VECTOR MEASURES A N D CONTROL SYSTEMS
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VECTOR MEASURES A N D CONTROL SYSTEMS
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NORTHHOLLAND 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
.
NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC.  NEW YORK
0 NORTHHOLLAND
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.
NorthHolland ISBN for this Volume: 0 7264 0362 6 American Elsevier ISBN: 0 444 1 1 040 2
Publishers : NORTHHOLLAND PUBLISHING COMPANY
 AMSTERDAM
NORTHHOLLAND 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 vectorvalued 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 vectorvalued measures. Several aspects of the theory are missing, however, including chapters on construction of vector measures, RadonNikodym 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 PostGraduate 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 semivariation 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. Vectorvalued 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. Nonatomic vector measures 7. Examples of bangbang 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. Setvalued 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 . Nonatomic control systems
160
5. Timeoptimal 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 realnumber 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 seminorms on X , in the sense that the family {x : p(xt.r
0 and every p
E
P, is a
subbase 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 seminorms 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 seminorm 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 seminorm (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 equicontinuous if
X'
and only if there is a continuous seminorm 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 seminorm 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 seminorm p induces P the norm x * pl (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 family P of sainorms. 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 equiconinuous 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(&IKWi)
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 nonempty subsets Wi, i
E
I , of a
X converges then, f o r any continuous seminorm 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 (OrliczPettis). 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 )
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 nonvoid 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 KreinWilman 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 nonempty 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 nonempty 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 KreinMilman 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 realvalued 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 aalgebra 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 ndimensional 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 aalgebra of Bore1 s e t s i n T .
W e denote t h e s e t of a l l Smeasurable realvalued 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, Smeasurable 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 (setwise) order.
I t i s wellknown ( 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 oalgebra 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 pairwise disjoint sets such that h ( F )
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,
limnEn,
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 pmeasure 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 pmeasure then f o r any
Vo such that i f E P
then 1(x*, r n ) ( E ) I
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 oalgebra of subsets of s e t T , X i s a linear
space, p a serninon on X and m
S
:
t
X a pmeasure.
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 seminorm 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 seminorms 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 oalgebra of s u b s e t s of T and
A realvalued Smeasurable f u n c t i o n
f on T i s s a i d
t o be mintegrable 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 mintegrable 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 mintegrable 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 mintegrable 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 welldefined and the OrliczPettis Lemma implies that it is a vector measure. An
mintegrable function f is said to be mnull if its indefinite integral
is (identically) the zero vector measure. We say that two mintegrable functions f , g are rnequivalent or that they are equal rnalmost everywhere (m a.e.) if the function If
 gl
is rnnull. The class of all mintegrable functions
11.2
VECTOR MEASURES AND INTEGRATION
23
mequivalent 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 seminorm p on X we d e f i n e t h e
pupper 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 seminorms 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 seminorm p On
x,
E
P.
then
For any continuous seminorm p on X , t h e application
(iii)
f * p(m)(f), f P r o o f.
Let f
is mnull
E
E
L(m), i s a seminom on L h ) . Statement ( i ) i s obvious.
L ( m ) and l e t p be a
continuous
seminorm 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
 IflxTE, 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 nintegrable, 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 Ssimple}.
:
191 
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 pseudometric 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 IA1measure, t h e r e i s
k
1 which tends t o f IA1almost 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 )
* * . ,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 nonnegative, 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 Aintegrable 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 aalgebra 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 aalgebra of subsets of a s e t T .
Then there e x i s t s
a compact, Hausdorff space T , a oalgebra 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 oalgebra 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 ) semidistance d, on S (defined by d A ( E , F ) = lhl(E
S) has a unique continuous extension onto a semidistance 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 semidistances 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 semidistances 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 Anull, 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 nonempty.
[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
Aa.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
+
lAITE(hn)
+
Consequently hn
0.
For a family of measures A
[O,ll
( A ) such t h a t
IAl(G). Then IXIE(hn)  IX(TE(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 )
relative 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 seminorm
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. Vectorvalued 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,+fgl\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 Aequivalent 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 seminorms, where
A , i s t h e Bochner seminorm.
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 wellknown (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
VECTORVALUED FUNCTIONS
62
m
where
f is a Aintegrable realvalued 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 realvalued 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,.. Aintegrable 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 Aequivalent if and only if f n = 1 , 2 ,...
E
T , then f
and gn are Aequivalent 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 wellknown 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 selfexplanatory. 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 Aa.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 oalgebra 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 oalgebra 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
VECTORVALUED 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) Iintegrable 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 Aintegrable functions
T, the value of f(t) is an element of V ,
SO
it can be represented by an 2integrable function on C0.11 taking on values 0 o r 1 at 2almost 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 1almost 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 Smeasurable 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 nonclosed 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 seminorms 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 ) realvalued 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, rno1( { 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 oalgebra 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 seminorm
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)
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 mnegligible 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 nonLiapunov 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 antiLiapunov.
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 antiLiapunov 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 mTE
:
sTE
SE
+
X is Liapunov
[El,,, and [TEl are the maximal elements rn
X i s antiLiapunov.
+
:
of S ( m ) such that mE i s Liapunov and mrE i s antiLiapunov. P r o o f.
Let
mG i s antiLiapunov. 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 antiLiapunov.
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 antiLiapunov, 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 muniqueness 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 antiLiapunov 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, antiLiapunov vector measure then
there e x i s t s a family F of pairwise messentia22y 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 , antiLiapunov measure, then
t h e r e e x i s t s a nonmnegligible 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 nonzero 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
messentially 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 messentia2ly 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 [TElm.
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 Aintegrable 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 Aequivalent 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 Aequivalent 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 nonzero
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 Aintegrable 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 Aintegrable 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 nonzero Ameasure 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 realvalued 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
nonzero 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 nonempty
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(MCn,n, ( B ( C 0 , l l ) ) ) has a nonempty 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 nonempty 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 Aunique.
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 aalgebra 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 aalgebra 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 S1llieasurable 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
NONATOMIC 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 nonatomic, we can f i n d a closed, Liapunov measure m

1
:
S
+
X, with ml(S) =
co m(S).
6. Nonatomic 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 nonatomic vector measure and S i s messentially
countably generated (Section 11.6), then also
@om
i s a nonatomic vector measure.
V.6
LIAPUNOV VECTOR MEASURES
95
P r o o f. Let SO c S be a countably generated aalgebra as required by
the definition that S is messentially 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 nonatomic then LEMMA 3 . E
P r
is nonatomic.
I f X i s a nonnegative, nonatomic measure and i f X ( E )
S, * l i e s
E
@om
mCE)
o o f.
+
0, then m
Assume
+
i s nonatomic,
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 nonatomic
@om
i s nonatomic.
( X I ,
@om
+
instead of m gives the result.
X is called scalary nonatomic if, for every
m ) E ca(S) is nonatomic.
I f m is scalarZy nonatomic then t h e weak closure of m ( S )
LEMM4 5 .
P r
S
The nonatomicity of m implies the nonatomicity of
Then Lemma 3 applies to
E
0, n
.....
E
By Corollary 2 to Theorem 111.1.1, there exists a nonnegative
measure X equivalent to m.
x'
+
= m ( A ) , n = 1,2
I f X is metrizable a d m nonatomic, 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
NONATOMIC VECTOR MEASURES
96
elements xi E *
( ( ~ 1 ,
,..., x'
rn)(E),
E
X'.
S i n c e t h e ndimensional 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 nonatomic, 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 nonatomic measure.
If
S is messentially
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
aalgebra).
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 Xvalued 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 nonatomic. =
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
nonatomic 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 Xvalued 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 nonatomic, by Lemma 4 it i s s c a l a r l y nonatomic, 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
nonatomic the norm closure of the range of m i s norm compact and convex.
7. Examples of bangbang 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 "bangbang" 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 "bangbang" 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 nonatomic 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 bangbang 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 nonatomic 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 Cl,lI, say. The aim in this section is to consider the bangbang principle for systems in infinite dimensions with bounded controls, Analagous to the finite dimensional case such systems will only be "bangbang" 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 nonatomic, in general all we can say is that optimal controls can be approximated by bangbang controls, e.g. Theorem 6 . 1 o r 6 . 2 . If m is Liapunov then for every f E MC1,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 "banEbanE': principle, !Vith the aid of Theorem 1.1 we can identify a number of control systems with distributed parameters for which the "hangbang" 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 nonvanishing 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 nonvanishing 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 Xvalued 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 onedimensional. Let E be a set in S which is not mnegligible. 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
threedimensional 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 C1.11. 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 nonnegligible measurable set E there is a nonvanishing 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 C1.13 then it can be satisfied by a function f o with values in
fl,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 halfplane 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((cs) / 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 bangbang control f o
E
h411,11 ( 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 mnegligible set in B ( T ) . Noticing that E is non mnegligible if and only if dy for some z
E
f
0,
J, it is easy to see that a set which is non rnnegligible contains
arbitrarily small sets which are not mnegligible. 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 nonintersecting, 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
initialvalue problem,
where f is a measurable function taking values in C1.11, 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 rnnegligible 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 nonnegligible 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 Xvalued
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 C1,11. 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 bangbang.
Remarks The subject of this Chapter possibly starts with the work of Sierpiiski C751
who proved that the set of values of a nonatomic realvalued 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 IR2valued nonatomic measure has a compact and
convex range, The name Liapunov's Theorem derives from the fact that Liapunov proved in C511 that any nonatomic IRn valued measure has a compact and convex range. He gave a counterexample in C521 to show this is not necessarily the case for nonatomic 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
rnnull 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,
4.
$(s,+t,)
and
l;=l(tnsn) =
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 "bangbang" 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 summedup 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 mnegligible 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 nonzero 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 mequivalent and 112
E
113
EXTREME AND EXPOSED POINTS
V I .1
both belong t o Llo,l,(m). Further, m ( h ) = mTE ( 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 mequivalent,
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 KreinMilman 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 munique 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 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 "onempty, 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
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 setvalued 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
TIMFOPTIMAL 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 ndimensional 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 timeoptimal 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 infinitedimensional 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
AntiLiapunov 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
BangBang p r i n c i p l e , 98
m easu r ab l e, 9
BeppoLevi's theorem, 27
m  eq u i v al en t , 23
BP p r o p e r t y , 31
r n  i n t eg r ab l e, 2 1 rnnull, 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
pupper, 23
Co n t r o l system, 154 FLiapunov, 156
Lexicographic o r d e r , 145
nonatomic, 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
OrliczPettis lemma, 4 pmeasure, 16 psemivariation, 17 pvariation, 16 Rybakov's theorem, 121 Scalarly nonatomic vector measure, 95 Schauder basis, 61 Setvalued 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 antiLiapunov, 88 closed, 71 direct sum of a family of, 35 injective, 88 isomorphic, 32 Liapunov, 8 2 nonatomic, 32 scalarly nonatomic, 95 Zonoform, 130 Zonohedron, 129
INDEX