
-+
X i s a p-measure then, for any decreasing sequence
{ F n } of s e t s i n S with F =
with respect t o x* P r o o f. 5
E
U
P
m
nnzlFn, limw(x*,
O.
Since s u p { l ( r * , m)(Fk) - ( x * ,
p(m)(Fk - F), f o r a l l k = 1.2,..
Lemma 3 .
m)(Fn ) = ( x * , m ) ( F ) uniformly
.
~)(F)I
: X*
E
vo3 P
= ph(Fk
, t h e r e s u l t follows by Corollary 1 t o
-
F))
*
11.1
VECTOR MEASURES
20
If m
LEMMA 5.
f i n i t e set J
c
P r o o f.
...
i s a p-measure then f o r any
Vo such that i f E P
then 1(x*, r n ) ( E ) I <
n = 1,2,
: S + X
Z* E
I f not, f o r some
E
, such t h a t ) ( xz? , m ) l ( E
) ( x ~ +r n~) ( E. n ) J
n = 1.2
?i E ,
> 0
there e x i s t s a
S and ) ( x * , m ) l ( E ) = 0 , f o r every x*
E
f o r ever3
E,
E
J,
Vo.
P
> 0 )
E
t h e r e e x i s t x;
i
= 0, f o r
c
VoP and En
S,
E
1 , 2 , . . . , n , and
.
,...
m
I f F = U? E . then {F 1 is a decreasing sequence and i f F = nn=lFn then n j=n j n lim
( x * m ) ( F ) = (x;,
m ) ( P ) = 0 , f o r i = 1.2
ni' n f o r n = 1 , 2 , ... . This c o n t r a d i c t s Lemma 4.
,...n, while
I(X;+~,
m)(Fn)I 2
TllEOFEM 1. Suppose S i s a o-algebra of subsets of s e t T , X i s a linear
space, p a serni-non on X and m
S
:
-t
X a p-measure.
f i n i t e positive measure X on S such that X ( E ) A(E)
0, E
+ .
S , implies that p ( m ) ( E )
E
P r o o f.
..
For n = 1 , 2 , .
J n c Vo such t h a t I(x*, m ) l ( E j E
Vo.
P
p ( m ) ( E ) , f o r every E
, choose, according t o Lemma
I f J n = {x; : k
c:
s, and
0.
0. f o r x* E J,,
P
f o r every x*
-f
5
Then there e x i s t s a
1,2
,...,j ) ,
5 , a finite set
implies / ( x * , rn)(E)I < l / n , let
i s a p o s i t i v e measure such t h a t X n ( E ) = 0 i E p l i e s I(x*, m)(E)I < l/n, n f o r a l l x* E Vo and X n ( E ) 5 sup{l(x*, m ) I ( E ) : I* E U o 3 5 p ( m ) ( E ) , f o r m y P' P E E S. Then X
Now l e t m
X
I:
($)"An.
n=l
Then
we have
X(E)
5
p ( r n ) ( E ) , f o r every E
E
S. Further, i f A ( E ) =
0, E
E
s,
E,
11.2
VECTOR MEASURES AND INTEGRATION
then a l s o X ( F ) = 0 , f o r every F
n
1,2,.
..
, o r ] ( x * ,m ) ( F : l =
c
E,
F
S, hence l ( x * , m ) ( F ) I < l / n , f o r each
E
0 , f o r each x*
implies t h a t / ( x * , m ) k E ) = 0 , f o r each x*
p ( m ) ( E ) = 0.
21
E
Uo. This i s t o say, A ( E ) = 0
E
P
Uo and hence, by Lemma 1, t h a t
P
Corollary 2 to'Lemma 3 gives t h e r e s u l t .
Suppose X is a 1 . c . t . v . s .
and m : S + X a v e c t o r measure.
measures i s s a i d t o be equivalent t o m i f / h l ( E ) + 0 , E
E
A family h o f
S, f o r every X
E
A,
i f and only i f p ( n i ) ( E ) + 0 , f o r every continuous semi-norm p on X. COROLLARY 1.
and m
I f X i s a 1.c.t.v.s.
S
:
-+
X a vector measure then
there e x i s t s a f a m i l y o f positive measures equivalent t o m. COROLLARY 2.
I f X i s a m e t ~ z a b l e1.c.t.v.s.
and m :
s
-+
X a vector
measure then there e x i s t s a f i n i t e positive measure equiuaZent t o m. P r o o f. 1,2,...
E
The topology of X i s given by a countable family { p n : n = Let X
1 of semi-norms on X.
S, i f and only i f p , ( m ) ( E )
E
by Theorem 1.
+
The measure X =
be a p o s i t i v e measure such t h a t h n ( E ) -+ 0,
n 0, n = 1 , 2 , , . .
1"n=lanA'n
.
where a
n
I t s e x i s t e n c e i s guaranteed > 0, n = 1.2
,...
, a r e chosen
such t h a t li=lcinXn(T) < -, has t h e required p r o p e r t y .
2. I n t e g r a t i o n
Suppose X i s a l . c . t . v . s . ,
m : S
+
X a v e c t o r measure.
T a s e t , S a o-algebra of s u b s e t s of T and
A real-valued S-measurable f u n c t i o n
f on T i s s a i d
t o be m-integrable i f i t i s i n t e g r a b l e with r e s p e c t t o every measure ( x r , m )
xr
E
X',
and i f , f o r every E
E
S, t h e r e e x i s t s an element xE of X such t h a t
INTEGRATION
22
11.2
We denote E
S,
E
and
xr = Ifdm =
!r
If&
= rn(f).
We identify
The set o f all m-integrable functions is denoted by L h ) .
freely sets in S with their characteristic functions and so, with a slight abuse of notation, we write S
c
Lh).
Also, we consider the integration
mapping as an extension of the vector measure mapping, m it by m
:
L(m)
:
S + X, and denote
X. The use of the same letter for this extended mapping as
+.
for the original vector measure will not be a source of confusion. L ( m ) is a linear l a t t i c e of functions and the integration mapping
LEMMA 1.
m
:
L(m)
+
P r o
X i s linear. o
If f E L h ) and E
S, then
fx,
E
L(m).
f. The linearity is trivial. If f is m-integrable and E
then it follows from the definition that
f+ = f
E
v 0 and f - =
fx,
E
S,
is integrable, too. Consequently.
(-7) v o are integrable functions and
I f f is an m-integrable function, then the mapping n
n ( E ) = mE(f), E
E
so is :
S
-+
If1
t
f-.
X defined by
S, is called the indefinite integral of f with respect to m.
The definition of an integrable function gives that its indefinite integral is well-defined and the Orlicz-Pettis Lemma implies that it is a vector measure. An
m-integrable function f is said to be m-null if its indefinite integral
is (identically) the zero vector measure. We say that two m-integrable functions f , g are rn-equivalent or that they are equal rn-almost everywhere (m a.e.) if the function If
- gl
is rn-null. The class of all m-integrable functions
11.2
VECTOR MEASURES AND INTEGRATION
23
m-equivalent t o an m - in te g r a b l e f u n c t i o n f is d en o t ed by [flm For a f u n c t i o n f
E
L(m) and a continuous semi-norm p on X we d e f i n e t h e
p-upper i n t e g r a l p ( m ) ( f ) o f f by
where p ( n ) is t h e p - s e m i - v a r i a t i o n o f t h e i n d e f i n i t e i n t e g r a l n o f t h e f u n c t i o n
( i ) Let P be a separating s e t of continuous semi-norms on X .
LEMMA 2 .
function f (ii)
E
L(m)
Given f
E
i f and on& i f p(m)(f) = 0, for every p
L ( m ) and a continuous semi-norm p On
x,
E
P.
then
For any continuous semi-norm p on X , t h e application
(iii)
f * p(m)(f), f P r o o f.
Let f
is m-null
E
E
L(m), i s a semi-nom on L h ) . Statement ( i ) i s obvious.
L ( m ) and l e t p be a
continuous
semi-norm on
X. If n i s t h e
i n d e f i n i t e i n t e g r a l of f , t h e n
f o r ev er y x’ c XI.
Hence (1) f o l l o w s by Lemma 1.1.
C l e a r l y , /Ifldl(x‘, m)l and x ’
E
2
] g d ( x ’ , m ) , f o r ev er y g
E
L(m) w i t h 191
5
If1
X’. But i f E,T - E i s t h e Hahn decomposition o f 2’ f o r (x‘, m), and
A
INTEGRATION
24
- IflxT-E, then obviously
if we put g = I f l ~ ,
11.2
191 s
I f \ and j l f l d l ( z t J n)l =
By Lemma 1 this function g is n-integrable, so
j g d(z', m ) .
As p ( m ( g ) ) =
supC(xf, m(g))
:
zr
E
U o l J the equality (1) implies ( 2 ) .
P
It is well known that j l f l d l ( z f , m)l = supljg d h ' , m f S-simple}.
:
191 -<
I f l , g is
Characteristic functions of sets in S are obviously m-integrable
hence, by Lemma 1, every S-simple function is m-integrable. So we can write
for every
2' E
X'. Now (3) follows again from (1).
Any of the equalities (l), ( 2 ) , (3) implies that ptm)(lfI)
= p(m)(f)
and
also the statement (iii). Denote by T ( m ) the topology on L ( m ) which is defined hy the family of seminorms f * p(m)(f),
f e L(m), f o r every continuous semi-normp on X. The
resulting locally convex space is not necessarily Hausdorff. The quotient space of L(m) modulo the subspace of all m-null functions is denoted by L 1(m). 1
The resulting Ilausdorff topology on L cm) is denoted again by - t ( m ) . Clearly L 1 (m) = {Cfl, p(rn)(f), [fIm E
:
f
E
L(m)).
Further, if we put p(m)([fl,)
=
for f E Lh), then by Lemma 2 the application rflm * p(m)([flm), 1 1 L (m), is a well defined semi-norm on L (m), for any continuous semi-
norm p on X. It is clear also that this system of semi-norms defines the 1
topology ~ ( m on ) L (m). The topology T(m) on L 1( m ) is analagous to the usual L1-norm topology of 1 the classical Lehesgue space, Indeed if X = R1, then L (mr is the standard
VECTOR MEASURES AND INTEGRATION
11.2
25
Lebesgue space together with its topology defined by its norm. As
m(f) = 0 for every m-null function f , the integration mapping generates 1
naturally a mapping Cml
: L
(m) + X defined by [ m l ( [ f l m )
= m(f), f
E
L(m).
We
also call Cml integration. When there is no danger of ambiguity we abandon the pedantic notation [ml and write m : L 1 ( m )
+
X. Similarly, we often write f instead of CflmJ even
when an element of L1(m) is considered and not one of L h ) . A
set E
E
S is said to be m-null if its characteristic function is m-null.
This happens if and only if p(m)(E) = 0, f o r every continuous seminorm p on X. Two sets E,F are m-equivalent if their characteristic functions are m-equivalent, o r if E
F is m-null. If E
are m-equivalent to E .
E
s,
then [ E l , is the cLass of all sets F
We put S ( m ) = { [ E l ,
E
S which
: E E S).
Since we have identified the elements of S with their characteristic functions, the set S(m) is only formally different from will use freely S ( m ) =
{CxElm
: E
E
{CxElm
:
E
E
Sj.
We
S}.
The restriction of the topology and uniform structure
T(m)
to S ( m ) will
again be denoted by ~ ( m ) . Also, S ( m )
is a Boolean algebra; it is the quotient-algebra of S modulo
the o-ideal of m-null sets, Hence the operations in S ( m ) are defined by [ E l , u CFlm = [ E u FI,.
E,F
E
S, etc.
The other Lebesque spaces L p ( m ) can be introduced. There will be no m
opportunity to use them, save possibly L (m). It is defined in an obvious way. two functions f,g (whether integrable o r not) are said to be m-equivalent if the set { t
:
f ( t ) f g ( t ) } is m-null. Given a measurable function f. the class
of all measurable functions g which are m-equivalent to f is denoted by [fflm. Then Lm(m)
{Cflm
: f
CI
BM(S)}.' The essential supremum norm topology is the
natural topology on L m ( m ) . But, in the sequel, several other topologies will
11.3
INTEGRABILITY OF ROUNDED FUNCTIONS
26
be considered on L m ( m ) and on its subsets. Let V be a set of real numbers. The set of elements in L 1 ( m ) with I
representants taking values in V will be denoted by L ( m ) .
This is to say,
V
1 LV(m)
{Cfl,
:
f
E
L ( m ) , f ( t ) E V for t
identification we have L1
{O,l)
1
Lc 0,1I ( m )
(m) = S(m).
E
TI. In the sense of our previous Another case often used will be
*
Similarly L i ( m ) = ICfl,
:
f
E
BM(S), f ( t )
E
V, t
E
TI.
Clearly, we have
m
also LIo,l)(m) S h ) .
3 . Integrability of bounded functions
The class of integrable functions is not trivial since we have LEI4MA 1. If X i s a sequentially complete 1 . c . t . v . s .
and m
: S
+
X a
vector measure, then every bounded S-measurable f u n c t i o n f i s m-integrable and,
f o r every E
E
S and every continuous semi-nomn p on X.
P r o o f.
If f is a simple function then it i s clearly m-integrable
and the inequality (1) follows from the definition of p ( m ) o r from Lemma 2 . 2 . In general, let f be a simple function such that sup{lfk(t) k
t
E
TI
< l/k,
for every E
E
for k
1,2
,... .
- f(t)I :
By (1)
S, and every continuous semi-norm p on X, k , Z
= 1,2,
... .
Hence
the sequence {mE(fk))is Cauchy in X, and, by sequential completeness of X, convergent in X. As f is clearly (z',m)-integrable, f o r every z' E X', the
11.4
27
VECTOR MEASllRES AND INTEGRATION
function f i s m-integrable and m (f)= limk*mE(fk), E
E
E
S.
Lemma 2 . 2 implies
t h e i n e q u a l i t y (1) i n g e n e r a l .
The s i t u a t i o n i s , by Lemma 1, s i m i l a r t o one of i n t e g r a t i o n with r e s p e c t 1 t o a f i n i t e r e a l measure, v i z . t h e space L m ( m ) i s a subset o f L ( m ) .
follows a l s o t h a t if V i s a bounded s e t of r e a l s then
Lib)
It Hence, i f
= Li(m). 1
m
V clR i s bounded, without ambiguity w e w r i t e L ( m ) i n s t e a d o f Ly(m) o r LV(rn). Y 1 This n o t a t i o n is e s p e c i a l l y used i f not t h e n a t u r a l topology of L ( m ) i s used on L V ( m ) . A s l i g h t l y more general r e s u l t t h a t Lemma 1 i s
THEOREM 1.
Let X be a sequentially complete l.c.t.v.s.,
P r o o f.
E
-+
X a
Let n be t h e i n d e f i n i t e i n t e g r a l of g and k ( t ) = f ( t ) l g ( t ) Then by Lemma 1, h i s n - i n t e g r a b l e .
Clearly f i s (x', m ) - i n t e g r a b l e f o r every x' c
S
T, i s m-integrable.
whenever g(t) f 0 and k ( t ) = 0 otherwise.
for E
:
Any S-measurable function f
vector measure, and g an m-integrable function. suck t h a t If(t)I s g ( t ) , t
m
E
X', and i f we put xE = n E ( h ) ,
S, then
(x', xE) = \k d ( z ' , n ) = j k g d h ' , m ) E
E
If dh', m) E
The d e f i n i t i o n of i n t e g r a l gives t h a t f i s i n t e g r a b l e and mE(f) = n E ( h ) , f o r every E
E
S.
4 . L i m i t theorems
THEOREM 1. Let f n be non-negative m-integrable functions, n = 1,2,.
suck t h a t the series
..,
LIMIT THEOREMS
28
11.4
m
i s convergent,
Then the s e r i e s c,",,fn
~ ~ , , f n ( t f) o r m - a b o s t every t
f(t) m
converges m-almost everywhere m d i f E
T, then f is m-integrable, m(f)
and, f o r every continuous semi-norm p on X,
c,,,mCfn)
P r o o f.
for n = 1 . 2
...
Since
, t h e s e r i e s (1) converges i f and only i f t h e series m
(3 1
The convergence of t h e s e r i e s (3) implies t h a t , f o r every x'
does.
X', t h e
E
series m
converges.
By t h e c l a s s i c a l Beppo Levi theorem, tire s e t of p o i n t s t
which Cz=lfn(t)diverges i s l ( x ' , m ) l - n u l l . hence t h i s s e t is m-null.
f o r every
5' E
X'.
If f
c;,,fn
rn-a.e., m
E
2' E
T for
X',
then f i s (x', m)-integrable,
Furthermore, t h e s e r i e s cn=lmE(fn) = c z = l m ( f n
convergent, f o r every E
xE)
is
S, hence we can d e f i n e mE(f) t o be t h e sum of t h i s
s e r i e s and deduce t h a t f is m-integrable.
By Lemma 2 . 2 ,
This holds f o r every
E
11.4
VECTOR MFASIIRES AND INTEGRATION
29
= supIp(m(g
5
supIp(x)
By Theorem 1 . 1 . 2 , ( 2 ) follows. COROLLARY 1. Let f be m-integrabZe functions, n = 1 , 2 , .
. . such
that the
series
i s convergent.
Then the series for t
f ( t ) = I;,,f,(t)
E
lzZlfn converges m-abost
T, f i s m-integrabZe, m ( f )
everywhere and i f
W
1:
cn=,mCfn)
and f o r every
continuous semi-norm p on X, n lim p ( m ) ( f - 1 f i ) = 0. i=1 n-
(5 1
Let
P r o o f.
ly, n = 1 , 2 , .
..
,
p, f- be n n
the positive and negative parts of f n respective-
Since
(4) implies that the series of functions
m
lnZ1G[and
similarly
l;=lfi)
satisfies the conditions of Theorem 1. The result follows by applying Theorem 1 to each of the series l;=lfi,
l:=lc,
in turn, and combining the results.
COROLLARY 2.
Suppose {fn} i s a monotone sequence of m-integrable functions
bounded by an m-integrable function. function f with f n norm p on X .
+
Then there e x i s t s an m-integrable
f m-a.e. and p ( m ) ( f
-
fn)
-+
0 f o r every continuous semi-
30
Since the sequence
P r o o f.
If 1
is monotone and bounded it converges
to a measurable function; let f = limf m-a.e.
By Theorem 3.1, f is m-integrable.
Now suppose that Ifn) is increasing. If g, )gnl
5
-
fn+l
Theorem 3.1,
f n , n = 1,2,.
lg
11.5
SUFFICIENT CONDITION
A
..
, then
]lg,l
is a measurable function and
llgnl
2
5
- fl m-a.e., and
f
is integrable. It is immediate that m [ l g
)
=
lm(gn)
weakly.
Since any gn can be replaced by the zero function, the Orlicz-Pettis Theorem gives that Em($,)
is summable. This means that the assumptions of Theorem 1
are fulfilled for the series ~~=,(f,,, - f n ) . THEOREM 2.
The result readily follows.
If {fn) i s a sequence of m-integrable functions converging
m-a.e. t o a function f and i f there is an m-integrable function g with I f n l
n = 1,2,.
m-a.e.,
.. , then f is m-integrabZe,
m(fn)
-+
m ( f ) and p(m)(f
g
- fn)
0,
f o r every continuous seminorm p on X.
P r
o o
f. Put
n+r gn = lim .V f i , r+- t.=n
...
n = 1,2, 1,2,,.,
.
,
The sequence {g,
-
fnl
5
gn
p(m(f n = 1,2,..,
.
= lim Y”
n+r
A
-
-
5
are m-integrable, n =
hn) decreases to 0. By Corollary 2 , p(m)(g,
-
k,)
Theorem 3.1 gives that f is integrable
kn it follows by Lemma 2.2,
- f,))
fi,
i=n
By Theorem 3 . 1 the functions gn and k
0, for every continuous semi-norm p .
and as I f
hn
p(m’(f - f,’
5
p!m)(gn
that
-
hnj,
This gives the result.
5. A sufficient condition for integrability In the definition of an m-integrable function it does not suffice t o only
+
31
VECTOR MEASURES AND INTEGRATION
11.5
r e q u i r e t h a t f be
(
x', m)-integrable f o r every x r
E
X' . Indeed, t h e i n t e g r a l
of such functions need not e x i s t . EXAMPLE 1.
X =
CO.
Suppose
p x E , f o r every E
Suppose t h a t 5" i s t h e s e t of a l l n a t u r a l numbers, S = 2', 9 :
T +IR i s defined by p f t ) = l / t , t
S, then m : S
E
x'
Z1
(XI,
m)-integrable, f o r every
+
6
X i s a v e c t o r measure.
T.
and
I f we p u t m ( E ) =
From t h e f a c t t h a t
follows e a s i l y t h a t t h e f u n c t i o n f : T +IR such t h a t f ( t ) = t , i s 2' E
X', but f i s not m-integrable.
I t i s conceivable, however, t h a t i f some completeness conditions are imposed on t h e space X, then any f u n c t i o n which is (x', m ) - i n t e g r a b l e f o r each
x'
E
X', w i l l be m-integrable. A 1.c.t.v.s.
i s s a i d t o possess t h e B
such t h a t ~ ~ = l j ( x3:f ,) I <
m,
f o r every x 1
-
P p r o p e r t y i f given xn, n = 1,2.
E
X', t h e r e i s an element
3: E
...
X with
x = I;=lxn. Any s e q u e n t i a l l y weakly complete space has t h e B
-
P property.
According
t o C61, i f X i s an Banach space then X has t h e B - P p r o p e r t y i f and only i f i t does not contain an isomorphic copy of c o . THEOREM 1.
Suppose X i s a 2.c.t.v.s.
E
P property and m :
s
+
X
XI, i s m-integrable. P r o o f.
2' E
-
Then any function which i s (x', m)-integrable, f o r every
a vector measure. x'
with the B
X'.
Let f be a f u n c t i o n which i s
(XI,
m ) - i n t e g r a b l e , f o r every
We can assume t h a t f i s non-negative, otherwise consider i t s p o s i t i v e
and negative p a r t s . Choose an i n c r e a s i n g sequence {f n } of non-negative bounded S-measurable f u n c t i o n s tending point-wise t o f .
Let g1 = fl, gn = fn
- fnel, n
=
2,3,
... .
i s m-integrable, by Lemma 3.1, and, by Beppo L e v i ' s Theorem,
Each function g m
f o r every x '
xE
~r . 6
AN ISOMORPHISM THEOREM
32
E
m
X' and E
E
Since X has t h e B
S.
-
P p r o p e r t y we can f i n d an
X such t h a t
E
f o r every x'
E
X'.
Hence f i s m-integrable.
6. An isomorphism theorem
Two v e c t o r measures rn
:
S
+
X, ml
:
S1 + X a r e s a i d t o be isomorphic i f
t h e r e i s a a-isomorphism b o f t h e Boolean o-algebra Sl(ml) onto S ( m ) c a r r y i n g
rnl onto m.
That i s , m and m
Sh)
S(m) such t h a t b(Sl(ml)) E,F
m
m
E
a r e isomorphic i f t h e r e i s a mapping b : Sl(ml) + 1 . ;
b(CE1
= b([El 1 S,, k = 1 . 2
- [Fl,
"1
S1; b(Uk=lCEklml)= UkZlb(CE 1 1, Ek
E
,...
- b(CF1, ) , f o r 1 ; and )
ml
CmJ(b(CElm
1, f o r every E E S1. ml S i s c a l l e d an atom of S i f A 1 0 and i f E
)) =
1 A set A
E
CmlJ(CEl
e i t h e r E = A or E = 0. A set A
and i f E
E
E
E
S. E c A imply t h a t
I f t h e r e a r e no atoms of S t h e n S i s c a l l e d non-atomic.
S i s c a l l e d an atom o f a v e c t o r measure rn
S, E c A imply t h a t e i t h e r m ( E ) = 0 o r m(E)
an atom of m i f and only i f CAI,
:
S + X i f m(A)
m(A).
*
0
Clearly, A i s
i s an atom of t h e o-algebra S ( m ) .
I f there
a r e no atoms o f m then m i s c a l l e d non-atomic. The o-algebra S i s s a i d t o be m - e s s e n t i a l l y countably generated if t h e r e e x i s t s a countably generated o-algebra SO
c
S such t h a t , f o r every E
E
s
there
VECTOR MEASURES AND INTEGRATION
11.6
is F
S o with E
E
LEMMA 1.
of 2'.
f
:
E
CFI,.
I t i s t o say, S ( m ) = {CEJ, : E
+
E
Sol.
Let P! be a s e t and S o a countabty generated o-atgebra of subsets
Then there e x i s t s an a t most countabte s e t F o
T
33
c
[ I , ? ) and
a function
[ 0 , 1 l u F o such that
and, for every y
P r o o f. assume t h a t E
E
For the s e t f - l ( { y } ) i s an atom of S .
Let S1 be generated by t h e family { E
n
*
0 for n = 1,2,.,.
:
n = 1,2,..
.
1 ; we
and t h a t no two of t h e s e s e t s coincide.
Every non-empty s e t of t h e form
where
E~
= 1 o r -1 f o r every n, and E-'
T - E , i s an atom of S o .
Let U be t h e union of a l l atoms (2) such t h a t f i n i t e number of i n d i c e s n. countable, hence Let T o = T
(I E
- U,
E~
= -1 f o r a l l but a
C l e a r l y t h e family of t h e s e atoms i s a t most
So. S o n T o = { E n To : E
E
Sl. For every t
E
T o define
,
then
m
f(t)
1 (1 -
n=l Then
f is
XE (t))2-n.
n
a f u n c t i o n on T o taking values i n C 0 , l ) .
Given a n a t u r a l number n and an i n t e g e r k with 0
where E
j
S
k
< 2
n
= 1 or -1 a r e determined by t h e dyadic expansion o f k .
Indeed i f
1 1 .6
AN ISOMORPHISM THEOREM
34
= 0 o r 1, t h e n
where a
i
= (-1lai.
E
J-
Conversely, f o r any sequence
E ~ , E ~ , . . . , Eof ~
numbers 1 and -1 t h e r e i s a
k w i t h 0 < k < 2n so t h a t (3) h o ld s . The r e l a t i o n (3) i m p l i e s t h a t f o r e v e r y F bel o n g s t o S o n To, and a150 t h a t f o r e v e r y E
F
E
E E
B(C0,l)) t h e s e t f-l(F)
So n T o t h e r e i s a set
B(C0,l)) with f-lfF) = E .
I f we extend t h e d e f i n i t i o n of f o n t o t h e whole of T by choosing any c o n s t an t v al u e, ta k e n from C1,2),
on e v e r y atom of So which is a s u b s e t of
U, c a r i n g only t h a t t h e v a l u e be d i f f e r e n t on d i f f e r e n t atoms, we o b t a i n t h e f u n c t i o n f on 2' as d e s i r e d . TIEOREM 1.
and m
: S
-+
Let T be a s e t , S ao-algebra of subsets of T , X a 1.c.t.v.s.
X a vector measure.
essentially countably generated.
Assume that m i s non-atomic and t h a t S i s mThen there e x i s t s a measure ml
: B(C0,l))
-+
X
which is isomorphic t o m.
P r o o f. So(m)
S(m).
Choose a c o u n t a b ly g e n e r a t e d o - al g eb r a So
c
S such t h a t
Let F o c C 1 , 2 ) be a c o u n ta b l e s e t and l e t f : T
-+
C0,l)
L.
Fo
b e a mapping c o n s t r u c t e d a c c o r d in g t o Lemma 1; i n p a r t i c u l a r , (1) h o l d s. Define t h e v e c t o r measure ml e v er y F
E
: B([O,l))
-r
X by p u t t i n g r n l ( F )
m(f
-1
(F)), f o r
B(C0,l)).
Now n o t i c e t h a t E o
$ - l ( F 0 ) i s an m-null s e t .
Indeed,
EO i s a co u n t ab l e
union of atoms of S o and s i n c e m i s non-atomic t h e atoms o f S O must be m - n u l l . Hence i f we d e f i n e b(iFlm ) 1
o-isomorphism o f B([O,l))
[f-'(F)Jm,
modulo ml-null
f o r ev er y F
E
B(CO,l)),
we o b t a i n an
s e t s o n t o t h e whole o f So(m) = S(m)
c a r r y i n g ml o n t o m. That i s , m and m 1 are isomorphic.
VECTOR MEASURES AND INTEGRATION
11.7
35
7 . Direct sum o f vector measures
Let I be an arbitrary set and, f o r each i algebra of subsets of T i , and.m
i disjoint union of the sets Ti, i
"id'(2' i T
Si, for every i
E
+
o-
X a vector measure. Suppose T is the
I. Usually it is constructed as 2' =
We consider on 2 the o-algebra S of all sets E
THEOREM I.
m : S
E
I. let T. be a set, Si a z
{ i l ) , but to avoid a too pedantic notation, we will simply write
x
UieTi.
E n Ti
: S +
E
x
c
T such that
I.
E
If the series
& p i ' s i ) is
convergent then the application
defined by
is a vector measure. P r o o f.
Let E
E be their union.
E
S, n = 1,2,.
..
, be pairwise disjoint sets and let
Noticing that the convergence of the series & E . m . ( S i ) is
unconditional since 0
E
m2. ( S 2. ) f o r every i
1 mi(E
m(E) =
E
I,we can write
m
n Ti) =
1 1 mi(En
n Ti) =
id n=l
id
m
The vector measure m : S vector measures m i' i
E
+
X defined by (1) is called the direct
sum of
I. Let f be
The integration with respect to a direct sum generalizes (1). a function on T and let f . be its partialisation to Ti,
for each i
E
I. The
following Theorem follows from the definitions almost immediately. THEOREM 2 .
The function f is S-measurable if and only i f fj is
si-
REMARKS
36
measurable, f o r every i
E
I.
If f i i s mi-integrabZe & ' i f f i o i t h respect t o mi,
ni : Si
for every i
there e x i s t s the direct sum n i f and only i f the series
I1
:
1.ze
S
E
-+
-+
X i s the indefinite integraZ of
I , then f i s integrable i f and onZy i f X of the vector measures niJ
i
I , i.e.
E
The vector measure n i s
.(S . I i s convergent. z z
the indefinite integraZ o f f . I f f i s m-integrable and f
2 0
then
Remarks While the finite dimensional vector measures (on a o-algebra) necessarily have finite variation, this is not true in infinite-dimensionalspaces. Khat is more, the famous theorem of horetzky and Rogers r171, p. 93, guarantees the existence of a measure with infinite variation with values in any infinitedimensional Banach space, Hence authors introduced some kind of o-sub-additive function to estimate the measure; most often used was sup{plm(F))
:
F
c:
sE}.
The definition of semi-variation given in Section 1 is, most likely,due to Bartle, Dunford and Schwartz [ql (used, of course, also in r171).
Its advantage
is due mainly to the properties stated in Lema 2 . 2 . The relation between p ( m ) and sup{p(m(F))
:
F
E
EI was studied in the case
S
when p is the Euclidean norm on IRn by Kaufman and Rickert r311. Rickert and Schwartz C721.
They obtained bounds for ~~ml~(E)/sup{~~rn(F)~~ : F e
Theorem 1.1 was essentially first proven in C41. follows that of C211.
C661,
sE}.
The proof given here
In applications it often suffices to have a measure X
37
VECTOR MEASURES AND INTEGRATION
I1
with r e s p e c t t o which t h e v e c t o r measure i s a b s o l u t e l y continuous, i t i s not necessary t o have p ( r n ) ( E ) s A @ ) ,
for a l l E
S.
E
Theorem 1.1 has an i n t e r e s t i n g sharpening which w i l l be taken up i n Chapter V I . The d e f i n i t i o n o f t h e i n t e g r a l (of a s c a l a r valued f u n c t i o n with r e s p e c t t o a v e c t o r measure) given i n t h i s Chapter i s perhaps t h e most obvious, and e a s i e s t t o work with. r e c e n t l y (C491,
S u r p r i s i n g l y , i t appeared i n t h e l i t e r a t u r e q u i t e
C501 and c a s u a l l y i n 1381).
The i n v e s t i g a t i o n w i l l be sup-
plemented by a d i s c u s s i o n of completeness of t h e space of i n t e g r a b l e f u n c t i o n s i n Chapter V when t h e r e l e v a n t techniques a r e a v a i l a b l e . I t can be shown (C491 Theorem 2 . 4 ) t h a t i n a Banach space t h e d e f i n i t i o n
of i n t e g r a t i o n given i n Section 1 coincides with t h a t of C171.
For a d e f i n i t i o n
of t h i s i n t e g r a l f o r measures on a & - r i n g , and a d i s c u s s i o n o f i t s r e l a t i o n t o o t h e r methods of i n t e g r a t i o n we r e f e r t o C501.
111.
FUNCTION
SPACES
I
One of t h e most e f f e c t i v e ways of i n v e s t i g a t i o n t h e p r o p e r t i e s of a v e c t o r measure i s t o consider it a s a mapping on t h e space of i n t e g r a b l e f u n c t i o n s (and i t s s u b s e t s ) .
From t h e p r o p e r t i e s of t h e domain and from t h e p r o p e r t i e s
of t h i s mapping can be i n f e r r e d t h e p r o p e r t i e s of t h e measure.
I n t h i s Chapter
we develop t h e necessary s t r u c t u r e s which w i l l be t h e b a s i s of our knowledge of t h e domain of t h e i n t e g r a t i o n mapping.
1. Topologies
Suppose T i s a s e t and S a a-algebra of s u b s e t s of T.
Let A c c a ( S ) .
A
real-valued S-measurable f u n c t i o n f w i l l be c a l l e d A-integrable i f it i s X i n t e g r a b l e f o r each X if
j\f-
E
A.
gldlAJ = 0, f o r a l l
Two such f u n c t i o n s w i l l be s a i d t o be A-equivalent X
E
A.
The s e t of a l l A-integrable f u n c t i o n s
which a r e A-equivalent t o f i s denoted by L
1 (A) =
{[fl,
: f
Cf1
A'
Define
i s A-integrable].
Furthermore, f o r any s e t V c I R , put
I f t h e r e is no danger o f ambiguity an element denoted simply by
c l a s s e s of s e t s i n S. E
A.
1
of L ( A ) w i l l be
f.
I t i s an advantage t o i d e n t i f y L'
f o r every X
Cfl,
I0,l)
( A ) with t h e s e t of a l l A-equivalence
S e t s , E.F i n S a r e A-equivalent i f lhl(E A F) = 0 ,
The c l a s s of s e t s i n S which a r e A-equivalent t o E
denoted LEIA. 38
E
S is
FUNCTION SPACES I
111.1
39
1 We consider L ( A ) a s a l i n e a r l a t t i c e and S ( A ) a s a Boolean algebra under rglA
t h e following o p e r a t i o n s , f o r [ f d A , aCfJA = tcxfi,,
Cfl,
A
CgiA =
Cf
A
E
1
L ( A ) d e f i n e Cfl,
g l A , Cfl,
t
Cf
rgl,
t
gl,,
Cf v gl,. and f o r 6 , F E S
v Cgl, =
put [El, u [FI, = [E u FIA and [ElA n CFIA = CE n P I A , [ E l , - CFI, = CE - F l , Each measure X
E A
induces i n t h e n a t u r a l way a well-defined mapping [XI
+IR, given by CXl(if1,)
L1(A)
= X(f),
Cfl,
E
1
We use t h e same symbol,
L (A).
[XI, f o r t h e r e s t r i c t i o n of t h i s mapping t o S ( A ) .
:
By an i n t e g r a l I on t h e
1
1
l a t t i c e L ( A ) we mean a l i n e a r mapping I : L ( A ) +lR with t h e p r o p e r t y t h a t i f
[f,lA
J.
0 , then
LEMMA 1.
I(CfnlA) J. 0.
I t follows e a s i l y t h a t
The map LXA is a - a d d i t i v e on S ( h ) and i t i s an i n t e g r a Z on L 1 ( A ) .
The s e t o f a l l elements
Cfl,
of L ' ( A ) with bounded r e p r e s e n t a t i v e s f w i l l
Since every bounded measurable f u n c t i o n i s c a ( S ) - i n t e g r a b l e ,
be denoted by L m ( A ) .
every bounded measurable function r e p r e s e n t s an element of L " ( A ) .
The n o t a t i o n
m
L ( A ) i s c l e a r , v i z . , given V c I R , L i ( A ) i s t h e s e t of a l l members [fl, o f V Lm(A) such t h a t f(t) E V , f o r every t
E
2'.
v
i s a bounded s e t of r e a l numbers then c l e a r l y t h e r e i s no d i f f e r e n c e 1 m between L v ( A ) and L y ( A ) . In such a case it s u f f i c e s t o denote t h i s s e t by If
Lv(A).
Let A be a s u b s e t of c a ( S ) such t h a t s u p { f l f l d l h l : A
E
A} <
-,
f o r every
A-integrable f u n c t i o n f and such t h a t t h e a p p l i c a t i o n
1
i s well defined on L
A).
1 C l e a r l y p A i s a semi-norm on L ( A ) .
1 The topology and t h e uniform s t r u c t u r e on L ( A ) defined by t h e semi-norm pA i s denoted by p ( A )
Clearly it i s t h e topology such t h a t t h e family of s e t s
TOPOLOGIES
40
for every
E >
111.1
0, is a fundamental family of neighbourhoods of the zero element
of L'(A). More generally, let A be an index set and, for every be such that sup{jlf]d]Al
X
:
E
<
Aa}
o E A,
let ha
c
ca(S)
-, for every A-integrable function f.
Suppose also that the number pa(CflA) = pA (Cfl,) = sup{~lfldlXl
:
X
E
Aa}
CL
does not depend on the representative f of the element Cfl, CfI, E LI ( A ) .
of L 1 ( A ) , for any
Then the topology determined on L 1 ( A ) by the family {pa norms is denoted by
for every
L > 0
p(Aa
E
of semi-
The family of sets
: a L A).
and every a
: a E A)
A , form
a basis of neighbourhoods of zero for
this topology. Usually each Aa will consist of only one element of ca(S).
r
c
ca(S), the topology p ( { A l
: A
E
For a set
r) on L'(A), if it has a meaning, is 1
denoted by r ( r ) . Clearly, T ( A ) = ? ( C A I ) is the classical topology of L ( I A 1 ) norm. The standard topology considered on L 1 ( A ) is T ( A
.
It is, according to
the given definitions, the topology determined by the family of semi-norms Ip, (1)
: A
E
A},
where
pA([fl,)
= ~ I ~ I ~ I ~A I3 .E, L'W,
or, alternatively, the topology whose basis of neighbourhoods of zero is the
FUNCTION SPACES I
111.2
41
family of s e t s
f o r every
Stil
> 0
E
{Cfl,
:
Cfl,
and h
E
A.
E
L1(A), l l f l d l h l < €1,
a d i f f e r e n t type of topology i s needed.
Let
r
c
c a ( S ) be a s e t f o r
which t h e f u n c t i o n a l
Cfl,
Cfl, * lfdv, is well defined, f o r every v
E
r.
E
1 L (A),
The weakest ( c o a r s e s t ) topology and uniformity
on L'(A) reading a l l t h e s e f u n c t i o n a l s continuous i s denoted by o(l'). topology i s given by t h e family o f semi-norms {q, : w
E
r),
This
where
2 . Some r e l a t i o n s between t o p o l o g i e s As u s u a l , t h e weak topology
o(X,Xl)
on a 1 . c . t . v . s . X r e f e r s t o t h e
weakest topology c o n s i s t e n t with t h e d u a l i t y between X and X', t h e continuous dual of X , i . e . t h e weakest topology under which t h e a p p l i c a t i o n x * x
E
X, i s continuous, f o r every x ' 1
t h e space L ( A ) i s ? ( A ) .
E
XI.
( X I ,
z),
The n a t u r a l topology considered on 1
I n o r d e r t o determine t h e weak topology on L ( A ) it
1 i s necessary t o i d e n t i f y t h e dual space t o L (A), t h e s e t of a l l T(A)-contin-
1 uous l i n e a r f u n c t i o n a l s on L ( A ) .
THEOREM 1. For every T(A)-COntinUOUS linear functional
e x i s t s a measure 11
E
1
on L ( A ) there
c a ( S ) , a measure h E A . m d a constant k such that 1111
k l h l and
dCf1,)
Ip
=
5
SOME RELATIONS
42
Conversely, i f II
E
111.2
ca(S) i s such that there i s a h
E
lpl
A and k 2 0 w i t h
2
klAl,
1
then (1) defines a T(h)-COntinU#US linear functional 9 on L ( A ) . The weak topoZogy on L 1( A ) i s the topology O ( i - 1 , where r i s the s e t of a l l measures
p E
ca(S) f o r which there i s h
e A
and k
2 0
w i t h 1p1
2
klhl.
P r o o f. Since T(A) is the topology defined by the family of semi-norms tpA : h
E
A } where pA(CflA) =
(Ifldlhl, [fl,
E
1 L (A),
a linear functional
1
L ( A ) is T(h)-continuous if and only if there exists a k 2 0, and h
IP
on
A such
E
that
I f , for every E
that
E
S, we put u ( E ) = I P ( [ X ~ ~ then ~ ) , (2) gives that p
E
r
ca(S) and
1111 5 k l h ] .
The converse, namely that every functional li
E
1
is linear and T(A)-COntinUOUS on L ( A ) ,
IP
defined by (l), for some
is also easy to establish.
Now the statement concerning the weak topology on L'(A) is an immediate consequence of the definition of o ( r ) . From the definitions it is to be expected that the topologies p(A are stronger (finer), in general, that T(A). every measure A
E
A belongs to some Aa.
: a E A)
This is clearly the case if
Instances when the reversed relation
hold are potentially of considerable importance. Such a case is described in the following. THEOREM 2 .
Assume t h a t A
c
ca(S) and t h a t , f o r each A
E
A , A A c ca(S) i s
a bounded s e t of measures unifomnly absolutely continuous with respect t o A. Let A = uAEAAA.Then the application
FUNCTION SPACES
111.2
43
i s a continuous nlapping from L c o , 1 3 ( ~ )equipped with the topology ~ ( hi n) t o with i t s topology p ( A A
Llo,l,(A)
:
A
E
A).
The proof of t h i s Theorem w i l l follow immediately from t h e following Lemma i n which A c o n s i s t s of a s i n g l e element.
We n o t e here t h a t i f A i s a bounded
family of measures uniformly a b s o l u t e l y continuous with r e s p e c t t o A , then
S(A)
) on t r i v i a l l y t h e ~ ( h topology
i s s t r o n g e r than t h e p ( A ) topology.
The
Leiiuna, i n e f f e c t , shows t h a t t h e r e l a t i o n extends t o t h e closed convex h u l l of S(h).
LEMMA 1. Suppose that A
c a ( S ) and that A
E
c
c a ( S ) i s a bounded f a m i l y of
measures u n i f o n l y absolutely continuous with respect t o vergence on L
C0,lI
P r o o f.
v
E
Then r(A)-con-
h.
(A) implies p(A)-convergence.
Let a > 0 be a number such t h a t llvll = I u / (7') 5 a f o r every
A.
Suppose fn while sup{/lfn
E
MLo,l,(S),n
- fldlu1
:
v
E
= 1,2 A]
+
,...
,f
E
MCo,l,(S)
0 i s not t r u e .
/Ifn
and
- fldlvl
+
0,
( I t s u f f i c e s t o consider
sequences and not general n e t s s i n c e both p ( A ) and ~ ( h a) r e pseudo-metric Then t h e r e e x i s t s an
topologies.)
E > 0,
an i n c r e a s i n g sequence Ink} of n a t u r a l
numbers and a sequence I v k l of measures i n A such t h a t
... .
/If
- fldlukl
t E,
for
nk
By assumption t h e r e e x i s t s a 6 > 0 such t h a t Iukl(E) < ~ / 2
a l l k = l,Z,
f o r a l l k and a l l E a subsequence of {f
E
S with A(E)
< 6.
Since f n + f i n IA1-measure, t h e r e i s
k
1 which tends t o f IA1-almost everywhere. We can assume
k, t h a t it i s
If
I i t s e l f [otherwise it would s u f f i c e t o s e l e c t t h e corresponding
nk subsequence of {vkl. 6 and f,
k
By
Egorov's Theorem, t h e r e is a s e t E
-+f uniformly on T - E6 '
6
E
S with J h ] ( E 6 )<
111.2
SOME RELATIONS
44
S
( t )- f ( t ) l : t
J ~ k l ( E 6+ a sup{]f
E
T - E6).
nk
( t )- f ( t ) l
Now, sup{]f
nk Consequently / I f n
t
:
T - E6 1
E
- f ] d ] v k ]<
k
<
~ / 2 a ,f o r k s u f f i c i e n t l y l a r g e .
for a l l k sufficiently large.
c,
This contradicts
our i n i t i a l assumption.
Let A
THEOREM 3 .
c a ( S ) and l e t Q be the set of a l l measures 11 E ca(S)
c
such that there e x i s t s A
A with p
E
Q
A.
Then the s e t s L c o , l l ( A ) and L r o , l l ( 0 )
coincide, and also, the topologies T ( A ) and
P r o o f.
f o r every p
If
l]f - g l d l h l
T(Q)
= 0 f o r every A
coincide on L L o , l , ( A ) =
A then
E
Q, s i n c e f o r every Y , E Q t h e r e i s a X
E
fI]
o t h e r hand A
c 0,
llf - g l d l p l
= 0 , f o r every v
and so,
- gldlAl E
Q.
E
(If
A with p
0 , f o r every A
E
That is Lco,i,(A) = L
- gIdlp1
* A.
= 0
On t h e
A , i f and only i f
c0.11
( 0 ) as s e t s .
The i n c l u s i o n A c 0 g i v e s t h a t T ( A ) is a weaker topology than r(i2). Conversely, assume t h a t a n e t {fa) of elements of L I o , l l ( A ) = L L o , l , ( Q ) tends t o f
E
L c o , l l ( A ) i n T(A),
but not i n
T(Q).
does not tend t o f i n ~ ( p )( i . e . i n L1(Ip))). Clearly f a
with p 4 A .
+
Then f o r some p
E
Q, {fa)
By d e f i n i t i o n t h e r e i s a A
f i n ~ ( h ) . Now t h e choice of A
E
A
i n t h e Lemma 1
yields a contradiction. 1 1 I f A and s1 a r e as i n Theorem 3 , then obviously L ( A ) and L ( O ) need not
coincide.
Also
7(Q)
can be a s u b s t a n t i a l l y s t r o n g e r topology than T ( A ) even
on sets where both t h e s e t o p o l o g i e s a r e d e f i n e d . I t can happen t h a t on S ( A ) topologies t h a t d i f f e r g r e a t l y on t h e whole 1
space L ( A ) , coincide. L E M 2.
For i n s t a n c e
Let A c c a ( S ) and Let 0 be the s e t of measures
p E
ca(S) for
FUNCTION SPACES I
111.3
which there e x i s t s a A
A with
E
IF]
5 ]A].
45
Then on S ( A ) , the topoZogies T ( A )
and a(@) coincide. P r o o f.
than T ( A ) ,
1 Since u(Q) i s c l e a r l y a weaker topology (on t h e whole of L ( A ) )
it s u f f i c e s t o show t h a t any n e t {CEalA1acAo f elements o f S(A) which
i s u(@)-convergent t o an element [ElA Let h
E
E
S(A) converges a l s o i n r(A) t o [El,.
Then, from t h e d e f i n i t i o n of 0 and t h e topology o(@), i t follows
A.
that )A1(Ea n F )
+
IXlCE n F ) ,
have )Al(Ea - E )
-t
0.
= [ h l ( E - E n Ea) =
a E A,
f o r every F
E
Choosing F = F - E , we
S.
Choosing F = E g i v e s IAI(E, n E )
1x1 ( E ) -
have IAl(Ea a E ) = IAl(E
IAl(E, n E )
+
IAl(E), or IAl(E
- Ea)
Combining t h e s e two r e s u l t s we
0.
- Ea) + JAI(Ea - E )
+
.+
0, a
E
A.
I f A c o n s i s t s of a s i n g l e measure, A , say, then r ( X ) is t h e L1(IAI) topology.
1 Lemma 2 shows t h a t on t h e subset S(A) of L ( J A I ) , which we can a l s o
regard a s a subset of L m ( A ) , t h e norm topology o f L1 c o i n c i d e s with a topology which i s ( i n appearance) weaker than t h e weak* topology. 1
1
t h e weak topology of L (]A]),
norm topology of L ( ] A [ ) ,
Hence on S(A), t h e and t h e weak* topology
of Lm( I A I ) a l l coincide.
3 . Completeness The b a s i s of sucess of modern i n t e g r a t i o n theory is perhaps t h e fact t h a t 1
t h e space L ( A ) i s complete (with r e s p e c t t o i t s n a t u r a l norm).
A slight
extension of t h i s f a c t i s c a ( S ) i s a countable s e t , then L1(A) i s T(A)-COmplete.
THEOREM 1.
If A
P r o o f.
F i r s t l y observe t h a t a s A is countable, T ( A ) i s a m e t r i z a b l e
topology.
c
Indeed, it i s determined by a countable family of semi-norms, and
COMPLETENESS
46
111.3
such topologies are well known to be metrizable. Now the problem of completeness is reduced to the question of whether every r(A)-Cauchy sequence {Cf,lA1 1
an element of L (A). {Ak :
k
This is easily settled by the diagonal process.
... 1 . Since {lfnlA} 1,'2,. . . . Let {[f 1 1 be In A
= 1.2.
every k =
1
of elements of L ( A ) is T(A)-convergent to
is r(A)-Cauchy, it is r(hk)-Cauchy, for a subsequence of ICfnlA1 such that { f i n ]
is X -a.e. convergent. Once a subsequence a subsequence {Cfk+l
Let A =
n l A j , of {CfknIA1,
I[fknlA}
has been selected choose
such that {fktl n} is hktl-a.e. con-
vergent. Then {fn n 1 converges hk-a.e. to a function f, for every k = 1 , 2 , Clearly f is hk-integrable and {Cf J
... .
1 converges in r ( A ) to Cfl,. 'k
This result can't be extended much further. Indeed, consider the following EXAMPLE 1. Let T = CO,lI, S the system of Bore1 sets of C 0 , l l . and A the family of all finite measures carried by finite and countable sets. Then it is easy to establish that a function is A-integrable if and only if it is bounded, and two functions can't be A-equivalent unless they coincide everywhere on 5".
Thus L1( A ) is the set of all bounded measurable functions. It 1
is further clear that if a net of uniformly bounded functions in L ( A ) tends point-wise to a function (measurable o r not) then it is r(A)-Cauchy.
If the
limit is not measurable, however, then the net is not convergent in L'(A).
In this situation the best we can hope for are conditions guaranteeing completeness. The following Theorem reduces the question of completeness of 1 1 L ( A ) to the question of completeness of a meager subset of L (A).
THEOREM 2 .
Suppose A
if S(A) is T(A)-complete.
c
ca(S).
Then L1( A ) is T(A)-complete if and only
111.3
FUNCTION SPACES I
47
P r o o f.
Since S ( A ) i s ( i d e n t i f i e d with) a T(A)-closed subset of L 1( A ) , 1 t h e completeness o f L ( A ) implies t h a t of S ( A ) .
s(A)
Conversely assume t h a t
i s T(A)-complete.
1 Cauchy n e t of elements i n L ( A ) .
Cauchy i n L 1(IAl). t h a t limiEIEfi7, Let E:
For each A
E
{CfilA}iEr
Let
A the net
be a r(A)-
{Cfil,3i61 is T ( A ) f, E L 1((At) such
Since L 1C l A l ) i s r(A)-complete t h e r e i s an =
= {t :
If,],
i n the T ( A ) metric.
fA(t)5 y ] f o r y
E
R, A
E
A.
Given a f i n i t e number of
..
E A, j 1,2,. , k , t h e r e e x i s t s a set i n S belonging t o every j , j = 1,2 ,...,k . In f a c t , i f 11 = lk ( A I , then {CfilpIiE1 is r ( u ) j-1 j
measures A
~j 1 Cauchy, hence t h e r e i s a l i m i t of t h i s n e t i n L (1111). i n t n e m e t r i c ~ ( l i ) . Since ( A .( 5 p, we have f 3 u klence {t : f,(tj 5 y~
E
i ~ y X . 1,~j =
1,2
E
,j
CfA 1, j
,...,k .
CfUl,,
Let
= limiEICf.i
= 1,2,..
t11
.,k.
j
~j Let K be t h e system of a l l f i n i t e s u b s e t s o f A d i r e c t e d by i n c l u s i o n . K E
K,
IC
= {A
j
: j = 1,2
,...,k } ,
Then {LelAIKEK i s T(A)-Cauchy. t h a t f o r K'
3 K ~ , K " 3
choose
K~
{A].
IAI(I$
A
=
EY) =
K
Let E y for
0,
x
E
3 be such t h a t
E
,...,
[ E y h l x , j = 1,2 k , be a r b i t r a r y . J j I n f a c t given X E A , t h e r e i s a K~ E K such l e t:5i
we , have {A((<,
~
r$JA
= limKEKCE;lA. I t follows t h a t
E A.
z > y, z E 2.
i n t e r s e c t i o n of a l l EZ f o r
E
E
IR put
t o be t h e
q) = 0 ,
A
Z , and (A](<
@
A
EZ) =
since
0, for
9 z
E
A.
Let f ( t ) = i n f I z : z
E
Z, t
E
If}, t
every y E J R . The r e l a t i o n IAl(FyA A-a.e., hence f,
limiEr[filA
For y
We have I A l ( @
i s t h e i n t e r s e c t i o n of E: f o r a l l z > y, z E
For t h i s it i s enough t o
A E z l , ) = 0.
Suppose 2 i s a countable dense subset of R.
and A
For
= Cfl,
E
Cfl,,
A E A.
E
8) = 0 A
2'.
Then
fl
= {t : f(t) 2 y ) , f o r
implies t h a t f ( t ) = f,(t) holds
In o t h e r words f i s A-integrable and
i n t h e T ( A ) topology.
Z
48
COMPLETENESS
As
L
c0,1:
(A)
111.3
is a T(A)-closed subset of L 1 ( A ) we have immediately the
COROLLARY. S ( A ) is r(A)-compZete if and only if Lro,l;(A)is -c(A)-compZete. LEMMA 1 . Suppose A
ca(S!.
c
If, for a bounded subset A of ca(S), p ( A )
is defined on S ( A ) , then S ( A ) is p(A)-compZete. SimiZarZy if p ( A ) is defined 1
1
on L ( A ) , P r
then L ( A ) is p(A?-compZete. f.
Suppose { En 1 is a pfA)-Cauchy sequence in S f A ) . Since we only have to prove the existence of a convergent subsequence of I E 1 we may suppose -n m that pA(En+l A En)' < 2 , n = 1.2 ,.,. , Put E = limsupEn = (n~=IUm=nEm), Then E
E
S. B
o o
A
En c UiZn!Em a Em+,), and p A ( E A En)
is p A ( E A E n ) .+ 0 as n
5
Z,"=,pAfE,,,
A
Em+l)
-n+l
< 2
.
That
+ m.
1
Suppose p ( A ) is defined on all of L ( A ) , and {f n } is a p(A)-Cauchy sequence 1 in L ( A ) . In line with our remarks above we can suppose pA(fn+l - fn' = ~ u p { / l f ~ +- ~fnld(61 fn+l
: 6 E A}
- fn, ~1 = 1,2,...
.
< 2-n,
.
for n
1,2,,.,
Define g1
= f i *gn+1
Then Z;=,/1gnldl6l converges and so, by Beppo Levi's
exists 6-a.e. and is 6-integrable, for each 6 Theorem f = 1" n = lgn
E
A.
But
m
sup{jlf
- fnld161
: 6 E A } = supI/l
Z
1 gmld(61 : 6 m=n+l
E
A]
m
5
sup{
l(gmld(61 : 6
E
A}
0
as n
5
2-n
m=n+l
for n = 1,2
,... .
Consequently pA(f
- fn)
*
+
m.
THEOREM 3. Let Aa c ca(S), a E A , A = UarAA a* Asswne that p(Aa : a 1 1 is defined on L ( A ) . Then L ( A ) is p(Aa : a E A) compZete if and only if S(A)
E A)
is. P r o o f. This Theorem can be proved in a similar way to Theorem 2 .
fact, if instead of
~ ( h convergence )
for some X
E A,
we substitute p(Aa)
In
111.4
FUNCTION SPACES I
convergence f o r some a
49
A , and apply Lemma 1, t h e proof follows e a s i l y .
E
4. L a t t i c e completeness THEOREM 1.
1
The spaces L ( A ) and L m ( A ) are r e l a t i v e l y a-complete linear
l a t t i c e s , and S(A) i s a Boolean o-algebra. The application f * CflA is a linear l a t t i c e o-homomorphism of the linear
l a t t i c e of a l l A-integrable functions (resp. a l l bounded S-measurable functions) 1
onto L ( A ) (resp. onto L m ( A ) ) .
The application E
++
[ElAJ i s a a-homomorphism
of the Boolean o-algebra S onto the Boolean algebra S ( A ) . The proof follows immediately from t h e c l a s s i c a l theorems s i n c e each element h
E
A i s a-additive.
THEOREM 2 .
I f S ( A ) i s r(h)-complete, then S(A) is a complete Boolean
1
algebra and L ( A ) and I . - ( A ) are r e l a t i v e l y complete linear l a t t i c e s . P r o o f.
Let A be a family of elements of S ( A ) .
has a l e a s t upper bound i n S A ) . inclusion.
For every h
E
W e can assume t h a t A is d i r e c t e d upwards by
A , t h e n e t { [ E l h : [El,
i t s l e a s t upper bound i n S ( h
.
We have t o prove t h a t A
E
A1 i s convergent i n S(A) t o
Hence t h e n e t ICE],, : [ElA E A) i s r(A)-Cauchy
and so, by assumption, i t i s convergent.
C l e a r l y , t h e l i m i t i s t h e l e a s t upper
bound. m
The proof concerning L ' ( A ) and L ( A ) i s s i m i l a r . There i s an i n t e r e s t i n g o b s t a c l e t o t h e converse of t h i s Theorem r e l a t e d t o t h e e x i s t e n c e of mepsurable c a r d i n a l s . A s e t T i s s a i d t o have measurable c a r d i n a l i f t h e r e e x i s t s
negative o - a d d i t i v e measure T on t h e system 2
T
a f i n i t e non-
o f a l l s u b s e t s of T which
111.4
LATTICE COMPLETENESS
50
vanishes on every f i n i t e s e t but does not vanish i d e n t i c a l l y . EXAMPLE 1.
Let T be a s e t with measurable c a r d i n a l .
algebra of a l l s u b s e t s of 2'.
Let
by t h e p o i n t t , i . e . 6 t ( E ) =
single s e t .
xE ( t ) ,E
{ E l , f o r every E
clearly [ E l ,
u be a u - a d d i t i v e p r o b a b i l i t y measure on S which
For every t
vanishes on every f i n i t e s e t .
c
Let S = 2 4 be t h e u-
E
E
S.
T, l e t 6 b e t h e Dirac measure c a r r i e d t Set A = : t E !7'1 u { u 1 . Then,
S, i . e . every A-equivalence c l a s s c o n s i s t s of a
I t follows t h a t S(A), being only formally d i s t i n c t from S , i s a com-
However, S(A) i s not T(A)-complete.
p l e t e Boolean a l g e b r a .
s e t s d i r e c t e d by i n c l u s i o n i s T(A)-Cauchy. ~ ( 1-Cauchy 6 convergent t o T , f o r every
t
t
The n e t of a l l f i n i t e
I t c a n ' t be r(h)-convergent s i n c e it i s
E
T, while it is T(p)-convergent t o
p.
Fortunately, t h e c a r d i n a l i t y of a set having measurable c a r d i n a l must be t o o l a r g e t o occur i n any reasonable a p p l i c a t i o n .
Hence, i n p r a c t i c a l l y a l l
cases t h e next Lemma and Theorem a r e v a l i d . To shorten t h e formulation, a s e t A
c
S(A) i s s a i d t o be d i s j o i n t i f
[El, n CFIA = [@I,, f o r any two d i f f e r e n t elements [ E l A , CF1, of A. L E b W 1. Assume that S ( h ) i s a complete Boolean algebra and t h a t there
is no d i s j o i n t s e t A Then f o r any X [El,
E
A , Chl(CE1,)
S ( A ) having measurable cardinal.
c E
A and any d i s j o i n t s e t
* 01
A
c S!A),
the s e t 0 = { [ E l A
:
of elements i n A w i t h non-zero measure EhJ i s a t most
countable and
P r o o f.
Let h
E
A ; assume without l o s s of g e n e r a l i t y t h a t X 2 0 .
Let
FUNCTION SPACES I
111.4
A be a d i s j o i n t s e t of elements o f S ( A ) .
51
S in ce I A J i s o - a d d i t i v e on S ( A ) and
S ( A ) complete Boolean a l g e b r a , t h e series
i s convergent f o r e v e r y c o u n t a b le s e t E c A .
Dn
number IZ, t h e s e t empty).
Consequently
{[ElA
:
[ElA
E
I t f o l l o w s t h a t , f o r ev er y n a t u r a l
A , [Al(IEI,,)
2
l / n l is f i n i t e (possibly
D = Urn D i s a t most c o u n t a b l e . n=l n
For every s e t E c A, l e t
i s a f i n i t e , u - a d d i t i v e , non-negative measure on t h e system of a l l s u b s e t s
Then
of A which c l e a r l y v a n is h e s on e v e r y f i n i t e s e t .
S i n c e , by assumption, A does
no t have measurable c a r d i n a l , ~ . lv a n i s h e s i d e n t i c a l l y .
A l l s t a t e m e n t s t h en
fol l o w. THEOREM 3 .
A
c
I f S(A) i s a complete Boolean algebra and i f no d i s j o i n t s e t
S ( A ) has measurable cardinal, then S ( A ) i s T(A)-compZete.
P r o o f.
Let {CEalAIacA be a ~ ( A ) - C a u c h yn e t o f el em en t s of S ( A ) .
e ver y f i n i t e set K
c A,
T ( K ) i s a complete metric topology on S ( K ) .
n e t I[EalKlucA i s T(K)-convergent; l e t i t s . r ( K ) - l i m i t be CEKIK.
for ev er y
h E A,
T(K)-limits
For
Hence t h e
Now l e t ,
be t h e union i n S ( A ) o f a l l elements [EKIA co r r esp o n d i n g t o
[ E 1 f o r a l l f i n i t e sets K K K
c A
containing A .
Finally, l e t
111.4
LATTICE COMPLETENESS
52
To prove t h a t [ E l , i s t h e T ( h ) - l i m i t of {CEalA3aEA,it s u f f i c e s t o show t h a t
[El, = CE,IU, f o r every
!J E
A, s i n c e CE 1
lJi-1
i s the T ( p ) - l i m i t
of {CEaIpIaEA.
= CE 1 f o r every f i n i t e s e t K c A with LI E K . Since [ p l K V' CE,,l,,. Further i f i s completely a d d i t i v e by Lemma 1, (1) g i v e s t h a t [ F 1 Clearly,
!J!J
1
f
!J
then [ E I h
,!J
) I X = [EXIX.hence (1) g i v e s t h a t CFXI!J3 CF 1 i n S ( ? J ) . Then lJlJ
t h e complete a d d i t i v i t y of p on S ( A ) g i v e s , by v i r t u e o f ( 2 ) . t h a t CEl
n
= !J
C F I = [ P I =LEI. XEA I. li P P iiu THEOREM 4 .
I f there are no measurable cardinals, L1(A) i s T(A)-COmpZete
i f and only i f it i s a r e l a t i v e l y complete linear l a t t i c e . P r o o f.
1 By Theorem 3 . 2 , L ( A ) i s r(A)-complete i f and only i f S(A) i s .
1 Further, L ( A ) i s a r e l a t i v e l y complete l i n e a r l a t t i c e i f and only i f S ( A ) i s a
complete Boolean algebra (C571, Theorem 42.9). Hence t h e r e s u l t follows by Theorem 3 . THEOREM 5.
If 0
c
r
c A c
c a ( S ) and if S ( A ) i s -r(A)-compZete then S ( r ) i s
r
T ( )- eomp Zete ,
P r o o f.
Let {CEalr}aEAbe a ~ ( r ) - C a u c h yn e t of elements of S ( r ) .
every non-empty f i n i t e s e t K c
r,
For
t h e n e t {CEalKlaEA is ~(K)-Cauchyand, s i n c e
t h e topology T(K) i s a complete metric topology on S ( K ) , t h i s n e t { C E a l K I a E A i s
T ( K ) convergent i n S ( K ) ; l e t i t s .r(K)-limit be CEKIK. Now, f o r every X
A . l e t F A E S be a set such t h a t
E
i . e . [ F A ] , is t h e union i n S ( h ) of a l l elements CEKIA corresponding t o T(K)l i m i t s [E I
K X
Theorem 2 .
for a l l f i n i t e sets K Also l e t E
E
c
r
S be such t h a t
containing 1. This s e t e x i s t s by
FUNCTION SPACES I
111.4
in S(A).
Theorem 2 again quarantees t h e e x i s t e n c e of E .
is t h e T ( r ) - l i m i t of ICEa 3 f o r every h Let X
r,
E
E
1
We s h a l l show t h a t [ E l r
For t h i s it suffices t o show t h a t CEl, = CE,I,,,
s i n c e [EXlx i s t h e T(X)-Iimit o f ICEulllaEA. The r e l a t i o n (3) implies t h a t [FXIX= [ E l l h .
I?.
CEKIX,o r lAl(E, ClXjl,
53
A
E ) = 0, f o r any f i n i t e s e t K K
i s completely a d d i t i v e on StA),
c
r
with A
E
Indeed, CExIX =
K , and, by Lemma 1
so
and s o [FXlx = CE,I,. Further t h e e q u a l i t y [ E I u , X l 1x = CE X 1X g i v e s t h a t
f o r every
p E
r.
As
C l A l l , i s completely a d d i t i v e on S(h), by (4) we o b t a i n
WEAK COMPACTNESS
54
111.5
This means that l A l ( E A FA) = 0, or [El, = [FAIA. S o , the equality [El,
IFXI,
is proved.
5. Weak compactness
In this section we give another two conditions for the T(A)-COmpkteneSS of L1(A).
r
3
A.
One is expressed in terms of a(I')-compactness of L
[O,ll
( A ) for some
If A is a linear lattice in ca(S), the other ties the problem of
completeness of L1(A) to the representability of linear functionals on A by bounded measurable functions. MEOREM 1. Let A
c
ca(S) and let
r
be the set of a21 measures
f o r which there exists a constant ;C and a measure A with l p l
5
1
The space L ( A ) is T(A)-complete if and only if L L o , l , ( A )
ii 6
ca(S)
klAl. is a(I')-compact.
P r o o f. If Y stands for the 1.c.t.v.s. L 1 ( A ) with the topology T(A) then, according to Theorem-2.1,the statement of the Theorem can be reformulated as follows: Y is complete if and only if its subset LLo,ll(A) is weakly compact. Since in any 1.c.t.v.s. a weakly compact set is complete, the o(T)-compactness of L r o , l I ( A )
implies the T(h)-completeness of L
[O,ll
and consequently,
(A),
1
by Theorem 3 . 2 and its Corollary, the r(A)-completeness of the whole of L ( A ) . 1
Conversely, assume that L ( A ) is T(!i)-COmplete. Since Lco,l,(A)
is a closed
subset of Y = L1(A), the set W = LLo,ll(A) is itself complete. Since it is a convex set, by Theorem 1.1.1 it suffices to show that the natural projection of
W into the space Y/p-l(O), for every semi-norm p from a family determining the topology of Y, is a weakly compact set in Y / p
-1 1 (0). The topology ? ( A ) on L ( A )
is defined by the semi-norms [f], *pA(CflA) = lAl(f),
A
E
A.
[fl,
E
1
L ( A ) , for every
1
Clearly, Y/pil(0) = L (A) and the projection of W = L
c0,11
(A)
into this
FUNCTION SPACES I
111.5
space is L
C0,ll
(A), for every A
E
A.
55
It is well known that LCo,l,(A)is a
weakly compact set in L 1(A) (e.g. C171 IV.8.11). COROLLARY 1. Suppose there e x i s t s a A
E
A
Hence the Theorem i s proved.
i s t h e s e t of aZZ measures p
Cl
w i t h p < A.
ca(S) f o r which
E
Then S ( A ) i s r(A)-complete i f and onZy if
LLo,ll(A)i s o(Q)-compact. P r o o f.
Obviously S(A) = S(n) and Lco,ll(A)= L r o , l , ( Q ) as sets.
Clearly, S ( A ) is T(l\)-COmplete if and only if it is T(Cl)-complete, and s o by 1
Theorem 3 . 2 , if and only if L (a) is T(Cl)-complete. I.I E
ca(S), f o r which there is a k
2
0 and a A
E
Cl
Since the set of measures
with 1 1 ~ 1
1
L ( a ) is T(Cl)-complete if and only if L
Lo,l,(Cl)
LC 0 , l l
S
k l A l , is
Cl
itself,
( A ) is o(Q)-compact.
I f S ( A ) is -t(l\)-complete t h e n LCo,ll(A)i s o(A)-compact.
COROLLARY 2 .
1 P r o o f. If S(A) is r(A)-complete, then L ( A ) is r(h)-complete and
LEo,ll(A) is o(r)-compact.
The topology o(A) is weaker (coarser) than o ( T ) ,
hence LCO,ll(A)-compact. The continuity of a functional defined on ca(S) is understood with respect to the norm
IJ
* IIpll = l d ( T ) ,
r
a linear space for every
II E
c
p E
ca(S).
A continuous linear functional 9 on
ca(S) is said to be represented by f
E
BM(S) if IP(IJ) = p(f),
r.
The set of all continuous linear functionals on a linear lattice is a relatively complete lattice. For such a functional v we write for every for every
r,
IJ E
p 2 0, we
p 2 0, IJ E
THEOREM 2 .
functionaZ on
r
Let
have
~(11)2
0. Similarly
Ip 5
1 if
v
r
c
ca(S)
2 0,
if
v ( p ) 5 ~ ( 1 ) IJ(T),
r.
r
c
ca(S) be a linear Z a t t i c e .
Every continuous l i n e a r
i s represented by an element of BM(S) i f and onZy i f Lco,l,(r)
111.5
W,AK COMPACTNESS
56
is a(I')-compact. If this is the case, then 0 5 9 5
1 is represented by a function i n
every functional 9 such t h a t
BM i 0 , l l ( S ) .
If BM(S) gives t h e whole o f t h e dual o f 'l then L C o , l , ( r ) i s
P r o o f.
a(r)-compact by t h e Banach
-
Alaoglu Theorem s i n c e t h e o ( T ) topology on L m ( A )
i s then t h e weakstar topology.
Assume t h a t LCo,ll'r) i s o ( r ) compact, and l e t f u n c t i o n a l on
r.
< 1, hence 9
llipll
Without loss of g e n e r a l i t y
1.
5
We a r e going t o prove t h a t t h e r e i s an f f o r every p
r.
E
Since L
any f i n i t e c o l l e c t i o n
i
with ~ ( p . )= pi(f),
C0,ll
(r)
1.2
...,k .
a compact, convex subset of IR
As
, if
the
t o it, t h e r e would e x i s t numbers a l , E
BM
( S ) such t h a t ~ ( u =) ~ ( f ) C0,ll i s o ( r ) compact it s u f f i c e s t o show t h a t , f o r E
u 1 > * * . ,uk of elements of r, t h e r e i s an f k
f
be a continuous l i n e a r
As 9 can be w r i t t e n a s t h e d i f f e r e n c e of two p o s i t i v e continuous
l i n e a r f u n c t i o n a l s , assume t h a t 9 i s p o s i t i v e . assume a l s o t h a t
9
E
BMCo,ll(S!
,...,uk(f)) : f E BMCo,lI(S)l is v e c t o r ( @ ( y l ) ,. .. , v ( k~) ) d i d not belong
{(ul(f)
. . . ,ak
> sup{&aipi(f) k such t h a t &aiq(pi) k
:
On t h e BMCo,l,(S)l o r t h a t v(~laiui) k . > s ~ p I k( ~ ~ a ~ u :~fi )E ( BMCo,lj(S)l. f)
o t h e r hand
f o r any
u
E
r,
hence f o r p
1k1ai
i'
Given a non-negative, not n e c e s s a r i l y f i n i t e , measure A on S , t h e space
L1(A) of a l l A-integrable functions i s considered n a t u r a l l y included i n c a ( S ) 1
every element o f L (A) i s represented by i t s i n d e f i n i t e i n t e g r a l a s an element of c a ( S ) . 1
then L ( A )
For i n s t a n c e , i f A i s a l o c a l i z a b l e measure (C731, Theorem 5.1) Sl, where
a i s t h e s e t of a l l measures
v
E
c a ( S ) such t h a t v
A.
:
FUNCTION SPACES I
111.6
57
COROLLARY. A measure space ( T , S , A ) i s l o c a l i z a b l e i f and only i f LCo,ll(h)
P r o o f.
1
Since L[o,ll(L( A ) ) = L [ o , l l ( A ) as sets, the result follows
from the Theorem and the definition if a localizable measure space (Section 1.3).
6. Completion Theorem 3 . 2 reduces the question of T(A)-COmpleteneSS of L ' ( A ) t o that u f In this section we show that if S ( A ) is n o t T(i\)-COmplete, then we can
S(A).
find a a-algebra S of subsets of a compact, llausdorff space T , and a family of A
measures
A c
^
ca(S) such that S(A) is T(A)-complete, and S(A) can be identified ^
^
with a dense subset o f S ( A ) .
THEOREM 1. Let S be a a-algebra of subsets of a s e t T .
Then there e x i s t s
a compact, Hausdorff space T , a o-algebra S of subsets of T and a subspace A c ca(S) such t h a t t h e following statements hold A
Ci)
T c T ; S = S n T = {E n T : E
(ii)
S(A) is T(A)-complete. >
(iii) To each E that E
c
S , an element E
E
.
.
E
S}.
A
E
S can be assigned i n such a mann'er
E , E = T n E , and t h a t the map E * [ E l A i s a uniform isomorphism o f
S onto a dense subset of S ( A ) .
(iv)
For each A
E
ca(S) there is exactly one A
where E corresponds t o E as i n (iii), f o r each E h -+ A
S.
A such t h a t A ( E ) =
The correspondence
i s a l i n e a r isometry of Banach spaces ca(S1 and A .
P r o o f. T
E
E
r(ca(S)).
Let B b e the completion of the space S with the uniformity
Since the operations of intersection and symmetric difference
COMPLETION
58
111.6
( a l s o the union) a r e uniformly continuous on S, they can be extended by c o n t i n u i t y onto B.
Hence we can consider 8 t o be a Boolean a l g e b r a which i s a complete
uniform space i n a uniform s t r u c t u r e which we denote by
T'
and S i s a dense
subset of i t By S t o n e ' s theorem 8 i s isomorphic t o t h e Boolean algebra B of a l l closed and open s u b s e t s of a compact Hausdorff space T.
The space T i s constructed as
t h e s e t of a l l homomorphisms of t h e Boolean algebra B i n t o a two element Boolean Since f o r each t
algebra { O , l } c a r r i n g t h e maximal element of 8 i n t o 1.
T,
E
i s uniquely extendable t o such a homomorphism, t w i l l be iden-
t h e measure cSt
t i f i e d with t h a t homomorphism and hence T with a subset of T. Let S be t h e o-algebra of s u b s e t s of T generated by B . Every h
c a ( S ) i s a uniformly continuous f u n c t i o n on S , s o t h a t i t may
E
be extended uniquely by c o n t i n u i t y t o a continuous f u n c t i o n h' onto B . t h e a d d i t i v i t y o f h implies t h a t of A'.
For each F
E
B, let F
8 be t h e set
E
A
which corresponds t o F under t h e isomorphism of B onto 6 . .
F
E
B.
.
Moreover
*
Denote A ( F ) = A'@),
A
This d e f i n e s an a d d i t i v e f u n c t i o n X on B , and s i n c e B c o n s i s t s of closed
and open s u b s e t s of a compact space, X is a c t u a l l y o - a d d i t i v e , and has a unique S e t A = {A :
a - a d d i t i v e extension, denoted again by A , onto t h e whole of S. E
ca(S)l. Each E
E
S i s a t t h e same time a member of €3.
representing E .
Let E be t h e member of
The c o n s t r u c t i o n of B and i d e n t i f i c a t i o n of T a s a subset of *
T g i v e s t h a t B n !7' = E . For every X E,F
E
E
A
Furthermore A ( E ) = X ' ( B ) = X ( E ) f o r each A
E
c a ( S ) semi-distance d, on S (defined by d A ( E , F ) = lhl(E
S) has a unique continuous extension onto a semi-distance d i on
uniform s t r u c t u r e
ca(S).
7'
a.
on 13 i s given by t h e family of semi-distances d i , A
We t r a n s f e r t h i s s t r u c t u r e onto B i n t h e following way. we f i n d t h e i r r e s p e c t i v e r e p r e s e n t a n t s F,G i n
R and
A
F).
The E
For a r b i t r a r y F,G
,.
A
ca(S'. E
8,
^
w e put d A ( F , G ) = di(F,C).
59
FUNCTION SPACES I
111.7
From t h e c o n t i n u i t y o f extensi0r.s of semi-distances d and measures A it i s seen A
,.
-
1
6
.
.
= lhl(F
t h a t d,(F,G)
A
A
G ) , o r i n o t h e r words d, = d i .
Eience t h e r e s u l t i n g uniform
i s i d e n t i c a l with T ( A ) .
s t r u c t u r e on
A
n
We prove f i n a l l y t h a t B ( A ) = S(A) i n t h e sense t h a t , f o r every E ,
.
is F
E
S , there
A
E
8 such t h a t E
E
[FIA. For t h i s purpose l e t A be t h e system o f a l l sets e
A
,
.
E
E
S f o r which t h e r e e x i s t s an F
.
.
E
.
8 with E
Obviously 8
[FI,.
E
,
{ E n } be a monotonic sequence of elements i n A, with En E = lid
Then { E n } i s T(A)-Cauchy f o r each A
n'
.
with F
E
.
1
8.
Clearly E
E
*
*
.
n
E
Let
8 and l e t
This is t h e same a s saying
A.
.
.
and so A = S .
[Fl,,
n
If A c c a ( S ) and A = { A
THEOREM 2 .
n
A
A.
By t h e completeness of 8, i t has a l i m i t [ F I A
t h a t {[FnlA} i s T(A)-Cauchy. n
E
CFnlA, F
E
L
c
ca(S) : A
E
E
A } i n the notation of
A
Theorem I, then S ( A ) i s .r(A)-complete and S ( A ) i s a dense subset of i t . P r o o f.
Since A
and S(A) i s T(A)-complete, Theorem 4 . 4 gives t h a t
c A,
A , .
A . .
S ( A ) i s T(I\)-complete.
The denseness of S ( A ) i n S ( A ) follows from ( i i i ) o f A
Theorem 1, s i n c e t h e mapping [ E l ,
+
A , .
A
A
IEI-,E
E
A
*
S , is a continuous map from S(A)
A
onto S ( A ) mapping S = S ( c a ( S ) ) onto S ( A ) .
7 . Extreme and exposed p o i n t s
For any E
P r o o f. and s o exL
[O,ll
f(t)
5
1
- E}
S , t h e element
Then t h e r e e x i s t s an
E > 0
i s not A-null, f o r some A
We have ChlA, [glA
E
Lco,ll(A),
c o n t r a d i c t s t h e e x t r e m a l i t y of
[xEIA
i s an extreme p o i n t of L
Accordingly l e t [fJ,
( A ) i s non-empty.
[fl, 4 L f o , l ) ( A ) . I.
E
[hlA
[fJ,.
*
E
E
exLIO,ll(A) and suppose
such t h a t t h e s e t E = { t :
A.
CglA and
[ O , I P
Put g = f
- EX^.
V I A =%([gl,
h =
f +
E S
EX
+ ChlA). This
E'
EXTREME AND FXPOSED POINTS
60
Suppose A
LEMMA 1.
p oi nt s of L
L0,ll
{h : h If h
Let E
then IAl(hfE)
Ihl(x, fE) = IhI(E). while i f h
xE
E
-
x ~ i s- i n~B M ( S ) .
xE
t
- \AIT-+(h)
IAl,(h)
=
11 n L
C0,ll
Consequently H supports L
A-a.e.
/Al(E).
Also,
(A) then -IAlT-+(h) = lXl,(l C0,ll
(A) only a t
xE
-
h),
and so
expLCo,lJ(A). Further, suppose Ih } i s a sequence of elements of L
IXI(hn fE) +
Thus
L1(A), Ihl(hfE) = lhl(S)l is a hyperplane i n L (A). Lco,lJ(A).
that i s h =
xE
S. Then fE =
t
1
t E
Then the s e t s of exposed and strongly exposed
ca(S).
(A) re2ative t o t h e ~ ( h topology ) coincide w i t h S(A).
P r o o f.
X
E
111.7
+
lAIT-E(hn)
+
Consequently hn
0.
For a family of measures A
[O,ll
( A ) such t h a t
IAl(G). Then IXIE(hn) - IX(T-E(hn) + I X l ( E ) , and so I X I E ( l -+
xE
s t r o n g l y exposed p o i n t of L l o , l l ( X ) .
which L
C0,lJ
c
If A
c
SO
xE
hn)
+
is a
The r e s u l t now follows from Theorem 1.
c a ( S ) it i s easy t o c o n s t r u c t examples f o r
( A ) has no exposed p o i n t s .
THEOREM 2 .
i n t h e T ( A ) topology, and
-
However,
c a ( S ) then the strongly extreme points of L L o , l , ( A )
r-elative t o the r(A) topology coincide with S ( A ) . P r o o f.
of ca(S).
F i r s t l y we can suppose by Theorem 2 . 3 t h a t A i s a s u b - l a t t i c e
Suppose E
t
S and CxEIA i s not a s t r o n g l y exposed p o i n t of L C o , l l ( A ) '
Then t h e r e must e x i s t a neighbourhood V o f CxEIA, which we can t a k e of t h e form V = {Cfl, E LLo,ll(A) : p,(Cfl,
p A , such t h a t Cx,lA
{CfalAlacA, V)
t
with CfaJA
- CxE 1A
€1
f o r some continuous semi-norm
o t h)e r G ( L ~ ~ -, v). ~ ~ In (A +
IxEl,,
words t h e r e e x i s t s a n e t
i n t h e T ( A ) topology and [falAE C O ( L [ ~ , ~ - , ( A ) -
. If we d e f i n e VA = V/pi1(0), pA a s above, then t
co(L (A) C0,ll
- VA),
a
E
A.
pA(CfaJh - CxEIA) +
0 and
This means t h a t CxEIA is not a s t r o n g l y
FUNCTION SPACES I
111.8
61
r e l a t i v e t o t h e T(X) norm topology.
extreme p o i n t of L I o , l l ( X l
As t h i s c o n t r a d i c t s
Lemma 1 t h e r e s u l t follows.
8. Vector-valued f u n c t i o n s Let T be a s e t and S a a - a l g e b r a of i t s s u b s e t s .
1 F i r s t l y , t h e v a l u e s of elements of L ( A ) can
3 can be extended i n two ways.
be taken i n any Banach space i n s t e a d o f R
1
.
L e t H be any r e a l Banach space with t h e norm 1 I *!I.
l e t L ( H , A ) be t h e s e t of a l l f u n c t i o n s f : T with r e s p e c t t o every A
E
A.
The r e s u l t s of s e c t i o n
-f
For a set A
c
ca(S),
H Which a r e Bocher i n t e g r a b l e
Two f u n c t i o n s f,g i n L ( H , A ) w i l l be c a l l e d A-
equivalent i f J,+f-gl\a\Xl = 0, f o r every X e A .
Let L ~ ( H , A = )
{c~I,,: f
L(H,A)~,
where [flAis t h e c l a s s of a l l f u n c t i o n s i n L ( H , A ) wLich a r e A-equivalent t o f. On t h i s space we d e f i n e again t h e topology and u n i f o r m i t y T ( A ) t o b e t h e one
determined by t h e family { p , : A
f o r any X
E
E
A ) of semi-norms, where
A , i s t h e Bochner semi-norm.
of elements of H i s c a l l e d a Schauder b a s i s f o r t h e
A sequence
space H, i f every element y e H can be expressed uniquely i n t h e form N
m
... .
where cn ( y ) a r e r e a l numbers, n t h a t i f we put ( y n ‘ , y ) = cn ( y ) , y
H , n = 1 , 2 ,...
E
I t i s well-known (C771 Section 2 . 2 2 ) ,
H , then yh i s a bounded l i n e a r f u n c t i o n a l on
.
THEOREM 1.
every element f
I,”,
Let H be a Banach space with .Schauder basis E
L ( H , A ) can be written uniquely as
!&en
IIl.8
VECTOR-VALUED FUNCTIONS
62
m
where
f is a A-integrable real-valued function f o r each n
f. For any function f
o
.. .
if S ( A ) is r(A)-compZete then so is L 1 ( H , A j .
Moremer,
P r o
= 1,2,.
m
o f real-valued functions If,},=,
: T +
H there exists a unique sequence
such that (1) holds. Furthermore, there exist
constants kn, n = 1,2 ,... , such that If,(t)I
2
knllflt)II, t
T, n =
E
Consequently, if f is 5'-measurable then so is fn, for n = 1.2,.. A-integrable for some X
ca(S), then so is f,,
E
n = 1,2,.
.. .
.
1,2
,... .
, and if f i s
It follows
further that,
The inequality (2) and the uniqueness of the representation (1) implies that if g is another function in L ( H , A ) and g ( t ) = Il,lgn(t)en, t and g are A-equivalent if and only if f n = 1 , 2 ,...
E
T , then f
and gn are A-equivalent for every
.
Assume now that S ( A ) is T(A)-complete. 1
net in L (H,A). Let fact) = ~ ~ = , < ( t ) e n Jt
C C f ~ l A 1 a c A is ~(A)-Cauchyin L
1
(A),
Let {[fal,),,,
k T, a
E
A.
be a r(A)-Cauchy By (2),
for every n = 1,2,.,.
.
the net
Now, by Theorem
1
... .
3.2, this net is r(A)-convergent to an element [f 1 of L ( A ) , n = 1 , 2 , n h We will show that Cfl, E L 1 ( H , A ) and that Define the function f by (1).
Cf1,
is the T(A)-limit of the net {LfalA)aEA. Choose
h
E
A.
1
It is well-known that the space L ( H , h ) is T(h)-complete
("261 Theorem 3.6.1).
x
Consequently, the net { C f a l x ~ a E A must have a limit [f 1,
p(t)
[GI,
lzzlfi(t)en, t E T , the net converges, by 1 h ( 2 ) , to Ifn;, in the topology T ( X ) of L ( A ) , and SO Cf,lA = Cfnlx, n = 1,2,... X 1 Since this holds for every It follows that If], [f I,, or Cfl, E L ( H , X ) .
in this space.
If
2:
.
FUNCTION SPACES I
111.8
63
1
The notation L v ( H , A ) is almost self-explanatory. If V stands for all elements Cfl that f(t)
V for t
E
E
of L'(H,A)
A
c
1
H , then L V ( H , A )
with representatives f
L ( H , A ) such
E
T.
COROLLARY 1. If V
1 H is a closed set then L V ( H , A ) i s a T(A)-cZosed subset
c
of L'(H,A). 1 P r o o f. This is a consequence of the fact that L ( H , A ) is a closed V
subset of L'(H,h)
f o r each h
In fact, by the Chebyshev inequality any
A.
E
r(A)-convergent sequence of elements of L1 ( H , A ) , has a A-a.e. convergent subsequence, and
A
E
so
it easily follows that L1(H,X) is V
~ ( h closed )
in L'(H,A),
A.
COROLLARY 2 .
If V
c
H is a closed set and if S(A) is r(A)-conrplete then
1 L ( H , A ) is T(A)-COmpZete. V
Let us now turn to the second generalization of the results of Section 3 . Let B = B ( l 0 , l l ) be the Bore1 o-algebra on C 0 , l l and let 1 be Lebesgue measure on S
F
1 E
B. In the next Theorem we use the notation T1
T
S o 8 , i.e. S, i s the o-algebra generated by all sets E
B.
Moreover, for any h
measures A and 1.
1
1
c
)
F with E
E
S and
ca(S), we denote X1 = X o 1, the product of the
ca(S), then A1 = {Al
The mapping E
P r o o f.
subset of S (A
If A
E
x
CO,ll, and
x
++
E
x
whose relative T(A')
[0,11,
E
:
E
A
E
A};
so
A1
c
ca(Sl).
S, identifies S ( A ) with a closed
topology is T ( A ) .
Hence the T(A~)-
111.8
VECTOR-VALUED FUNCTIONS
64
completeness of Sl(hl) implies T(h)-COmplet€m3SS of S ( h ) . The interesting part of this Theorem is the converse statement, Assume that S f h ) is r(h)-complete and that {TEaJhl}aeA is a T ( A 1 )-Cauchy net in
S,(A,). We will apply Theorem 1 to the space H = L'!rO,ll), (the classes of) I-integrable functions on [ O , l l ; H is denoted by II*II,
the standard space of
the natural Lebesgue norm on
Tt is known ( r 7 7 1 Example 2,3) that the sequence {?z,);=~
of Haar functions form a Schauder basis for H .
For any set F e S1 and t
Clearly F~
E
E
T we define Ft = {y
:
y e CO.11,
(t,y)E F 1 .
B.
p
H be the function defined by fOL(t) = x t Ea 1 T , where x t represents an element of H . Then rfalA E L V ( H , A ) , where V is Ea
For each a
t
E
A , let
E
: T
+
the set of elements in H whose representatives are characteristic functions of sets in B .
The set V is closed in H .
1 The net {TplA}aEA is T(A~-Cauchy in L ( H . h ) ,
- fB (t)lldlhl4t)
pf%: T
=
J(/lXEt(Y) T
=
t
12(Ea A T
O
t E 1 3 ) d l A l ( t )= Ih,l(E:
Now, by Corollary 2
x
t(y)ldZ(9))dlhI(t)
Ei)
+
0,
a,B e
A. =
A.
Theorem 1, there exists an element rfl,
to
E
El3
a A
Indeed, for any a
From rfl,
which is the T(h)-limit of the net
E
1 Lv(H.h)
we construct an
element of S ( h l ) which is the ~(h~)-limit of the net ICE 1 I a E A . a A1 By Theorem 1, f(t) = ~~=,f,(t)h,, t
f,, n
= 1,2,.,,
.
For every t
E
E
4,with unique A-integrable functions
T, the value of f(t) is an element of V ,
SO
it can be represented by an 2-integrable function on C0.11 taking on values 0 o r 1 at 2-almost every y
E
r0.11.
Denote its value at y
It is known (C771 Example 2 . 3 ) that f,(t)
1
/$(t)(y)
E
C0.11 by f(t)(y).
h n ( y ) d y , n = 1.2,...
.
A
FUNCTION SPACES I
111.8
65
c l a s s i c a l r e s u l t about Haar functions (L771 Example 2 . 3 ) s t a t e s t h a t , given
t
E
T,
m
f(t)(y)
= lim ni-
f o r 1-almost every y
E
E = f(t,y) Clearly E
c
CO.11. :
t
1 fn(t)hn(y)
n=l
Define m
T, y
E
[0,11, I i m
E
1 ~,(t)h,(y)
m ~ n=l o
S1.
The proof w i l l be f i n i s h e d by showing t h a t IXII(Ea A E ) every A1
E
= 1).
A.
Let A 1
E A1,
A1 = X
@
Z with X
E
A.
-+
0, a
E
A , for
Then
I t i s i n t e r e s t i n g t o observe t h a t t h e s e t o f elements [ElA of S,(A,) 1
which have r e p r e s e n t a t i v e s E = {(t,y) function f : T
-+
C 0 , l J i s closed i n
: 0 S
S,(Al).
y s f(t)) f o r some S-measurable F u r t h e r t h e r ( h l ) topology on t h i s
s e t i s t h e same as t h e T ( A ) topology on Lco,ll(A). Hence Theorem 2 g e n e r a i i z e s t h e Corollary t o Theorem 3 . 2 . Theorem 2 i t s e l f can be g e n e r a l i z e d . and t h e family of s e t s o f f i n i t e A Lebesgue measure on t h e whole of A
1
x
2 measure f o r every A
(-m,m).
E
x
(-a,-),
A , where Z i s
Then, i f t h e s e t of t h e s e measures i s
t h e topology T(A ) can be n a t u r a l l y defined and t h e corresponding Theorem 1
stated. T
Q
We could consider t h e space T
Using Theorem 2 and t h e decomposition o f T
[ n , n + 11, n = 0,+1,+2,,..
x
(--,-I
into sets
t h e proof of t h i s extended Theorem can be given.
We do not go i n t o d e t a i l s a s we w i l l have no opportunity t o use t h i s Theorem
I11
REMARKS
66
i n t h e sequel.
Remarks The o r i g i n s of t h e technigue of considering a v e c t o r measure a s a mapping on a s u i t a b l e space is hard t o t r a c e back. i n t h i s d i r e c t i o n d e r i v e s from 1221.
I t was taken over i n C41 and a v a r i a t i o n
I t i s very c o n s i s t e n t l y used i n C21 which w i l l
of t h e approach used i n C173.
be r e f e r r e d t o more i n Chapter V I . there.
Possibly t h e most important stimulus
The proof of Lemma 2 . 1 e s s e n t i a l l y appears
This p o i n t ofview i s a l s o c o n s i s t e n t l y used i n C141.
The topologies ? ( A ) and a(A) were defined i n L391, and t h e Corollary t o Theorem 3 . 2 was proved t h e r e .
The r e l a t i o n of t h i s Corollary t o s p e c t r a l
theory may be worth n o t i c i n g . Section 4 r e l a t e s t h e concept of T(A)-completeness t o t h e concept of a l o c a l i z a b l e measure space C731.
Some i d e a s from t h i s s e c t i o n appear i n
various contexts i n t h e l i t e r a t u r e , i n p a r t i c u l a r 1561.
Theorem 5 . 1 and i t s
Corollary appear i n C391, and Theorem 5 . 2 i n C40l. The t r i c k i n i t s proof was suggested t o us by P . Dodds.
Theorem 6 . 1 i s again from C391.
Theorem 6 . 2 i s
c l o s e l y r e l a t e d t o Theorem 3 . 4 i n S e g a l ' s fundamental paper C731. The method of t h e proof of Theorem 7 . 1 d a t e s back t o C30l. afterwards by s e v e r a l authors.
Lemma 7 . 1 i s from C2l.
I t was used
CLOSED VECTOR MEASURES
IV.
Equipped with t h e information i n Chapter I11 we f i r s t l y r e t u r n t o t h e study of t h e p r o p e r t i e s of t h e i n t e g r a t i o n mapping with r e s p e c t t o a v e c t o r measure.
Then t h e concept of a closed measure i s introduced.
I t is perhaps
t h e c e n t r a l concept of t h e whole t e x t , and w i l l be used i n a l l subsequent Chapters.
Closed v e c t o r measures a r e those f o r which most of t h e c l a s s i c a l
theory of L
1
spaces c a r r i e s over, e s p e c i a l l y r e s u l t s concerning completeness.
The phenomenon of non-closed measures i s observable only i f t h e range space i s not m e t r i z a b l e .
1 . P r o p e r t i e s o f t h e i n t e g r a t i o n mapping
Suppose X i s a 1 . c . t . v . s .
rn)
{(XI,
: X'
E
XI}.
and m :
S + X i s a v e c t o r measure.
Let X1.m
=
T h e n X ' o m c c a ( S ) and t h e following Lemma follows d i r e c t l y
from t h e d e f i n i t i o n s . LEMMA 1. oCY'Om)
The integration mapping m
: L
1
(m)
+
X i s continuous betmeen the
topology on L 1( m ) and the weak topoZogy on X .
By Corollary 1 t o Theorem 11.1.1 t h e r e i s a family of measures, A c c a ( S ) , equivalent t o m. exists a h UCY'om)
E
A with
Q
A.
1
E
c a ( S j f o r which t h e r e
1
Then L ( m ) = L ( A ) as s e t s , and s i n c e X ' a m c n, t h e
1 topology i s weaker than t h e o ( G ) topology on L ( A ) .
THEOREM 1.
the
Let 0 be t h e s e t of a l l measures
o(Qj
The integration mapping m
:
1 L (A)
-+
Consequently,
X is continuous betmeen
1
topology on L ( A ) and the weak topoZogy on X.
This Theorem can be strengthened i f t h e i n t e g r a t i o n mapping i s r e s t r i c t e d 67
11'. 1
PROPERTIES OF INTEGRATION
68
to bounded subsets of its domain. Namely, THEOREM 2. The integration mapping m the T ( A ) topology on L r o , l , ( A )
:
LCO,l,fh)
+
X i s continuous between
and the Mackey topology on X .
P r o o f. Suppose P is a family of semi-norms determining the topology of X and A = {A
: p E P I a corresponding family of equivalent measures to m. P [Corollary 1 to Theorem 11.1.1).
Assume that there exists a net {fa),,,
f,
+
f in the
T ~ A ) topology
on Lr0,,,(m!, but p ( m ( f a
converge to zero, f o r some p n = 1,2,.
..
of members of L
E
P.
such that
C0,ll
- f,)does not
Then there must exist a subsequence {fn :
1 of the net { f a } a E A with p ( m ( f n
- f))
ft
But f n
0.
+
f in the T ( X )
P
topology and so by Lemma 111.2.1 it converges in the uniform T ( A ) topology, where A = { ( z t , m )
and so p ( m ) ( f n
-
: 3:'
E
f ) + 0.
Uoj.
P
By Lemma 11.2.2,
-
Consequently p ( m ( f n
f)
-+
0 and this contradiction
gives the result. As the
o(G)
and ~ ( h ()=
T(m))
topologies coincide on S(A) = S ( m ) (Lemma
111.2.2) we have COROLLARY 1. The integration mapping i s continuous from the o(n) topology on S ( h ) t o the Mackey topoZogy on X.
LEMMA 2. If
p
i s a f i n i t e measure, then for every sequence Cxnl of elements
of GvCS) converging t o
converges t o {f
:
f
E
t,
the sequence o f s e t s {f
Lro,ll(lpl), ~
( f )=
XI
:
f
E
L ~ ~ , ~ , ( I L I I ) ,v ( f ) =
i n the Hacsdorff metric on the
space of (T(Y)) closed subsets of Lco,l,(lvl).
tJ
IV. 1
CLOSED VECTOR MEASURES
P r o o f.
69
L e t d be t h e Hausdorff metric on t h e c l o s e d s u b s e t s of
Lco,l;(lull)and l e t
denote t h e r e s t r i c t i o n of t h e i n t e g r a t i o n mapping
po
L ~ ~ ~ , ~ , ( I ! A } ) . Suppose
T',
t
B = p ( T 1 , and a = u ( T - 1 .
2'-is t h e Hahn decomposition of T r e l a t i v e t o p, set
Then G u ( S ) = ~ o ( L c o , l J ( I p I ) )
s u f f i c e s t o show t h a t , f o r any y
E
E
-1
uo ( { y ) ) such t h a t Iul(6 -
I f y = x, we t a k e 11, = v . similarly.
Ca,BI.
It cle,rly
Ca,B1,
In f a c t (1) w i l l follow i f we can show t h a t f o r any y t h e r e e x i s t s a J,
u to
@)
= 1x
I f not we may suppose
E
Ca,BI and any
-
yI.
z> y
9 E
-1 uo (hl),
as t h e converse follows
Since x > y , x > a and t h e f u n c t i o n J, = P t
is well defined.
s ( x T -
- 9)
Also
The Lemma then follows from our e a r l i e r remarks.
THEOREM 3 .
If m
:
k
S +IR , i s a vector measure and k a p o s i t i v e integer,
then the integration mapping m
:
LCo,ll(m)+ m(LCo,13(m))with the
on i t s domain and t h e usual topotogy on i t s range, i s open.
T(m)
topotogy
CLOSED VECTOR MEASURES
70
P r o o f.
Suppose
IV.2
X i s a f i n i t e , p o s i t i v e measure equivalent t o rn,
(Corollary 2 t o Theorem I I . 1 . 1 ) , and l e t mo denote t h e r e s t r i c t i o n of t h e i n t e g r a t i o n mapping rn t o L Co,ll(rn) each m
= Lco,l,(X).
i is a ( f i n i t e ) real-valued measure, i = 1,...,k .
r n k ( f ) ) f o r each f
6
converges i n T(A)
to y
-1 rno ({y,))
I f some sequence { y f of elements of rn ( L
Lco,ll(X).
n
0, a s
n
,...,k , + m.
=
(Y,,;
+
m.
k
and y
)k
( y i ) i = l , then w e have ynJi -1
and so by Lemma 2 , d(mi tyn,i}
n LLo,ll(A),rni
+
-1
{yij n L[o,ll(h))
c
L
L0,ll
(XI (C461) and so,
Then d(rn,l({yn)),rn,l({y}))
+
0 , as n
+
m.
Now i f rno is not open, t h e r e e x i s t s an open subset 0 of Lco,17(A) such
t h a t rno(0) is not open i n rn ( L 0
C0,ll
Consequently w e can f i n d a sequence
(A)).
) converging t o an element z of elements {a: f i n r n o ( L ~ o , l l ( ~- ~rno(0)
E
rn,,(O).
-1 The s e t rno ( { X I ) n 0 cannot be empty, and i f f belcngs t o i t , t h e r e must be
a closed b a l l with c e n t r e f and r a d i u s
..., d(f, rno-1( { x n f ) 2
192,
(A))
yi f o r each
The operation ( A , B ) + A n B is continuous with r e s p e c t t o d, f o r
any closed s e t s A,B
as n
C0,ll
converges t o rno?{yf) i n t h e Hausdorff m e t r i c , d, on t h e c l o s e d sets of
For, if Y, 1,2
0
then we claim t h a t t h e sequence of s e t s
E rno(LCo,ll(h)),
50 , l $ A ) . i =
,..., mk), where Then m(f) = ( m l ( f ) ,...
Suppose rn = frn,
E,
E,
s a y , contained i n 0.
For each n =
whence
which c o n t r a d i c t s t h e f i r s t s e c t i o n of t h e p r o o f .
Hence rno i s open.
2 . Closed v e c t o r measures Suppose S i s a o-algebra of s u b s e t s o f a s e t T. X a l . c . t . v . s . ,
and
+
IV. 2
rn
:
CLOSED VECTOR MEASURES
S -+ X a v e c t o r measure.
71
In Section 1 1 . 2 , t h e Boolean a - a l g e b r a S ( m ) was 1
S ( m ) is a subset of t h e space L (m), and so we can consider t h e
introduced.
topology and uniform s t r u c t u r e
T h ) .
o r r a t h e r , i t s r e l a t i v i z a t i o n , on S ( r n ) .
I f S ( m ) i s a complete uniform space with r e s p e c t t o t h e uniform s t r u c t u r e
~ ( r n ) ,then t h e measure rn i s c a l l e d a closed v e c t o r measure. Referring t o t h e d e f i n i t i o n of t h e uniformity
T(rn),
a n e t fCEalrnlaEA of
elements of S ( m ) i s T(rn)-Cauchy i f and only i f , f o r every continuous semi-norm
p on X , and every f o r any a t
E
A,
> 0, t h e r e e x i s t s an a.
E
all
E
A such t h a t a.
2 a t , a.
A such t h a t p ( r n ) ( E a l a Eall) <
E
S
E,
a".
Equivalently, a n e t { [ E a l r n ) a E A i s T(rn)-Cauchy i f and only i f , f o r every neighbourhood U of t h e zero element i n X, t h e r e i s an
a.
E
A such t h a t
c U , wherever a' E A , a" E A and a. S a t , a0 2 a". "@Eat a Eall ) S i m i l a r l y , {CEalrn}a6A i s ?(rn)-convergent t o [El, E Sfm) i f and only i f
p(rn)(E, a E ) + O , a
E
A , f o r every continuous semi-norm p , o r , e q u i v a l e n t l y , i f
and only i f , f o r every neighbourhood U t h e r e i s an a.
U, f o r every u e A ,
a.
E
A such t h a t n ( S E a
E)
c
5 a.
The d e f i n i t i o n of a closed v e c t o r measure rn r e q u i r e s t h a t every
T
(rn) -Cauchy
n e t of elements of S(m) be r(rn)-convergent i n S ( r n ) . I t i s c l e a r t h a t i n t h e given d e s c r i p t i o n o f ~(rn)-convergenceit i s not necessary t o consider a l l continuous semi-norms p on X, o r a l l neighbourhoods of 0; a fundamental family P of semi-norms, o r a fundamental family of neighbourhoods s u f f i c e s . Let P be a fundamental family of semi-norms on
X. F o r every p
E
P, let
{(XI, rn) : x t E VO). Then, by Lemma 11.2.2, t h e topology and uniformity P P ~ ( r n ! i s t h e same a s p ( A : p E P ) , C l e a r l y , S ( m ) = S(A) as s e t s , where A
A
P
U
Then rn i s closed i f and only i f S ( m ) = S(A) i s p ( A
pEPAp. I f , f o r every p
E
P. A
P
E
P
:
p
E
P ) complete.
c a ( S ) i s a non-negative measure equivalent t o
CLOSURE OF A VECTOR MEASURE
72
IV.3
0 if and only if A ( E ) -+ 0, E E S ) and A = { A : p E PI, P P the topology and uniformity ~ ( m )coincides with T ( A ) on S ( m ) = S ( A ) . It then
p ( m ) (i.e. p ( m ) ( E )
+
follows that m is closed if and only if S ( A ) is r(A)-complete.
3 . Closure of a vector measure
From many points of view it is important to know that any vector measure can be extended in some sense to become closed. To achieve this the underlying space T has to be extended. The next Theorem implies that for many purposes it suffices to consider only closed measures. THEOREM 1. Let T be a s e t , S a a-algebra of subsets of T , X a quasi-complete l.c.t.v.s.,
and m
:
S
X a vector measure.
.+
Hausdorff space T , a a-algebra :
s
s
Then there e x i s t s a compact
o f Bore1 subsets of T and a vector measure
such that
+
(i)
TcTandS=SnT;
(ii)
m i s closed;
(iii)
t o every E
A
E
S there corresponds a unique E
A
E
S such t h a t E = E n T , *
E * E,E
E
A
S , i s an i n j e c t i v e a-homomorphism of S i n t o S , and m ( E ) = m ( E ) , E
E
A , .
(vi)
the mapping [El, * L E I 2 E
E
S, i s m i n j e c t i o n of S ( m ) i n t o S ( m )
vhich i d e n t i f i e s S ( m ) with a T(m)-dense subset o f
(v)
-
s(m);
A , .
co m ( S ) = CO m ( S ) .
P r o o f.
Let P be a fundamental family of semi-norms on X. Let A
a non-negative measure equivalent to p ( m ) for each p
P; A
be
P PI.
= IX
:p E P Choose T and 5' as in Theorem 111.6.1. Since m ( o r , rather [ m l ) is a bounded E
and uniformly continuous function on S(A) there exists a unique continuous A , .
extension Cml onto s ( A ) .
By continuity [ m l is a-additive on S(A)
.
Define
S;
CLOSED VECTOR MEASURES
IV.4
,
m ( E ) = [ r n J ( [ E ] i ) , f o r every E
E
S. Clearly rn
.
73
A
S + X is a vector measure.
:
Statement (i) follows from Theorem 111.6.1 (i); (ii) from Theorem 111.6.1 (ii) and the definition of a closed measure. A , .
Evidently m ( S )
c
*
m(S)
A
A , .
A , .
m(S).
c
But m ( S ) A
the closure of m ( S ) in X. Hence A
r n ( S ) , since rn(S
c
is part of
*
m(s).
m(S)
^
A
.
.
COROLLARY 1. The s e t , rn(S), o f values of m on S i s p a r t of the closure of the s e t m ( S ) i n the topology of X . A
P r ^
o o
*
f. m is a continuous function on S ( m ) and S ( m ) is a dense subset
A
of S ( r n ) . 1 4 . Completeness of L ( r n )
We now show the importance of closed measures t o the theory of integration The relation of the following result to the classical
started in Chapter 11.
(scalar) Lebesqua theory of integration is clear. THEOREM 1. Suppose S i s a o-akebra, X a 2.c.t.v.s. measure.
and rn
:
S
-+ X
a vector
1
If X i s quasi-comptete (resp. complete) then L ( r n ) i s quasi-complete
(resp. complete) i n the .r(rn) topology if and only i f m is closed. 1 P r o o f. Suppose L ( r n ) i s quasi-complete. As S ( m ) is a .r(rn) closed and 1
bounded subset of L ( m ) , S ( m ) must be r(m)-complete, and so m is closed. For the converse suppose that
X is a quasi-complete 1.c.t.v.s. with topology
determined by a family P of semi-norms. (The case f o r a complete space X follows similarly.) If [fl,
Let A
P
=
{(XI,
E
m ) : z'
1 L ( m ) and p
E
OO},
P
p
E
E
P, from Lemma 11.2.2 we have
P, and
A
Up E p A p .
By (11, Cfl,
=
[flA, for
LATTICE COMPLETENESS
74
IV.5
1 any m-integrable function f, and t h e r ( m ) topology on L ( m ) is i d e n t i c a l with the p ( A
P
:
p
E
P ) topology.
1 Suppose C c L ( m ) is r(m)-closed and bounded, and {TforlmIccEA is a r ( m ) -
i s closed S(A) = S ( m ) i s 1
L ( A ) i s p(A [fujA
-+
P
:
p
E
p(A
:
P
p
:
P
p
:
P
p
E
PI-Cauchy.
As
m
1 Consequently t h e r e e x i s t s a [ f l A E L ( A ) with
P) topology.
E
is p(A
P)-complete, and so, by Theorem 111.3.3,
E
P)-complete.
[fiAin the p ( A
{CfalhIaEA
Then
Cauchy n e t of elements of C.
To prove t h a t {Cfalm}atA is
convergent i t only remains t o show t h a t f i s m-integrable. As C i s r(m)-bounded, t h e n e t {mE(fa)IaEA i s bounded and Cauchy i n
any E
E
S.
Since X i s quasi-complete t h e r e e x i s t s an
x8 i n t h e topology of X.
xE
E
X, for
X such t h a t m E ( f a )
+
Then
(x', m (f E Hence I E f d ( x t , m ) = (x', 3c ), E E
))
E
fa d ( z ' , m )
=
a
-+
(z',
xE).
E
I'
S,
E
X' and so f i s rn-integrable.
5. L a t t i c e completeness THEOREM 1.
If m
:
S
+
X i s a closed vector measure then S i m ) i s a complete
Boolean algebra. I f S ( m ) i s a complete Boolean algebra and i f no d i s j o i n t s e t A c S ( m ) has
measurable cardinal then m i s a closed vector measure. The proof follows from Theorem, 1 1 1 . 4 . 2 and 111.4.3 and from t h e d e f i n i t i o n o f a closed v e c t o r measure.
THEOREM 2 . rn : S
I f there are no measurable cardinals the vector measure 1
.+
X i s closed i f and onZy if L ( m ) is a r e l a t i v e l y complete linear l a t t i c e .
P r o o f.
1 L (m) i s a r e l a t i v e l y complete l i n e a r l a t t i c e i f and only i f
75
CLOSED VECTOR MEASURES
IV. 6
S ( m ) i s a complete Boolean algebra (C571 Theorem 4 . 2 . 9 ) . Let X be a quasi-complete l.c.t.v.s.,
THEOREM 3.
and m
: S +
X a vector measure.
completely additive.
S a o-algebra of subsets,
Then the induced measure Cml
Moreover, f o r any d i s j o i n t s e t A
c
:
S(m)
-+
X is
S ( m ) , the s e r i e s
i s convergent i n X. P r o o f. By considering t h e c l o s u r e m o f m , i f necessary, we can assume t h a t m i s closed.
Then, by Theorem 1, i t s u f f i c e s t o show t h a t Cml i s completely
a d d i t i v e on S ( m ) .
F i r s t l y note t h a t t h e mapping [ml : S ( m )
ous.
+
X i s r(m)-continu-
Secondly, i f A i s a d i s j o i n t s e t i n S ( m ) with t h e union [El,,
then t h e n e t
of a l l unions of f i n i t e s u b s e t s of A i s r(m)-convergent t o CEI,. 'EiEOREM 4 .
measure.
Let X be a quasi-complete 1.c.t.v.s.
Then the integration mapping [ m l
:
1
L (m)
+
and m
: S
-+
X a vector
X i s a regular vector
1
integral on L ( m ) , i . e . i f {CfalmlaEAi s a monotonically decreasing net of 1
elements of L ( m ) tending t o COI,,
then Cml(Cfalm)+ 0 , a
E
A.
6 . Weak compactness of t h e range The concept of a closed v e c t o r measure i s h i g h l y r e l e v a n t i n t h e study of t h e range of a v e c t o r measure. L E M A 1.
m
:
In t h i s s e c t i o n w e i n v e s t i g a t e t h e connection.
If X i s a sequentially complete l . c . t . v . s . ,
S + X a vector measure then
S a a-algebra and
IV. 6
WEAK COMPACTNESS
76
P r o o f. m(f)
E
If f
E
M[o,ll(S)t a k e s on f i n i t e l y many values only, then
co m ( S ) , by Abel's p a r t i a l summation.
A s every f
E
MCo,ll(S)can be
uniformly approximated by such f u n c t i o n s , (l), follows by passing t o t h e l i m i t . THEOREM 1. L e t T be a s e t , S a o-algebra of subsets of T , X a quasi-
and m : S
complete l.c.t.u.s.,
fie set
c0 m ( S )
-+
X a vector measure.
i s weakly compact.
If the vector measure m is closed then
P r o o f.
let X
P
E
Assume t h a t m i s closed.
For any continuous semi-norm p on X
ca(S) be a measure equivalent t o p ( m ) and l e t A
{A
:
P
p
E
PI.
The
assumption t h a t m i s a closed measure means t h a t S ( m ) = S(A) i s T(h)-COmplete. Then Corollary 1 t o Theorem 111.5.1 implies t h a t LLo,ll(A) i s o(n) compact, where i2 i s t h e s e t of a l l measures p p
<
Xp.
The mapping
Cfl,
Cfl,
rn(f),
+
E
c a ( S ) such t h a t t h e r e is a A E
P
E
A with
.LLo,ll(A) i s well-defined and continuous
i f LLo,l,(A) i s given t h e u i f i ) topology and X i t s weak ( i . e . u ( X , X O ) topology (Theorem 1 . 1 ) .
Hence { m ( f ) : f
t
MCo,l,(S)}
= {m(f) :
convex, weakly compact subset of X containing m ( S ) .
-
co m ( S ) .
CfIA
E
Lro,l,(A)l
is a
I t follows t h a t it contains
Lemma 1 gives ( 2 ) .
I f t h e measure m i s hot closed, by Theorem 3 . 1 ,
co m ( S ) =
A
m ( S ) i s s t i l l weakly compact.
Indeed,
*
m ( S ) , where m i s t h e c l o s u r e of m.
Since m is a
,.A
closed measure, t h e s e t = m ( S )
i s weakly compact from t h e f i r s t p a r t of t h e
proof. The assumption t h a t t h e v e c t o r measure m i s closed i s c r u c i a l f o r t h e v a l i d i t y of ( 2 ) . strict.
For measures which a r e n o t closed t h e i n c l u s i o n (1) can be
This phenomenon is i l l u s t r a t e d i n t h e following
IV.6
CLOSED VECTOR MEASURES
77
EXAMPLE 1. Let T = [0,11 and S t h e Borel s u b s e t s of T , Let
X be t h e space
of a l l real-valued f u n c t i o n s on T with t h e topology of point-wise convergence ( i . e . t h e spaceIR[oyll).
For each E
of E , considered a s an element of X.
T with values i n C 0 , l l .
S , l e t m ( E ) be t h e c h a r a c t e r i s t i c f u n c t i o n
E
m ( S ) c o n s i s t s of a l l f u n c t i o n s on
Then
Hence, f o r t h i s measure, t h e i n c l u s i o n (1) i s s t r i c t .
While t h e r e i s an i n t i m a t e r e l a t i o n between t h e v a l i d i t y of (2) and t h e closeness of t h e measure rn, t h e e q u a l i t y (2) does not imply t h a t t h e measure m I n t h e next example a non-closed measure rn : S
i s closed.
-+
X is constructed
such t h a t rn(SE) i s a (closed) weakly compact, convex s e t equal t o { r n E ( f ) : f
f o r every E
M[o,ll(SjI,
S.
E
Let T = l 0 , l l
EXAMPLE 2.
x
[0,11, B t h e a-algebra of Borel sets i n C0.11.
Let S be t h e a - a l g e b r a of s u b s e t s E of T such t h a t I?= f V : u belongs t o B f o r every u a l l points u
E
belongs t o 8.
E
C0,ll f o r which t h e r e e x i s t s a v
way a v e c t o r measure m : S
For every E
f
Given E
A
E
-+
Clearly m ( E )
C0,ll with (u,v)E E , a l s o
E
X(8),
X, f o r every E
f o r every u E
[0,11,
S , and i n t h i s
: CG
X,
E
0 5 2(u) 5
m(E)(u), u
E
[O,ll) =
0, f o r every u c E
x1(E)
A
nl(F).
E
S a r e m-equivalent i f and only i f
CO,II. If E,F a r e m-equivalent,
=
=
After t h e s e remarks it i s easy t o show t h a t rn i s
In f a c t , l e t G
c
[ O , l l be a s e t not belonging t o B and
l e t A be t h e family o f a l l f i n i t e s u b s e t s of G d i r e c t e d by i n c l u s i o n . x
E
X is d e f i n e d .
S, rn(SE) = {x
not a closed measure.
net {CU
El
E
MLo,lj(S)},which i s a compact, convex subset of X. I t i s c l e a r
E
P) =
f o r every u
E
S , l e t rn(E)(u)
E
from t h e d e f i n i t i o n t h a t two s e t s E.F
x(EU
(u,v)
Let X be t h e space of a l l f u n c t i o n s on C0.11 w i t h t h e topology
where X i s Lebesgue measure.
:
CO,ll,
E
CO,lI, and t h e p r o j e c t i o n n,(E), t h a t i s t h e s e t of
of point-wise convergence.
{mE(f)
E
Then t h e
rO,lllrn}orcA is T(rn)-Cauchy but cannot be r h ) - c o n v e r g e n t i n S(m).
o
IV.7
SUFFICIENT CONDITIONS
78
7.
Sufficient conditions for closedness
The importance of closed vector measures can he seen from the preceeding theorems. Consequently we give some sufficient conditions for a measure to be closed. Most measures met in applications are covered by some of these results. THEOREM 1. I f the space X i s rnetrizable and S a u-algebra, then every measure m : S
+
X i s closed. By Corollary 2 to Theorem 11.1.1, there exists a non-negative
P r o o f.
Then S ( m ) = S(X) as sets and S ( m ) is r(rn)-complete
measure X equivalent t o m .
Clearly S(X1 is a T(X)-closed subset of
if and only if it is r(X)-complete. 1
L (X) and so ?(A)-complete.
THEOREM 2 .
I f rn : S
-P
X i s a closed vector measure and g : T
m-integrable function, then t h e measure n : S
+
-+
I R an
X , the i n d e f i n i t e i n t e g r a l of
g with respect t o rn, i s a l s o closed.
P r o o f. Suppose A rn c ca(S) is an equivalent family of measures for m. As rn is closed S ( A ) is T(Arn)-cOmplete. Define 61, t { p t ca(S) : \I % X for rn some A E A m } , Then by Corollary 1 to Theorem 111.5.1 LCo,ll(Arn) is ~ ( 6 1 , ) compact. he a family of measures equivalent to n. Then An c Qm. Suppose n T n is the set of all measures y t ca(S) for which there exists a constant k and Now let
a measure h
t
A
A
n with I y (
5 klh/.
By Theorem 111.2.1, the weak topology on
1
L ( A n ) is the u ( r n ) topology. Since Ann'
rn
c
Qrn the map i : L(Arn) + L ( h n ) ,
) = Cfl , f E M(S), is well-defined and continuous if L ( A r n ) rn 'n is given the o ( Q m )topology and L(An! the weak (i.e. a ( r n ) ) topology. Consequently
defined by i(Cf1,
LLo,ll(An)is weakly compact, and
so
~(h~)-complete.This means that S ( A n )
79
CLOSED VECTOR MEASURES
IV. 7
must be T ( A )-complete, o r t h a t n i s closed.
Suppose ( T , S , X ) is a localizable meamre space, X a l.c.t.v.e.,
THEOREM 3.
X a vector measure Such that (xr, m ) < A, for each
and m : S
I'
X'.
E
Then
m is closed.
P r o o f. let
rm
m be a family of measures equivalent t o m, and
As before, l e t A
be t h e s e t of a l l measures p
k and some y
E
A
m
c a ( S ) f o r which
E
S
1
.
Regarding L (1) a s a family of measures, it follow:
from Corollary 2 t o Lemma 1 1 . 1 . 3 and Lemma 11.1.1 t h a t Am
L1(X).
klyl f o r some c o n s t a n t
Consequently t h e i d e n t i t y mapping i : L c o , l l ( h )
+
c
easily
1
L (A), and s o Tm
c
LLo,ll(Am)i s continu-
1
ous i f LCo,ll(A)is given t h e o(L (1)) topology and Lco,l,(Am\m) t h e weak ( i . e .
(A) i s a(L'(X))-compact,
u(rm)) topology.
Since X i s localizable, L
Theorem I I I . 5 . 2 ) ,
and t h e argument follows as i n t h e proof o f Theorem 2 .
c0.11
The direct swn m of vector measures mi
THEOREM 4 .
closed if and only if each measure mi, i
E
i n S such t h a t E n T
io * j,j
= 0,
ii
E
si + X ,
i
E
I , is
I , is closed.
Suppose t h a t m is closed and choose io
P r o o f.
:
(Corollary,
E
I. If E i s t h e s e t
I, and E n Ti = Ti,,
then S(mE)
0
S . (m. ) , and t h e r(m) topology is i d e n t i c a l t o t h e r ( m . ) topology on t h e s e 2o 2 0 2O s e t s . Clearly, mi i s closed. 0
Conversely, suppose each m Then {CEo n Tilm
i
i ' i
E
I, i s closed and ICEolmjaEAi s .r(m)-Cauchy.
I a E A i s T(m.)-Cauchy and s i n c e S 2. ( m%. ) i s r(m.)-complete, t h i s
n e t must be r(m.1-convergent t o some s e t [E
iIm
i
, EZ
E
Si. Let E =
U be a closed convex neighbourhood of 0 i n X. Then t h e r e i s an U f o r every F
S, F
E
and
A such
with oo S a,O. Given a f i n i t e s e t B m(F n (Ea a E B ) n J c I, and a set F c EaA E , F E S , we have m(F) lim 6 4 UiEjpi)E U, f o r a 0 5 a. In o t h e r words, f o r any s e t F c E a A E, F E s , that m ( F )
E
E
c
Ea
E
a.
Ui.pz,
80
IV
REMARKS
m(F n
= limJcp(F n U i c T i )
E
U, for a l l
CL E
A with a.
_<
a, where t h e
l i m i t i s taken over t h e n e t of a l l f i n i t e subsets d i r e c t e d by i n c l u s i o n .
Hence
[Elrn i s t h e d e s i r e d l i m i t .
Remarks I n p r a c t i c e t h e main r e s u l t of S e c t i o n 1 i s Theorem 1.1 i n s p i t e of i t s transparency.
Theorem 1 . 2 is i n s p i r e d by C21.
Lemma 1 . 2 and Theorem 1 . 3 are
from L31. The concept of a closed v e c t o r measure was introduced i n C391.
The
r e l a t i o n between closed v e c t o r measures and t h e problem of c o n s t r u c t i o n (extension) of a v e c t o r measure can be noticed i n C371.
Theorem 4.1 could i n i t s e l f
be s u f f i c i e n t motivation f o r introducing t h e concept of a closed measure. In t h e Theorem on t h e c l o s u r e of a measure (Theorem 3 . 1 ) i t s u f f i c e s t o merely assume t h a t rn : S w i l l then be a - a d d i t i v e .
-t
X i s f i n i t e l y a d d i t i v e and bounded.
The c l o s u r e rn
( c . f . Uhl 1833 where he extends a f i n i t e l y a d d i t i v e
measure by a compactification of t h e underlying space and o b t a i n s a a - a d d i t i v e measure.)
Theorem 3.1 i s from [391.
Section 5 brings i n again t h e connectiort with l o c a l i z a b l e measures. Theorem 5 . 4 is r e l a t e d t o C561. Theorem 6 . 1 has a long h i s t o r y .
Its origin is i n the f a c t that a scalar-
valued ( r e a l o r complex) measure on a a-algebra has bounded v a r i a t i o n , hence t h e s e t of values i s a bounded s e t .
I t was proved i n C43 t h a t t h e range of a
Banach space valued measure is r e l a t i v e l y weakly compact. work t o [221 i s c l e a r and acknowledged.
The r e l a t i o n of t h e i r
The f a c t t h a t t h e range of a quasi-
complete 1 . c . t . v . s . valued measure i s r e l a t i v e l y weakly compact was proved i n [78] by use of James' Theorem.
The p r e s e n t proof i s from 1391, where t h e
CLOSED VECTOR MEASURES
IV
r e l a t i o n ( 6 , 2 ) was noticed.
81
I t i s worth observing t h a t i n a Banach space t h i s
r e l a t i o n i s always t r u e . I t could be of i n t e r e s t t o know whether a measure has r e l a t i v e l y compact
range with r e s p e c t t o a topology s t r o n g e r than t h e weak one.
In t h i s d i r e c t i o n
t h e r e s u l t s of 1821 s t a t e t h a t a v e c t o r measure with bounded v a r i a t i o n and with v a l u e s i n e i t h e r a r e f l e x i v e Banach space o r i n a s e p a r a b l e dual Banach space has r e l a t i v e l y norm compact range. Bochner i n t e g r a l s i n any Banach space. Theorems 7 . 2 and 7 . 3 a r e from [431.
The r e s u l t . is t r u e f o r i n d e f i n i t e
LIAF'UNOV VECTOR MEASURES
V.
The theme, stemming from t h e famous Theorem cf Liapunov which s t a t e s t h a t a non-atomic finite-dimensional space valued measure has compact and convex range, i s followed i n t h i s Chapter.
The problems of extension of t h i s Theorem
t o i n f i n i t e dimensional spaces a r e i n v e s t i g a t e d and workable conditions f o r t h e v a l i d i t y of such extensions a r e given. Besides i t s i n t r i n s i c elegance, Liapunov's Theorem provoked i n t e r e s t due t o i t s a p p l i c a t i o n i n Control Theory.
Some r e s u l t s along t h i s l i n e are a l s o
i n d i c a t e d i n t h i s Chapter.
1 . Liapunov v e c t o r measures A v e c t o r measure
m
:
S
-+
X i s c a l l e d a Liapunov v e c t o r measure i f m ( S E ) i s
convex and weakly compact f o r each E
E
S.
Since we assume t h a t X i s quasi-complete
t h i s condition i s equivalent t o t h e requirement t h a t m ( S E ) be convex and closed (Theorem IV.6.1). Liapunov v e c t o r measures do not have t o be closed.
In f a c t , t h e measure
constructed i n Example IV.6.2 is Liapunov but not c l o s e d . THEOREM 1. I f m : S
+
X i s a closed vector measure then the following
E
S which i s not m-null, there e x i s t s a f u n c t i o n f i n
properties are equivalent. (i)
For any s e t E
0.
BM(S) not m-null on E such t h a t m E ( f ) (ii) v
E
For every f u n c t i o n u i n RM(S) not m-null, there e x i s t s a f u n c t i o n
BM(S) such t h a t uv i s n o t m-null but m(uv) (iii) m
mE : L (m,)
FOP e a e q s e t E
E
0.
S which i s n o t m-null the i n t e g r a t i o n mapping
+ X i s not i n j e c t i v e . 82
LIAPUNOV VECTOR MEASIJRES
v.1
83
m i s a L i a p u n o u vector measure.
(iv)
P r o o f.
C l e a r l y ( i ) and ( i i i ) a r e e q u i v a l e n t , and ( i i ) i m p l i e s ( i ) .
Suppose t h a t ( i ) h o l d s and t h a t u e x i s t a non rn-null s e t E, and some
E
BM(S) i s n o t rn-null.
E
> 0 , such t h a t
lu(t)l >
Then t h e r e must E,
for t
E
E.
By
( i ) we can f i n d a bounded measurable f u n c t i o n f n o t m-null on E such t h a t
fdm = v
I,
0.
Set v ( t ) =
B M ( S ) , and f o r t
E
I,
uudm
fdm =
E
0.
f(t)/u(t)f o r t
E, u(t)u(t)
E , and v ( t ) = 0 o t h e r w i s e .
E
f(t) and s o uu cannot be rn-null.
Then Also
Hence ( i i ) h o l d s . Then t h e r e e x i s t s a non rn-nu 1 s e t E , such t h a t
Suppose ( i i i ) i s f a l s e .
t h e i n t e g r a t i o n map mE : Lm(rnE)
-+
X is i n j e c t i v e .
Hence m ( S E ) = r n E ( S ( m E ) ) i s
s t r i c t l y co n t ai n e d i n m E ( L C O , l I ( r n E ) ) . As rn i s c l o s e d , by Theorem IV.6.1, mE(Lco,l,(rnE))
= Z m ( SE ) and s o m ( S E ) cannot b e c l o s e d and convex.
Hence ( i v )
is false. Suppose ( i i ) h o l d s .
We o n l y show t h a t m ( S ) is convex and
We prove ( i v ) .
weakly compact, a similar argument can b e used t o show t h a t m ( S ) i s convex E and weakly compact f o r each E
S.
E
Let A b e any f a m i l y of measures e q u i v a l e n t t o rn, and s e t Q = {LI E cafS) : p Q A,
f o r some A
E
A}.
Let f
Theorem 111.5.1 g i v e s t h a t L
E
Lco,ll(A). As m i s c l o s e d . C o r o l l a r y 1 t o
( A ) i s o(Q)-compact. ByTheorem IV.l.l t h e ' C0,ll
i n t e g r a t i o n map i s c o n ti n u o u s from L
i t s weak topology.
Hence t h e s e t H = { g
compact, and s o h a s extreme p o i n t s . t h e r e must e x i s t a set F
co m ( S )
= m(L
C0,ll
(A))
E
C0,ll
E
L
C0,ll
(A) : m(g)
I f we can show t h a t exH
S such t h a t m ( f )
= rn(F).
rn(f)I i s ~ ( 0 ) c
S(A), t h en
Then, as rn i s c l o s e d ,
rn(S(A)) = r n ( S ) , and so rn w i l l b e Liapunov.
Accordingly, suppose f o exL
( A ) w i t h t h e a(Q) t o p o l o g y i n t o X w i t h C0,ll
E
exH, b u t f,
4
S(A).
By Theorem 111.7.1, S ( A ) =
( A ) and s o t h e r e must e x i s t a bounded, S-measurable f u n c t i o n u , n o t
v.7
LIAPUNOV VECTOR MEASURES
84
A-null such that
fo
f u
E
L
C0,l-l
(A).
But u cannot he m-null and
so
by (ii) there
exists a bounded measurable function v , which can be chosen with u ( t ) t
E
T , such that uv is not m-null, and m(uu) = 0. Then
cannot be an extreme point of H . COROLLARY 1. If rn : S
every
3: E
-f
fo
t uu
E
F!,
E
r-1,11.
and so $0
This contradiction gives the result.
X i s a closed Liapunou vector measure then, f o r
Co m ( S ) ,
P r o o f.
The statement is proved in the section "(ii)
implies (iv)" of
the proof of Theorem 1 . Before continuing it may be illustrative to show on an example the way Theorem 1 works, We will see later that many examples are variants of the following Let T be a set, S a a-algebra of subsets of T, and A
EXAMPLE 1. Let T1 = T
x
E
ca(S).
C O , l l , and S1 = S B 8, where B is the Rorel o-algebra on r 0 , l l .
Define a vector measure m
:
1
1
Sl + L ( A ) by m(E)(s) = / O ~ E ( 3 : , y ) d y ,E
E
S,,
3: E
T.
We show this measure rn i s Liapunov. Suppose E
E
S, is not m-null.
For ( z , y )
1 a(3:)
(regarding 0/0 = 0). (z.y)
E
2; -
=
(
I0 y XE(3:'Y)dY 1 /
Further, f o r ( r , y )
E , set f ( x , y ) = 0.
E
E define
E
1
(
XE(S.Y)dY
)
0
E, put f ( z , y ) = y
-
a(3:),
and f o r
Then it can be easily shown that f I s a hounded,
S -measurable function cn TI, which is not rn-null. However 1
v. 2
85
LIAPlJNOV VECTOR MEASURES
2. Consequences of t h e t e s t As i s t o be expected Theorem 1.1 has many consequences.
By t h e dimension of a l i n e a r space we mean t h e c a r d i n a l number of i t s Hamel b a s i s with r e s p e c t t o t h e f i e l d of r e a l numbers.
A source of Liapunov measures
could be t h e following
If m
THEOREM 1.
with [Elm z
0
: S
-+
X i s a cZosed vector measure and i f , f o r every E
E
S
the dimension of the Zinear space L m ( m E ) i s greater than t h a t of
X , then m is Liapunov.
We can now give a simple proof of t h e c l a s s i c a l Liapunov Theorem C511.
I f the space X i s f i n i t e dimensionaZ
COROLLARY l.(Liapunov's Theorem.)
and the measure rn
:
S
-+
X i s non-atomic, then rn i s Liapunov. m
P r o o f . The non-atomicity of m implies t h a t t h e dimension of L ( m E ) i s i n f i n i t e f o r every E
Let m
LEMMA 1. 4, :
X
+
E
S with [El,
: S -+
;z
0.
X be a Liapunov measure, Y another l.c.t.v.s.
Y a continuous Zinear mapping.
Then the vector measure
@om
: S
and +
Y is
Liapunov . P r o o f.
The statement follows immediately from t h e d e f i n i t i o n of a
Liapunov measure, and t h e f a c t t h a t EXAMPLE 1.
Lebesgue measure on S .
..., E
l,Z,
p(E) =
E
S.
x
i s a l s o weakly continuous. C0,ll.
Defir?e measures p
n
S i s t h e Bore1 a - a l g e b r a on T w i t h :
S +IR by ! J ~ ( E =) &yn& dy, n =
W e s h a l l show t h a t t h e v e c t o r measure
(y ( E ) , y (E), ... 1,
For a s e t E
C0,ll
Suppose T
4,
E
E
E
S , i s Liapunov.
S d e f i n e m ( E ) = g where
:
S
* cO given by
CONSEQUENCES OF THE TEST
86
v.2
U
for almost all y
E
Then g
C0,ll.
E
L1([O,ll) and the application rn : S
is, by Example 1.1, a Liapunov vector measure. Further, for any g put v =
9
co
E
and the mapping
: L
Q
1
(0.11)
-+
co
L (r0,ll)
..
is linear and continuous. It
follows from Fubini's Theorem that p ( E ) = Q ( m ( E ) ) . E 1
L'([O,ll)
= Wg), where
(ip,)
n = 1,2, Then
E
+
1
E
S. Lemma 1 gives that
is Liapunov. Suppose rn
:
Sa
-+
X,
a
A are vector measures, and m
E
:
S
+
X i s their
direct sum (Section 1 1 . 7 ) . T!1EOREM 2 .
Each measure m a , a
E
A , i s Liapunov i f and only i f m i s
Liapunov. P r o o f.
each m a
E
A.
I f m is Liapunov it follows easily from the definition that
is Liapunov, for a
E
A.
Conversely, suppose each m
is Liapunov,
Let 2 be the topological product of the sets m a ( S a ) , a
E
A , each
equipped with the weak topology of X. By the Tikhonov Theorem 2 is compact, and it is obviously convex. But by definition m ( S ) is the image of Z under the map carrying elements
(3:
of
2
into
1acAz a'
The definition of the direct sum
ensures that this map i s well-defined, continuous and linear. Hence m ( S ! i s weakly compact and convex.
By a similar argument we can show m(S,)
weakly compact and convex for each E LEMMA 2 .
Suppose m : S
+
E
is
S, i.e. rn is Liapunov.
X i s a closed vector measure, u a bounded S-
measurable function, and n : S + X the i n d e f i n i t e integral of u with respect t o
m.
I f m i s Liapunov, then n i s Liapunov, and conversely, i f u i s bounded may
v.2
LIAPUNOV VECTOR MEASURES
87
from zero and n is Liapunov, then m i s Liapunov. F i r s t l y n i s c l o s e d by Theorem IV.7.2.
P r o o f.
and
Cfl,
L m ( n ) . Then [fulm # 0 and so
E
Suppose m i s Liapunov
by Theorem 1.1 t h e r e e x i s t s a bounded
S-measurable f u n c t i o n h w it h Cufhlm # 0 and m ( u f h ) = 0 .
n(fh) =
#
Hence C f i I n
0 and
and t h e r e s u l t f o ll o w s by Theorem 1.1.
0,
For t h e second p a r t l / u i s bounded, measurable, and f E ( l / u ) d n = m ( E ) , E
Let m : S
THEOREM 3 .
and n
: S
.+
t o SE is Liapunov.
Conversely, suppose E = I t
I f m i s Liapunov
.. .
:
u(t)
and m r e s t r i c t e d
0)
2
Then i f n i s Liapunov, m is Liapunov. S e t E~ = {t : i s u ( t j
Suppose m i s Liapunov.
P r o o f.
S.
X be a closed measure, u an m-integrable function,
X the i n d e f i n i t e integral of u with respect t o m.
then n i s Liapunov.
0,+1,+2,.
+
E
Then m i s t h e d i r e c t sum o f t h e measures m
,i
sum o f t h e measures M
= O,+l,
...
Ei
.
,
< i t
11, i
=
and n i s t h e d i r e c t
Ei By d e f i n i t i o n each measure m
Liapunov, and ap p ly in g Lemma 2 , we see t h a t each n
'i
,i
is
Ei
= O,kl,
... must
be
The r e s u l t f o l lo w s by Theorem 2 .
Liapunov.
Conversely, suppose n i s Liapunov, and f i r s t l y c o n s i d e r t h e c a s e u ( t ) > 0 for all t
i =
2,3,.
E
S e t El = { t : u ( t )
T.
.. .
2
11,
Ei = { t : l/i 5 u ( t ) < l/(i - 111,
A s b e f o r e n ( r e s p . m) i s t h e d i r e c t sum o f t h e measures nE
( r e s p . mE ) i = 1 , 2 ,
i
...
i
, and t h e r e s u l t f o l l o w s by Lemma 2.
I n t h e g e n e r a l c a s e i f we d e f i n e E
1
= { t : u ( t ) > O } , F2 = {t : u ( t ) < O } ,
and E as g i v en , t h e r e s u l t f o l l o w s from Theorem 2 as n ( r e s p . m ) i s t h e d i r e c t sum o f t h e measures nE, nE1, nE2 ( r e s p . mEJ mEl, m E 2 ) . COROLLARY 1.
If {t
: u(t)
0)
i s m-negligible i n Theorem 3, then m i s
Liapunov if and only if n i s Liapunov.
88
LIAPUNOV DECOMPOSITION
v.3
3 . Liapunov decomposition
Clearly t h e extreme case of a non-Liapunov measure i s a measure rn : S such t h a t , f o r every E
S with LEI,
E
0. t h e r e i s a F
;z
and mF : L C O , l , ( m F ) + X i s i n j e c t i v e .
E
-+
X
SE such t h a t Elm f 0
Such measures w i l l be c a l l e d anti-Liapunov.
I t w i l l be shown t h a t any closed v e c t o r measure i s a d i r e c t sum o f a Liapunov measure and an anti-Liapunov measure.
If rn
THEOREM 1.
:
S
-+
X i s a closed vector measure there e x i s t s an m-
essentiaZly unique s e t E i n S such that the measure mE and mT-E
:
sT-E
SE
-+
X is Liapunov
[El,,, and [T-El are the maximal elements rn
X i s anti-Liapunov.
+
:
of S ( m ) such that mE i s Liapunov and mr-E i s anti-Liapunov. P r o o f.
Let
mG i s anti-Liapunov. elements [GI, of G .
G be t h e family of
a l l elements [GI, of S ( m ) such t h a t
Let P be a s e t i n S such t h a t CFIm i s t h e union of a l l I t s e x i s t e n c e i s guaranteed by Theorem IV.5.1.
The v e c t o r measure mF i s anti-Liapunov.
I f CFIm *,O, choose an a r b i t a r y s e t H
then it i s obvious. t h e r e is [GI,
E
In f a c t , i f G contains only r01,
G such t h a t CG n H I m
f
0.
c
F , CHI,
*
0.
Then
Since mG i s anti-Liapunov, G n H
contains a s e t on which t h e i n t e g r a t i o n mapping i s i n j e c t i v e .
Consequently H
contains a set on which t h e i n t e g r a t i o n mapping i s i n j e c t i v e . Let E
The maximality o f CFl, and Theorem 1.1 imply t h a t mE i s
T - F.
Liapunov. The m-uniqueness of E, i . e . t h e uniqueness of r E l m , f o l l o w s a l s o from t h e maximality of CT
-
Elm = [Fl,.
A vector measure m :
mapping m : Z m ( m ) Liapunov. measure.
-+
S
-+
X w i l l be c a l l e d i n j e c t i v e i f t h e i n t e g r a t i o n
X is injective.
An i n j e c t i v e measure i s obviously a n t i -
The vector measure i n Example IV.6.1 i s a c a s e of an i n j e c t i v e The following Theorem s a y s t h a t it i s , i n a sence, a t y p i c a l case.
LIAPUNOV VECTOR MEASIJRES
v.4
89
Every anti-Liapunov measure can be b u i l t up as a d i r e c t sum of i n j e c t i v e measures.
If m
THEOREM 2.
:
S
+
X i s a closed, anti-Liapunov vector measure then
there e x i s t s a family F of pairwise m-essentia22y d i s j o i n t s e t s i n S such t h a t , f o r every F i n F, the measure mF i s i n j e c t i v e , and the union in S ( m ) of a l l !FIm
f o r F i n F i s [TIm. If m : S
P r o o f.
+
X i s a n o n - t r i v i a l , anti-Liapunov measure, then
t h e r e e x i s t s a non-m-negligible s e t G i n S such t h a t mG i s i n j e c t i v e .
The
r e s u l t follows by exhaustion based on t h e Theorem IV.5.1. The family
F
i n Theorem 2 need not be unique, as it can e a s i l y be shown by
examples. We say, a s i n C351, t h a t t h e space X has t h e p r o p e r t y ( C ) i f any family of
i t s elements summable, by t h e n e t o f a l l f i n i t e subfamilies ordered by i n c l u s i o n , contains a t most countably many non-zero terms.
The c l a s s of spaces with
property ( C ) i s e f f e c t i v e l y l a r g e r than t h e c l a s s of m e t r i z a b l e spaces. I f t h e space has property ( C ) then t h e family f of Theorem 2 is a t most countable.
I f F i s countable t h e elements of F can be made a c t u a l l y d i s j o i n t .
Theorems 1 and 2 combine t o g i v e t h e following d e s c r i p t i o n of t h e s t r u c t u r e
of closed measures. TBEOREM 3 .
I f m : S + X i s a closed vector measure then there e x i s t s an
m-essentially unique s e t E i n S and a f a m i l y F of p a i m i s e m-essentia2ly d i s j o i n t s e t s i n S such that mE i s Liapunov, E n F = 0, mF i s i n j e c t i v e for every F
E
and the union of F i n S h ) is [T-Elm.
4 . Moment sequences
The aim of t h i s s e c t i o n is t o p o i n t t o an i n t e r e s t i n g source of Liapunov
F,
MOMENT SFQUENCFS
90
v.4
measures by showing t h e r e l a t i o n between t h i s concept and t h e moments s f an incomplete system of f u n c t i o n s . I f ( T , S , A ) i s a measure space we c a l l a sequence valued A-integrable f u n c t i o n s on T complete on a s e t E
I, fqR dA = 0 ,
...
f o r each n = l , Z ,
E
= 1.2.
S if f
E
... 1
of r e a l -
BM(SE) and
, implies t h a t f i s A-equivalent t o 0.
sequence i s not complete on E i f t h e r e e x i s t s f f q n dA = 0. f o r n = 1,2,.
on E with
{vn: n
E
This
BM(SE) not A-equivalent t o
0
.. .
I f t h e f u n c t i o n s of t h e sequence C'P,
:
n = 1,2,
... 1
2
belong t o L (1) then
it can be e a s i l y shown t h a t t h e sequence i s complete on a s e t E
E
S of non-zero
2
measure i f and only i f t h e L -closed l i n e a r span of t h e f u n c t i o n s {vnl equals
Let X = Rm be t h e product of countably many copies of t h e r e a l l i n e . t h e product topology X i s a complete 1 . c . t . v . s . . n o t ) {qn : n = I , ? ,
m
:
... 1
In
Now any sequence (complete o r
of A-integrable f u n c t i o n s on T d e f i n e s a v e c t o r measure
S + X by m(E) =
(1)
(
I qldA, I Ip:,dh ,...
),
E
f o r every E
E
S.
In many s i t u a t i o n s , including p r a c t i c a l l y a l l c l a s s i c a l ones, we can reduce m
t h e space X t o a proper subspace o f R For example, i f t h e system
{vn
:
and considerably s t r e n g t h e n t h e topology. 2
n = l,Z,.., 1 i s orthonormal i n L (A),
it is
n a t u r a l t o t a k e X = 12. For any function f
E
BM(S). t h e i n t e g r a l m ( f ) with r e s p e c t t o t h i s measure
i s t h e moment sequence of f with r e s p e c t t o t h e sequence of f u n c t i o n s {'P,
n
1,2.
... I .
THEOREM 1. 12,
...
:
Suppose ( T , S , A ) i s a localizable measure space and
1 a sequence of A-integrable functions.
{Ipn
:n =
Then the vector measure
m
91
LIAPUNOV VECTOR MEASURES
v.4
: S +
X defined by (1) i s closed.
incomplete on every s e t E
I f the sequence I'P,
:
..
n = 1.2,.
1 is
S of non-zero A-measure then t h i s vector measure i s
E
Liapunov.
P r o o f.
The v e c t o r measure is closed by Theorem IV.7.3 s i n c e it Then t h e proof follows from Theorem 1.1.
obviously has a d e n s i t y .
I t i s not d i f f i c u l t t o c o n s t r u c t t h e s i t u a t i o n modelled i n Theorem 1, e . g . Example 2 . 1 .
However, it is more i n t e r e s t i n g t h a t t h i s Theorem a l s o a p p l i e s t o
some c l a s s i c a l systems of orthogonal f u n c t i o n s on t h e i n t e r v a l C 0 , l I with r e s p e c t t o Lebesgue measure. A sequence
on [ O , l l
{vn
:
...
n = 1,2,
1 of real-valued Lebesgue i n t e g r a b l e f u n c t i o n s
i s c a l l e d a Riesz system i f t h e r e e x i s t s c o n s t a n t s
f o r any numbers el, e2,
... , eN and N
A1
and A1 such t h a t
.. .
= 1,2,,
The following Theorem d e s c r i b e s s i t u a t i o n s where Theorem 1 i s a p p l i c a b l e . The proof i s given i n [643 and 1 7 4 1 . THEOREM 2.
Let
{'P,
:
...
n = 1,2,
be an orthonorma2 sequence of functions
1
2
i n L ( [ O , l l ) and l e t Y be the L -cZosed linear span of the functions
{'P~;.
Y', regarded as a subspace of Lm, i s separable, and i f f o r every E
C0,ll with
non-zero Lebesgue measure
{"IE
:
n = 1,2,.
..
c
If
1 i s a Riesz system, then {vn1 i s
incomp2ete on E. Let {wnl be any subsystem of the WaZsh functions (C771 p. 398) i n L2(C0,11) with the property t h a t , f o r any natural numbers bl,. ..,bk,
0 "1
22
"k
il ,..., ik,
MOMENT SEQIJENCFS
92
f o r my k = 1.2,
... .
v.4
Iwn 1 i s incomplete on any s e t of positive measure.
Then
I f 19n1 i s a lacunary subset of e i t h e r the Haar functions, or the Sckauder 2
functions i n I; ( L O , l l ) ,
then {qn1 is incomplete on every
(C771 Example 2 . 3 ) .
set of positive measure. Let T = [0,11,
EXAMPLE 1.
1,2,.., 1 be t h e sequence o f Rademacher f u n c t i o n s .
Let {rn : n
( jErl dZ, hr2dZ,...
m(E) =
m
:
S
+
Z2.
S = K ( C 0 , l l ) and l e t l be Lebesgue measure.
) , f o r every E
E
Then s e t t i n g
S , d e f i n e s a v e c t o r measure By Theorem 2 and Theorem 1
This measure i s closed by Theorem I V . 7 . 3 .
t h i s measure i s Liapunov, s i n c e t h e Rademacher f u n c t i o n s a r e a lacunary subsystem of t h e Walsh system s a t i s f y i n g Theorem 2 . E
E
I f follows t h a t m ( S ) = { m ( E ) :
K ( C O , l l ) } i s a weakly compact, convex subset of 2,. I t i s , perhaps, of i n t e r e s t t o n o t i c e t h a t t h e s e t m ( S ) has a non-empty
i n t e r i o r i n 2,.
Indeed, i t is a c l a s s i c a l r e s u l t of Banach, s e e C291, p. 250,
t h a t f o r every element x that m(f)
= z.
E
Z 2 , t h e r e i s a continuous f u n c t i o n f on 1 0 , l l such
I t follows t h a t l 2 = U i - l m ( M (K([o,ll))). [-n,nl
As Z2 i s a
complete metric space, t h e Baire Category Theorem g i v e s t h a t t h e r e e x i s t s an
n such t h a t t h e s e t m(MC-n,n, ( B ( C 0 , l l ) ) ) has a non-empty i n t e r i o r . and t r a n s l a t i o n m(M
C0,ll
( K ( [ o , l l ) ) ) has non-empty i n t e r i o r .
i s closed and Liapunov implies t h a t m ( S ) = m(M
C0,ll
By c o n t r a c t i o n
The f a c t t h a t m
~K(C0,lI))).
Theorem 3 . 3 applied t o t h e s i t u a t i o n considered i n t h i s S e c t i o n g i v e s t h e following r e s u l t . THEOREM 3 .
I f ( T , S , A ) i s a Localizable measure space and {vn
:
n = 1.2,
... 1
is a sequence of integrable functions, then there e x i s t s a countable f a m i l y of pairwise disjoint sets iEi jq
n
:
i =
0,1,2,.
.. 1 from S ,
with Uiz0Ei
= T , such that
1 i s complete on each Ei, i = 1,2 ,... , but not complete on any F m
sets E o and UizlEi
are A-unique.
E
SE
0
.
fie
93
LIAPUNOV VECTOR MEASURES
v.5
5. Liapunov Extension THEOREM 1. Suppose T i s a s e t , S a a-algebra of subsets of T , and m : S
v'
and a closed Liapunov vector measure rn : S1 1 co r n ( S ) .
&l'
+
-
f
E
E 1,
S
,where
Define T
1
Et = Iy
1
C0,ll
X such t h a t rn1 ( S1 1 =
ml(Sl) =
Let 8 be t h e i3orel s e t s on t h e i n t e r v a l C0,ll and 2 : 8 +IR
Lebesgue measure.
for E
X
Then there e x i s t s a s e t T a a-algebra S of subsets of 1 1
a closed bector measure.
P r o o f.
+
:
= T x CO,ll,
(t,y, E E l , t
S1 =
vn)) = z r n ( S ) , as m i s closed.
Then E = { ( t , y )
T1
E
: 0 5
y
S
Clearly rn ( S
T.
E
+ X by
S o 8 , and m1 : S
1
1
j
c
I
IT fdm
For t h e converse suppose f
f c t ) } i s S1-llieasurable and ml ( B ) =
IT
E
:
Llo,l,(m).
fdm.
By
Theorem 111.8.2, m i s closed and we show i t i s Liapunov using Theorem 1.1. 1 Suppose f
E
BM(S,).
f
Then s i n c e both
e x i s t , by F u b i n i ' s Theorem we have f o r any
Now suppose E
E
S
1
f(t,y)dm,(t,y)
T1
is not rn - n u l l . 1
X I
E
and & , l i f ( t , y ) d y d m ( t )
XI,
By analogy with Example 1.1 we can
f i n d a bounded, S -measurable f u n c t i o n f such t h a t
1
1
f f(t,y)x,(t.y)dy
= 0
0
for a l l t
E
T.
I t can e a s i l y be shown t h a t
[fl,
f
1
Theorem 1.1 then g i v e s t h e r e s u l t .
0 , however
j E fh1=
0.
94
NON-ATOMIC VECTOR MEASIIRES
V.6
I t may be worth a remark t h a t by Theorem 11.6.1 [and its converse, which can be proved s i m i l a r l y ) , t h a t i f S i s m - e s s e n t i a l l y countably generated, and
m non-atomic, we can f i n d a closed, Liapunov measure m
-
1
:
S
-+
X, with ml(S) =
co m(S).
6. Non-atomic v e c t o r measures In t h i s s e c t i o n we w i l l i n v e s t i g a t e t h e compactness and convexity of t h e range under weaker assumptions than Section 1. LEMMA 1.
I f S i s countably generated and
then there e x i s t s an atom B of S such t h a t B For any s e t E
P r o o f.
n = 1,2,
... 1 be
c
c
if A i s an atom ofm, A and m ( B ) = m ( A ) .
T , put E1 = E , and
a countable family generating S.
A belongs t o i t , and t h a t i n f a c t El
Put
= A.
(Section 11.6)
E
1
EV1 = T - E .
Let { E n :
Obviously we can assume t h a t = 1 and determine i n d u c t i v e l y
n t o be 1 o r -1 i n such a manner t h a t
E
This i s p o s s i b l e s i n c e A is an atom of m.
then B
E
S and m ( B ) = m ( A )
f
0.
Hence B
Moreover, i f we put
*
0.
I t follows from t h e construction
t h a t B i s an atom of m. Let Y a 1 . c . t . v . s . and l e t
LEMMA 2 .
:
X
-f
Y be a continuous l i n e a r map
If m i s a non-atomic vector measure and S i s m-essentially
countably generated (Section 11.6), then also
@om
i s a non-atomic vector measure.
V.6
LIAPUNOV VECTOR MEASURES
95
P r o o f. Let SO c S be a countably generated a-algebra as required by
the definition that S is m-essentially countably generated. If m has no atoms then neither has the restriction mo of m to S O , But, if A were an atom o f @Om0
then, by Lemma 1, A would contain an atom B of S O . Since B is also an
atom o f
B must be an atom of m o .
@ O m o ,
The last observation, completeing the
proof, is that if @om0 is non-atomic then LEMMA 3 . E
P r
is non-atomic.
I f X i s a non-negative, non-atomic measure and i f X ( E )
S, * l i e s
E
@om
mCE)
o o f.
-+
0, then m
Assume
+
i s non-atomic,
that A is an atom of m, Using the classical result
that the range of A is an interval, we can construct inductively sets E such that A have X ( E n ) We
3
-+
En
3
En+l, X ( E
0 but not " ( E n )
A vector measure m : S
X', the measure
coincides with o o
We will
+ m,
@om
to be non-atomic
@om
i s non-atomic.
( X I ,
@om
-+
instead of m gives the result.
X is called scalary non-atomic if, for every
m ) E ca(S) is non-atomic.
I f m is scalarZy non-atomic then t h e weak closure of m ( S )
LEMM4 5 .
P r
S
The non-atomicity of m implies the non-atomicity of
Then Lemma 3 applies to
E
0, n
.....
E
By Corollary 2 to Theorem 111.1.1, there exists a non-negative
measure X equivalent to m.
x'
+
= m ( A ) , n = 1,2
I f X is metrizable a d m non-atomic, then
P r o o f.
X.
) = 2 c n ( ~ ) ,m ( E n )
will now give another sufficient condition for
LEMMA 4 .
0,
m(S).
f. Suppose 3:
E
m ( S ) , and we are given a natural number n , and
NON-ATOMIC VECTOR MEASURES
96
elements xi E *
( ( ~ 1 ,
,..., x'
rn)(E),
E
X'.
S i n c e t h e n-dimensional s p a c e valued measure
..., (x;,
m)(E)), E
(C o r o l l ar y 1 t o Theorem 2 . 1 ) exists a set E
E
V.6
E
S, i s non-atomic, by Liapunov's Theorem
i t s r a n g e i s compact and convex.
S w i th (xi, r n ) ( E ) =
("1, x), i =
1,2,.
. . , n.
Thus t h e r e
Thus ev er y weak
neighbourhood of x c o n t a i n s a n element o f m ( S ) . Applying Lemmas 2,4,5, THEOREM 1.
Let m : S
we th e n have
+
X be a non-atomic measure.
If
S is m-essentially
countably generated o r i f X i s metrizable, then the veak closure of m ( S ) coincides w i t h
m(S) .
N e i t h e r of t h e s e c o n d i t i o n s can b e o m it ted a s t h i s example shows. EXAMPLE 1.
Choose a s e t T , and a - a l g e b r a S such t h a t S h as no atoms
( e . g . l e t S b e t h e product o f a n uncountable f am i l y of c o p i e s o f a n o n - t r i v i a l
a-algebra).
For every t
E
T , l e t X t s t a n d f o r t h e sp ace o f r e a l numbers w i t h
i t s u s u a l topology, and l e t X.be t h e t o p o l o g i c a l p r o d u ct of t h e sp aces X t , Hence X i s a Monte1 s p a c e .
w el l d ef i n ed X-valued measure on
E
S.
x ( t ) , f o r each x
a p u r e l y atomic measure. S i n Lemma 1 i s e s s e n t i a l .
2'.
I t follows t h a t m w i l l b e a
S.
A s S h as no atoms, m must b e non-atomic. =
E
Define m ( E ) t o b e t h e c h a r a c t e r i s t i c f u n c t i o n of E
c o n s i d er ed a s an element o f X, f o r each E
X d e f i n e d by ( x i , x )
t
E
But i f z' i s t h e f u n c t i o n a l on
t
X and some t
E
T , t h en x i o m w i l l b e
T h i s measure can b e used t o show t h a t t h e assumption on The measures m , and x'om show a l s o t h a t t h e assump-
t
t i o n s on S i n Lemma 2 and on X i n Lemma 4 a r e e s s e n t i a l . Lastly the cl o s u re of m ( S ) i n
X i s , a t t h e same time, t h e weak c l o s u r e o f
m ( S ) and c o n s i s t s of f u n c t i o n s t a k i n g v a l u e s 0 and 1 o n l y .
97
LIAPUNOV VECTOR MEASURES
V.6
Under c e r t a i n assumptions t h e r e s u l t of Theorem 1 can b e s t r e n g t h e n e d .
Let X be a Banach space which i s e i t h e r a r e f l e x i v e space or
THEOREM 2 .
a separabZe dual space.
If m
: S
+
X is a vector measure w i t h bounded v a r i a t i o n ,
then t h e strong closure of the range o f m i s norm compact.
Further, if m is
non-atomic then the cZosure of t h e range of m is compact and convex.
P r o o f.
Then m is a b s o l u t e l y co n t i n u o u s
Suppose h i s t h e v a r i a t i o n o f m.
with r e s p e c t t o h s o , by C631 p . 3 0 and C161 Theorem 2 . 1 . 4 , t h e assumptions on
X
guar an t ee t h e e x i s t e n c e o f an X-valued f u n c t i o n f which i s Rochner i n t e g r a b l e w it h r e s p e c t t o X and f o r which m ( E ) =
1,
fdh, E
S.
E
1 S e l e c t a sequence { f n } o f s i m p l e f u n c t i o n s i n L ( X , A ) converging t o f i n t h e Bocher norm, i . e . j,lf and I n ( g ) =
-
fnlldh
gf A , n = 1 , 2 , .
T n l&lder's i n e q u a l i t y
t he y a r e bounded o p e r a t o r s .
..., each
.. .
0,
n
For g
-+ a.
E
Lm(A) s e t I(g) =
-+
m.
j T gfdh,
Then I , I n : Lm(h) + X are l i n e a r and, by
I n a d d i t i o n (1) shows t h a t In converges t o
uniform o p e r a t o r topology as n
n = 1,2,
+
I in the
S i n c e each f i s a si m p l e f u n c t i o n ,
In h a s f i n i t e dimensional r an g e, and so i s a compact o p e r a t o r .
Consequently I i s compact, and as t h e set
Ix,
:
B
E
Sl i s co n t ai n ed i n t h e u n i t
b a l l of Lm(A),
i s a norm precompact s e t i n X.
T h i s proves t h e f i r s t a s s e r t i o n .
If m i s non-atomic, by Lemma 4 it i s s c a l a r l y non-atomic, and s o t h e weak c l o s u r e o f m ( S ) i s weakly compact and convex.
However t h e norm c l o s u r e of m ( S )
i s norm compact, hence t h e norm c l o s u r e of m ( S ) e q u a l s t h e weak c l o s u r e of m ( S ) ,
EXAMPLES
98
and t h e r e s u l t follows.
COROLLARY 1. Suppose (T,S.A) is a measure space, X is a Banach space, and
f
: T
+
X i s Bocher integrable with respect t o A,
m :S
-+
X defined by m(E) = ,fE f d h , E
E
then the vector measure
S , has precompact range, and if h i s
non-atomic the norm closure of the range of m i s norm compact and convex.
7. Examples of bang-bang c o n t r o l A very important f e a t u r e of f i n i t e dimensional l i n e a r c o n t r o l systems i s
t h e "bang-bang" p r i n c i p l e [ 2 4 l .
Namely, any p o i n t , , which is reachable by a
n
c o n t r o l taking values i n some compact,convex s e t U ofIR c o n t r o l taking values on t h e extreme p o i n t s of U.
, is
reachable by a
H e r e a f t e r we r e s e r v e t h e I t i s well
use of t h e term "bang-bang" t o d e s c r i b e p r i n c i p l e s of t h i s type.
known t h i s r e s u l t is a consequence of Liapunov's theorem s t a t i n g t h a t t h e range of any f i n i t e dimensional non-atomic v e c t o r measure i s compact and convex. However i n need not hold.
i n f i n i t e dimensional c o n t r o l systems t h e bang-bang p r i n c i p l e This i s an exact p a r a l l e l t o t h e f a c t t h a t Liapunov's theorem
need not hold f o r i n f i n i t e dimensional non-atomic v e c t o r measures. Most i n f i n i t e dimensional extensions of t h i s p r i n c i p l e t o d a t e consider systems whose c o n t r o l s t a k e values i n some compact, convex s e t U i n a Banach o r H i l b e r t space, e.g. CSSl, Chapter 3 , 817.3.
I n t h i s s e t t i n g a l l t h a t can be
s a i d i s t h a t any p o i n t reachable by a c o n t r o l t a k i n g values i n U i s reachable by a c o n t r o l t a k i n g values on t h e boundary of U, which may be a very l a r g e s e t In p a r t i c u l a r , suppose t h e c o n t r o l i s a f u n c t i o n ( x , y ) * f ( x , y ) of two v a r i a b l e s , which for f i x e d x i s regarded as an element of a H i l b e r t space.
I f U is t h e
u n i t b a l l i n t h e H i l b e r t space and i f t h e c o n t r o l i s r e s t r i c t e d t o t a k e values on U then t h e known Theorems s t a t e t h a t any p o i n t reachable by such a c o n t r o l
LIAPUNOV VECTOR MEASURES
v.7
99
is reachable by a control having values with norm exactly one. Since many elements of norm one in the Hilbert space correspond to unbounded funtions, such Theorems do not apply if the controls f are restricted to take their values in the interval C-l,lI, say. The aim in this section is to consider the bang-bang principle for systems in infinite dimensions with bounded controls, Analagous to the finite dimensional case such systems will only be "bang-bang" if the vector measure determining them has convex range which is compact in a suitable topology, in particular if the vector measure is Liapunov. Perhaps it is worth noticing that if the vector measure determining the system is just non-atomic, in general all we can say is that optimal controls can be approximated by bang-bang controls, e.g. Theorem 6 . 1 o r 6 . 2 . If m is Liapunov then for every f E MC-1,lI(3)there exists an f o such that
/$chi
= /2focbn.
E
M
t -l,dS)
Clearly, any control system which is given by a
Liapunov vector measure will satisfy the "banE-banE': principle, !Vith the aid of Theorem 1.1 we can identify a number of control systems with distributed parameters for which the "hang-bang" principle holds, It is stated in sufficient generality to be flexible in applications, While the range space is not necessarily a Banach space, situations when it is not metrizable occur rarely, hence the problem of proving that the measure is closed does not occur. It is important, as can be seen in several of the applications given here, that the measure in the Theorem is not assumed to possess a density, There are cases where the Theorem 1.1 is immediately applicahle. EXAMPLE 1. If in a problem of the linear Theory o f rlasticity, the validity of St. Venant's Principle is accepted, then the relation between
v.7
EXAMPLES
100
deformations ( o r s t r e s s e s ) and t h e f o r c e s c a u si n g them i s a t r a n s f o r m a t i o n e x p r e s s i b l e as i n t e g r a t i o n w i t h r e s p e c t t o a Liapunoy v e c t o r measure. Assume t h a t a s e t T r e p r e s e n t s a p a r t o f an e l a s t i c body where some f o r c e s
For s i m p l i c i t y assume f i r s t t h a t such a f o r c e i s g i v en by a
are applied.
s i n g l e bounded B(T)-measurable f u n c t i o n f, which we i n t e r p r e t as t h e d e n s i t y o f The corresponding s t r e s s e s i n a p a r t A o f t h e body a r e g i v en
the applied force. by a k - t u p l e
ip
= (v 1’””
P k ) o f continuous f u n c t i o n s .
Let X b e t h e sp ace o f
S in c e t h e r e l a t i o n between f and t h e corresponding
a l l such k - t u p l e s .
v
is
assumed l i n e a r and continuous i n a c e r t a i n s en se, t h e r e e x i s t s a measure
rn
:
B(T)
+
X such t h a t
f o r any f
/+%n,
9
E
M ( B ( T ) ) and co r r esp o n d i n g stress
9.
I n some s i t u a t i o n s , e s p e c i a l l y when t h e s i z e o f T i s r e l a t i v e l y small compa r ed with t h e d i s t a n c e from T t o A , S t . Venant’s P r i n c i p l e ( s e e 1201) i s assumed t o hold.
That is, i t i s assumed t h a t t h e s t r e s s caused by f o r c e s s t a t i c a l l y
e q u i v al en t t o ze r o a r e none, o r a t l e a s t , a r e n e g l e c t e d . measurable set E
I t i s c l e a r t h a t any
T o f non-vanishing measure s u p p o r t s a f u n c t i o n f which is t h e
c
d e n s i t y of a non-vanishing f o r c e s t a t i c a l l y e q u i v a l e n t t o z e r o , i . e . with v aq i sh i n g z e r o t h , and f i r s t moments. t h e measure m i s Liapunov. d e n s i t y f wi t h 0
fo(t) E
t 0, 1} ,
5
Under t h e s e assumptions, t h e Theorem g i v e s t h a t
Hence t h e same s t r e s s e s a s a r e caused by a f o r c e with
f(t) 5 1, t
f o r every t
E
E
T, a r e caused by a f o r c e with d e n s i t y fo such t h a t
z‘.
In a more g e n e r a l (and more r e a l i s t i c ) s i t u a t i o n t h e f o r c e s a r e g i v en by an n - t u p l e f = (fi,
...,G of measurable
functions representing d e n s i t i e s of
d i f f e r e n t components o f f o r c e s and t o r q u e s .
If fi,
....f‘ n a r e
interpreted as
components o f a c o n t r o l f, we have a s i t u a t i o n co r r esp o n d i n g t o a c o n t r o l system w it h n - d i m en s i o n a l c o n t r o l s .
If U i s a compact, convex set i n I R
n
and S t . Venant’s
P r i n c i p l e i s assumed t o h o l d , t h e n any system o f s t r e s s e s caused by f o r c e s f w i t h f(t) E 0 f o r every t
E
T, w i l l a l s o b e caused by f o r c e s f, such t h a t fo(t)
LIAPUNOV VECTOR MEASURES
v.7
is an extremal point of U , for every t
E
101
T.
In this example, it is an immediate consequence of St. Venant's Principle that the vector measure o f the.contro1 system mediating between forces and stresses is Liapunov. Of course, this is no way contributes to the discussion on the validity o r the extent of validity of St. Venant's Principle itself. The solution of many problems in Mathematical Physics is given in the form of an integral transform.
If interpreted as integration with respect to a
vector measure it is interesting to give conditions for such a measure to be Liapunov. Many examples can be shown to be variations of the measure described in the following LEMMA 1. Let I and J be i n t e r v a l s , T = I
x
suppose K i s a Lebesgue integrable function on T .
1
J . S = B ( T ) , X = L ( J ) , and
The mapping which associates
with every bounded measurable function f on T the element of X defined by
f o r almost a l l y
E
which i s Liapunov.
J , i s integration with respect t o an X-valued measure m,
For any s e t E
given by the formula (1) f o r f = P r o o f.
E
S the value m(E) of m i s the element
IP
\o
of X
-+
X
xE.
Using Fubin's Theorem, it is easy to show that if m
is defined by putting m ( E ) = E
E
:
S
to be the element of X obtained in (1) for f =
S, then m is a vector measure and (1) holds if and only if
=
xE,
I f h a
We show that rn is Liapunov by using Theorem 1.1. For siniplicity we assume that I is one-dimensional. Let E be a set in S which is not m-negligible. For y smallest number o r
--
satisfying the condition
E
J , let a ( $ ) be the
v.7
EXAMPLES
102
j K ( z , y ) xE(z'y)cI;c -
Now we define f ( z , y ) z
E
I, y
E
J, z
EXAMPLE 2 .
E
1 if
z E I, y
J. z
E
=
K(x,y) xz(z,y)& In(a(y)rm)
I n ( --,a (Y ) )
E
(-m,a(y)),
and f ( z , y )
0.
= -1 i f
Then f i s a bounded measurable f u n c t i o n and
(a(y),m).
/Gfd"
= 0.
Consider t h e problem of c o n t r o l l i n g a q u a n t i t y obeying t h e
three-dimensional wave equation, by varying t h e i n i t i a l c o n d i t i o n .
W e a r e given
t h e i n i t i a l value problem n
u
tt
u(p.0)
cLAu, 0,
p
3
JR ,
u t f p , O ) = f(p), p
3I
where t h e c o n t r o l f i s a measurable f u n c t i o n taking v a l u e s i n C-1.11. point p o
E
lR3 and a l o c a l l y i n t e g r a b l e f u n c t i o n q on
CO,m),
Given a
we wish t o choose f
s o t h a t t h e s o l u t i o n u of ( 2 ) s a t i s f i e s u ( p o , t ) = q ( t ) , f o r almost a l l t
E
IO,m).
By symmetry we can assume t h a t p o i s t h e o r i g i n of our coordinate systems, p o Then i f we write f ( p ) = f ( p . e , A )
using s p h e r i c a l coordinates f o r p , t h e Poisson-
Kirchoff formula gives t h e value 2n n
u(0.t) =
//
f ( c t . 8 , A ) s i n ti de d h ,
0 0
of t h e s o l u t i o n u at t h e o r i g i n f o r almost every t i m e - i n s t a n t t 5: 0.
Hence we wish t o have
f o r almost every t
5
0.
I f t i s confined t o a bounded time i n t e r v a l J then, by Lemma 1, t h e r e l a t i o n between f and IP i s given by i n t e g r a t i o n with r e s p e c t t o a Liapunov
0.
LIAPUNOV VECTOR MEASURES
v.7
103
vector measure with values in L I L T ) . The same argument as in Lemma 1 shows that, for every non-negligible measurable set E there is a non-vanishing function
f such that
/I f(ct,e,h)sine de dh
= 0,
E
for every t
0. Hence the solution ( 3 ) is given by integration with respect to
2
a Liapunov measure with values in L1loc (CO,=l). It follows that if (3) is satisfied by a function f with values C-1.13 then it can be satisfied by a function f o with values in
f-l,lj.
Let 11, 12, J be intervals, T = II x 12,S = B(T).
LEMMA 2 .
Let X be the
space of continuacs functions on J with the topology of Zocally uniform convergence. Let L be a bounded continuous function on I2
x
J and 2et M be
a Lebesgue integrabte
function on Il x 12. The m a p p i n g which associates with every bounded measurable function f on T the element IP of X defined by (4 1
IP(Z)
=
/I L(y,z) M(x,y)
f ( Z r 3 ) dk dYJ
T
f o r every z
E
J J is integration with respect t o a Liapunov vector measure
m:S+X. For any E
E
S, the value m ( E ) of the measure m i s the element IP of X
obtained i n (4) f o r f =
P r
o o
xE.
f. If, for any E
E
S, we define n ( E ) = g where
g(y) =
/
M(x,y) XE(".Y)
h.
12
for almost all y
E
I2' then g
6
1 1 L ( I ) , and the application n : S + L (I2)is, 2
by Lemma 1, a Liapunov vector measure. If, further, for m y g
E
L ' ( 1 2 ) , we put
104
EXAMPLES
@ ( g ) , where
9
E
J , then 9
E
I
L ( y , z ) g ( y ) dy , I2 X and the mapping @ : L 1 ( I 2 )-t X is linear and continuous. q(z) =
z
v.7
, E It follows from Fubini’s Theorem that m(E) = @ ( n ( E ) ) for
E
s. Lemma
2.1
/$dm
gives that m is a Liapunov measure. Since (4) holds if and only if 9 the Lemma is proved. EXAMPLE 3 .
This is an example of a process governed by the diffusion
equation. Consider the problem,
(51
of finding the distribution of temperature u in the half-plane LO,-)
x
(--,m)
dependent on time, if the initial temperature was zero everywhere and the edge x
0 at the point y and at time
Having fixed a line y for every y
E
(-m,-)
and
tion along the line y
=
= c
T E
t is kept at temperature f ( y , t ) .
and an instant of time t we wish to choose f ( y , ~ ) ,
(O,t),in such a way that the temperature distribu-
c at time t is prescribed, That is, given a continuous
function 9 of one variable we wish to choose f in the interval (--,-I so
x
(0,t)
that if u is the corresponding solution of (5) then u ( z , c , t ) = IP(X)for
every z
E
(a,-).
In this setting,
105
LIAPUNOV VECTOR MEASURES
v.7
for x
> 0,
y
E
> 0.
t
(-m,-),
Hence we wish to determine f so that
1I
@ ( 2 )=
L(S,T)
M(V,T)
dr,
f(V,T)
dT,
C0,tlx(--,m) 3: >
0, where L ( ~ , T ) =
( l / ( t - T ) )
2
exp(-x / b k ( t - r ) ) .
From Lemma 2 the relation between f and where m
:
IP
M(q,T) =
is given as
ip
=
2
exp(-(c-s) / b k ( t - r ) ) .
jfl&,
T = [O,t3
x
(--,m),
B(T) + X is a Liapunov measure with values in the space X of continuous
functions on
CO,m)
with locally uniform convergence. If the control function
f is constrained to belong to Ml - l , i l ( B ( T ) ) , say, and if a result IP
is a temperature distribution @ along the line y such a function, then
IP
= c
E
X, that
at time t , is reachable by
is reachable by a bang-bang control f o
E
h41-1,11 ( B ( T ) )
in the same time. The kernels of many problems are, of course, not necessarily factorized in the form which permits direct application o f Lemma 2.
It can happen, however,
that by a suitable transformation they can be reduced to a form where Lemma 2 is applicable. A sufficient condition for this is given in the following. LEMMA 3 .
Let T
c
1~~ be an open s e t , J an internal.
Let K
:
T
x
J +R
1
be a continuous function such that (x,y) * K(x,y,z) i s continuously differentiable
i n T for every z
E
J.
Assume that there are continuous functions a and b i n T
s a t i s f y i q the Zinear homogeneous equation,
for every ( x , y ) ( 6 ) has
E
T and z
E
J.
Assume further1 that the space of solutions
0.f
dimension 1.
Let X be the spuce of continuous ftmctioris on J equipped iJith the topozory of locaZ1y uniform convergance, and l e t M be a Lebssgue iztegrcble functior. on T.
106
v.7
EXAMPLES
The mapping which associates to every function f E M ( B ( T ) ) the element q of X defined by (7)
z
E
J , is integration with respect to a Liapunov measure m : B ( T )
for any E
f
B(T), the value m(E) of m is the element
E
v
-+
X, where,
of X obtained i n (7) for
= XE.
P r o o f. We shall only prove that the measure in question is Liapunov. Let E be a non m-negligible set in B ( T ) . Noticing that E is non m-negligible if and only if dy for some z
E
f
0,
J, it is easy to see that a set which is non rn-negligible contains
arbitrarily small sets which are not m-negligible. Let z o be a fixed point in J .
Assume that E is a subset of an open disc
D in which the partial derivative K (x,y,zo),
Y
say, is bounded away from zero.
Then the manifolds
for 5
E
(--,-),are non-intersecting, and by condition ( 6 ) , the family of
manifolds (8) is identical to
for 5
E
(--,-),
any z
c J.
(Of course, a member of this family can be obtained
for a different value of 6 in the presentation (8) then in (S), the set
(8)
can be empty.)
and for some 5
LIAPUNOV VECTOR MEASURES
v.7
107
Let
1) E
be a family bf orthogonal trajectories to (8).
(-m,m),
assumptions made that the mapping (z,y) *
is invertible in D.
for (5.9
E
It follows from the
defined by
(g,q),
Let the inverse transformation be
C,an open set which is the image of D under (11).
are identical, there exists a continuous function L
K(a(S,o),B(S,o),z) = L ( 5 . v ) . for
(5.1)) E C,
z
E
: C x J
Since
(8)
and (9)
+R2 such that
J.
Hence using the substitution (12) we obtain
11 K ( t , y , z )
(13)
M(z,y) f(x,y) dx dy
E =
L ( S . 2 ) M a ( C S 1 ) ) . B ( 5 , v ) ) g(5.1))
J ( 5 . v ) dS &,
F
where F is the image of E under (ll), g ( 5 , s )
f(a(5,1)).@(5,1))), and J ( S . 1 1 )
the absolute value of the Jacobian of (12).
For the right hand side of (13)
Lemma 2 is applicable. Hence g , and consequently f , can be chosen
SO
is
that (13)
vanishes. It is clear that a similar statement to this Lemma can be made in higher dimensions. In practice the transformation is often seem directly, and the Lemma may not have to be used.
EXAMPLES
108
EXAMPLE 4 .
v.7
Consider the following control problem governed by the heat
equation in the plane. We wish to choose the initial temperature f(3:,y) (r,y)
E
x
(-m,m)
(-m,m),
at the origin, f o r t
E
to give a desired time distribution of temperature q(t) C 0 , t o l a given time interval. That is we are given the
initial-value problem,
where f is a measurable function taking values in C-1.11, and we are given a function q continuous in C0,tol. The aim is to choose f so that for the corresponding solution u of (14) the relation u ( O , O , t )
q ( t ) , for t
E
CO,tol,
will hold, From Fourier's solution to this problem we have
51 / m
u(O,O,t)=
-m
t
E
2
m
f(Z&t3:,
2&ty)P
2 -y clx dy,
-m
C0,tol. The relation between f and
9 is given by integration with respect
to a vector measure m on B ( I R 2 ) with values in C(C0,tol). We show this measure
rn is Liapunov. Any set E
2
E
B(1R ) is not rn-negligible if and only if E is not negligible
with respect to Lebesgue measure on IR B ( I R L ) , and a function f
E
E
3:
= p cos
g
E
e, y =
M(B(IR2))
E
2
.
M ( B ( I R L )),
Consequently given a non-negligible set by transforming to polar coordinates
p sin 8 . we can find a non Lebesgue null set F
with
E
B(IR2) and
v. 7
t
E
LIAPIINOV VECTOR MEASURES
CO,tol.
109
The r e s u l t follows by applying Lemma 2 , and then transforming
back t o Cartesian coordinates. A similar r e s u l t holds i f t h e time i n t e r v a l i s not bounded.
t h e temperature
q(t),
f o r every t > 0 .
Then P i s considered as an element of
t h e space X of continuous f u n c t i o n s on C0,m) uniform convergence.
We can p r e s c r i b e
w i t h t h e topology of l o c a l l y
Again t h e r e l a t i o n between f and q i s given by an X-valued
Liapunov vector measure. EXAMPLE 5.
As an example of a c o n t r o l system governed by an e l l i p t i c
p a r t i a l d i f f e r e n t i a l equation, consider t h e following problem of s t e a d y s t a t e heat conduction i n a s e m i - i n f i n i t e s o l i d M = I(x,y,z)
,IR3 : z 2 0) whose
s u r f a c e temperature i s c o n t r o l l e d t o be f(x,y),
E
(z,y)
(-m,m)
x
(-m,m),
for
some measurable f u n c t i o n f with values i n C--1,11. The temperature i n t h e body s a t i s f i e s t h e boundary value problem
C W
Au = 0 i n M ,
We wish t o c o n t r o l t h e s u r f a c e temperature t o o b t a i n a d e s i r e d temperature ~ ( z ) a t p o i n t s ( O , O , z ) , z > 0 , i n s i d e t h e body.
I f u i s t h e s o l u t i o n of ( 1 5 ) , then
Hence, given t h e continuous f u n c t i o n IP i n ( 0 , m ) we have t o determine t h e f u n c t i o n
f so t h a t
f o r every z > 0.
Lemma 3 or transformation t o p o l a r c o o r d i n a t e s shows t h a t t h e
r e l a t i o n between f and q i s given by i n t e g r a t i o n with r e s p e c t t o a Liapunov
110
V
REMARKS
vector measure on B(IR')
with values in the space of continuous functions on
(0,m) under the topology of locally uniform convergence.
Consequently, the
system is bang-bang.
Remarks The subject of this Chapter possibly starts with the work of Sierpiiski C751
who proved that the set of values of a non-atomic real-valued measure is
an interval (see also CIS1 and C19l).
Neymann and Pearson C591 is probably the
first instance of an application of this fact. Buch 191 reproved the result of C751
and also showed that an IR2-valued non-atomic measure has a compact and
convex range, The name Liapunov's Theorem derives from the fact that Liapunov proved in C511 that any non-atomic IRn -valued measure has a compact and convex range. He gave a counter-example in C521 to show this is not necessarily the case for non-atomic infinite dimensional measures. A simpler example to this effect is in C611. There are several proofs of Liapunov's Theorem now available, C231. C331. [71.
An outstanding new proof was given by Lindenstrauss C531. whose paper
revived interest in the subject. A condition for an infinite dimensional measure with density to be
Liapunov was given by Kingman and Robertson C331 (see also C861). is from
C441.
Theorem 1.1
It is based on the idea of Kingman and Robertson. The idea
contained in the Corollary to Theorem 1.1 goes back to Karlin C30l.
In fact,
the ideas of Karlin are reflected in practically all proofs of Liapunov's Theorem and its generalizations that came after. They are inherent in C331, C531
and in Theorem 2 . 1 and its Corollaries. The condition (i), (ii) and (iii) in Theorem 1.1 need not be sufficient
LIAPIINOV VECTOR MEASURES
V
for
61
111
t o be Liapunov i f they a r e s a t i s f i e d not f o r a l l s e t s E which a r e not
rn-null but only f o r s e t s i n a smaller family, even i f t h i s family i s T(rn)-dense in S(rn).
1
For example, l e t T = (O,l), S = B ( T ) , X = L ( 0 , l ) based on t h e
Lebesgue measure 2.
Let r l , r 2 ,
be numbers i n T such t h a t F = UE=l(sn,tn),
8, <
... be
t,,
G = T-F; so Z(G)
t h e measure rn : S
-+
a l l r a t i o n a l numbers i n T; l e t s,,t,
T, =
>
4.
$(s,+t,)
and
l;=l(tn-sn) =
4.
Let xo be a f i x e d element of X.
X by m(E) = Z(EnP)zot
xEnG,
E c S.
Let Define
Then rn
E i s not
i n j e c t i v e whenever E i s a union of i n t e r v a l s b u t rnG i s i n j e c t i v e . Section 3 i s based on C421.
Tweddle C791 be obtained t h e r e s u l t contained i n
Theorem 3 . 3 f o r v e c t o r measures having a d e n s i t y with r e s p e c t t o a a - f i n i t e measure. A s p e c i a l case of Theorem 5 . 1 on Liapunov extension i s Theorem 1 . 6 i n [ E l .
There a r e many authors who proved Theorem 6 . 1 , o r r a t h e r Lemma 6 . 5 , i n special cases.
F o r i n s t a n c e 1 4 8 1 , C711, [ E l l ,
C271, C391.
Theorem 6 . 2 and
i t s Corollary i s due t o Uhl [ 8 2 1 . Section 7 was i n s p i r e d by an attempt t o extend t h e approach and r e s u l t s concerning t h e c o n t r o l of systems with a f i n i t e number of degrees of freedom, t o systems governed by p a r t i a l d i f f e r e n t i a l equations. I t seems t h e f i r s t mathematically f e a s i b l e formulation of t h e "bang-bang" p r i n c i p l e i s i n C51 and C471.
Of course, i n f i n i t e dimensions t h e r e i s much
more l i t e r a t u r e concerning t h e s u b j e c t .
In p a r t i c u l a r , we r e f e r t o C241 where
t h i s s i t u a t i o n i s well summed-up and t h e r o l e o f Liapunov's Theorem i s c l e a r l y shown.
EXTREME AND EXPOSED POINTS OF THE RANGE
VI.
In t h i s chapter t h e p r o p e r t i e s of t h e closed convex h u l l of t h e range of a v e c t o r measure a r e examined f u r t h e r , e s p e c i a l l y from t h e p o i n t of view of t h e extremal s t r u c t u r e .
The r e s u l t s i n t h i s d i r e c t i o n have i n t e r e s t i n g measure-
t h e o r e t i c a l consequences.
There a r e a l s o a p p l i c a t i o n s t o c o n t r o l theory, as
t h e uniqueness of c o n t r o l s i s r e l a t e d t o t h e extreme p o i n t s of t h e a t t a i n a b l e set.
1. Extreme p o i n t s
We s t a r t with a c h a r a c t e r i z a t i o n of t h e extreme p o i n t s of t h e closed convex h u l l of t h e range of a v e c t o r measure m : S
X i n terms of t h e i n t e g r a t i o n
i .
mapping. I f x is an extreme point o f the s e t m ( L
THEOREM 1.
e x i s t s a unique element belongs t o LIo,ll(m)
Cfl,
of L
C0,lJ
( m ) ) then there co,13 ( m ) such that x = m(f) and this eZement
Sh).
I f x belongs t o m(LiO,ll(m))and if x i s reached by m by a unique element
of LCo,l,(m)and if this element belongs t o L { o , . i ) ( m ) = S ( m ) , then x is an extreme point of m ( L C o , 1 3 ( m ) ) . P r o o f.
such t h a t E
6
CfIm
Suppose t h a t x E
E
(m)) and t h a t x = m(f) f o r some f em(L C0,ll
Lco,l,(m) - L{o,ll(m). Then t h e r e e x i s t s an
S which i s not m-negligible such t h a t
assume t h a t m(E)
f
0;
E,
s f(t)5 1 -
E,
> 0
for t
E
and a s e t E.
We can
i f n o t , we can choose a subset o f E with non-zero measure.
Define functions g,h by g ( t ) and h i t ) = f ( t ) t
E
E
for t
E
h(t) E.
f ( t ) ,f o r t
E
T - E , while g ( t ) = f ( t ) -
Then t h e functions g,h Erenot m-equivalent and 112
E
113
EXTREME AND EXPOSED POINTS
V I .1
both belong t o Llo,l,(m). Further, m ( h ) = mT-E ( f ) t mE(f
-
= x t
E)
E
m(E).
Similarly, m ( g ) = x -
E
and, t h e r e f o r e , m ( g )
m ( h ) , t h i s c o n t r a d i c t s t h e extremal c h a r a c t e r o f x .
f
Consequently x = % ( m ( g ) t m ( h ) ) . Since m ( E )
m(E).
Moreover, i f m ( E ) = m(F), with E , F then x =
m(4(xE
xF ) ) ,
t
E
*
0,
S , and i f E,F a r e n o t m-equivalent,
which i s not p o s s i b l e s i n c e
C4(xE t
y.
F
) Im does not
belong t o S ( m ) = LIo,ll(m). Suppose now t h a t x = m ( E ) , f o r some E
f
E
L E O l j ( m ) implies t h a t f
LCo,l,(m),then 4(g + h )
m(g) = m(h) =
2,
CxElm.
S, and t h a t i f x = m(f) with
Then, i f x = %(m(g) t m ( h ) ) , with g,h
Consequently g E
E
E
C0,ll
E
CxElm.
Hence
( m ) ) are contained i n the range
w z m ( S ) i f and onZy i f {Cflm : m ( f ) = x,
LCo,l,(m)l i s a singleton belonging t o S ( m )
P r o o f.
CxEIm, h
E
e m ( L r 0.1 1(m)
The extreme points of m ( L
I f m i s closed then x
m ( S ) of m. E
[xElm.
which means t h a t x
COROLLARY 1.
Cfl,
E
E
E
L{o,ll(m).
The only e x t r a information needed i s given i n Theorem IV.6.1.
The assumption t h a t m i s closed i s needed f o r t h e extreme p o i n t s of
co m ( S )
t c belong t o m(S). The Example IV.6.1 e x h i b i t s a v e c t o r measure m such
that
m ( S ) has many extreme p o i n t s not belonging t o m ( S ) .
The v e c t o r measure
m is not closed, of course. COROLLARY 2 .
co m ( S ) are
:
S + X , the extreme points of
contained i n the closure ( i n the topology of X ) of m ( S ) .
P r o o f. p o i n t s of
For any vector measure m
Let
h
:
S
-t
X be t h e c l o s u r e of m.
By Corollary 1 a l l extreme
G h ( S ) belong t o h ( S ) , b u t , by t h e Corollary t o Theorem IV.3.1, G(S)
i s contained i n t h e closure of m ( S ) . COROLLARY 3 .
I f m i s a closed measure which i s e i t h e r Liapunov or i n j e c t i v e ,
VI.1
EXTREME POINTS
114
then x i s an extreme point of -& m ( S ) i f and onZy if there e x i s t s a unique eZement LEIm of S ( m ) with m(E)
P r
o o
x.
f. The necessity of the condition follows from Theorem 1. Conversely,
suppose there is just one element [ E l , of S ( m ) with x family of equivalent measures for m.
Let n = {p
E
m ( E ) , and let A be a
ca(S)
: P
4 A for some A
E
A}.
By the Corollary 1 to Theorem V.l.l all extreme points of the set {Cfl m : Cfl, E LCo,ll(m)and m(f) = slbelong to S ( m ) , so the only extreme point o f this set is Since this set is convex and o(n)-compact, the Krein-Milman Theorem
[El,.
implies that it consists of the single element [ E l that x
E
Then Theorem 1 implies
exm(lC0,13(m)).
LEMMA 1. For any vector measure m
co m t S )
m'
:
S
+
X , the s e t s m ( S ) , % ( S ) , and
have the same supporting hyperplanes.
P r o o f. Given any
T with respect to
( X I ,
Y'E
m).
XI, let Tt and T- be the Hahn decomposition of
Then
and sinilarly for the inf. It is known (1681 p. 753 that the extreme points of a weakly compact convex set in X need not be strongly extreme. However, if the set is the closed convex hull of the range of a vector measure, the situation is more favourab1e . THEOREM 2 .
co m ( S ) i s
If m
:
s -+
strongZy extreme.
X i s a vector measure, then every extreme point o f
115
EXTREME AND EXPOSED POINTS
VI.2
P r o o f.
Since the ranges of a vector measure and o f its closure have
the same closed convex hull, from the outset we will assume that rn is closed. Suppose x
Then there exists a neighbour-
exco m ( S 1 and x 4 st.exzm(S).
E
hood Y of z in G m ( S ) , in the relative topology of G m ( S ) as subset of X, and such that x
_ co(co
E
In other words, there exists a net { x a I a E A
m(S) - V).
k"
a
a
of elements of co(Z m ( ~ )- V ) converging to z. Let za = lj=lyj y j , lj=lyj k a a = 1, y g
m ( S ) - V , for all j and
E
a E
A.
yg 2 0,
Since m is closed, by
Theorem IV.6.1, there exists f? E Lco,ll(m) such that m(f?) = y" 3 3 i* For every ka y a fa a E A, define f a = j; and so f a E LCo,ll(m).
ljzl
Suppose A is a family of measures equivalent to m, and p
< A for some X
E
E
LIO,lI(A) in the
{p
ca(S)
E
o(Q)
of the net {CfalA!aEA,
:
( A ) and the weak topology
[O,ll 0 weakly in X, and so m(f) = x. As x
Theorem 1, there exists a set E The set WE = {Cfl,
E
S with
E
L[o,l,(A)
converging
topology. Further, since the integration
mapping is continuous with the a(a) topology on L on X, m(fg - f) -+
=
Then LEo,l,(A) is a(Q)-compact (Corollary 1 to Theorem
A).
III.S.l), and so there exists a subnet {CfslAl
to some Cfl,
Ci
: m(f)
[fl, E
=
E
ex=
m ( S ) , by
CxEIA and also m(E)
= x.
V } is a T(A)-neighbourhood o f
[xEIA in L[o,ll(A) by the continuity of integration mapping, (Theorem
IV.l.Z)..
On the other hand we have just proved that CxEIA belongs to the o(Q)-closure of the set co{CfI,
convex, Cx,lA
E
LLo,ll(A)
:
m(f) E z m ( S ) - V 3 .
Since this set is
-
is in its T(A)-closure, i.e. CxElAe C O ( L ~ ~ , ~ ~-(WE'. A)
This
contradicts Theorem 111.7.2.
2. Properties of the set of extreme points
THEOREM 1. ~f m
:
s
-f
x
i s a vector measure then, on ex
CO m ( S ) , a t 1
topologies consistent with the duality between X and X' coincide.
PROPERTIES OF EXTREME POINTS
116
VI.2
-..
P r o o f,
If $ is t h e c l o s u r e o f m, t h e n , by Theorem IV.3.1, co m ( S ) =
-
co m ( S ) , hence we can assume without l o s s of g e n e r a l i t y t h a t m i s c l o s e d . . C l e a r l y , i t s u f f i c e s t o show t h a t t h e ( r e l a t i v e ) Mackey t o p o l o g y on
m(S) is n o t s t r o n g e r t h a n t h e ( r e l a t i v e ) weak t o p o l o g y u ( X , X ’ ) .
ex
n e t o f elements o f
As ev er y
X which does n o t converge i n t h e Mackey topology t o an
element x, has a s u b n e t , no s u b n e t o f which converges t o x , it s u f f i c e s t o show
corn($)converging weakly
t h a t ev er y n e t IxolIaEAof elements o f ex
x
E
t o an element
m(S), has a subnet Mackey convergent t o x.
ex
Let E
E
S be t h e m-unique s e t such t h a t x
= m(Ea), a
be such t h a t x = m(E). These s e t s e x i s t by Theorem 1.1.
E
A , and l e t E
E A
LLo,ll(A) i s u(R)-compact.
w it h p
* A.
E
ca(S) f o r
Then, by C o r o l l a r y 1 t o Theorem 1 1 1 .5 .1 ,
Hence t h e n e t
{CxE I A l a E A has
a s u b n e t , which we
a
can suppose i s t h e n e t i t s e l f , which converges i n o ( Q ) t o an element LCo,ll(A)).
S i n ce t h e mapping
w i t h i t s weak and L
=
: LCo,l,(A)
CxE 1A
sequently
-+
rfl, o f
X i s co n t i n u o u s i f X i s equipped
( A ) with i t s o ( Q ) topologies, x
m(E) i n t h e weak topology o f X.
m(f) = x
IxE 1m
C0,Il
m
= m(Ea) = m ( x E )
=
Cx,l,
-+
a
On t h e o t h e r hand, by Theorem 1.1,
i s t h e unique element o f L r o , l , ( A ) w i t h x = m(XE) = m(E).
Cfl,
S
Let A be a f am i l y o f
measures e q u i v a l e n t t o m, and l e t fi be t h e s e t o f a l l measures p which t h e r e e x i s t s a I
E
Con-
and t h e n e t ICEalAIaEA converges i n u ( 0 ) t o CEI,,. Now
C o r o l l a r y t o Theorem IV.1.2 i m p l i e s t h a t Im(Ea)laEA converges i n t h e Mackey topology t o m(E). THEOREM 2 .
I f X i s a Banach space and m : S + X a vector measure then
through every extreme point of
CO m ( S ) passes a supporting hyperplane.
P r o o f , The c l o s e d l i n e a r span o f we may assume t o be X i t s e l f .
-
Theorem 4 , co m ( S ) =
As
exp z m ( S ) .
m(S) i s a Banach sp ace, which
m ( S ) i s weakly compact and convex, by C11 Then by Milman’s Theorem (C321 p . 132)
117
EXTREME AND EXPOSED POINTS
VI.2
the s e t of exposed p o i n t s o f G r n ( S ) must be weakly dense i n t h e s e t of extreme p o i n t s of G r n ( S ) . Hence, by Theorem 1 , t h e exposed p o i n t s of G r n ( S ) a r e norm dense i n t h e extreme p o i n t s . Let a:
-
such t h a t IIzn
Choose a sequence la: 3 of exposed p o i n t s of
miS).
ex
E
3cll
I f U' denotes t h e closed u n i t b a l l of X', then f o r
0.
+
miS)
... t h e r e
e x i s t s an a:;
every n = 1,2,
E
U' exposing
c0 m ( S )
a t zn.
I f necessary
we can choose a subsequence, which we w i l l again denote by {x'l, such t h a t n (a:;, z n ) = sup
(3c'
co
'n
r n ( S ) ) . As X i s t h e closed l i n e a r span of a weakly compact
t h e sequence { s ' 3 c o n t a i n s a subsequence {a:!) which i s
s e t , b y 111 Theorem 2
n
weak* convergent t o some a:' Let
Bi
= (t!, a:. 2
=
2
E
U'. We w i l l show t h a t a:' supports
sup (xi, co r n ( S ) ) ,
y =
(XI,
t) and
6 = sup
m ( S ) a t a:. (t',
corn(S)).
I t s u f f i c e s t o show y = R . Suppose A z! +
E
i s a measure equivalent t o rn (Theorem 1 1 . 1 . 1 ) .
ca(S
x' i n t h e weak* topology on X', we have
f o r every A
E
S.
(t' r n ) ( A ) = (ti, rn(A))
i'
Since + (XI,
rn)(A),
I t follows from t h e Vitali-Hahn-Saks Theorem C171, t h a t
... .
uniformly f o r i = 1,2,
Since xi, t
E
ex z r n ( S ) , t h e r e e x i s t , by Theorem
i ' E i n S such t h a t a:i = r n ( E . ) , i :: 1,2, ... , and 3: = rn(EZ. The sequence {m(E.)l of extreme p o i n t s of c0 m ( S ) converges weakly t o m ( E ) , 1.1, rn-unique s e t s E
so, according t o t h e proof of Theorem 1, t h e r e e x i s t s a subsequence { r n ( E . ) } of 3
{ r n ( E i ) } such t h a t A ( E . A E ) 3
+ 0.
by ( l ) , and R j + (a:!J r n ) ( E ) = y. 3
Then
118
VI . 2
PROPERTIES OF EXTREME POINTS
Clearly, t h e r e e x i s t s a s e t F
E
S such t h a t
(XI,
-
m )(F)
max(x", com(S)) =
z ' i n t h e weak* topology, (x!, m ) ( F ) 3 ( X I , m Y F ) 8 . Now 6 2 3 i so, by t a k i n g limits we have y = l i m p . Z 1.2,. (x' r n ) ( F ) f o r a l l j j' 3 lim(x!, m ) ( F ) = B . Clearly 6 ;r y , and s o 6 = y . 3
6. Since zj'
3
..,
When { B a I a E A i s a n e t of s u b s e t s of a topological space X, r e c a l l t h a t t h e l i m i t i n f e r i o r ([461 p . 3 3 5 , 337) of { B a } i s defined t o be t h e s e t of a l l z
such t h a t ( B
Suppose X i s a Fr&het space, and m
such that the integration mapping m : T(m)
X
1 eventually i n t e r s e c t s every neighbourhood of x.
THEOREM 3.
has i t s
E
+
x
: S +
a vector measure
CO m ( S ) i s open when ~ ~ ~ , ~ ~ ( m
m ( S ) the r e l a t i v e topoZogy of X .
topology and
Then the extreme
points of z m ( S ) form a cZosed s e t ,
P r o o f. t o some
x
E
X.
Let {x
1
be a sequence of extreme p o i n t s of z m ( S ) converging
-
By Theorem IV.6.1, co m ( S ) i s weakly compact, hence closed i n
t h e topology o f X , and s o x
m ( S ) . As xn is an extreme p o i n t of
.E
and m i s closed (Theorem I V . 7 . 1 . ) t h e r e e x i s t s an m-unique s e t E z
n
= m ( E 1, n = 1 . 2 , . n
..
(Corollary 1 t o Theorem 1 . 1 ) .
L C0,ll
(A) and z m ( S ) a r e m e t r i z a b l e and m
S such t h a t
E
Let X be a measure
equivalent t o m ( t h i s e x i s t s by Corollary 2 t o Theorem 11.1.1).
LIO,ll(m)
m(S),
: L
C0,ll
The s e t s
(m)
+
i s open, s o , by C761 Theorem V , we have
= liminf{xE
n+and hence t h e sequence
{xE 1 n
L{o,ll(A)
i s T ( A ) closed and
converges i n T(A>
xE
E
n
1
n t o , say, f
L{o,il(A), f o r each n
E
LCo,ll(A). However, 1,2,
...,
so f =
xE.
VI.2
119
EXTREME AND EXPOSED POINTS
A-a,e.,
f o r some E
E
Since we have proved t h a t
S.
{X }, by Theorem 1.1, x
E
if
:
f
E
m(f)=r}
Lro,ll(A),
=
exGm(S).
E
Combining t h i s with Theorem IV.1.3 we have t h e
The extreme points of the mnge of every f i n i t e dimensional
COROLLARY 1.
measure form a closed s e t . When X i s i n f i n i t e dimensional, t h e extreme p o i n t s of t h e range do n o t n e c e s s a r i l y form a closed s e t . EXAMPLE 1.
Let T
C-l,Il, S be t h e o-algebra of Rorel s e t s of T , 1
1 Lebesgue measure on S, and X = L ( [ O , l I ) .
I t i s easy t o show t h a t G m ( S ) =
t
E
Co,lI,
0 5 a 5
COROLLARY 2 .
+
ax
C0,lI
:
f
E
M(S), f ( t )
E
+
X by
c0.11,
1 3 , and
Thus t h e sequence Ixco,ll however i t s l i m i t
!f
Define a measure m : S
x co,13 If m
- l,nl } i s contained in t h e extreme p o i n t s of G m ( S ) . i s not : S +
X i s a vector measure s a t i s f y i n g the conditions
of Theorem 3 which i s e i t h e r Liapunov or i n j e c t i v e , then for each E
E
3,
In particular ( 2 ) holds i f m i s non-atomic and X i s f i n i t e dimensional.
P r o o f.
Clearly
m(S) =
m(SE) +
m(STmE). Suppose t h a t
VI . 3
RYBAKOV'S THEOREM
120
m(Z)
e x z m ( S E ) , f o r some Z
E
E
SE.
By v i r t u e of Theorem 3 t h e extreme p o i n t s
of z m ( S ) form a [weakly) closed s e t and s o , by t h e Corollary t o Theorem 1 . 8 . 2 , must e x i s t a s e t W
E
ST-E such t h a t m(W)
W ) E e x z m ( S ) . Then m ( Z )
m(2 u
Conversely suppose m(Y)
E
= m((Z u W) n El.
To show m(Y n E )
ex= m ( S ) .
E
m ( S T - E ) and m ( 2 ) + m(W) =
ex=
e x z m(SE), a s m
E
i s Liapunov o r i n j e c t i v e , it i s s u f f i c i e n t t o show t h a t i f t h e r e e x i s t s s e t s W,Z
4 m(W) + 4 m ( Z ) ,
SE such t h a t m ( Y n E )
E
4m(W)
t
4 m(z) + m(Y
4 ( m ( W ) + m(Y - E ) ) + L,(m(Z)
- E) =
(Y - E ) ) + + m ( ~u (Y - E ) ) .
then m ( W ) = m ( Z ) .
Now m ( Y ) =
t m(Y - E l ) = %(m(W u
CO m ( S ) we must
Since m ( Y ) i s an extreme p o i n t of
have m ( W u (Y - B)) = m ( Z u (Y - E ) ) , o r t h a t , m ( W ) = m ( Z ) .
Rybakov's Theorem
3.
THEOREM 1. Let m : S
-f
X be a vector measure and l e t x'
E
XI. The vector
measure m i s absolutely continuous with respect t o the scalar measure
if and only i f there e x i s t s a nwnber which is reached by
(XI,
m)
m ) o n l y once on
(XI,
S(rn). P r o o f. then E
(XI,
A
Let F
E
S be such t h a t i f E
S and (x', m ) f E ) = ( x f , m)(F)
E
F i s rn-negligible. S and
I( XI,
E
m ) (E) =
m ) ( F ) . Hence N = E
(XI,
If E = P
m ) l ( N ) = 0.
Assume t h a t N
A
i s absolutely continuous with r e s p e c t t o
A
N , then, c l e a r l y
P i s m-negligible.
This means t h a t m
m).
(XI,
Conversely, l e t m be a b s o l u t e l y continuous with r e s p e c t t o (x', m ) .
Let
t h e s e t s T+ and T-, elements of S, r e p r e s e n t t h e Mahn decomposition of T with respect t o
E
E
(XI,
(XI,
m ) , i.e. Tt n T-
S , E 5 Tt, and m)(E) =
( X I ,
(XI,
m)(E)
S 0
rn)(Tt) f o r any E
B . Tt u Tf o r every E E
s,
then
T. ( ~ 1 , r n ) ( E ) E
](XI,
2 0 f o r every
S , E 5 T-. I t follows t h a t i f m ) I ( E A)''3
= 0.
By absolute
EXTREME AND EXPOSED POINTS
VI .3
c o n t i n u i t y , T f a E i s m - n e g l i g i b l e , and so [ E l , m- n eg l i g i b l e if E values
(XI,
[ T ' I , .
S is such t h a t (x', n ) ( E ) = (x',
E
m )(!i'-) i s reached by
m)(T--),(XI,
121
S i m i l a r l y , 7'- a E i s m)(T-).
Hence each of t h e
m ) o n l y once on S ( m ) .
(XI,
COROLLARY 1. The vector measure m i s absolutely continuous with r e s p e c t
to
m ) i f and only i f
(XI,
m ) achieves both i t s m a x i m and m i n i m only
( X I ,
once on S h ) . There e x i s t s an x'
COROLLARY 2 .
with respect t o
E
X' such t h a t m i s absolutely continuous
m ) i f and only i f the range m ( S ) of t h e vector measure rn
(XI,
has an exposed p o i n t .
P r o o f. H = Ix :
(XI,
xo
Suppose
2 ) = a),
E
exp
f o r some x'
ds). Then there is a s u p p o r t i n g h y p er p l an e E
XI,
R, w i t h H n m ( S ) = {sol. S i n c e
a E
xo i s a l s o an extreme p o i n t o f m ( S ) , by Theorem 1.1, t h e v a l u e x o i s t ak en by m only once on S ( m ) , hence a i s ta k e n by
(XI,
m ) o n l y once on S ( m ) .
Conversely i f m i s a b s o l u t e l y continuous w i t h r e s p e c t t o a = max{(x', (XI,
rn)(F), F
m)(E) E
:
E
E
S1 i s t a k e n by (x', m ) o n l y once on S h ) .
S. Then x0 = m ( F ) i s an exposed p o i n t o f m ( S ) .
c l e a r t h a t m ( S ) n Ix
E
X
m ) , then
(XI,
:
(XI,
Let a =
Indeed, it i s
x ) = a} = {xol.
The n e x t r e s u l t i s Rybakov's Theorem C701. THEOREM 2 .
I f X i s a Banach space and m : S
there e x i s t s an x'
P r o o f.
E
X a v e c t o r measure t h e n
X' such t h a t m i s equivalent t o
( X I ,
m
)
m).
(XI,
C l e a r l y i t i s s u f f i c i e n t t o f i n d an x 1
a b s o l u t e l y continuous w i t h r e s p e c t t o The s e t
-f
E
XI such t h a t rn i s
.
m ( S ) i s weakly compact (Theorem IY.6.1) and so from [ I 1 Theorem
4 i t h as an exposed p o i n t .
By Lemma 2 . 1 an exposed p o i n t o f
m ( S ) i s an
VI . 4
EXPOSED POINTS
122
exposed p o i n t of m ( S ) .
The r e s u l t follows by Corollary 2 t o Theorem 1.
The statement of Theorem 2 need not hold when X i s not a normed space.
In
t h e following example a measure with values i s a F r i c h e t space i s given f o r which t h e Theorem does not hold, EXAMPLE 1. Let
Let T = [0.11. and S be t h e o-algebra of Bore1 s e t s i n C 0 , l l .
Fn =
where l i s t h e Lebesgue measure.
m(S) =
and l e t m(E) = ( Z ( E n Fn))iz0, E
f o r n = 1,2,...,
m
rn(S) = nn=,CO,($)n].
exposed p o i n t s .
Then m : S +lRm i s a v e c t o r measure.
E
S,
Clearly
As observed i n [ 3 4 1 p.96, t h i s s e t has no
Corollary 2 gives t h e r e s u l t .
4 . Exposed p o i n t s of t h e range
In general i f t h e closed convex h u l l &A,
of a s e t A i s weakly compact,
A and G A n e e d not have t h e same exposed p o i n t s .
However f o r t h e range of a
measure we have .-
THEOREM 1.
The range of a vector measure rn, i t s weak closure, m ( S ) , and
i t s closed, convex hull a l l have the same exposed points. P r o o f.
exp
m(S)
c
I f expm(S)
exp m ( S ) and
Let exp m(S)
f
0.
0, t h e r e s u l t follows by Lemma 1.1 a s expm(S)
exp m ( S ) c
exp m ( S ) .
By Lemma 1.1, m(S) and z m ( S ) have t h e same supporting
hyperplanes, and as m i s closed (Theorem 3.2) ex(&m(S))
c
m ( S ) (Corollary 1
t o Theorem 1.1). Lemma 1 . 2 . 1 immediately gives t h a t exp m(S) = exp The proof of
exp m ( S ) =
c
m(S).
exp m ( S ) follows s i m i l a r l y .
I t has been shown C681 t h a t t h e exposed p o i n t s of every weakly compact convex s e t need not be s t r o n g l y exposed.
However,
VI . 4
EXTREME AND EXPOSED POINTS
123
THEOREM 2. The exposed points of m ( S ) , of t h e weak closure of m(S), and of the closed convex hull of m ( S ) are strongly exposed.
P r
o o f.
As in the proof Theorem 1 we can suppose that exp
m(S) # 0,
and also that m is closed. Firstly we prove that exp m ( S ) exp m(S).
c
st.exp z m ( S ) . Accordingly let m(E)
Then by Theorem 3 . 2 there exists an z '
E
X' such that
equivalent to rn. Suppose {x,laEA is a net in z m ( S ) such that ( X I ,
m(E) ).
Then E
* T+
( X I ,
x
m ) ] is ) +
Let T+, T- be the Hahn decomposition of T relative to (a!',m). is m-negligible by Corollary 1 to Theorem 3.1. Since m is closed
there exists a net
,},fI
of elements of . L ~ o , l ~ ( ~ m ( x) l' ), such that m ( f , ) = x , .
Then ( x ' , x , ) = j f , d ( x I , m). Let decomposition of
Now,
] ( X I ,
E
( X I ,
m). Then
( X I ,
m)
( X I ,
m)+
-
( X I ,
m)- be the Jordan
124
VI.4
EXPOSED POINTS
and so, by (l), ITlxE LLo,l,(l(x', m ) l )
-t
- f,(dl(xl, m)(
-+
Since t h e mapping m :
0.
X i s continuous when ~ ~ o , l l ( l ( xm t) ,l ) has i t s
topology and X i t s given topology (Theorem I V . l . 2 1 , of X, i . e . m(E) c s t . e x p z m ( S ) , and s o exp m ( S )
m(S)
1 . 2 . 1 , we know t h a t s t . e x p
c
xa c
s t . e x p m ( S ) , s o , exp m ( S ) = s t . e x p
c
c
o f.
Choose E
E
S and l e t Y
I t i s easy t o s e e t h a t G m ( S ) =
X a vector measure, then
c
exp G m ( S ) ) .
E
S b e such t h a t m(Y)
corn(.$,) t z m ( S T - E ) and
E
exp
G m(S).
t h a t x' c XI exposes
a t m(Y) i f and only i f i t exposes z m ( S E ) a t m ( E n Y ) and z r n ( S T v E )
a t m((T - E ) n Y ) r e s p e c t i v e l y .
Thus m ( E n Y )
T o prove t h e second i n c l u s i o n , l e t E
-+
S we have,
norm cZosure of Im(E n Y ) : m ( Y )
an x 1
m ( S ) , and s o t h e r e s u l t
exp m(S), and a l s o t h a t exp m(S)c
THEOREM 3 . I f X i s a Banach space, and m : S
co m ( S )
m(S)
The proof of exp m(S) = s t . e x p m(S) then follows as b e f o r e .
s t . e x p m(S).
P r o
By Lemma
z m(S).
W e can analogously show exp m ( S )
f o r every E
m)l)
m(E) i n t h e topology
st.exp G m ( S ) .
s t . e x p m(S). Further, by Theorem 1, exp m ( S ) = exp follows f o r m(S) and
+
T(J(Z',
x
X t which exposes Z r n ( S E ) a t x.
E
E
exp G m ( S E ) .
exp z m ( S E ) .
Then t h e r e e x i s t s
Define
and
Since
2'
exposes z m ( S E ) a t x, it is easy t o s e e t h a t K = x + KE.
weakly compact, convex, extremal subset of
z m ( S ) , and
K is a
so K has an extreme p o i n t
y and y
E
ex
m(S).
Further, by Theorem 1.2.1 there exists a z
that y = z t z , and as m is closed, G
E
125
EXTREME AND EXPOSED POINTS
VI . 4
3:
E
ex KE such
= m ( F ) and z = m ( G ) , for some sets
F
SE,
ST-E. (Corollary 1 t o Theorem 1,l).
As noted in the proof of Theorem 2.2 exp
m ( S ) i s norm dense in ex
c0 m ( S ) ,
so there exists a sequence {yn} of exposed points of z m ( S ) converging to
In addition, by Corollary 1 to Theorem 1.1, each y , = m ( H n ) for some Hn 1.2,
n
E
...
y.
S,
E
, and following the proof of Theorem 2 . 1 we can construct a subsequence
{m(Hi)} of { m ( H n ) ) such that [Hi], Then [Hi n El,,,
-+
[(F u G)
11 E l m
+
[F u GIrnin the
T(m)
topology on S(rn).
[FI, in the r ( m ) topology, and as rn is a
continuous mapping from S ( m ) with its r ( m ) topology to the norm topology on X, we get IIm(Hi n E )
-
m(F)II
-+
Since m ( H . )
0.
exp z r n ( S ) for every i = 1,2, ... ,
E
It follows that z = m(F) belongs to the norm closure o f I m ( Y n E ) exp
:
m(Y)
E
CO m(S)I. THEOREM 4. If X i s a Banack space, m : S
which exp r n ( S ) ( = exp
-f
X is a v e c t o r measure f o r
r n ( S ) ) is weakly closed, then f o r e v e q E
E
S we have
P r o o f. By Theorem 3 it is sufficient t o prove that the set on the right hand side is norm closed. Let m ( En ) suppose that Ilm E n n E ) - zll
+
0
exp z m ( S ) for n = 1,2,
E
for some z
E
m ( S ) . The set exp
... and
G m ( S ) is
weakly closed by hypothesis, and so weakly compact, and further, as the weak and norm topologies coincide on exp = m ( S )
(Theorem 2.1) this set is norm
compact. Adding this to the arguments used in the previous Theorem, there must exist a subsequence { r n ( E i ) ) of { r n ( E n ) ) with Ilrn(Ei) - m ( F ) II -+ 0 for some
F
E
S , where m ( F )
E
exp G r n ( S ) , and CEilm
As before, CE3 n Elrn words
3c
-+
CE n Fj,,
-+
CFIm in the ~ ( m topology ) on S ( m ) .
and s o Ilm(Ei n F ) - m(E n F)II * 0. In other
= m ( E n F) for some set F with m ( F )
E
exp
rn(S).
V I .4
EXPOSED POINTS
126
When exp m ( S ) i s not weakly closed t h i s Theorem need not hold, even i n f i n i t e dimensions. Let rn
EXAMPLE 1.
1
be a non-atomic measure on t h e Borel s e t s of C0.lJ whose
range i s t h e closed u n i t d i s k i n n
2
.
(e.g. C641), and l e t rn
2
be a non-atomic
measure on t h e Borel s e t s of L1.21 whose range i s t h e segment from ( 0. 0) t o ( 1 , O ) . Define a v e c t o r measure rn : B m2(E n i1.21).
llz
-
(1,O)ll
B(C0.21) + I R
2
by m ( E ) = ml(E n L 0 , l l )
The range of rn i s t h e convex h u l l of {x : IIxlI
5 11.
5
1) u
+
:
Now ( 0 . 1 ) i s an exposed p o i n t of rn(SLOSi1)and i t i s easy t o
s e e i t is not of t h e form m ( E n F ) f o r any exposed p o i n t m ( F ) of t h e range of
rn, F
E
S.
We have already seen t h a t i f X i s a Banach space t h e range of an X-valued measure has an exposed p o i n t . THEOREM 5 .
The following is a much deeper r e s u l t .
If X i s a Banach space and rn :
s
+
X a vector measure, then
the s e t of exposing functionals of the range of rn forms a residua2 G6 s e t in XI.
P r o o f.
By Theorem 1 . 2 t h e r e e x i s t s a1 l e a s t one x: exposing
For any o t h e r f u n c t i o n a l
5'
i t i s easy t o v e r i f y (C701) t h a t a l l but countably
many elements o f t h e segment f o r xb t o x' i n XI expose of exposing f u n c t i o n a l s of
rn(S).
m ( S ) i s dense i n X'.
Theorem 4 . 2 t h a t every exposing f u n c t i o n a l of
r n ( S ) , and s o t h e s e t
I t has been proved i n
m(S) s t r o n g l y exposes t h a t s e t ,
and so Theorem 1 . 2 . 3 gives t h e r e s u l t . COROLLARY 1.
the s e t of those xt G6 s e t i n
If X i s a Banach space and rn E XI
:
S
+
X a vector measure then
f o r which (x', m ) i s equivalent t o rn form a residual
XI.
Since t h e i n t e r s e c t i o n of countably m a n y r e s i d u a l s u b s e t s o f XI i s again
VI
EXTREME AND EXPOSED POINTS
127
r e s i d u a l i n X' we o b t a i n COROLLARY 2 .
I f X is a Banach space and f m . 1 is a sequence of X valued
measures, then those x'
i
= 1,2,.
..
E
X'for which
( X I ,
m.
z
)
is equivaZent t o mi, f o r every
, f u m s a residual G6 s e t i n XI.
Remarks Theorem 1.1 of t h i s Chapter d a t e s back t o t h e famous paper of Liapunov I t appears ( p o s s i b l y disguised) i n many papers i n connection with t h e
C511.
uniqueness of optimal c o n t r o l s .
e . g . C601, C241.
Corollary 1 i s i n C391.
content of Corollary 3 was s t a t e d by Anantharaman i n C3l.
The
H e a l s o proved t h a t
i f X i s a m e t r i z a b l e space with weak* separable dual and i f m i s non-atomic, then t h e r e s u l t of Corollary 3 s t i l l holds.
Lemma 1.1 i s a l s o from C31, as
i s Theorem 1 . 2 f o r m e t r i z a b l e space valued measures. Theorem 2 . 1 i s s t a t e d i n [21, unaer some e x t r a c o n d i t i o n s which a r e e a s i l y removable i n o u r c o n t e x t .
Theorems 2 . 2 and 2 . 3 a r e d i r e c t l y from C21.
2 t o Theorem 2 . 3 was proved i n f i n i t e dimensions i n 1281.
Corollary
I t may be of i n t e r e s t
t h a t Tweddle proved i n 1803 t h a t i f m ( S ) i s convex ( f o r i n s t a n c e i f m i s Liapunov) then m(E) ex m(S,)
n ex m(S,-,)
E
ex m ( S ) , E
E
S, i f and only i f m ( S & n
(ST-E) =
= {ol.
Theorem 3 . 1 i s adapted from C21 and 131, where t h e i d e a of reducing Rybakov's Theorem t o t h e e x i s t e n c e of exposed p o i n t s of t h e range, f i r s t appeared.
Rybakov's Theorem i t s e l f appears i n C701.
The proof t h e r e i s more
direct. Theorems 4 . 1 , 4 . 2 , 4 . 3 and 4 . 4 a r e Anantharaman's C31.
Corollary 1 t o
Theorem 4.5 i s due t o Walsh r851, and Corollary 2 t o Drewnowski C151.
VII.
THE RANGE OF A VECTOR MEASURE
The problem considered i n t h i s Chapter i s one of s y n t h e s i s :
express a
given s e t as t h e closed convex h u l l of t h e range of a v e c t o r measure.
I t is
shown t h a t t h e s e t s which can be expressed i n such a form a r e e x a c t l y zonoforms, limits i n a sense of zonohedra.
This geometric condition i s p o t e n t i a l l y of
considerable i n t e r e s t s i n c e i t r e l a t e s t h e theory of v e c t o r measures with t h a t of conical measures, negative d e f i n i t e f u n c t i o n s , i n f i n i t e l y d i v i s i b l e p r o b a b i l i t y laws and o t h e r t h e o r i e s ,
There i s a l s o an e x t r i n s i c i n t e r e s t i n t h e problem
s i n c e i t can be i n t e r p r e t e d as one of c o n s t r u c t i o n of a c o n t r o l system with prescribed a t t a i n a b l e s e t .
1. The problem
The problem t o c o n s t r u c t , f o r a given s e t K such t h a t m ( S ) = K i s unreasonably ambitious. one.
c
X , a v e c t o r measure m
5
K.
+.
X
More t r a c t a b l e i s t h e following
If K i s a convex s e t , K c X , f i n d a v e c t o r measure m
co m ( S )
: S
;
S
+.
X such t h a t
The following lemma e x p l a i n s p a r t i a l l y t h e d i f f i c u l t i e s involved
i n searching f o r a c t u a l ranges of v e c t o r measures r a t h e r than t h e i r convex h u l l s ; t h e closed convex h u l l i s a c t u a l l y t h e range of another measure.
If the space X is quasi-compZete then, f o r any vector measure
LEMMA 3 . m : S
-+
X, there exists a closed vector measure ml
co m ( S ) .
P r o o f.
co m ( S )
A
If
4:S
:
S1
+.
X such that ml(S1) =
+.X i s t h e c l o s u r e of m then, by Theorem IV.3.1.,
*
= z m ( S ) .
Now we can use Theorem V.5.1.
To introduce t h e geometric c r i t e r i o n f o r s o l v a b i l i t y of t h e problem j u s t 128
VII.1
THE RANGE
129
That i s , l e t t
posed l e t us consider f i r s t t h e case of a d i s c r e t e measure. be p o i n t s of T and x elements of X, j = 1 . 2 , j
....n ,
E
E
S.
Let u s assume, without l o s s of g e n e r a l i t y , t h a t f o r every k = 1 . 2 ,
E
... , M ,
there
..
with j f k , j 1.2.. .n. j X belongs t o t h e closed convex h u l l of t h e range of m i f
i s a s e t i n S containing t k and no o t h e r p o i n t t Clearly, a p o i n t x
j
and l e t
and only i f
where 0 s a j
1, j = 1 , 2 , . . . , n
5
co m ( S ) This means t h a t
.
In o t h e r words,
n =
1
C0,llZ j=l j’
-
m ( S ) i s a sum of segments, i . e . co m(S) i s a zonohedron i n
t h e sense of Coxeter C131. Let T o
c
X be a s e t containing a l l p o i n t s xj, j = 1 , 2 ,
...,n ,
and l e t So
be a a - a l g e b r a of s u b s e t s of T o s e p a r a t i n g p o i n t s x ( f o r every k t h e r e i s j E E So with x E E and z . 4 E k , f o r j f k ) . Let u E c a ( S o ) be defined by
k
k
k
3
n
Then (1) can be i n t e r p r e t e d as
x = where v
E
ca(S,), 0
5 V
It
IJ,o r
dv(t),
THE CONCIAL MEASIlRE
130
VII.2
The measure u in this formula is not the only possible. For example (assuming that SO is large enough), if
where 8 . are positive numbers, then (2) will hold if u is replaced by u1. 3
This observation suggests that
should be interpreted as a conical measure
on X. Moreover with this interpretation the relation (2) holds for any vector measure, not necessarily one which is discrete. This is the content of a Theorem we are going to prove. Let u be a conical measure on X, u with u
5
u,
has the resultant
The sets K
U'
for u
E
T(U)
E
Mt(X), such that, every u
E
Mt(X)
belonging to X. Then we put
Mt(X), and their translates are called zonoforms.
They are generalizations, in a natural way, of zonohedra. The relation (2) is extended to any vector measure in the following THEOREM 1. Let X be a quasi-complete l.c.t.#.s. ezists a uector measure m some u
E
:
s + x with K
=
CO m ( ~ if )
and let K
and only
c
X.
if K
There =
xu,
for
M+(x).
This is a basic characterization Theorem. Its proof will be given in the next Section. Two Theorems proven there will correspond to its "only if?! and "if" parts, respectively.
2. The conical measure associated with a vector measure Let X be a quasi-complete 1.c.t.v.s. Let T be a set, S a a-algebra of
THE RANGE
VII. 2
subset T and rn : S
+
131
X a v e c t o r measure.
The v e c t o r measure m d e f i n e s , by d u a l i t y , t h e l i n e a r mapping x' from X' i n t o c a ( S ) ,
*(XI,
rn)
This mapping can be extended t o a l i n e a r l a t t i c e
homorphism of h ( X ) i n t o c a ( S ) . I n f a c t , f o r every
where 1
5
j
k and x' j
S
2'
h ( X ) , written as
E
XI, f o r 1 s i
E
S
k , put
t h e l a t t i c e o p e r a t i o n s on t h e right-hand s i d e being t h o s e o f c a ( S ) . This formula d e f i n e s unambiguously a unique l i n e a r l a t t i c e homomorphism
am
: h(X)
ca(S) such t h a t
-+
@ (2')
rn
=
( X I ,
r n ) , f o r every r '
E
X' ( s e e
The symbol @m w i l l have t h i s meaning throughout t h i s Chapter.
Theorem l),
Denote by u = A ( m ) t h e c o n i c a l measure on X defined by u ( z ' ) f o r every 2'
l i n e a r and, s i n c e E
h(x).
@rn(z')(T),
h(X).
E
The f u n c t i o n a l u = A h ) r e a l l y i s a c o n i c a l measure.
2'
C401,
rn i s a l a t t i c e homorphism,
rn ( z ' )
@
@
9rn ( z ' ) ( T ) 2 0 , f o r
2' 2 0 , so ~ ( 2 ' )=
2'
E
Z
I t i s obviously
0 f o r every
h(X),
2'
t 0.
The conical measure u = A ( m ) a s s o c i a t e d with rn d e f i n e s a zonoform which
i s equal t o t h e closed closed convex h u l l of t h e range of rn. I f u = A h ) i s t h e conical measure corresponding t o the vector
THEOREM 1.
measure m
:
S
+
X then the resultant r ( u ) e x i s t s and belongs t o X , f o r every
conicaZ measure v P r o o f.
E
If
M+(X) such t h a t v
k
A(;)
and
m(S) =
u, and
Ku = z r n ( S ) .
i s t h e c l o s u r e of rn then, by Theorems IV.3.1. -
u
5
A
.*
. . A
co m ( S ) = h ( f ): f
E
MIo,l,(S)},
and IV.6.1,
Hence we can assume
132
VII .2
THE CONICAL MEASIIRE
t h a t m i s cl o s ed and prove t h e e q u a l i t y i m ( f ) : f 1, E
M~o,l,(S))= { r ( v )
E
u
:
u,
S
M+(X)}.
Let f th en v
5
E
Mco,13(S). Define n
u and m ( f ) = n(T) =
Conversely, l e t v
E
: S -+
W ( X ) ,v < u .
@m(y'- Z ' A y ' ) = 0. f o l l o ws t h a t 0 u(y'-
:
m
A
r
=
rn
Consequently, u ( z ' -
v ( z ' - z'
y') 5 u(z'-
= Q m ( k ( X ) ) . For p
E
r
k ( X ) and @ m ( y ' ) = ' p , ( z ' )
E
2' A
z ' A
@ (2')
rn
y')
= @ m ( y ' ) and Q ; , ( z ' - z '
u ' ( y ' - z'
y ' ) = 0 and 0 A
r.
d e f i n e d f u n c t i o n on
i s l i n e a r and t h a t 0
A
y f ) = 0.
v(y'-
5
z ' A
It
y')
5
we p u t 9 ( p ) = v ( z ' ) , where z ? e h ( X ) i s a cp
i s an unambiguously
If f o ll o w s from t h e d e f i n i t i o n o f q t h a t q :
5 9(u) S
y') =
A
y') = u(y').
a r n , ( z ' ) . I t was j u s t proved t h a t
f u n c t i o n such t h a t p
If v = A ( n ) ,
h(X) + ca(S) is a l i n e a r l a t t i c e
@ (2') A @ (2')
rn
z'
If y',
y ' ) = 0, and hence, v ( z ' ) = u ( z '
z'A
Let
L
y') =
A
S.
E
r(v).
then v ( y ' ) = v ( z ' ) . In f a c t , since homomorphism, @m(z'
X by n(E) = m E ( f ) , E
~(l), f o r e v e r y
p E ,'l
u
r
+IRA
2 0.
By assumption, m i s a c l o s e d measure, consequently MCo,l,(S) i s o ( r ) - co m p act and C o r o l l ar y 1 t o Theorem 1 1 1 . 5 . 1 g i v e s t h a t t h e r e i s an f q ( v ) = p(f),
f o r every p
In p a r t i c u l a r ,
(XI,
THEOREM 2 .
v
E
M t ( X ) with v
E
r.
That i s , @ m ( z ' ) ( f =) v ( z ' ) ,
m ) ( f ) = u ( x ' ) , f o r e v e r y X'
E
MCo,ll(S) w i t h
f o r ev er y z '
X I , or m(f)
E
E
k(X).
= r(u).
I f u i s a conical measure on X w c k t h a t r ( u ) c X , f o r euery L
u, then there e x i s t s a s e t T, a a-aZgebra S of subsets of
T and a closed vector measure m : S + X suck t h a t u = A h ) .
P r o o f.
Assume f i r s t t h a t X i s complete i n i t s weak topology o ( X ' , X ) .
By ill] o r C121, Theorem 3 8 . 1 3 , u i s a D a n i e l l i n t e g r a l on h ( X ) . By t h e s t a n d a r d t h eo r y of D a n i e l l i n t e g r a l , t h e r e i s a l i n e a r l a t t i c e
1
L
1
( u ) containing
1
h ( X ) and a D a n i e l l i n t e g r a l u1 on L ( u ) s u c h t h a t L ( u ) i s a complete sp ace under t h e seminorm z 1 ,+ IIzlIIu
u l ( ] z ' l ) , z'
E
L
1
(u).
The q u o t i e n t s p a c e L
1
(U)
THE RANGE
VII.2
of L1( u ) modulo t h e c l a s s of f u n c t i o n s z ' e
133
L1( u ) such t h a t u
(
12'
I)
= 0 is
1
a Eanach space and, more s p e c i f i c a l l y , i t i s an CALI-space. Theorem (C171, Theorem IV.4.2.),
By Kakutani's
t h e r e e x i s t s a s e t T, a o-algebra S of s u b s e t s
of T and a non-negative (possibly i n f i n i t e ) measure A on S such t h a t L 1( u ) i s l i n e a r l a t t i c e isomorphic and i s o m e t r i c t o the space L 1( T , S , A ) .
Moreover, i f
h i s inYiiiita2, i t i s a d i r e c t sum of f i n i t e measures, hence i t i s l o c a l i z a b l e .
So, f o r every z '
E
L1( u ) , t h e r e corresponds an element of L 1( T , S . A ) such t h a t
t h i s correspondence i s a l i n e a r l a t t i c e homomorphism and i f f z l i s a r e p r e s e n t a n t of t h e element corresponding t o z '
E
L 1( u ) , then u l ( z l ) = If,,dh.
In p a r t i c u l a r
(11 f o r every z '
For any
E
Z'
h(X). E
h ( X ) and E
E
I
S, l e t uZ,(E) = E f z 'dh, where fzl i s a represen-
1
t a n t o f t h e element of L ( T , S , h ) corrcsponding t o z ' . not depend on t h e choice of t h e r e p r e s e n t a n t .
Obviously, u z l ( E ) does
1 Since t h e mapping of L ( u ) onto
1 L ( T , S , A ) i s a l i n e a r l a t t i c e homomorphism, t h e correspondence z'
E
h ( X ) , preserves t h e l i n e a r l a t t i c e o p e r a t i o n s .
2'
* vzl(E),
Now, t h e space
X , being
complete i n i t s weak topology, can be i n t e r p r e t e d a s t h e space X'* of a l l l i n e a r forms on t h e dual X' of X.
m(E)
E
Hence, f o r every E e S , t h e r e e x i s t s an element
X such t h a t z ' ( r n ( E ) ) = pz,dE), f o r every x'
z ' ( r n ( E ) ) = p z l ( E ) , f o r z' every x '
E
E
h ( x ) . Since
p
5'
E
XI, and, more g e n e r a l l y ,
i s a r e a l - v a l u e d measure on S, f o r
XI, t h e O r l i c z - P e t t i s lemma gives t h a t m
: S +.
X i s a v e c t o r measure.
I t i s c l e a r from t h e d e f i n i t i o n of rn and from (1) t h a t u = A h ) .
By Theorem
IV.7.3, rn i s closed.
Now l e t X be an a r b i t r a r y quasi-complete 1 . c . t . v . s .
Every element of h ( x )
i s a f u n c t i o n on X which is uniformly continuous with r e s p e c t t o t h e weak topology of X. hence i t i s a r e s t r i c t i o n of a unique f u n c t i o n on t h e weak
completion X'*of X.
E
E
Every
has a r e s u l t a n t b e lo n g in g t o XI*, s o , we have proved t h a t ,
element of &(X'*) f o r every u
This e s t a b l i s h e s an i d e n t i f i c a t i o n of h(X) with h ( X ' * ) i n
Consequently, t h e s p a c e s W ( X ) and Mt(X'*) a r e i d e n t i c a l .
an obvious way.
if u
VII.3
THE RELATION
134
@(XI*) t h e r e is a v e c t o r measure m
M + ( X r * ) = @(X)
i s such t h a t Ku
c
: S
+
X'* w i t h u =
X, th en by Theorem 1, m ( S )
A h ) . c
But
z m ( S ) = Ku,
o r t h e v al u es o f m belong t o X, i . e . m : S + X.
3. The r e l a t i o n between m and A ( m ) The shortcoming of Theorem 2 . 2 i s t h a t i t i s p u r e l y e x i s t e n t i a l i n character.
The p o s s i b l y more i n t i m a t e r e l a t i o n between t h e v e c t o r measure m and
t h e corresponding c o n i c a l measure A ( m ) i s l o s t due t o t h e a b s t r a c t n e s s i n t h e c o n s t r u c t i o n of rn.
The r e l a t i o n between m and A(m) i s q u i t e t r a n s p a r e n t i n t h e
c a s e when rn h as a d e n s i t y w i t h r e s p e c t t o a non-negative measure on S .
Then
A h ) can be e x h i b i t e d i n terms o f i n t e g r a t i o n with r e s p e c t t o a measure on C ( X ) ,
t h e c y l i n d r i c a l o - a lg e b r a on X . LEMMA 1.
rn
: S
-f
Let (!7',S,A) be a measure space,
6
:
T
+
X a A-integrable function,
X the i n d e f i n i t e integral of 6 with respect t o A and
corresponding conical measure.
For every E
E
CU!.
let v(E)
u = A h )
the
A(d-'(E)).
Then
v i s a measure on C ( X ) such t h a t
for every z '
E
P r o o f.
h(X).
For every z'
i n t e g r a l of t h e f u n c t i o n
2'06
E
h ( X ) , t h e measure @ m , ( ~ ' i) s t h e i n d e f i n i t e
w i th r e s p e c t t o A .
d e f i n i t i o n , u(z') = @m(z')(T) =
1 ~ ~ dA. 0 6
Hence, i f u = A h ) , t h en , by
On t h e o t h e r hand, v({x :
~ ' ( 2 E)
B})=
THE RANGE
VII.3
A({t :
El)
(zto4)(t) E
that Iz'dv =
12106
f o r any z 1
E
135
h ( X ) and any Bore1 s e t B c
d. I t
follows
di.
I f u can be represented, a s i t o f t e n can be, a s an i n t e g r a l with r e s p e c t t o a non-negative measure on a o-algebra of s u b s e t s of a s e t T
c
X, then t h e
c o n s t r u c t i o n of t h e v e c t o r measure m with u = A ( m ) resembles more t h e c l a s s i c a l method of i n t e g r a t i o n by p a r t s .
I n t h e next Theorem t h e e x i s t e n c e of such
r e p r e s e n t a t i o n w i l l be assumed and i n t h e subsequent t h e e x i s t e n c e i s guaranteed. Every function z 1
h ( X ) i s t h e r e s t r i c t i o n t o X of e x a c t l y one f u n c t i o n
E
belonging t o h ( X ' * ) , where X I * i s t h e weak completion of X . d i s t i n g u i s h i n n o t a t i o n between z '
E
So, we do n o t
h ( X ) and t h e corresponding element of
h(X'*) and we a l s o w r i t e h ( X ) = h ( X t * ) . THEOREM i .
Let u be a conical measure on X such that, for every v
v s u, the reeultant r ( v ) e x i s t s and belongs t o X .
Let T
c
E
M+(X),
XI* be a s e t , S a
o-algebra of subsets of T and A a non-negative possibly i n f i n i t e measure on S such that every
2'
E
A(z').
h ( X ' * ) i s A-integrable and u ( z l )
Then the identity-function on T i s A-integrable, the integrals (1)
m(E)
belong t o X , f o r every E
E
/t dA(t) E
S , and so defined vector measure m : S
+.
X satisfies
u = A(m).
P r o o f.
f o r every E
E
S.
Since XI* i s weakly complete m(E) e x i s t s and belongs t o XI*, This defines a v e c t o r measure m : S
of Lemma 1 gives t h a t u = A ( m ) . m(E)
E
X, f o r every E
E
S.
A conical measure u
E
By Theorem 2 . 1 ,
-+XI*.
co m ( S )
The d e f i n i t i o n
Ifu c X. I n p a r t i c u l a r ,
Hence m i s X-valued. @(X) i s s a i d t o be l o c a l i z e d on a compact s e t T if
VII.3
THE RELATION
136
t h e r e i s a non-negative f i n i t e r e g u l a r Bore1 measure h on T such t h a t u ( z ' ) = h ( z ' ) , f o r every z'
THEOREM 2 . T
c
E
h ( X ) . We'say t 5 a t X l o c a l i z e s u C21, D e f i n i t i o n 3 0 . 4 .
If u is a conical measure on X localized on a compact set
X, then there e x i s t s a closed vector measure m on the a-algebra S of Bore2
s e t s i n T with values in X such that u = A ( m ) .
P r o o f, t
++
t. t
every E
E E
I f h localizes u define m : S
-t
XI* by (1). Since t h e integrand
T, i n (1) i s continuous with compact domain, m ( E ) belongs t o X , f o r S.
The v e c t o r measure m i s closed by Theorem IV.7.3, s i n c e ( T , S , A )
i s a l o c a l i z a b l e measure space i f h i s a f i n i t e measure.
Another s u f f i c i e n t condition f o r a p p l i c a b i l i t y of Theorem 1 is given i n terms of t h e "size" of t h e space X i n t h e next C o r c l l a r y .
I t could be o f i n t e r e s t
t h a t i f t h e space X i s not 'Yo0 large" then t h e v e c t o r measure m such t h a t
u
= A ( m ) can be taken
THEOREM 3.
on t h e same domain f o r every u
E
Mt(X).
Let the weak completion of the space X be IR'
with c a r d I
S
H 1'
Then there e x i s t s a s e t T and a a-algebra S of subsets of T such that, f o r every u with v
E
5
Mt(X), such that the resultant r ( v ) e x i s t s i n X for every v u, there e x i s t s a closed vector measure m
P r o o f.
By Theorem 1.4.1, t h e r e i s a s e t T
subsets of T such t h a t , f o r every u t h a t u ( z ' ) = A ( z ' ) . for every z t
E
E
: S -+
c
E
M+(X)
X such that u = A ( m ) .
XI* and a a-algebras S of
M+(Xl*) t h e r e i s a measure
A on S such
h ( X ' * ) and t h e measure space ( T , S , X ) i s a
d i r e c t sum of f i n i t e measure spaces.
Then t h e v e c t o r measure m : S
-+
X defined
by (1) s a t i s f i e s u = A ( m ) , by Theorem 1, and it i s closed by Theorem IV.7.3 s i n c e , according t o C731, a d i r e c t sum of f i n i t e measure spaces i s a l o c a l i z a b l e measure space.
THE RANGE
VII . 4
137
4 . Consequences of t h e t e s t
By Theorem 1.1 t h e problem o f c h a r a c t e r i z a t i o n of s e t s o f t h e form Z m f S ) , f o r some v e c t o r measure m : S +.X, i s reduced t o t h e problem of c h a r a c t e r i z a t i o n of zonoforms.
There e x i s t s a considerable body of r e s u l t s about zonoforms
a p p l i c a b l e here.
Some of them w i l l be quoted i n t h e next two Theorems.
F i r s t l y , t h e question can be r e s t r i c t e d t o s e t s having 0 f o r t h e i r c e n t r e
of symmetry. I f S i s a o-algebra of subsets of s e t T and m : S
LEMMA 1.
measure, then
4 m(T) is the centre
o f symetry of
+
X a vector
mfS).
Any translate of z m ( S ) containing 0 i s the closed convex hull o f the
range of another vector measure.
P r o o f.
For convenience, by Lemma 1.1, we can assume t h a t
mfS)
m(S). If F
E
S and i f we put m l ( E ) = m(E - F )
-
m ( E n F), f o r E
S - + X i s a v e c t o r measure such t h a t rnl(S) = m ( S ) The choice of F
E
S with m ( F ) =
4 m(T) gives
-
E
S , then ml :
m(F).
t h e symmetry of m ( S ) around
4m(T). I f K i s a weakly compact ( o r , a t l e a s t , bounded) s e t , K c X , d e f i n e IlxfIIK = s u p I I ( x ' , x )I :
3: E
K ) , f o r every x'
E
XI.
For t h e following c r i t e r i o n we r e f e r t o C111, Th6orBme 64, o r C12 THEOREM 1.
Let K be a ueakzy compact, convex subset of a 1.c.t.v
having 0 for the centre of symetry. n
Then K
i8
s. X
a zonofonn if and onzy if
138
VII.9
CONSEQUENCES OF THE TEST
f o r any choice o f z j 1.2,...
E
XI. ai
E
IR, i
.
= 1,2... .n, suck t h a t
n
= 0, n =
*
Let p be a real number, p 2 2 .
EXAMPLE 1.
Banach space $(A).
The closed u n i t b a l l of t h e
f o r m y o - f i n i t e measure A , i s t h e range of an LP(A)-valued
measure. I f K i s t h e u n i t b a l l , by r e f l e x i v i t y of #(A),
K i s weakly compact.
The
dual space of L p ( A ) being L 4 ( A ) , with q = p / ( p - 1 ) . t h e norm Ilx'llK o f an element z1
E
L 4 ( A ) i s equal t o i t s n a t u r a l norm i n Lq(A). Hence, by Theorem 1 , i t
s u f f i c e s t o show t h a t
f o r any x! e L ~ ( A ) a, . e IR, i = 1.2 2
2
,...,n .
l:=lai
= 0, n = 1.2
In our case
This statement i s known f o r q i n t h e i n t e r v a l C1,21. 1 < q 5 2.
,... .
I t can be proved e . g . by observing i t f i r s t f o r t h e case when
"1
a r e simple functions (see C251)and then passing t o t h e l i m i t . A n e t { K a I a E A of weakly compact, convex s e t s i n
X i s s a i d t o converge t o
a weakly compact convex s e t K i f
f o r every z ' A net
E
XI. We w r i t e K
= lim
K
aeA a'
{uaIcrEAof conical measures on X i s s a i d t o converge (vaguely) t o a
c o n i c a l measure u i f u ( z ' )
limaeAua(z'), f o r every z '
E
h ( X ) . We write
u = l i m aeAUa' The following Theorem can again be found i n 1111 (ThkorBme 6 9 ) . THEOREM 2 .
If {uaIacA i s a n e t of conical measure8 on X and u
= limaEAua,
139
THE RANGE
VII.4
then KU = limUGAKU
.
Every zonoform i s the l i m i t of a net of zonohedra.
(I
I f {uuIaEA i s a net of symetrical conicaZ measures and i f K i s a weakly com-
pact, convex set such that K such that u
= limorEAuff and
K
= lim =
K a 4 u
' then there i s a conicaZ measure
u
Ku.
A convex, weakly compact s e t K i s a zonoform i f i t s image under any
continuous affine mcrp on any f i n i t e dimensional space i s a zonoform. A convex, weakly compact set which i s a l i m i t of zonoforms is a zonoform. A projective l i m i t or a Cartesian product of zonoforms i s a zonoform. A closed face of
EXAMPLE 2 .
a zonofom is a zonoform.
Theorem 2 again shows t h a t t h e u n i t b a l l i n Z p ( I ) , with p
i s t h e range of a v e c t o r measure by reduction t o f i n i t e dimension.
2 2,
Either the
l i m i t with r e s p e c t t h e n e t of f i n i t e s u b s e t s o f I can be used, or continuous
a f f i n e maps onto f i n i t e dimensional spaces. EXAMPLE 3 .
The u n i t b a l l i n Z m ( I ) i s geometrically s u i t a b l e t o be t h e
range o f a v e c t o r measure, but i f f a i l s t h e t o p o l o g i c a l t e s t , v i z . it i s n o t weakly compact.
However, t h e u n i t b a l l i n l " ( I ) i s t h e range of a v e c t o r measure
taking values i n l m ( I )equipped with i t s weak* topology.
I t suffices t o take
t h e c h a r a c t e r i s t i c f u n c t i o n t o be t h e value of t h e measure defined on a l l s u b s e t s of I and use Theorem V.5.1.
Remarks Theorem 1.1 f o r measures with values i n f i n i t e dimensional spaces was apparently f i r s t proved by Rickert i n C67.l.
H e proved, t h a t f o r any IRnvalued
measure m t h e r e i s a non-negative measure A on t h e u n i t sphere
s"-1 such
that
t h e range of m and t h a t of t h e i n d e f i n i t e i n t e g r a l of t h e i d e n t i t y with r e s p e c t t o A have t h e same closed, convex h u l l s .
He a l s o gave c o n d i t i o n s f o r two
VII
REMARKS
140
measures with values i n R n t o have t h e same closed convex h u l l s of t h e i r ranges.
n To show t h a t t h e r e i s a measure having f o r i t s s e t of v a l u e s a b a l l inIR i s a simple m a t t e r .
I t s u f f i c e s t o t a k e t h e i n d e f i n i t e i n t e g r a l of t h e i d e n t i t y
Since t h e range of t h i s f u n c t i o n with r e s p e c t t o t h e s u r f a c e measure on $-I. measure i s r o t a t i o n a l l y i n v a r i a n t i t i s a b a l l .
Rickert f i n d s its r a d i u s i n 1661;
t h i s i s a l e s s t r i v i a l matter. The a t t e n t i o n t o geometric aspects of t h e range of a v e c t o r measure was turned by Bolker i n C8l.
He gives a good complete survey of r e s u l t s concerning
n
t h e range o f R -valued measure and connections with n e g a t i v e - d e f i n i t e f u n c t i o n s . Lemma 1.1 i s an analogue t o a statement i n h i s paper.
The statement of Theorem 1.1 and i t s proof i n t h e given g e n e r a l i t y i s from L401.
This Theorem has value only i f a body of r e s u l t s concerning
zonoforms i s a v a i l a b l e . For t h i s we r e f e r t o Choquet Clll o r 1121, where f u r t h e r references a r e given.
I t could perhaps be of i n t e r e s t t h a t Theorem 1.1
can give r e s u l t s about zonoforms.
For example, t h e extremal s t r u c t u r e of
zonoforms can be studied using v e c t o r measures.
For i n s t a n c e , it follows from
Theorem VI.2.3 t h a t t h e s e t o f extreme p o i n t s of a zonoform i n
If
i s closed.
Statements t h a t every extreme (exposed) p o i n t of a zonoform is s t r o n g l y extreme (resp. s t r o n g l y exposed) (Theorem I V . 1 . 2 and VI.4.2) a r e p o s s i b l y new. Lemma 4 . 1 i s due t o Halmos C231.
For a Banach space valued measure with precompact range Anantharaman proved i n C3l t h a t i t s closed convex h u l l i s t h e l i m i t i n Hausdorff m e t r i c of zonohedra.
In t h e converse d i r e c t i o n , t h e convergence i n Hausdorff m e t r i c ,
more generally i n Hausdorff uniformity, of compact, convex s e t s implies t h e convergence i n t h e sense used i n Theorem 4 . 2 .
Indeed, t h i s convergence i s
equivalent t o convergence i n t h e Nausdorff uniformity derived from t h e weak topology of t h e space X .
141
THE RANGE
VII
As t o Example 4.1, it can be conjectured t h a t t h e u n i t b a l l i n
p
< 2,
i s not t h e range of a v e c t o r measure.
#,
for
This c o n j e c t u r e i s c o r r e c t , i f
L p ( X ) i s of i n f i n i t e dimension ( i . e . t h e measure h does not reduce t o a f i n i t e
number of atoms), s i n c e J. D i e s t e l has shown t h a t t h e closed convex h u l l o f an #(X)-valued measure c a n ' t contain an i n t e r i o r p o i n t .
( I f it d i d
9 would
be
simultaneously a subspace of LA and a q u o t i e n t of C ( S ) and hence a H i l b e r t space.) If
#
i s of f i n i t e dimension then we can ask whether t h e u n i t b a l l i s e x a c t l y
t h e (closed convex h u l l of the) range o f a v e c t o r measure. question i s not y e t answered.
I t seems t h a t t h i s
YIII.
FUNCTION SPACES 11
The problems c o n s i d e r e d i n Chapter I11 a r e now t ak en up ag ai n .
The
d i f f e r e n c e i s , h e r e we d e a l w it h sequences of f u n c t i o n s and measures i n s t e a d S i m i l a r t ech n i q u es as i n Chapter I11
o f i n d i v i d u a l f u n c t i o n s and measures.
a r e used except f o r t h o s e problems r e l a t e d t o measurable s e l e c t i o n s o f s e t T h i s c h a p t e r forms a b a s i s t o t h e f o l l o w i n g one on c o n t r o l
valued f u n c t i o n s .
systems i n t h e same way as Chapter 111 was t o t h e t h e o r y of cl o sed measures.
1. S e t - v a lu e d f u n c t i o n s Let Embe t h e c o u n t a b le p r o d u c t o f the real l i n e t r e a t e d a s a 1 . c . t . v . s . under t h e product topology, i . e . t h e topology o f co - o r d i n at ew i se convergence.
Id"'
Similarly let the real line.
i =
1,2,
...
d") i s
Every element z ' of
, where
dm)i s
that
be t h e t o p o l o g i c a l d i r e c t sum o f co u n t ab l y many c o p i e s o f
a l l b u t f i n i t e l y many o f t h e
the dual of the 1.c.t.v.s.
zt
o f t h e form z ' = (z;), 3:;
E
IR,
I t i s well known
are zer o .
El-, w i t h t h e p a i r i n g
m
m
where
3:
(IR")'
=
(zi)E IR
, and
z ' = (xi) E
A"'.
Ir, o t h e r words w i t h o u r u s u a l n o t a t i o n
nf-'.
Suppose S i s a o - a l g e b r a of s u b s e t s of a s e t T.
h function f : T
+
IRm
w i l l be c a l l e d S-measurable i f each o f i t s components i s S-measurable, t h a t i s i f
f
(fi),fi
i = 1,2,. on IR".
.. .
:
T
-t
IR, i = 1 . 2 ,
...
, then f
fi
i s S-measurable i f
E
E.4(S) f o r each
Let hl(Rm,s) d e n o te t h e v e c t o r sp ace of a l l S-measurahle f u n c t i o n s
S i m i l a r l y dMmM,S) i s t h e s e t o f measurable f u n c t i o n s
which a r e uniformly bounded, i . e . s u p { l l f ~ l , : 142
i
1.2, . . . I <
f = (fi) :
-.
T
+
IRm
VIII.1
FUNCTION SPACES I1
F
A set-valued function
defined on T whose values are subsets of IR" will
be called bounded if there exists a compact set W
t
E
143
c
IRmsuch that
F(t)
c
R.
We call
V,
T. For such a set-valued function p we put,
Elements of BMFmm,S) will be called measurable selections of F. Denote by C C R m the family of all compact convex subsets of a set-valued function
F
:
T
+.
C C R m measurable, if, for every
Z'
E
@RJ'
, the
mapping
is measurable. Equivalently F is measurable if, for every
(e:)
Z'
dm),
E
the mapping m
t
++
sup{ 1 zizi i.1
: z
(xi) E F(t)1, t
E
T.
is measurable. We remark that if function f, then
F
F
is single valued, i.e. F ( t ) = f(t), t
E
T, for some
is measurable if and only if f is.
We will now show the existence of measurable selections of a measurable set-valued function. L E W 1. Suppose
{FnI is a sequence of measurable set-valued functions (t) c Fn(t), t E T , n = 1,2,..) Fn+l m CCIRmdefined by F(t) = nnZlFn(t),t E T , i s
mapping T i n t o CCB?, ujiich are decreasing ( i . e . Then the set-valued function
F
:
T
+
measurable . OD
P r o o f. Let To = nnzlft : F n ( t ) s ( O , F n ( t ) ) = --I
E
S we have T0
E
* PI.
Since It
S. Clearly To = It
:
:
Fn(t)
F(t)
*
01.
PI
=
{t
:
Suppose
x'
VIII . l
SET-VALLIED FUNCTIONS
144
E
(IFm)' and
t
E
T O . We w i l l show t h a t
(in fact this i s t r i v i a l i f
8
s ( x ' , F n ( t ) ) = (x', xn), n = 1,2
x), n
( X I ,
+
m.
Fn ( t ) a r e chosen so t h a t We can f i n d x E F ( t ) such t h a t ( I ' , z n ) *
T O ) . Suppose
,... .
I
n
E
Then
The converse i n e q u a l i t y i s c l e a r .
If F is measurable, z '
LEMMA 2 .
E
(IR")'
and if f :
T
-f
IR is a measurable
function, then the set-valued function
i s measurable.
P r o o f. -f(t)
5
Iaz' : a 2'
F i r s t l y , T O = I t : H(t) # 01 = {t : f(t) 5 s(z',F(t)) and
s(-z'.F(t))l E
E
S. Now suppose x'
E
(ROD)' and t
E
To. I f x'
IR} then x' = aoz' and s ( ~ ' , H ( t )=) a o f ( t ) f o r some a0
4 Ilz',and
E
Rz' =
R . Otherwise
E
we w i l l show t h a t
This Lemma w i l l then follow a s t h e f u n c t i o n a * s ( a z ' + I' , F ( t ) )
-
af(t), a
E
i s continuous with r e s e p c t t o a , and so we need only t a k e t h e infimum over a l l r a t i o n a l numbers a . To prove (1) suppose x
I'
E
(TRm)'.
E
H(t) and consider t h e l i n e a r map z ' *
(I',
x),
I t i s bounded on (IRm)I by s(.,F(t)), and i f we consider i t s
r e s t r i c t i o n t o IRz'
@
IRE' a s an extension of i t s r e s t r i c t i o n t o IRz', by a
IR,
145
FUNCTION SPACES I1
VIII.2
Lemma p.220 in r581 we have, ( X I ,X )
s infIs(az' + z I , F ( t ) )- af(t)
: a
E
IR}.
In other words,
s(z',H(t))5 inf{s(az'
+
x',F(t))
-
crf(t) : a
E
El.
Conversely, suppose h is the linear map defined on IRz' by h (c r z ' ) = af(t), a
E
IR, and h' an extension of h to I R z ' " R z ' defined by
As h' is bounded above by
s ( * , F ( t ) )on IRz'@Ipx', from the Hahn-Banach
Theorem, h' can be extended to a linear map on (IRm)' which remains bounded above by s ( . , F ( t ) ) .
(z',5 )
=
In other words we can find an 5
h'(x') and (x', 5 )
imply that 5
E
5 s(z',
(IF?")'*
E
F(t)), z' E ( I R m ) ' .
= IRm such that
These inequalities
H(t), and that
This proves (1) and the LemTa follows by o u r remarks earlier in the proof,
2. Ffeasurable selections Suppose {eAl is a countable separating subset of ( R defines a lexicographic order on E mas follows. If z , y
I=
{< :
call z
m
Each such set
) I .
m
E
IR , and if the set
( e . z - y ) # 0 ) is non-empty, denote its smallest element by io, and
< y
if xi
<
0
y the set {ell is y i , If I is empty, then ~ = because 0
separating. Clearly any non-empty, compact set I<
m
c
I!? has a largest element
in this ordering. We call such an element the lexicographic maximum of K
146
MEASURABLE SELECTIONS
VIII . 2
ordered by { e A l , THEOREM 1. If F : T
-+
C C R m i s a measurable, non-empty set-valued function,
{ e ; } any countable, separating subset of (IRrn)', and i f f(t) i s the lexicographic maximm of F ( t ) ordered by { e h l , f o r each t f ( t ) E exF(t), f o r each t
P r o
o
E
E
T, then f is measurable and
T.
f. Clearly, f i t )
E
exF(t), t d . Set F , ( t )
= F ( t ) and f o r n > 1 put
.
is measurable, and as Fn 3 Fn+,, n = 1 , 2 , , , , their intern section will be measurable by Lemma 1.1. However, n" F ( t )= {f(t)), t E T. n=O n By Lemma 1.2 each F
If F
THOEREM 2.
:
T + CCIRm is a measurabze, non-empty set-valued function,
there e x i s t s a countable f a m i l y Ifn) of measurable seZections of F such that { f n ( t ) : n = 1.2
P r o o f.
,... 1 i s dense Let D
=
i n F ( t ) , for each t
Q(m) be the subset of
E
T.
dm) whose elements have rational
co-ordinates. Then D is a countable dense subset of (IR")'.
Choose an
5' E
D
and order D so that x ' is the first element. Suppose f x , ( t ) is the lexicographic maximum of F ( t ) subject to this ordering, t
E
T. Then fxlis measurable by
Theorem 1, and
Let Ifn
:
n = 1.2. . . . I be the family of all barycentres of finite combinations
of the functions f,,, set K = { f n ( t )
:
x'
E
D with rational coefficients. For a fixed t
n = 1,2,. , . } .
Then
E
T.
contains the convex set A consisting of
all barycentres of the points f x , ( t ) , x'
E
D.
Clearly 7
=
I
or 7 is conva.
VIII.2
FUNCTION SPACES I1
We know 17
147
F(t). To prove the converse we will show that all closed half-spaces
c
and F(t) are the same, and then the result will follow from the
containing
Hahn-Banach Theorem. Suppose 3:' E
E
(JR")', and
> 0.
E
There exists y'
E
D such that I ( x ' - y ' , s)l<
F(t). Then f ,(t) E K and (3:',f ,(t))2 (y',fyl(t)) - E = Y Y - E = s(z',F(t)) - 2 ~ . Since this holds for all E > 0 , sup{(z', x ) :
for all
3: E
s(y',F(t))
= s(s',F(t)).
3: E
LEMMA 1. A non-empty set-valued function F only i f {t
F(t) n U # !a3
:
S f o r every open s e t
E
T
:
u
-+
c
i s measurable if and
CCR
mm,
P r o o f. If F is measurable, let Ifn) be the dense sequence of measurable selections of F constructed in Theorem 2. for each n = 1 , 2 , 3 ... and I t Conversely suppose z 1 sup{(z'.
3:)
:
3: E
:
Then if U is open in lRm, fil(U) E S
F(t) n U # 03
U ~ , l f ~ l ( U )E
( R m ) ' ,then the function, t
E
E
F(t) n U # 03 where U = m
open in lR
E
lR
,
: (r',x ) > a}, a
{3:
E
:
s(x',F(t))
> a)
IR. Since U is clearly
the result follows by the hypothesis. Let F,G
LEMMA 2.
t
E
=
S. Because of the compactness of F(t), the supremum in
the definition can be replaced by maximum and so I t :
* s(z',F(t))
F(t)I, is measurable if and only if, for each a
{t : s(z',F(t))> a}
{t
S.
:
T
-+
CCIRm be measurabk.
I f we define H ( t ) = F(t) n G(t),
is "on-empty, then it is measurable.
T , and H
P r o o f. By Lemma 1 we need only show that for any open set U c R
{t
:
F(t) n G(t) n U # 01
If g
:
E
5'.
:
F(t) n (g(t) + V ) n U # 0)
T
choose measurable functions f
:
dense in F(t) for each t
Then
3
E
T.
,
The proof falls into two parts.
T + IRm is a measurable function and U,V are open sets in IR
show that {t
m
-t
IR
m
E
m
we
S. The fact, by Theorem 2 , we can
such that
If 3.(t) : j
= l,Z,..
.
is
SEQUENCES OF MEASURES
148
It
:
F(t) n (g(t) t V ) n U
* 01
VIII.3
iii(fjl(~) n (f. 3
=
-
Let Vn be open s e t s i n Rmhaving only 0 i n c m o n and Vn Suppose g
i
i = 1,2,.
.. 3
It
:
T
:
-+
IRm a r e measurable f u n c t i o n s i
i s dense i n G ( t ) , f o r each t
*
F(t) n G(t) n U
nzl
01 =
E
iclIt
T. :
1,2,.
g)-l(V))
3
S.
E
Vn+l, n = 1,2,.
.. such t h a t
{gi(t)
. .
:
Then
F(t) n (gi(t) + V n ) n U
*
01
which belongs t o S by t h e f i r s t p a r t of t h e proof.
3 . Sequences of measures
Let cca(S) be t h e s e t of a l l sequences p = (pi) o f measures
i=
1,2,.
..
m
, with ,&=ll
For a s e t A
pil
c cca(S).
(TI <
ui
i
1,2
equivalent i f
,... .
If -
A-equivalent t o f
d e f i n e Ai
= {Ai
:
A = (Ai)
E
A 3 f o r every j = 1,2,..
fi i s Ai-null
Two functions f = (fi), g = ( g i ) i n M(IRw,S) a r e A-
91 is A-null. E
ca(S),
-.
We c a l l a function f = (fi) i n M(Rm,S) A-null ( o r A-negligible) i f f o r each
E
M(IRm,S)
by
As b e f o r e we denote t h e s e t of a l l g
Cfl,,
E
M(Rm,S)
and d e f i n e ,
For a bounded set-valued mapping F from T i n t o t h e s u b s e t s of
f we
put,
On Lm(Rm,A) d e f i n e t h e l o c a l l y convex topology ~ ( h by ) t h e family of semi-
norms {pA
: A E A)
where m
pA([f’A) =
-1/IfiIdlA{l, 2-1
.
VIII.3
for
f
149
FUNCTION SPACES I1
= (fi)
BM(Rm,S) and h = ( A . 1
E
2
A.
E
As before t h e same symbol T ~ A )w i l l
be used t o denote t h e uniform s t r u c t u r e on Lm@YA) and i t s s u b s e t s .
By t h e
weak topology on Lm(IRm,A) we mean t h e o ( Y , Y ‘ ) topology, where Y‘ i s t h e s e t of a l l T ( A ) continuous l i n e a r f u n c t i o n a l s on Y = Lm@lm,A). The o t h e r topology we w i l l use on t h e space Lm(IRm,A) i s t h e o ( r ) topology where F i s a subset of c c a ( S ) .
u
each
= (u.1 2
E
r,
+
+
such t h a t t h e maps ‘$1,
I t i s defined a s follows.
m
[fl,
&=ljfidpi,
E
If
r
c
cca(S) is
Lm(IRm,A), a r e well defined f o r
then t h e o ( T ) topology i s t h e weakest topology on L”(R”,n)
making a l l t h e s e maps continuous. SupposeIN r e p r e s e n t s t h e n a t u r a l numbers, and d e f i n e % t o be t h e o-algebra of s u b s e t s E of T
x
{t
IN f o r which each s e c t i o n E ( i )
an element of S, i = 1,2,
... .
F o r each h
(h.)
z
E
:
t
E
T, (t,i) E E l i s
cca(S) t h e r e e x i s t s a
measure X IN: SIN + IR defined by
LEMMA 1. If A E cca(S) and f : T
+
CCJRmfs a bounded s e t - v a l u e d function,
then LF@Im,h) is a reZativeZy weakly compact s u b s e t of Lm(IRm,i).
P r o o f.
Consider t h e space L m
m
(b) under
1
t h e L )A,(
-Lebesgue norm..
m
The elements of L (R,A) can be i n t e r p r e t e d i n an obvious way a s elements of Lm(Q.
The r e s u l t follows from t h e boundedness of
F and t h e well-known c r i t e r i a
1 f o r weak compactness i n L -space ( e . g . C173 Theorem IV.8.9). THEOREM 1 .
and
F
Suppose A c cca(S).
If S(Ai) is T(Ai)-complete, i = 1,2,... ,
: T + C C I R m i s a bounded, measurabZe s e t - v a t u e d function,
then L F ( ~ : h )
i s a weakty compact, convex s u b s e t of L ~ ( R ~ , A ) .
P r o o f.
Consider t h e space Y = Lm(lRm,A) with t h e topology T(A) defined
SEQUENCES OF MEASURES
150
by the family of semi-norms Ip,
:
X
VIII .3
A } defined above.
E
For A
E
Y, = Y / p - l ( O )
A,
h
is the space Lm(IRm,X); denote the natural projection of Y onto Y A by
W = L F ( I R m , A ) , then n,(W) = LF(IRm,X), 1
T,.
Since W is convex, t o prove that
A.
E
If
W is weakly compact, it suffices to prove that W is complete and, for every X
E
A,
the set n,(W) is weakly compact in Y, (Theorem 1.1.1).
Because of the boundedness of F , we can assume without loss of generality that F ( t )
c
7, t
m
T, where I = I I ~ , ~I. I ~= ,C O , l l , i = 1.2 2
E
.... .
“hat is,
considering 7 to be, at the same time, the constant set-valued function t t
E
T, we assume LF(IRm,A)
The completeness of W
c L7(IRm,A).
proved by showing that L r (1R;A)
is complete and that LF(R:A)
++
= LF(IRm,A)
7
wi 1 be
is closed in it
Every S(Ai) is -t(Ai)-complete and so, by the Corollary to Theorem 111.3.2, LCo,ll(Ai)is T(hi)-C0mplete.
is a product of sets LIo,ll(Ai),so
Now L7(R:A)
it follows that L7(IRm,A) is T(h)-COmplete. Suppose that Cfl,
We
belongs to the r(A)-closure of LF(lRYA) in L”(IR”,A),
have to show that CfI,
E
LF(IRm,A).
L ~ ( I R ~ , Af)o,r every A
E
A.
This will be done by showing that
[?IX
E
Let X = (X.) be given. We assume that CfI, belongs to the r(X)-closure of 2 We can select a sequence ([fnl,) such that Cfnl,
LF(IRm,A).
n = 1.2
,...
i = 1,2,...
, and ,
c(t) +
fi(t)as n
m,
There is to be found a g
t
E
T, and g
G
:
T - t CCIRm by G ( t ) = {(xi) :
E
-+
Cfl,,
E
except for t
( j $ ( t )for )
(xi) E F(t) and xi
1,2,...
This does not necessarily hold if t
E
Ei, where I h2. l ( E .%) = 0
.
E
When t
fi(t),t
T. Fix t
E
F(t) for
4 Ei),
t
E
T.
T, consider the
4 E i s we have j $ ( t )
+
fi(t).
Ei, however, we can choose a subsequence
(c(t))
(fl(t)) which converges in each co-ordinate. Since z F(t), for every n = 1,2. ... , and F ( t ) is closed, the limit of
of
E
To do this define a set-valued function
i.e. CflA = Cgl,.
each i
LF(Rm,X),
BM(IR”,S) such that g ( t )
We first show that G ( t ) # 0. for each t sequence
E
E
is a point of
the subsequence
151
FUNCTION SPACES I1
VIII , 3
belongs t o F ( t ) , s o i t i s an element of G ( t ) . Let
Now we show G i s measurable.
r =
1,2
,...,k .
Let z t
E
ir b e n a t u r a l and
r e a l numbers f o r
a
ir
(IRm)l be d e f i n e d by
(x', 2 ) =
k
1a.
xi'
r=l ' r r f o r any form.
3:
= (xi) E lRw .
m
I t i s c l e a r t h a t e v e r y element o f (IR ) ' i s o f s u c h a
Then
where $r i s t h e c h a r a c t e r i s t i c f u n c t i o n o f E . f u n c t i o n o f i t s complement, r = 1,2,
...,k.
o b v i o u s l y a measurable f u n c t i o n o f t . LC E
and
(pr
is t h e c h a r a c t e r i s t i c
r The second term o f t h i s e q u a l i t y i s
The t h i r d term i s e q u a l t o sup{(yi,
2) :
F ( t ) ) , where 3;; E (lRm)l i s d e f i n e d by
Ei , r = 1 , 2 , ..., k l . S i n c e t h e s e t {St : t E 2'1 i s f i n i t e r and t h e s e t of p o i n t s t f o r which St i s t h e same ( a c o n s t a n t ) i s m e a s u r a b l e ; we
where St = {r : t
E
e a s i l y see t h a t t h e t h i r d term i n (1) i s m e a s u r a b l e .
So G i s m e a s u r a b l e .
According t o Theorem 2 . 1 t h e r e e x i s t s a measurable f u n c t i o n g : T such t h a t g ( t ) every t
E
E
T, and
G ( t ) , f o r every t
If],
E
7'.
By t h e c o n s t r u c t i o n , g ( t )
E
+
lRm
F ( t ) , for
= CgiA. T h i s p r o v e s t h a t t h e s e t W i s c l o s e d and h e n c e
complete. m
By Lemma 1, t h e s e t n,(W) = L F ( R , A ) f o r every A
E
A.
YA,
To show t h a t it i s weakly compact i f s u f f i c e s t o show t h a t
it i s weakly c l o s e d .
S i n c e it i s a convex s e t , t h i s i s e q u i v a l e n t t o i t s b e i n g m
closed.
i s r e l a t i v e l y weakly compact i n
But i n p r o v i n g LF(IR,A)
i s .r(A)-closed,we proved a t t h e same time
VIII.4
EXTREME POINTS
152
This f i n i s h e s t h e proof.
LF(IRm,A) i s r(X)-closed.
I f S ( A i ) i s T(Ai)-CO”pZete f o r each i = 1,2,
COROLLARY 1.
...
, and F : T
+
CCJRm i s a bounded, measurable set-valued function, then LF(IRm,A) i s o(A)-compact.
P r o o
The weak topology on Lm(Rm,A) i s t h e topology o ( r ) where
f.
t h e s e t of a l l sequences P = (11,) and a X = (X.)
E
E
c
r,
is
c c a ( S ) f o r which t h e r e e x i s t s a constant k ,
A with lpil s k ) h i ) f o r each
a s s e t s and s i n c e A
r
i = 1,2 ,...
.
m
LF(IRm,A) = L F ( R
o!A) i s a weaker (coarser) topology than o(I’).
,r)
The
r e s u l t follows from t h e Theorem. COROLLARY 2 . pi Q X i ,
Suppose il i s the s e t of a l l sequences
i = 1,2 ,... , f o r
each i = 1.2,.
.. , and
F
:
some
T
-+
)I
= (11.)
z
E
cca(S) with
I f S(Ai) is T(hi)-COmpk?te f o r
X = (1.) z E A.
is a bounded, measurable set-valued function,
CC$
then L~(IF?,A) is u(n)-campact.
,... .
Let 51 = (Q.), j = 1 , 2 Then s ( A ) = (Q.) as s e t s , and 3 5 3 S ( A . ) i s r(h.)-COmplete i f and only i f it i s r(R.1-complete, f o r each j = 1,2, 3 3 3 Further L F ( l f , A ) = LF(]Rm,51) a s s e t s and so t h e r e s u l t follows by Corollary 1. P r o o f,
4 . Extreme p o i n t s
THEOREM 1.
Suppose
set-valued function.
P r o o f.
E
and F
: T + CCE?
i s a bounded, measurable
m
Then exLF(IRm,A) = LexF(R , A ) .
Clearly LexF(Rm,A)
t h e converse i n c l u s i o n ,
t
A c cca(S)
Let Cfl,
E
c
exL (Rm,A), hence it s u f f i c e s t o prove
F
exLF(IRm,A).
T f o r which t h e r e e x i s t s an element u ( t )
E
Ix
Let B+
be t h e s e t of a l l
E
5
n,j F(t) :
i - -f 3(.t )5
l/nl
= G+ . ( t )and an element v ( t ) E F ( t ) such t h a t f ( t ) = $ ( u ( t ) + v ( t ) ) ; and l e t n 13 BG- . ( t )be analoguously defined s e t s with 2 - f .(t) Z l/n replaced hy n,j’ n , j j 3
...
153
FUNCTION SPACES I1
VIII
f . ( t )- zj 5 l/n; j = 1,2,..., n = 1,2,.... 3
.
The s e t s Band B, n,j ,i a r e A 3. - n e g l i g i b l e f o r ev er y j 1 , 2 , . . and n = 1 , 2 , + i s n o t A . - n e g l i g i b l e , f o r some j and n, d e f i n e H : B+ I f , s ay B+ n ,j 3 n ,j CCIRm by H ( t ) = 2f(t - G+ , ( t ) ,t E Bn Let u : T + P m h e a measurable
.. ..
,j*
nJ
f u n c t i o n such t h a t u t) E H ( t ) n F(t), f o r t
t
4 Bn,j.
Further,
i s not A-equivalent t o u , s i n c e
f
i
Denote B j = ui=l(BA,j LJB;,~), g i b l e , f o r ev er y j = 1 , 2 , with t
4 B3. } .
f o r ev er y
f o r ev er y t E
E
T.
i3A,j and u ( t ) = f ( t ) , f o r
u(t),t c T. Then f = + ( u + v ) and f
i s n o t ! , . - eq u i v al en t t o u 3 j*
f o r e v e r y j = 1,2,...
Let F ! t ) = {I
E
F(t)
: z
j
S o , B . is A.-negli3 3 :
f . ( t ) f o r ev er y j 3
m
Let e ! b e t h e j - t h c o o r d i n a t e f u n c t i o n a l on IP. , i . e . ( e ' . , ~=) x 3
IR ,j = 1,2,.
or d er ed by t h e f a m i ly
every t
....
m
I E
-
et u ( t ) = 2 f ( t )
E
{el,
3
:
...
3 j' Let f(t) he t h e l e x i c o g r a p h i c maximum o f f ( t )
...1 .
j = 1,2,
-
Then, by Theorem 2 . 1 , f ( t ) E e x F ( t ) ,
T, and t h e d e f i n i t i o n o f F g i v e s t h a t a l s o f(t) E e x F ( t ) , f o r S i n c e f i s A-equivalent t o f , we have
Cfl,
E
LexF(iR
m
,A).
Remarks The c o n t e n t o f S e c t i o n 1 and 2 i s from V a l a d i e r C831.
The t r e a t m e n t
t h e r e i s more comprehensive; i n most s t a t e m e n t s t h e s e t - v a l u e d f u n c t i o n can t a k e v a l u e s i n a s e p a r a b l e F r e c h e t s p a c e , o r any 1 . c . t . v . s .
whose d u al c o n t a i n s a
c ou n t ab l e s e p a r a t i n g set of f u n c t i o n a l s .
S e c t i o n s 1 and 2 are o b t ai n ed from
V a l a d i e r ' s work by s p e c i a l i z i n g t o IRm.
The t e c h n i q u e s i n S e c t i o n 2 c l o s e l y
n
resemble t h o s e o f Olech c 6 0 ! who used them f o r C C m -valued set f u n c t i o n s .
Ca s t ai n g C l O l gave an a l t e r n a t i v e p r o o f o f t h e measurable s e l e c t i o n Theorem i n t h e IRn c a s e .
S e c t i o n s 3 and 4 f o l l o w t h e corresponding s e c t i o n s i n Chapter 111.
The r e s u l t s a r e from C411.
IX . CONTROL
SYSTEMS
We co n s i d er t h e f o l lo w i n g model f o r a c o n t r o l system w i t h i n f i n i t e l y many de g r ees of freedom s t e e r e d by a sequence o f i nd ep en d en t l y o p e r a t i n g c o n t r o l s .
For i = 1 , 2 ,
...
, a measure m
i i s given on s u b s e t s o f T (a time i n t e r v a l ) with v a l u e s i n a l o c a l l y convex t o p o l o g i c a l v e c t o r space X, t h e s t a t e sp ace of t h e system.
The t o t a l e f f e c t o f a l l c o n t r o l s fi, i = 1 , 2 , .
..,
i s e i v e n by
m
For every t
E
T, a s e t F ( t )
lRm i s g iv e n which r e p r e s e n t s t h e r e s t r i c t i o n
c
on t h e choice of c o n t r o l s a t t h e i n s t a n t t , i . e . fi are chosen s o t h a t (fi(t))E
F ( t ) , f o r ev er y t
E
T.
I n t h i s c h a p t e r t h e p r o p e r t i e s o f t h e a t t a i n a b l e s e t of t h e c o n t r o l system (1) w i l l be co n s id e r e d , along with c e r t a i n a p p l i c a t i o n s t o time o p t i m al c o n t r o l . I t i s perhaps worth remarking h e r e t h a t we have a l r e a d y shown i n S e c t i o n V . 7
t h a t c e r t a i n c o n t r o l problems governed by p a r t i a l d i f f e r e n t i a l e q u a t i o n s can be placed i n t h e form (1).
1. A t t a i n a b l e s e t
Let S be a a - a l g e b r a o f some s e t 7'.
R e c al l t h a t BM(JRm,S) i s t h e s e t o f
a l l sequences (fi)o f f u n c t i o n s i n BM(S) w i t h ~ u p ~ l l f :~ i l l ~ 1,2,.
.. 1
c m.
t h e g e n e r a l s e t t i n g o f t h e c o n t r o l problem c on si d er ed h e r e , elements o f BM(Rm,S)
w i l l be called controls.
A sequence of c lo s e d v e c t o r measures mi m
c a l l e d a c o n t r o l system i f l i = l m i ( S ) 1.2.
..., t h i s
: S +
X , i = 1 .2 ,
i s convergent i n X .
convergence i s u n c o n d i t i o n a l . 154
We write m =
...
S i n ce 0
h.). 2
, w i l l be E
m.(S), i = 2
In
CONTROL SYSTEMS
IX.1
155
LEMMA 1. If m = ( m i ) i s a controZ system and f = (fi)E EM(pm,S)
control, then the series Ei=imi(fi P r o o f.
Let m n J :
SIN
3
converges.
X be t h e d i r e c t sum o f t h e measures m
i = 1 , 2 , ... , [Section 1 1 - 7 1 . I f f , fm(t,i) = f . ( t ) , t
z
hence it i s m m
E
i s t h e f u n c t i o n on T
T, i = 1 , 2 ,... , t h e n
fm
i
: S +
X,
W d e f i n e d by
x
i s bounded and Sm-measurabIe
I t s i n t e g r a l w i th r e s p e c t t o m
integrable.
is a
can be shown t o
IN
be t h e d e s i r e d s e r i e s . Accordingly, f o r any E
E
S , we can d e f i n e
m
For a bounded s e t - v a l u e d f u n c t i o n F : T.
and m T , ( f ) = m ( f ) .
+
CCIRm s e t
and
For our g e n e r a l c o n t r o l problem, A ( m ) r e p r e s e n t s t h e a t t a i n a b l e s e t o f
F
t h e c o n t r o l system. THEOREM 1. If m = (mi) is a control system and F : T
3
CCIRma
bounded,
measurable set-valued function, them A F ( m ) is a convex, weakly compact subset
of
x. P r o o f.
Suppose P i s a fundamental f am i l y o f semi-norms d et er m i n i n g
t h e topology of X, and l e t :A 1,2
,...,
= {(p.) 2
E
c a ( S ) be a measure e q u i v a l e n t t o p ( m i ) ,
chosen such t h a t X = ):A( E
c c a ( S ) : for some p
E E
cca(S).
P, p i
.(
S e t A = ):A({
17, f o r i = 1,2
:
,...
p
E
i =
PI, and
, where ($1
E
A).
IX.2
EXTREME POINTS
156
Then it i s easy t o show t h a t t h e mapping m : MF(lRm,S)-+ X defined by m([fl,)
~~,lmi(fi),f = (fZ.)
m(f)
E
=
M F (Rm,S), i s well defined and i s continuous i f
MF(IRYS) i s given t h e u(n) topology and X i t s weak topology.
Since A ( m ) =
F
t h e r e s u l t follows by Corollary 2 t o Theorem VIII.3.1.
m(MF(IR:S))
By comparison with t h e Example IV.6.1 it i s easy t o see t h a t t h e assumption
t h a t each measure mi i s closed i s c r u c i a l .
2 . Extreme p o i n t s of t h e a t t a i n a b l e s e t
Suppose m
(m.) i s a c o n t r o l system.
i s m.-null f o r every i
1,2,..
. .
A set E
E
s
i s c a l l e d m-null i f it
S i m i l a r l y two c o n t r o l s f,g
c a l l e d m-equivalent i f fi and gi a r e m.-equivalent f o r i = 1,2, 2
of a l l c o n t r o l s m-equivalent t o a c o n t r o l f i s denoted Cfl,, obtained equivalence c l a s s e s L(IRm,m).
F
E
BMCR~,S) are
... .
The c l a s s
and t h e s e t of so
For a bounded set-valued f u n c t i o n
m
:
T
+
CCIR
t h e meaning of LF(lRm,rn) is c l e a r .
We say t h e r e i s an m - e s s e n t i a l l y unique c o n t r o l with a given p r o p e r t y i f t h e s e t of a l l c o n t r o l s with t h e property belong t o t h e same m-equivalence c l a s s . We say f has m-essentially a property i f t h e r e i s a c o n t r o l g with t h e property m-equivalent t o f. Given a set-valued f u n c t i o n F : T c a l l e d F-Liapunov i f A ( m 1 = AexF(rnE) F E THEOREM 1.
Suppose F : T
+
+
CCE"
, a c o n t r o l system m
f o r every E
CCRm is
E
= (m.) i s
S.
a bounded, measurab'le set-va'lued
function, and rn = (m.) a controZ system. 2
An element x
f
E
E
X is an extremal p o i n t of A F (m) if and only if x
3:
m(f) and
MF(Rm,S) impZy t h a t f be'longs m-essentially t o MexF@Rm,S).
I f x i s an extremal p o i n t of A F ( m ) then t h e r e is an m-essentially unique
IX.2
f
E
157
CONTROL SYSTEMS
MF(Rm,S)f o r which x = m ( f ) . Moreover, m E ( f ) i s an extreme point o f A F ( m E )
f o r every E
S.
E
I f the controZ system m i s F-Liapunov then the onZy points x
an m-essentially unique f
E
A F ( m ) with
M (R",s) f o r which x = m ( f ) are the extremat points
c
F
of A F ( m ) .
P r o o f.
Assume t h a t x = m ( f ) , f = (fi) E MF(IRm,S), i s an extremal
p o i n t of A F ( m ) .
VIII.4.1,
I f f does not m - e s s e n t i a l l y belong t o M
ex F
(JRm,S), by Theorem
it does not m - e s s e n t i a l l y belong t o exLF(Rm,m) which g i v e s a
contradiction. I f x is an extremal p o i n t of A ( m ) and z = m ( g ) = m ( h ) , with g,h
E
F
then x
m ( f ) where f = m
m E ( f ) , for some E
p o i n t of A F ( m E ) . Then xE = Let xT-E = m,,(f).
*
y
z, but
x
+h. We have j u s t shown t h a t
f
S, and assume t h a t xE i s not an extremal
E
4 ( y E + z E ) , where
z E'
yE
E
Then y = y E + xTmEE A F ( m ) and z = zE + xT-E
= &(y t z ) .
E
F
E
zE.
AF(m),
This gives a c o n t r a d i c t i o n .
MexF(JRm,S) with x = m ( f ) .
A ( m ) , then t h e r e e x i s t s g
*
A F ( m E ) , and y E
If t h e c o n t r o l system rn is F-Liapunov then, f o r e v e r y x
exists f
F
belongs m - e s s e n t i a l l y
and hence g and h a r e m-equivalent.
t o MexF@ , S ) ,
Put xE
9~ t
M (IRYS),
E
E
AF(m), there
I f x i s not an extremal p o i n t of
MF(IRm,S) with x = m ( g ) and g not m - e s s e n t i a l l y
i n MexF(IRm,S). Hence t h e only p o i n t s i n A F ( m ) with m - e s s e n t i a l l y unique representation x = m ( f ) , f
m
E
MF@R ,S) a r e t h e extremal p o i n t s of A F ( m ) .
By Liapunov's Theorem(Corol1ary 1 t o Theorem V.2.1) non-atomic measures with values i n a f i n i t e dimensional space a r e Liapunov measures.
Hence t h e
s p e c i a l i z a t i o n of Theorem 1 t o t h e case T = [O,tl a given time i n t e r v a l
S
t h e system of Bore1 s e t s on T , X = l R k , k some p o s i t i v e i n t e g e r , m = (m
with
i
measures mi real-valued non-atomic, and zero f o r i = k+l, kt2,
... , and
IX.3
LIAPUNOV CONTROL SYSTEMS
158
k F(t) = ni=lIi, where Ii = C-1.11, i = 1 ,
...
. k , gives the well-known results
concerning uniqueness of the control to reach certain points (see r 2 4 1 Theorems 14.1 and 14.2). The requirement that m is F-Liapunov in the last statement of Theorem 1 must not be omitted as the following well-known examples shows. EXAMPLE 1. Let S be the a-algebra of Bore1 sets in T = C0,11, and
F
:
T
+
CCIR1 defined by F(t) = C0.11, t
usual Lebesgue measure. Define m : S
+
E
T.
Let X = L 1( 0 , l ) with respect to
X by m(E) =
xE,
E
E
S , xE considered
as an element of X. Then it is well-known that m is non-atomic but not Liapunov, hence not F-Liapunov for our chosen
r/f& f
: f E E
F. But every x
E
AF(m)
M C o , 1 3 ( S ) I has an essentially unique representation x = /fdm with since the map f *
MCo,l,:S),
lfdm is essentially the identity.
3 . Liapunov control systems
THEOREM 1. I f m = ( m . ) i s a control system, then m i s F-Liapunov for
every measurable, bounded set-valued function F (A)
:
T
+
CCIP.m,
i f and only i f
f o r every u i n BM(?Rm,S) not m-equivalent t o 0, there e x i s t s a bounded,
measurable function v , with uv not m-equivalent t o P r o
o
0
and m(uv) = ly=lmi(uiv) =
0.
The proof of AF(mE) = AexF(mE)
f. Suppose condition (A) holds.
follows in the same way as the proof of Theorem V.l.l by applying Theorem VIII. 4.1 instead of Theorem 111.7.1. Conversely suppose u
E
Bb~(IR",S) and l u l ,
#
0.
not m-negligible consider the set-valued function
For each E
E
S which is
CONTROL SYSTEM
IX. 3
Then F
:
T
-+
159
CCIRm i s bounded and measurable, and so AexF(m) = ImF(u) : F
E
is convex and weakly compact by hypothesis. Consequently the measure n : S defined by n ( F ) = m,(u), F
Then
S , is Liapunov.
define v =
we
0, however m ( u v )
x
- xp-F.
n(v) =
+
X
if T is n-negligible choose
Now
I f T is not n-negligible there exists a set F
1.
v
E
SE}
E
S with n ( F ) = %?(T).
It is easy to see that uv is not m-equivalent to
0.
COROLLARY 1. Let the space X have the property that (B) 1,2
f o r any control s y s t a of Liapunov measures n = ( n i ) , ni
,...,
t h e measure n
;
S
:
m
+
X defined by n ( E l = CiZlni(E), E
S
-+
i =
X,
S, i s also
E
Liapunov. Then every control system m = (m.) of X-valued Liapunov vector measures i s F-Liapunov, f o r each bounded, measurable set-valued function F P r o o f.
We show that condition (A) holds.
m-equivalent to 0. Define a measure n : S
+
Let u
E
:
T
-L
CCIRm.
RM(lRm,S) be not
X by n ( E ) = m , ( u ) ,
E
E
S.
By (B)
and Theorem V . 2 . 3 , n is Liapunov, and so we can choose a hounded measurable function v satisfying property (A) as before. COROLLARY 2 .
If m = ( m i ) , mi
:
S
+
I R k J k a p o s i t i v e integer,
i s a control system of non-atomic vector measures, and F
: T
+
CCIR
i m
1,.2,.
a measurable,
bounded set-valued function, then m i s F-Liapunov. P r o o f.
It is sufficient to show that I R k has property (B) .
Let
n = ( n . ) be any control system of k-dimensional Liapunov (i.e. non-atomic) vector measures and define n : S
-+ l R k
m
by n ( E ) = CnZlni(E), E
E
S.
If k = 1, and each of the measures ni is positive, then n is non-atomic
and so Liapunov. In general, let X;
:
S + R b e an equivalent measure for ni, i
..,
...,
= 1,2,
160
NON-ATOMIC CONTROL SYSTEMS
chosen such t h a t (hi)
E
E
IX.4
m
cca(S).
E
The measure X : -3 * IRgiven by X(E) = IiZ1Xi(E),
S, w i l l be f i n i t e p o s i t i v e and non-atomic by our e a r l i e r remarks.
X(E)
+
0, E
E
Since
S , implies n ( E ) -+ 0 , Lemma V.6.3 gives t h a t n i s non-atomic and
so Liapunov.
I t i s easy t o c o n s t r u c t examples of i n f i n i t e - d i m e n s i o n a l spaces (e.g. Rm) f o r which property (a) does not hold.
However t h e question of whether a f i n i t e
sum of Liapunov measures is Liapunov i s unanswered.
4. Non-atomic c o n t r o l systems We weaken now assumptions on t h e c o n t r o l system m = (mi) and suppose only t h a t each measure m . be s c a l a r l y non-atomic (see Section V.6). THEOREM 1.
measures, and F
If m :
T
+
= (m.) Z
is a controZ system of scaZarZy non-atomic vector
CCIRma bounded, measurable set-vaZued functiox, then the
weak closure of AexF(m) equaLs A F ( m ) .
P r o o f.
A s F i s bounded and measurable, LF(IRm,m) i s weakly compact,
convex, and non-empty (Theorem V I I I . 2 . 1 and Theorem 1.1). Consequently, by m
Theorem VIII.4.1, exLF(Rm,m) = L e x F m ,m)
*
m
0.
I f LexF(IR ,m) i s a s i n g l e t o n Otherwise, svppose f,g
t h e r e s u l t i s c l e a r from t h e Krein-Milman theorem.
E
LexFORm,m) a r e d i s t i n c t , and s o we can d e f i n e a n o n - t r i v i a l v e c t o r measure
n : S
+
X by n ( E )
mE(g
- f ) = 1:z=1IE (g. z-
fi)dmi, B
E
S.
Following t h e
proof of Corollary 2 t o Theorem 3 . 2 , we s e e t h a t n i s s c a l a r l y non-atomic, and so t h e weak c l o s u r e of n ( S ) is convex (Lemma V.6.5). Now, m ( f )
= n(0)
t
m(f),
m ( g ) = n ( 2 ) t m(f).
and s o
IX.4
161
CONTROL SYSTEMS
f o r any A
E
Tnen An(@) t (1
C0,l.l.
n ( S ) . s i n c e t h i s set i s convex.
-
A ) n ( T ) belongs t o t h e weak c l o s u r e o f
So, t h e r e e x i s t s a n e t I E a I a E A , E
n ( E a ) converging weakly t o An(@)t (1 - A ) n ( T ) , a m ( h a ) , where h , ( t ) = g ( t ) i f t
ha
LexFmm,m), a
E
E
ex F
S, with
S e t t i n g n(E,)
A.
Ea and h , ( t ) = f ( t ) i f t
8
+ n(f: =
E a , we have t h a t
A , and t h a t m ( h ) converges weakly t o A m ( f )
In o t h e r words COA
by (1).
E
E
E
t (1
-
A)m(g),
( m J c weak c l o s u r e o f AexF(rn).
As A F ( m ) i s weakly compact and convex (Theorem l.l),
Theorem and Theorem 2 . 1 , A F ( r n ) = GAexF(m)
c
by t h e Krein-Milman
weak c l o s u r e o f A exF(m).
The
r e v e r s e i n c l u s i o n i s obvious.
i =
I f X is a Banach space, and each of t h e measures mi,
COROLLARY 1.
1,2,..
o f t h e control system m = (mi) i s of the form m i ( E ) = hZui dAi f o r some Bochner integrable f u n c t i o n ui
:
T
+
X , A . a p o s i t i v e non-atomic measure, ( A i )
cca(S),
E
and ~ ~ = l J T ~ ~ dAiu i i< i m, then A F ( m ) is equal t o t h e norm closure of ~ ~ ~ ~ ( m ) . P r o o f.
Suppose f,g
de f i n ed by n ( E ) = m E ( g E
E
I
(fi
1:z = 1j'E ( f i -
-t
X i s a v e c t o r measure
gi)ui d h i , E
E
S.
Let A(E) =
Ci,l~i(~). m
Then A i s a w e l l d e f in e d p o s i t i v e measure, and f o l l o w i n g t h e proof of
S.
Theorem V . 6 . 2
E
- f)
and n : S
BM@?,S)
E
-
each o f t h e o p e r a t o r s I
g i ) h ui ?AiJ
: L"(A)
h
i
m
E
: Lm(A) +
X d e f i n e d by l i c k ) =
L ( A ) , i = l,Z,... i s compact.
+X, I(h) = m((g
- f)h), h
E
Since the operator
L m ( A ) , i s t h e sum ( i n t h e uniform m
o p e r a t o r topology) o f t h e o p e r a t o r s Uiji=l, I must a l s o be compact, and so
n(S) = {I(x,I
: E
E
S j i s r e l a t i v e l y ( s t r o n g l y ) compact i n X.
is s c a l a r l y non-atomic, i = 1,2,
...
, t h e v e c t o r measure
y1
S i n ce each m
i
must be s c a l a r l y
non-atomic, and t h e weak c l o s u r e o f n ( S ) i n X must be convex and weakly compact by Lemma V . 6 . 5 .
Combining t h i s w i t h t h e above, it i s e a s i l y shown t h a t t h e
norm c l o s u r e o f n ( S ) i s convex. proof o f Theorem 3 .
The proof now f o l l o w s i n t h e same way as t h e
IX.5
TIME-OPTIMAL CONTROL
162
COROLLARY 2 .
Let X be a Banach space which i s e i t h e r r e f l e x i v e o r a
separable duaZ space and l e t m = (m.)be a control system of non-atomic measures
of bounded variation, such t h a t ( v ( m i ) ) l , l
E
Then A F ( m ) is equal t o t h e
cca(S).
norm closure of A e x F ( m ) . P r o o f.
-
mE(g
f ) . g,f
As b e f o r e c o n s i d e r t h e measure n : S
S/d(IRm,S), B
E
E
+
X d e f i n e d by n ( E ) =
I n t h i s c a s e , v ( n ) ( E ) 2 c l i = l v ( m i ) ( E ) ,E
S.
S,
E
f o r some c o n s t a n t c , and so n i s of bounded v a r i a t i o n , and non-atomic by Lemma V.6.3.
t h e s t r o n g c l o s u r e o f n ( S ) i s convex, and t h e proof
By Theorem V . 6 . 2 . ,
fol l o ws as b e f o r e .
5. Time-optimal c o n t r o l
Let X be a l . c . t . v . s . , fu n ct i o n z : [ O , t o l
+
t o > 0 be a f i x e d p o s i t i v e number, and suppose a
X i s g iv e n .
The f u n c t i o n z can b e i n t e r p r e t e d as t h e
t r a j e c t o r y o f t h e t a r g e t t h a t t h e c o n t r o l system i s t o r each .
For ev er y t
E
1 l e t m t be a c o n t r o l system on t h e Bore1 s e t s B ( l 0 , t J ) o f t h e i n t e r v a l C0,tJ. If F : i O , t o l + CCR”is a given bounded, measurable s e t - v a l u e d f u n c t i o n , C0,t
0
t h en t o s h o r t e n n o t a t i o n we a b b r e v i a t e A F ( m t ) t o A ( t ) . If t h e r e e x i s t s a minimum time t* f o r which t h e t a r g e t z ( t * ) belongs t o A(t*),
t h en t* i s c a l l e d t h e optimal time, and c o n t r o l s r each i n g z ( t * ) i n time
t * a r e c a l l e d optima? c o n t r o l s .
In t h i s s e c t i o n we g i v e some c o n d i t i o n s f o r
t h e e x i s t e n c e of t h e o p t im a l ti m e . Suppose a c o n t r o l system m = (mi) is given on B ( r O , t o l ) ,
i s a f i x e d measurable, bounded s e t - v a l u e d f u n c t i o n .
rot, f o r ev er y t
mt =
E
[O,tol,
and F : T
CCR
co
Define t h e c o n t r o l system
as t h e r e s t r i c t i o n o f m t o t h e i n t e r v a l [ O , t l ,
The a t t a i n a b l e s e t f o r t h i s system i s o f t h e form
( ( V I ~ ) ~ ~ , ~ , ) .
-f
i.e.
IX.5
CONTROL SYSTEMS
163
L e t m be a contro2 system on B(Co,t,l) and F : T
THEOREI! 1.
bounded measurab2e set-valued f u n c t i o n . d e f i m d above, f o r each t
t
Co, t , l .
continuous, and i f there e x i s t s a t'
Suppose the contro2 system mt i s as
If the target E
* CCRma
CO,t,l
z : Co,t,l
f o r which z ( t t )
+
X is ueakZy
E
A ( t ' ) , then
the optima2 time e x i s t s . S e t t* = i n f I t '
P r o o f.
show t h a t z ( t * )
t
: z(t') E
A(t')I.
There e x i s t s a n o n - i n cr easi n g sequence t
A(t*).
LO,t,l, and an a s s o c i a t e d sequence of c o n t r o l s
For each x'
t
The a i m o f t h e proof i s t o
=
+
(c)
with
X' we have, m
+*
F i r s t l y we s h a l l show t h a t t h e t h i r d term i n (1) t e n d s t o z e r o as n Let L O , t , l
LO,t,l,
x
( S ect i o n 1 1 . 7 ) .
+ m.
LN be t h e d i s j o i n t union o f co u n t ab l y many c o p i e s o f t h e s e t
and mIN : Sm
-+
X t h e d i r e c t sum of t h e measures mi, i = 1,2,. . .
I f En i s t h e s e t i n Sm whose p r o j e c t i o n o n t o C O , t , l
f o r ev er y component, th e n c l e a r l y I ( x ' , mm) ((E,)
+ 0 as n -+
as n
(4)
+ m,
t*, tn E
f o r some c o n s t a n t c, as t h e f u n c t i o n s
f o r each n = 1,2,...
m,
i s C t ,t*l
and s o
are u n i f o r m l y hounded
.
As z i s weakly continuous t h e second t e r m i n (1) t en d s t o zer o a s
so z(t*) must be t h e weak l i m i t o f a sequence belonging t o A ( t * ) . 1.1 t h i s s e t i s weakly c l o s e d , hence z ( t * )
E
A(t*).
n * -, and
By Theorem
164
TIMF-OPTIMAL CONTROL
IX. 5
I t would perhaps be worth n o t i c i n g t h a t t h i s r e s u l t can be extended t o
s i m i l a r systems d e f in e d on n-dimensional i n t e r v a l s . I n t n e n ex t Theorem we c o n s i d e r a c o n t r o l system o f t h e co n v o l u t i o n t y p e . Such systems o ccu r f r e q u e n t l y .
Let I
c
ni",J
c
E n be Bore1 measurable s e t s and l e t
Suppose t h a t F : k O , t o l and t h a t
K
:
LO,t,]
x
I
i n the f i r s t variable.
s
= 8 ( L0 ,t o :
X
I).
+
CCIR i s a bounded, measurable s e t - v a l u e d f u n c t i o n ,
x
J + IR i s a bounded i n t e g r a b l e f u n c t i o n , co n t i n u o u s
1 Suppose X = L ( J ) . and t h e c o n t r o l system mt
: S
-L
L1(J)
i s of t h e form
t m t ( E) ( y )
1 /K(t or
T , X , ~ ) X ~ ( T , Z ) ~d ZT ,
y
E
J,
t
E
iO,t,l.
In o t h e r words, t h e a t t a i n a b l e s e t i s
t A ( t ) = {g
f o r some f L e t mt,
THEOREM 2 .
If t h e t a r g e t z t'
E
:
[O,t,l
t -r
: g(y) =
L'iJ)
E
E
4IF\S); y
t
J),
~,x,y:f(x,~)dXd
E
T ,
Io,t,l.
1
L ( J ) is (norm) continuous, rmd t h e r e exists a time
t h e n t h e r e exists an o p t i m a l time t * .
E
A(t'),
For each t
E
LO,t,J,
able s e t , A ( t )
-
IK(t
or
LO,t,l, be t h e c o n t r o l spstem d escp i b ed above.
E
t o , t , l f o r which z ( t 1 ) P r o o f.
E
I
{IL,,t,xIfdmt
:
f
E
m t is a cl o sed measure and so t h e a t t a i n 1
MF(S)) i s weakly compact i n L ( J ) .
(Theorem I V . 6 . 1 . ) From t h e d e f i n i t i o n t * = i n f { t ' : z ( t ' ) no n - i n cr eas i n g sequence t lvj
+
F i s ) wi t h z ( t n ) ( y ) = / $ & K ( t n
E
A(t')},
and so t h e r e e x i s t s a
t*, and an a s s o c i a t e d sequence of c o n t r o l s fn
-
F i r s t l y consider t h e integral
-r,x,y)fn(T,x)dX
dT, n = 1,2,
...
,y
E
J.
E
IX
165
CONTROL SYSTEMS
tn
jjj
(2)
J I O
t
-
Now, j,*IK(tn
I f n ( ~ ) l IK(~,
T,z,~)
-
-
K(t*
- ~ ( t -* T,z,Y)~~T
T,z,~)
T,x,~)I~T
+.
0
c ~ y .
f o r a l l z,y as n
+.
m
by t h e
Dominated convergence Theorem, s i n c e K i s bounded, and co n t i n u o u s i n t h e f i r s t component.
But t h e i n t e g r a l (2) i s l e s s t h a n o r eq u al t o c
r,y) - K(t*
-
T,Z,LJ)~~T
0 a s n -+
-+
t /OOIK(tn -
T
f o r some c o n s t a n t c as F i s bounded, and so by
&dy,
Dominated convergence (2)
,/ I,
m.
However,
t
(3)
IlZ(t*)
- /
5 IlZ(t*)
5 IIz(t*)
-
-+ I l Z ( t , )
Z(t,)lll
-
jfn(T,Z)K(t*
O I
-
t*
/
/fn(T,Z)K(t*
or
tn - z(tn)II + II( j ’ f , ( ~ , ~ ) ~ ( t-,
O I
t* + II/
1 f,(T,Z)K(t*
-
&lll
T,X,y)dT
T,z,~)
-
T,Z,y)dT
-
T,Z,y)dT
&Ill
K ( ~ * - T , z , ~ ) ) ~ T& I l l
&Ill.
tn I Since the as n
-+
m,
Ifn}
a r e u n if o r m ly bounded, t h e l a s t term o f ( 3 ) t e n d s t o z e r o
and t h e second term t e n d s t o zero because z i s norm co n t i n u o u s.
F i n a l l y t h e t h i r d term o f (3) i s dominated by t h e i n t e g r a l ( 2 ) and so it must converge t o 0.
1 I n o t h e r words z(t*) i s t h e l i m i t ( i n L -norm) o f a sequence
of p o i n t s helonging t o A ( t * ) .
S in c e A ( t * ) is weakly compact and so L’-closed,
z(t*) E A ( t * ) .
Remarks The r e l e v a n c e of v e c to r - v a lu e d measures t o t h e problems o f t h e time-optimal c o n t r o l t h e o r y i s s a l i e n t l y e x h i b i t e d i n t h e monograph of Hermes and LaSel l e C241.
They c o n s i d e r systems with an a r b i t r a r y f i n i t e number o f d eg r ees of
freedom and s t e e r e d by a f i n i t e number of c o n t r o l s .
Consequently, t h e t h e o r y
IX
REMARKS
166
n
inv o l v es IR -valued measures.
The r o l e o f Liapunov's Theorem and i t s v a r i o u s
g e n e r a l i z a t i o n s i s a l s o c l e a r l y shown. dimensions i s well summed u p .
In [ 2 4 1 much o f t h e s i t u a t i o n i n f i n i t e
We r e f e r t o 1241 f o r b a s i c r e f e r e n c e s on t h e
s u b j e c t t u r n i n g s p e c i a l a t t e n t i o n t o O le c h ' s work C601. The p r e s e n t Chapter i s a c o n t r i b u t i o n t o t h e programme o f e x t e n d i n e I 2 4 1 t o the infinite-dimensional s i t u a t i o n .
Such an e x t e n s i o n i s m o t i v at ed by t h e
d e s i r e t o have t h e methods and r e s u l t s d e s c r i b e d t h e r e f o r systems o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , a v a i l a b l e f o r systems governed by p a r t i a l d i f f e r e n t i a l equations.
Admitting i n f i n i t e l y many c o n t r o l s ( i . e . t a k i n g c o n t r o l s i n ?!I
m
rn
i n s t e a d of IR ) i s a n a t u r a l g e n e r a l i z a t i o n which could b e o f i n t e r e s t s i n c e t h e space IR- is " f a i r l y u n i v e r s a l " . Theorem 1.1 h a s i t s o r i g i n i n K a r l i n ' s p ap er 1301, whose r e s u l t i s covered by ours i f X = Rn, Rm i s r e p l a c e d by If?,
and F i s a c o n s t a n t s e t - v a l u e d f u n c t i o n .
There a r e s e v e r a l a u t h o r s e x te n d in g K a r l i n ' s r e s u l t , 1601, L l O l , and o t h e r s . The o r i g i n o f Theorem 2 . 1 a l s o d a t e s back t o L241.
It is, clearly, related
t o Liapunov's r e s u l t i n h i s famous paper 1511, as p o i n t e d o u t i n t h e remarks t o Chapter V I .
The r o l e o f t h i s Theorem i n Co n t r o l Theory i s shown i n 1241.
The importance o f t h e e x i s t e n c e o f measurable s e l e c t i o n s i n t h e p r o o f o f Theorem 2 . 1 should be a p p r e c i a t e d .
I t permits extension of the r e s u l t t o the
c a s e where F i s n o t c o n s t a n t . The r e s u l t s of S e c t i o n s 3 and 4 a r e a d i r e c t g e n e r a l i z a t i o n o f t h e corresponding f i n i t e - d i m e n s i o n a l r e s u l t s e . g . 1101. The p r o o f o f Theorem 4.1 fol l o ws a t r i c k i n V a l a d ie r L 8 4 1 .
I t i s c l e a r l y r e l a t e d t o t h e r e s u l t s of
S e c t i o n V.6. One aim of S e c t io n 5 i s t o show how t h e geometric p r o p e r t i e s o f t h e a t t a i n a b l e s e t ( i t s compactness, convexity e t c . ) can be used i n Co n t r o l Theory. The theorems t h e r e c o n s i d e r o n l y two of t h e p o s s i b l e forms t h e c o n t r o l system
IX
can take.
CONTROL SYSTEMS
167
The relevance of these theorems to control of distributed systems
can be seen from the examples of Section V.7.
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NOTATION
INDEX
4
9
162
9
155
23 38
10
11 7
9 22 148 27
143 39
coA
4
-
61
coA
c
=
U
C(X)
24.26
11 38
d K ( x ') A(m)
A
P
7 131 71
ex A
4
ex F
152
exgA
5
exp A
4
iEJA
9
[ElA
38
iEJ,
25
177
NOTATION INDEX
178
9
st . e x p A
5
142
4
11
143
16
41
22
2
15 8 72 9
25 24
IN
149
40
17
2
23
v (rn)
P
39 131
Q"
X
P X'.
41
2
x*
X'* 1,8.11 142
1
1,134
x', x*
1
x;
8
39 40
16
8
X'vn
67
(x*, r n )
15
8
llX'llK 39 25 149
s @-a
53
137
INDEX
Anti-Liapunov measure, 88
Family o f e q u i v a l e n t measures, 2 1
Atom
F i n i t e measure sp ace, 10
o f a o - al g eb r a , 32
Function
o f a v e c t o r measure, 32
bounded measurable, 9 A - eq u i v al en t , 38
Banach s p ace, 1
A - i n t eg r ab l e,
38
Bang-Bang p r i n c i p l e , 98
m easu r ab l e, 9
Beppo-Levi's theorem, 27
m - eq u i v al en t , 23
B-P p r o p e r t y , 31
r n - i n t eg r ab l e, 2 1 rn-null, 2 2
Closed v e c t o r measure, 71 Cl o s u r e o f a v e c t o r measure, 72
I n j e c t i v e vector. measure, 88
Complete weak s p a c e , 11
I n tegra1
Conical measure, 10
i n d e f i n i t e , 22
l o c a l i z e d on a compact
on a l a t t i c e , 39
s e t , 135
P e t t i s , 10
r e s u l t a n t o f , 11
p-upper, 23
Co n t r o l system, 154 F-Liapunov, 156
Lexicographic o r d e r , 145
non-atomic, 160
Liapunov v e c t o r measure, 82 L o c a l i z a b l e measure sp ace, 9
Daniel1 i n t e g r a l , 1 2
L o c a l l y convex t o p o l o g i c a l
Denting p o i n t , 14
v e c t o r sp ace ( l . c . t . v . s . ) ,
D i s j o i n t union of s e t s , 35
Mackey t o p o l o g y , 3
Dominated convergence
property (Z),
theorem, 30 Dual o f a l . c . t . v . s . ,
89
series i n , 3 1
weak t o p o l o g y , 2
Equicontinuous f a m i l y o f l i n e a r
Measurable c a r d i n a l , 49
functionals, 2
Measure sp ace, 9
Equivalent measures, 9 Exposed p o i n t , 4
Optimal control, 162
Extreme p o i n t , 4
Optimal time, 162 179
1
180
Orlicz-Pettis lemma, 4 p-measure, 16 p-semi-variation, 17 p-variation, 16 Rybakov's theorem, 121 Scalarly non-atomic vector measure, 95 Schauder basis, 61 Set-valued function, 143 St. Venant's principle, 100 Strongly exposed point, 4 Strongly extreme point, 4 Supporting hyperplane, 5 Uniformly absolutely continuous family of measures, 9 Vector measure, 16 anti-Liapunov, 88 closed, 71 direct sum of a family of, 35 injective, 88 isomorphic, 32 Liapunov, 8 2 non-atomic, 32 scalarly non-atomic, 95 Zonoform, 130 Zonohedron, 129
INDEX
