Vector Measures and Control Systems

VECTOR MEASURES A N D CONTROL SYSTEMS

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NORTH-HOLLAND MATHEMATICS STUDIES

20

Notas de Matemstica (58) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Vector Measures and Control Systems

IGOR KLUVANEK and GREG KNOWLES The Flinders University of South Australia

1976

.

NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM OXFORD AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

0 NORTH-HOLLAND

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- 1975

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Y

PREFACE These notes are the result of our effort to present in a systematic way the theory needed for investigating the range of a vector-valued measure. The inclusion of the term control systems in the title has two reasons. We are convinced that we are dealing with those parts of the theory of vector measures which will allow the extension into infinite dimensional spaces of the results obtained f o r finite dimensional linear control systems using finite dimensional vector measures. This extension is motivated by the desire to have the techniques described in the monograph of Hermes and LaSalle o r the article of C. Olech (both quoted in the Bibliography), available for control sytems governed by linear partial differential equations. The second reason for mentioning control systems is that we have included results about slightly more general objects than vector measures, We call these objects control systems as they serve as a suitable model for many control problems. ITe believe that these notes could also serve as an introduction to the general theory of vector-valued measures. Several aspects of the theory are missing, however, including chapters on construction of vector measures, Radon-Nikodym theory, representation of linear maps, etc. These are or will be covered by the works o f other authors who have the necessary expertise. In particular, we have learned that J. Diestel and J . J . IJhl are preparing a text where many subjects not treated here will be presented, From the many colleagues who have assisted us directly, o r indirectly, we would like to mention Peter Dodds. He discussed with us many aspects of the work, especially those involving order. While engaged in this work one of us (Knowles) was supported by a Commonwealth Post-Graduate Studentship, and later by a Flinders University Research Grant.

Igor Kluvinek Greg Knowles

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TABLE

I.

OF

CONTENTS

PRELIMINARIES

1. Locally convex spaces

1

2 . Extreme and exposed p o i n t s

4

3. Measure spaces

8

4. Conical measures Remarks 11.

14 16

1. Vector measures; v a r i a t i o n and semi-variation 2. Integration

16 21

3 . I n t e g r a b i l i t y of bounded f u n c t i o n s

26

4. L i m i t theorems

27

5. A s u f f i c i e n t condition f o r i n t e g r a b i l i t y

30

6. An isomorphism theorem

32

Remarks FUNCTION SPACES I

35 36 38

1. Topologies

38

2 . Some r e l a t i o n s between topologies

41

3 . Completeness

45

4. L a t t i c e completeness

49

5. Weak compactness

54

6. Completion

57

7. Extreme and exposed p o i n t s

59

8. Vector-valued functions

61

Remarks IV.

10

VECTOR MEASURES AND INTEGRATION

7. Direct sum of v e c t o r measures

111.

1

66

CLOSED VECTOR MEASURES

67

1. P r o p e r t i e s of t h e i n t e g r a t i o n mapping

67

2 . Closed v e c t o r measures

70

3. Closure of a v e c t o r measure

72

viii,

CONTENTS

1

V.

VI.

VII

VIII

.

.

4. Completeness of L (rn) 5. Lattice completeness 6. Weak compactness of t h e range 7. Sufficient conditions for closedness Remarks

73 74 75 78 80

LIAPUNOV VECTOR MEASURES

82

1. Liapunov vector measures 2. Consequences of the test 3. Liapunov decomposition 4. Moment sequences 5. Liapunov extension 6. Non-atomic vector measures 7. Examples of bang-bang control Remarks

82 85 88 89 93 94 98 110

EXTREME AND EXPOSED POINTS OF THE RANGE

112

1 . Extreme points 2. Properties of the set of extreme points

112 115

3 . Rybakov's theorem

120

4. Exposed points of the range Remarks

122

THE RANGE OF A VECTOR MEASURE

128

1. The problem 2. The conical measure associated with a vector measure 3 . The relation between rn and A h ) 4. Consequences of the test Remarks

128 130 134 137 139

FUNCTION SPACES I1

142

1. Set-valued functions 2. Measurable selections 3 . Sequences of measures 4 . Extreme points Remarks

142 145 148 152 153

127

CONTENTS

IX.

ix

CONTROL SYSTEMS

154

1. A t t a i n a b l e s e t

154

2 . Extreme p o i n t s of t h e a t t a i n a b l e s e t

156

3 . Liapunov c o n t r o l systems

158

4 . Non-atomic control systems

160

5. Time-optimal c o n t r o l

162

Remarks

165

BIBLIOGRAPHY

169

NOTATION INDEX

177

INDEX

179

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I , PRELIMINARIES

There is no pretence of a systematic o r complete presentation in this Chapter; it is meant to serve two purposes, The conventions and notations used throughout are fixed here. Also there are collected some results of direct relevance to problems treated in subsequent Chapters to facilitate reference. Hence the Chapter is meant to be use& as an Appendix, to be consulted only when needed.

1.

Locally convex spaces

All vector spaces used will be real vector space5,i.e. the field of scalars will be Ip, the real-number field. If X is a linear space, X* will stand for the space of all linear forms on X .

Given x*

E

X* and z

E

X , ( z * , x ) is the value of x* at x.

If X is a topological vector space then XI is the subspace o f X * consisting of all continuous linear forms on X . again, for the value of x'

E

X' at

We use, of course, the symbol

3: E

( X I ,

x)

X.

Only locally convex topologies wilI be used, The term "locally convex topological vector space" is abbreviated to 1.c.t.v.s. It is well known that the topology of such a space is given by a family P of semi-norms on X , in the sense that the family {x : p(xt.r




0 and every p

E

P, is a

sub-base o f neighbourhoods of zero in P. I f P i s fundamental then this family forms a base of neighbourhoods. The family of all continuous semi-norms can be taken for P . A normed space is a 1.c.t.v.s. whose topology is given by

a

single semi-

norm which is separating, i.e. which is a norm. As usual, the norm of x denoted by llzll rather than p!x).

E

X is

A Banach space is a complete normed space. 1

2

LOCALLY CONVFX SPACES

If p is a semi-norm on a vector space X, then we put U = {x : x E Y , P p(x) i 11, and Uo = h* : x* E X * , /(x*,Z ) / S 1 , f o r every x E C } , If p is

P

P

a continuous semi-norm (in a given locally convex topology on X), then every element x*

E

~0

P

belongs to X'

,

In fact, a set F."

c

is equi-continuous if

X'

and only if there is a continuous semi-norm p on X such that W' A

c

Uo.

P topology on a 1.c.t.v.s. X is said to be consistent with the duality

between X andX' if an element x* only if x* belongs to X I .

E

X* is continuous in the topology if and

The weakest o f such topologies is called the weak

topology on X and is denoted by o f X , X ' I .

The strongest of them is called the

Mackey topology. I f X is a 1.c.t.v.s. and p a continuous semi-norm on Y , then p

Ix

:

x

X/p-'(O)

E

X , p(x) =

01

-1

(0)

is a closed vector subspace o f X. lrie denote by Y

the quotient space of X modulo p

-1

(0).

-1

Let

71

P

P

2

=

he the natural mapping

associating x k p ( 0 ) with any 3: E X . The semi-norm p induces P the norm x * p-l (O) * p ( x ) , x E Y , on X Then X becomes a normed space and P' P hence one can consider, say, the dual o f X the weak topology on X etc. P' P' of Y onto X

Y whose

THEOREM 1. Let W he a complete convex s e t in a l . c . t . v . s . topology is given by a famil-y P of sai-norms. TI

P

W ) is a weakly compact subset of X

If, f o r every p

E

P, t h e s e t

then I.I is weakly compact.

P'

This Theorem is stated only f o r the purpose of reference. Its proof is immediate from James' Theorem. However, such a deep theorem is not needed f o r the proof. Theorem 1 is an easy consequence of any weak compactness condition involving equi-coninuous families of linear functionals (e.g. r32! Theorem 17.12. (ii)). Let xn be an element of a 1.c.t.v.s. Y , for n = 1,2,. , the series l;=lxn is convergent and x

E

..

We say that

~ ~ , , z=n x, o r

X is its sum if limN+

if f o r every neighbourhood U of 0 there is a 6 such that

N

1,

xn

-x

E

U , for

PRELIMINARIES

1.1

3

every N > 6 . More g e n e r a l l y i f Wn

c

..

X , f o r n = 1,2,.

, we s a y t h a t t h e s e r i e s 1n=lWn i s

convergent i f l i = l x n is convergent f o r e v e r y c h o i c e of x

The s e r i e s of s e t s

I,,

m

1”n=13:n

n

E

Wn, n

1,2,.

.. We p u t

i s s a i d t o be u n c o n d i t i o n a l l y convergent i f t h e s e r i e s

{O,xnj i s c o n v e r g e n t .

This i s e q u i v a l e n t t o t h e e x i s t e n c e of an

element x such t h a t , f o r every neighbourhood U o f 0 , t h e r e i s a f i n i t e s e t of n a t u r a l numbers such t h a t ln,,xn numbers such t h a t

I

3

a f i n i t e set of i n d i c i e s c I c

I f Wi

c

K

c

E

X , i f , f o r e v er y neighbourhood U of

0 there is

E

X , for

-

I w i t h liE,xi

i

x

E

E

I.

The series

U , f o r ev er y f i n i t e s e t

I

such

I. X, i

THEOREM 2 .

p(&I-KWi)

of n a t u r a l

is said

E

I , we s a y t h e s e r i e s

convergent f o r ev e r y c h o i c e of x i

2.c.t.v.s.

I

1.Z E

Let x .

t o be convergent t o t h e sum x

K

U , f o r any f i n i t e s e t

E

K.

Let I be an i n d e x s e t .

that

-x

K

I f a series

E

liEIWi

Wi, i

1id W i E

I.

i s convergent if

1.2.6 f c i

is

We write

o f non-empty subsets Wi, i

E

I , of a

X converges then, f o r any continuous semi-norm p on X , l i m =

0,

where the l i m i t i s taken over the net of a l l f i n i t e subsets of

I ordered by inclusion. The proof of t h i s Theorem i s o m it te d as i t is o b t ai n ed by an easy ( al t h o u g h perhaps t e d i o u s ) argument of t h e 3~ t y p e .

( I f o r d i n a r y sequences a r e i n v o l v ed

see e.g. 1361.) LEMMA 1. L e t I and K be s e t s , X a 2.c.t.v.s.

k

E

K.

Let Wi = 1keKWik and l e t W =

liEIwi.

and Wik

Then W =

c

X, for i

lkEK& E I w i k .

E

I,

EXTREME AND EXPOSED POINTS

4

LEMMA 2 .

If

wi

c

X, i

E

I , are convex and

LEMMA 3 .

If

wi

c

X, i

E

I , are compact and

1.2

CiE1wi,

FI =

w

=

then

CiE1wi,

w

then

i s also convex.

w

is also

compact.

THEOREM 3 (Orlicz-Pettis). Let

x

A s e r i e s l;=lxn i s

be a 1.c.t.v.s.

weakly unconditionally convergent i f and only i f it i s unconditionally convergent i n any topology consistent with the duality between X and X A series IiEIWi,

where rJi

c

X, i

E

I .

I , i s weakly convergent i f and only i f

it i s convergent i n any topology consistent with the duality between Y and X I . The sequential part of this Theorem is classical, for Ranach spaces at least. The generalization represents no substantial problems.

2. Extreme and exposed points

If X is a l.c.t.v.s., A

c

X, we denote

the weak closure of A , coA

(resp. bcoA) the convex (resp. balanced) convex Iiull of A, =A

the closed

convex hull of A , and by exA the set of extreme points of A . A

point zo

E

A is called a strongly extreme point of A if xo is not in the

closed convex hull of A - U for any neighbourhood U of

TO.

The set of strongly

extreme points of A is denoted by st.exA. A point

such that

xo

( X I ,

E

A is called an exposed point of A if there exists an x '

xo )


i s c a l l e d a supporting hyperplane of A .

xo i s c a l l e d a support p o i n t of

The p o i n t

E

A (and t h e hyperplane).

X, A i s a

LEMMA 1. Suppose B i s a closed, convex subset of a 1,c.t.v.s.

subset o f B, and A,B have the same supporting hyperplanes. and st.expB

c

Then expB

c

expA

st.expA.

Moreover, i f B i s weakly compact and exB

c

then A and B have the same

A,

exposed points. P r o o f.

I t i s c l e a r t h a t expB c expA a s A and I? have t h e same support-

ing hyperplanes. If b

E

st.expB then b

E

exp4 and a r o u t i n e argument shows t h a t b

E

st.expA.

Let a

E

expA

For t h e second p a r t i t s u f f i c e s t o show t h a t expA c expB.

and H be a supporting hyperplane t o A such t h a t H n A = { a ] . Then B is supported by H , s o B n H i s a non-void weakly compact, convex s e t and s o has From t h e assumptions, ex(B n IT)

extreme p o i n t s ,

A.

c

Thus ex!B n Hj

c

A n H =

{ a > , and so t h e Krein-Wilman Theorem gives t h a t B n H = { a ] . In o t h e r words a

E

expB.

Let A and B be non-empty compact convex subsets of a ~ . c . t . v . s .

THEOREM 1.

X.

If x

E

ex(A

that x = a + b.

t

B ) then there e x i s t s a unique a

Further a

E

exA and b

E

exB.

E

A and a unique b E B such

ConverseZy, i f an element x of

A + B has a unique representation i n the form x = a + b, a E A,

b

E

B , and i f

EXTREME AND EXPOSED POINTS

6

a

L

exA and b

exB, then x

E

P r o o f. a

f

al, b

f

ex(A t B )

E

Suppose x = a t b = al t b Then a t b

bl.

s i n c e a t b = al

+

b

1

1

1.2

a1

f

t

1

where a,ul

b , f o r , o t h er w i se, a

%(alt b ) , x cannot be an extreme p o i n t of A t B . E

A and a u n iq u e b

A , and b,b,

B and

b = al - b,,

-

Consequently i f z B w i t h x = a t b.

E

E

and

S i n c e x = % ( at b l ) t

we must have a = al and b = b,.

t h e r e must e x i s t a unique a

E

ex(A t B )

E

As

C451 § 2 5 . 1 ( 9 ) ) , t h e f i r s t p a r t of t h e Theorem

ex(A t B ) c ex4 t exB(e.g. f o l lows.

Conversely, suppose t h a t x = a t b , where a determined, and a

bl,b2

E

E

exA, b

B wi t h al t b

1

t

a

exB.

E

2

t

b

2

If z

a

E

A, b E B , are u n i q u el y

ex(A t B ) t h e r e e x i s t alJa2

E

A , and

and x = +(a, t b,) t +(a2 t b 2 ) . On r e a r r a n g -

i ng x = %(al t a,) t %(bl t b ), which i m p l i e s t h a t a = % ( a l t u 2 ) and 2

b = +(bl b

E

t

b 2 ) , s i n c e t h e r e p r e s e n t a t i o n of x i s u n i q u e.

exB we have a

Since a

E

exA and

= a2 = a and bl = b , = b , which g i v e s a c o n t r a d i c t i o n .

The c a n c e l l a t i o n law c o n t a i n e d i n t h e f o l l o w i n g Lemma was proved i n 1651, Lemma 2 , f o r t h e c a s e when X i s a Banach s p a c e .

The st at em en t h o l d s i n g e n e r a l

(with similar p r o o f ) a s p o i n t e d o u t i n 1281 Lemma 1. LEMMA 2 .

Let A and B be d o s e d , convex subsets of a 2 . c . t . v . s .

suppose there is a bounded subset C of X such that A THEOREM 2 .

I.c.t.v.s.

t

C = B

t

X, and

C. Then A

B.

If A and B are non-empty compact, convex subsets of a

X , then ex& i s dense i n exA.

P r o o f.

Let

c=

=(ex

8 ) . Then

ex(A

t h e Krein-Milman Theorem (C45]), A t B c C t B Hence, by Lemma 2 , A = C. (6)), i . e . ex#

t

B) c

A

c

C

t

B c A

+ B ; or

t

B , s o t h a t by

C t B = A t B.

I t now f o ll o w s t h a t exA = exC c cl ( ex $ l )

i s a dense s u b s e t o f ewl.

(1451 5 2 5 . 1

PRELIMINARIES

1.2

COROLLARY.

7

I f A,R are compact,convex subsets o f a 2.c.t.v.s.

ex(A t B ) i s closed, then exA and exB are c2osed and ex#

P r o o f. a

t b.

Consider the mapping t : A

x

B

+

A

t

= exA.

B d e f i n e d by t ( a , b ) =

S i n c e t i s c o n t in u o u s , t - l ( e x ( A t R ) ) i s cl o sed i n A Let P

compact.

A denote t h e p ro j ecti o n of A

PA(t-'(ex(A

t

B onto A .

x

B , and s o

By Theorem 1, ex@

B ) ) , and hence t h i s s e t i s compact, and s o c l o s e d .

From t h e

= exA.

i s d e n s e i n exA, and so ex$

Theorem, ex$

x

X, and

I f X i s a Banach s p a c e and K i s weakly compact, convex set i n X , we l e t

and we d e f i n e dK : X'

-+

IR t o be t h e map

K (2') = diamK,,

d L E l W 3.

x'

,

E

X' .

For any weakly compact convex subset K of a Banach space X,

the ;nap dK i s continuous at every x' i n X' which strong22 exposes K .

P r o o f.

Let x'

E

X' be a s t r o n g l y exposing f u n c t i o n a l o f K .

K (x') =

i s a s i n g l e t o n and s o d Then t h e r e e x i s t s a n

dK(2') t n

E ,

E

-

Suppose t h a t dK i s n o t co n t i n u o u s a t z ' .

> 0 and a sequence

f o r e v e r y n = 1,2,

i n Kx, s u ch t h a t Han

0.

b,"

Then K z l

... .

{x;} converging t o

2'

f o r which

Thus, f o r each n, t h e r e e x i s t s an and bn

1 % ~ .Since

K i s weakly compact t h e r e e x i s t s a

?l

subsequence {a,} ( r e s p . { b j } ) of { a n } ( r e s p . {b,}) converging weakly t o some 3 2'11 0 , we have p o i n t a ( r e s p . b ) i n K . S i n c e K is bounded and llx'

j '

x ) - (x', x ) 1

i

-+

-

-+

0 , o r o t h e r words, t h e sequence { x ' . ( K ) } of compact

3 i n t e r v a l s converges t o t h e compact i n t e r v a l x'(K) i n t h e Hausdorff metric on

t h e cl o s ed s e t s of H1.

S e t B . = s u p ( z ' K ) , and 8 = s u p ( x ' , K), and s o B 3 j ' j

-+

B.

8

I.3

MEASURE SPACES

Further,

..

since a + a weakly. But, f o r every j = 1,2,. , uj j Consequently s t ( a ) = limB = 6, and since x ' ( a . ) 'j j 3 exposes K at a , we have IIa all + 0.

.

j

Similarly, r ' ( b ) so IIa

j

-

Ksl,

that x 3! ( a3. ) = x ' ( a ) , and x' strongly so

J

+

-

6 and Ilb.

3

-

bII

and since x' exposes K, a = b , and

+ 0.

b.11 + 0. This contradiction gives that 3

THEOREM 3 .

E

d

K

is continuous at

2'.

I f K i s a weakZy compact, convex su5set of a Banach space X

such that the s e t o f strongly exposing functionals of K i s dense i n X', then the s e t of exposing functionals of K i s residual i n X'. Further i f every exposing functional of K i s strongly exposing, then the exposing functionals from a G6 residual s e t i n X'.

P r

Let C be the set of points of continuity of dK, and let XL

o o f.

and Xi be the sets of exposing and strongly exposing functionals of K in X'. Since X'S is, by hypothesis, dense in X' and dI< vanishes at every point of Xs,

d

K

d

K

is zero at every point where it is continuous. But x' exposes K whenever (XI)

= 0.

Thus from Lemma 3, XI

c C c

Xi.

Since the points of continuity of any real-valued function form a G6 set,

XI contains the dense G 6 set C, and so X'e is residual in X'. For the second part of the Theorem XA

c

Xi, whence Xi

C.

3 . Measure spaces

Suppose T is a set and S is a a-algebra of subsets of T. For a set E put SE = { F

:

F

E

S, F

c

El.

If T is a Bore1 measurable subset of lRn, the

t

S

PRELIMINARIES

1.3

9

usual n-dimensional r e a l space, o r i f T i s a compact Hausdorff space, l e t B ( T ) be t h e a-algebra of Bore1 s e t s i n T .

W e denote t h e s e t of a l l S-measurable real-valued f u n c t i o n s on '2 by M(S), and by BM(S) t h e s e t of a l l bounded, S-measurable f u n c t i o n s on T. V

C R = IR',

For a s e t

M ( S ) ( r e s p . BMV(S))i s t h e s e t of a l l f u n c t i o n s f i n M(S) ( r e s p .

v

E M ( S ) ) with f ( t )

V, t

E

E

T. C l e a r l y , i f V i s a bounded s e t , then MV(S)

BMV(S). We consider t h e usual l a t t i c e operations on M ( S ) , namely, f o r f,g

f

v g

= 4Cf

+

g

+

If - g l )

*g

f

and

= 4(f

+

g

If

-

-

E

M(S),

gl).

By ca(S) we mean t h e Banach space of a l l f i n i t e countably a d d i t i v e ( r e a l valued) measures on S with norm IIpll = 1p1 ( T I , p

ca ( S ) , where IpI i s t h e

E

v a r i a t i o n of u. The n o t a t i o n p 4 A means t h a t p i s a b s o l u t e l y continuous with r e s p e c t t o A.

Two measures p,A a r e c a l l e d equivalent i f h

< 11

and p 4 A. A family

A c caCS) i s c a l l e d uniformly a b s o l u t e l y continuous with r e s p e c t t o h

denoted by A respect t o

* A,

L, E

if A(E)

+

0, E

E

S, implies t h a t u ( E )

E

ca(S),

+ 0 uniformly with

A.

When convenient, we a l s o consider ca(S) a s a l i n e a r l a t t i c e with r e s p e c t t o n a t u r a l (set-wise) order.

I t i s well-known ( e . g . C171. 111.7.6) t h a t t h i s

l i n e a r l a t t i c e i s r e l a t i v e l y complete (Dedekind complete). A measure space i s a t r i p l e (2',S,A) where S i s a o-algebra of s u b s e t s of

T and X i s a p o s i t i v e , p o s s i b l y i n f i n i t e , measure on S. i n t e g r a b l e f u n c t i o n with r e s p e c t t o A on a s e t E hE(f)

=

jGfdX; and h(f) = A T ( f )

=

E

The i n t e g r a l of an

S w i l l be denoted by

lfdh.

A measure space ( T , S , A ) i s c a l l e d l o c a l i z a b l e i f , f o r every continuous 1 on L ( T , S , A ) t h e r e i s a f u n c t i m g 1 ~ ( f )= j f g d ~ , f o r every f E L (T,S,A).

linear functional

For a s e t E

@

E

E

BM(S) such t h a t

S we denote CEIAthe c l a s s of a l l s e t s F

E

s

f o r which

CONICAL MEASURES

10

X(E

* F) =

0 and set S ( X ) = { r E I X : E

E

1.4

Then SfX) is a Boolean algebra

S}.

with respect to the operations induced by those of S .

The measure space

( T , S , h ! is localizable if and only if S(X) is a complete Roolean algebra.

Further ( T , S , X ) is a localizable if and only if the ring of elements of S(X) corresponding to sets of finite measure is relatively complete. This is, in fact, the original definition in f731 of localizability. If X f T )

c m

then ( T , S , X ) is called a finite measure space. A measure

space ( T , S , X ) is said to he a direct sum of finite measure spaces if there is a family

F

c

S of pair-wise disjoint sets such that h ( F )
6.

p(m)(En)

* p(rn)(E) +

if and only if X(E) = o implies p ( m ) ( E )

The "oply i f " p a r t i s obvious.

E.

-+

0, E

E

s,

0.

Conversely, suppose t h a t i f

S , then p ( m ) ( E ) = 0 and t h a t t h e r e e x i s t s 6 > 0 and a sequence

{ A n } i n S such t h a t p ( m ) ( A , ) > 6 and A(A,) m

by Lemma 3 , f o r every

I f A i s a f i n i t e p o s i t i v e measure then X ( E )

COROLLARY 2 .

P r o o f.

S

E,

limn-En,

n =

1.2.

... .

( 4 ) n . Thus i f

< ($)ntl,

for n = 1.2,.

.. .

Let

By monotonicity, p ( m ) ( B n ) > 6 , f o r a l l n, and then A ( B ) = 0 b u t , by Corollary 1,

B = limwBn

p(m)(B) 2 6. L E M 4.

If m : S

-+

X i s a p-measure then, for any decreasing sequence

{ F n } of s e t s i n S with F =

with respect t o x* P r o o f. 5

E

U

P

m

nnzlFn, limw(x*,

O.

Since s u p { l ( r * , m)(Fk) - ( x * ,

p(m)(Fk - F), f o r a l l k = 1.2,..

Lemma 3 .

m)(Fn ) = ( x * , m ) ( F ) uniformly

.

~)(F)I

: X*

E

vo3 P

= ph(Fk

, t h e r e s u l t follows by Corollary 1 t o

-

F))

*

11.1

VECTOR MEASURES

20

If m

LEMMA 5.

f i n i t e set J

c

P r o o f.

...

i s a p-measure then f o r any

Vo such that i f E P

then 1(x*, r n ) ( E ) I
0

there e x i s t s a

S and ) ( x * , m ) l ( E ) = 0 , f o r every x*

E

f o r ever3

E,

E

J,

Vo.

P

> 0 )

E

t h e r e e x i s t x;

i

= 0, f o r

c

VoP and En

S,

E

1 , 2 , . . . , n , and

.

,...

m

I f F = U? E . then {F 1 is a decreasing sequence and i f F = nn=lFn then n j=n j n lim

( x * m ) ( F ) = (x;,

m ) ( P ) = 0 , f o r i = 1.2

ni' n f o r n = 1 , 2 , ... . This c o n t r a d i c t s Lemma 4.

,...n, while

I(X;+~,

m)(Fn)I 2

TllEOFEM 1. Suppose S i s a o-algebra of subsets of s e t T , X i s a linear

space, p a serni-non on X and m

S

:

-t

X a p-measure.

f i n i t e positive measure X on S such that X ( E ) A(E)

0, E

+ .

S , implies that p ( m ) ( E )

E

P r o o f.

..

For n = 1 , 2 , .

J n c Vo such t h a t I(x*, m ) l ( E j E

Vo.

P

p ( m ) ( E ) , f o r every E

, choose, according t o Lemma

I f J n = {x; : k

c:

s, and

0.

0. f o r x* E J,,

P

f o r every x*

-f

5

Then there e x i s t s a

1,2

,...,j ) ,

5 , a finite set

implies / ( x * , rn)(E)I < l / n , let

i s a p o s i t i v e measure such t h a t X n ( E ) = 0 i E p l i e s I(x*, m)(E)I < l/n, n f o r a l l x* E Vo and X n ( E ) 5 sup{l(x*, m ) I ( E ) : I* E U o 3 5 p ( m ) ( E ) , f o r m y P' P E E S. Then X

Now l e t m

X

I:

($)"An.

n=l

Then

we have

X(E)

5

p ( r n ) ( E ) , f o r every E

E

S. Further, i f A ( E ) =

0, E

E

s,

E,

11.2

VECTOR MEASURES AND INTEGRATION

then a l s o X ( F ) = 0 , f o r every F

n

1,2,.

..

, o r ] ( x * ,m ) ( F : l =

c

E,

F

S, hence l ( x * , m ) ( F ) I < l / n , f o r each

E

0 , f o r each x*

implies t h a t / ( x * , m ) k E ) = 0 , f o r each x*

p ( m ) ( E ) = 0.

21

E

Uo. This i s t o say, A ( E ) = 0

E

P

Uo and hence, by Lemma 1, t h a t

P

Corollary 2 to'Lemma 3 gives t h e r e s u l t .

Suppose X is a 1 . c . t . v . s .

and m : S + X a v e c t o r measure.

measures i s s a i d t o be equivalent t o m i f / h l ( E ) + 0 , E

E

A family h o f

S, f o r every X

E

A,

i f and only i f p ( n i ) ( E ) + 0 , f o r every continuous semi-norm p on X. COROLLARY 1.

and m

I f X i s a 1.c.t.v.s.

S

:

-+

X a vector measure then

there e x i s t s a f a m i l y o f positive measures equivalent t o m. COROLLARY 2.

I f X i s a m e t ~ z a b l e1.c.t.v.s.

and m :

s

-+

X a vector

measure then there e x i s t s a f i n i t e positive measure equiuaZent t o m. P r o o f. 1,2,...

E

The topology of X i s given by a countable family { p n : n = Let X

1 of semi-norms on X.

S, i f and only i f p , ( m ) ( E )

E

by Theorem 1.

+

The measure X =

be a p o s i t i v e measure such t h a t h n ( E ) -+ 0,

n 0, n = 1 , 2 , , . .

1"n=lanA'n

.

where a

n

I t s e x i s t e n c e i s guaranteed > 0, n = 1.2

,...

, a r e chosen

such t h a t li=lcinXn(T) < -, has t h e required p r o p e r t y .

2. I n t e g r a t i o n

Suppose X i s a l . c . t . v . s . ,

m : S

+

X a v e c t o r measure.

T a s e t , S a o-algebra of s u b s e t s of T and

A real-valued S-measurable f u n c t i o n

f on T i s s a i d

t o be m-integrable i f i t i s i n t e g r a b l e with r e s p e c t t o every measure ( x r , m )

xr

E

X',

and i f , f o r every E

E

S, t h e r e e x i s t s an element xE of X such t h a t

INTEGRATION

22

11.2

We denote E

S,

E

and

xr = Ifdm =

!r

If&

= rn(f).

We identify

The set o f all m-integrable functions is denoted by L h ) .

freely sets in S with their characteristic functions and so, with a slight abuse of notation, we write S

c

Lh).

Also, we consider the integration

mapping as an extension of the vector measure mapping, m it by m

:

L(m)

:

S + X, and denote

X. The use of the same letter for this extended mapping as

+.

for the original vector measure will not be a source of confusion. L ( m ) is a linear l a t t i c e of functions and the integration mapping

LEMMA 1.

m

:

L(m)

+

P r o

X i s linear. o

If f E L h ) and E

S, then

fx,

E

L(m).

f. The linearity is trivial. If f is m-integrable and E

then it follows from the definition that

f+ = f

E

v 0 and f - =

fx,

E

S,

is integrable, too. Consequently.

(-7) v o are integrable functions and

I f f is an m-integrable function, then the mapping n

n ( E ) = mE(f), E

E

so is :

S

-+

If1

t

f-.

X defined by

S, is called the indefinite integral of f with respect to m.

The definition of an integrable function gives that its indefinite integral is well-defined and the Orlicz-Pettis Lemma implies that it is a vector measure. An

m-integrable function f is said to be m-null if its indefinite integral

is (identically) the zero vector measure. We say that two m-integrable functions f , g are rn-equivalent or that they are equal rn-almost everywhere (m a.e.) if the function If

- gl

is rn-null. The class of all m-integrable functions

11.2

VECTOR MEASURES AND INTEGRATION

23

m-equivalent t o an m - in te g r a b l e f u n c t i o n f is d en o t ed by [flm For a f u n c t i o n f

E

L(m) and a continuous semi-norm p on X we d e f i n e t h e

p-upper i n t e g r a l p ( m ) ( f ) o f f by

where p ( n ) is t h e p - s e m i - v a r i a t i o n o f t h e i n d e f i n i t e i n t e g r a l n o f t h e f u n c t i o n

( i ) Let P be a separating s e t of continuous semi-norms on X .

LEMMA 2 .

function f (ii)

E

L(m)

Given f

E

i f and on& i f p(m)(f) = 0, for every p

L ( m ) and a continuous semi-norm p On

x,

E

P.

then

For any continuous semi-norm p on X , t h e application

(iii)

f * p(m)(f), f P r o o f.

Let f

is m-null

E

E

L(m), i s a semi-nom on L h ) . Statement ( i ) i s obvious.

L ( m ) and l e t p be a

continuous

semi-norm on

X. If n i s t h e

i n d e f i n i t e i n t e g r a l of f , t h e n

f o r ev er y x’ c XI.

Hence (1) f o l l o w s by Lemma 1.1.

C l e a r l y , /Ifldl(x‘, m)l and x ’

E

2

] g d ( x ’ , m ) , f o r ev er y g

E

L(m) w i t h 191

5

If1

X’. But i f E,T - E i s t h e Hahn decomposition o f 2’ f o r (x‘, m), and

A

INTEGRATION

24

- IflxT-E, then obviously

if we put g = I f l ~ ,

11.2

191 s

I f \ and j l f l d l ( z t J n)l =

By Lemma 1 this function g is n-integrable, so

j g d(z', m ) .

As p ( m ( g ) ) =

supC(xf, m(g))

:

zr

E

U o l J the equality (1) implies ( 2 ) .

P

It is well known that j l f l d l ( z f , m)l = supljg d h ' , m f S-simple}.

:

191 -
0 be a number such t h a t llvll = I u / (7') 5 a f o r every

A.

Suppose fn while sup{/lfn

E

MLo,l,(S),n

- fldlu1

:

v

E

= 1,2 A]

+

,...

,f

E

MCo,l,(S)

0 i s not t r u e .

/Ifn

and

- fldlvl

+

0,

( I t s u f f i c e s t o consider

sequences and not general n e t s s i n c e both p ( A ) and ~ ( h a) r e pseudo-metric Then t h e r e e x i s t s an

topologies.)

E > 0,

an i n c r e a s i n g sequence Ink} of n a t u r a l

numbers and a sequence I v k l of measures i n A such t h a t

... .

/If

- fldlukl

t E,

for

nk

By assumption t h e r e e x i s t s a 6 > 0 such t h a t Iukl(E) < ~ / 2

a l l k = l,Z,

f o r a l l k and a l l E a subsequence of {f

E

S with A(E)

< 6.

Since f n + f i n IA1-measure, t h e r e i s

k

1 which tends t o f IA1-almost everywhere. We can assume

k, t h a t it i s

If

I i t s e l f [otherwise it would s u f f i c e t o s e l e c t t h e corresponding

nk subsequence of {vkl. 6 and f,

k

By

Egorov's Theorem, t h e r e is a s e t E

-+f uniformly on T - E6 '

6

E

S with J h ] ( E 6 )
* * . ,uk of elements of r, t h e r e i s an f k

f

be a continuous l i n e a r

As 9 can be w r i t t e n a s t h e d i f f e r e n c e of two p o s i t i v e continuous

l i n e a r f u n c t i o n a l s , assume t h a t 9 i s p o s i t i v e . assume a l s o t h a t

9

E

BMCo,ll(S!

,...,uk(f)) : f E BMCo,lI(S)l is v e c t o r ( @ ( y l ) ,. .. , v ( k~) ) d i d not belong

{(ul(f)

. . . ,ak

> sup{&aipi(f) k such t h a t &aiq(pi) k

:

On t h e BMCo,l,(S)l o r t h a t v(~laiui) k . > s ~ p I k( ~ ~ a ~ u :~fi )E ( BMCo,lj(S)l. f)

o t h e r hand

f o r any

u

E

r,

hence f o r p

1k1ai

i'

Given a non-negative, not n e c e s s a r i l y f i n i t e , measure A on S , t h e space

L1(A) of a l l A-integrable functions i s considered n a t u r a l l y included i n c a ( S ) 1

every element o f L (A) i s represented by i t s i n d e f i n i t e i n t e g r a l a s an element of c a ( S ) . 1

then L ( A )

For i n s t a n c e , i f A i s a l o c a l i z a b l e measure (C731, Theorem 5.1) Sl, where

a i s t h e s e t of a l l measures

v

E

c a ( S ) such t h a t v

A.

:

FUNCTION SPACES I

111.6

57

COROLLARY. A measure space ( T , S , A ) i s l o c a l i z a b l e i f and only i f LCo,ll(h)

P r o o f.

1

Since L[o,ll(L( A ) ) = L [ o , l l ( A ) as sets, the result follows

from the Theorem and the definition if a localizable measure space (Section 1.3).

6. Completion Theorem 3 . 2 reduces the question of T(A)-COmpleteneSS of L ' ( A ) t o that u f In this section we show that if S ( A ) is n o t T(i\)-COmplete, then we can

S(A).

find a a-algebra S of subsets of a compact, llausdorff space T , and a family of A

measures

A c

^

ca(S) such that S(A) is T(A)-complete, and S(A) can be identified ^

^

with a dense subset o f S ( A ) .

THEOREM 1. Let S be a a-algebra of subsets of a s e t T .

Then there e x i s t s

a compact, Hausdorff space T , a o-algebra S of subsets of T and a subspace A c ca(S) such t h a t t h e following statements hold A

Ci)

T c T ; S = S n T = {E n T : E

(ii)

S(A) is T(A)-complete. >

(iii) To each E that E

c

S , an element E

E

.

.

E

S}.

A

E

S can be assigned i n such a mann'er

E , E = T n E , and t h a t the map E * [ E l A i s a uniform isomorphism o f

S onto a dense subset of S ( A ) .

(iv)

For each A

E

ca(S) there is exactly one A

where E corresponds t o E as i n (iii), f o r each E h -+ A

S.

A such t h a t A ( E ) =

The correspondence

i s a l i n e a r isometry of Banach spaces ca(S1 and A .

P r o o f. T

E

E

r(ca(S)).

Let B b e the completion of the space S with the uniformity

Since the operations of intersection and symmetric difference

COMPLETION

58

111.6

( a l s o the union) a r e uniformly continuous on S, they can be extended by c o n t i n u i t y onto B.

Hence we can consider 8 t o be a Boolean a l g e b r a which i s a complete

uniform space i n a uniform s t r u c t u r e which we denote by

T'

and S i s a dense

subset of i t By S t o n e ' s theorem 8 i s isomorphic t o t h e Boolean algebra B of a l l closed and open s u b s e t s of a compact Hausdorff space T.

The space T i s constructed as

t h e s e t of a l l homomorphisms of t h e Boolean algebra B i n t o a two element Boolean Since f o r each t

algebra { O , l } c a r r i n g t h e maximal element of 8 i n t o 1.

T,

E

i s uniquely extendable t o such a homomorphism, t w i l l be iden-

t h e measure cSt

t i f i e d with t h a t homomorphism and hence T with a subset of T. Let S be t h e o-algebra of s u b s e t s of T generated by B . Every h

c a ( S ) i s a uniformly continuous f u n c t i o n on S , s o t h a t i t may

E

be extended uniquely by c o n t i n u i t y t o a continuous f u n c t i o n h' onto B . t h e a d d i t i v i t y o f h implies t h a t of A'.

For each F

E

B, let F

8 be t h e set

E

A

which corresponds t o F under t h e isomorphism of B onto 6 . .

F

E

B.

.

Moreover

*

Denote A ( F ) = A'@),

A

This d e f i n e s an a d d i t i v e f u n c t i o n X on B , and s i n c e B c o n s i s t s of closed

and open s u b s e t s of a compact space, X is a c t u a l l y o - a d d i t i v e , and has a unique S e t A = {A :

a - a d d i t i v e extension, denoted again by A , onto t h e whole of S. E

ca(S)l. Each E

E

S i s a t t h e same time a member of €3.

representing E .

Let E be t h e member of

The c o n s t r u c t i o n of B and i d e n t i f i c a t i o n of T a s a subset of *

T g i v e s t h a t B n !7' = E . For every X E,F

E

E

A

Furthermore A ( E ) = X ' ( B ) = X ( E ) f o r each A

E

c a ( S ) semi-distance d, on S (defined by d A ( E , F ) = lhl(E

S) has a unique continuous extension onto a semi-distance d i on

uniform s t r u c t u r e

ca(S).

7'

a.

on 13 i s given by t h e family of semi-distances d i , A

We t r a n s f e r t h i s s t r u c t u r e onto B i n t h e following way. we f i n d t h e i r r e s p e c t i v e r e p r e s e n t a n t s F,G i n

R and

A

F).

The E

For a r b i t r a r y F,G

,.

A

ca(S'. E

8,

^

w e put d A ( F , G ) = di(F,C).

59

FUNCTION SPACES I

111.7

From t h e c o n t i n u i t y o f extensi0r.s of semi-distances d and measures A it i s seen A

,.

-

1

6

.

.

= lhl(F

t h a t d,(F,G)

A

A

G ) , o r i n o t h e r words d, = d i .

Eience t h e r e s u l t i n g uniform

i s i d e n t i c a l with T ( A ) .

s t r u c t u r e on

A

n

We prove f i n a l l y t h a t B ( A ) = S(A) i n t h e sense t h a t , f o r every E ,

.

is F

E

S , there

A

E

8 such t h a t E

E

[FIA. For t h i s purpose l e t A be t h e system o f a l l sets e

A

,

.

E

E

S f o r which t h e r e e x i s t s an F

.

.

E

.

8 with E

Obviously 8

[FI,.

E

,

{ E n } be a monotonic sequence of elements i n A, with En E = lid

Then { E n } i s T(A)-Cauchy f o r each A

n'

.

with F

E

.

1

8.

Clearly E

E

*

*

.

n

E

Let

8 and l e t

This is t h e same a s saying

A.

.

.

and so A = S .

[Fl,,

n

If A c c a ( S ) and A = { A

THEOREM 2 .

n

A

A.

By t h e completeness of 8, i t has a l i m i t [ F I A

t h a t {[FnlA} i s T(A)-Cauchy. n

E

CFnlA, F

E

L

c

ca(S) : A

E

E

A } i n the notation of

A

Theorem I, then S ( A ) i s .r(A)-complete and S ( A ) i s a dense subset of i t . P r o o f.

Since A

and S(A) i s T(A)-complete, Theorem 4 . 4 gives t h a t

c A,

A , .

A . .

S ( A ) i s T(I\)-complete.

The denseness of S ( A ) i n S ( A ) follows from ( i i i ) o f A

Theorem 1, s i n c e t h e mapping [ E l ,

+

A , .

A

A

IEI-,E

E

A

*

S , is a continuous map from S(A)

A

onto S ( A ) mapping S = S ( c a ( S ) ) onto S ( A ) .

7 . Extreme and exposed p o i n t s

For any E

P r o o f. and s o exL

[O,ll

f(t)

5

1

- E}

S , t h e element

Then t h e r e e x i s t s an

E > 0

i s not A-null, f o r some A

We have ChlA, [glA

E

Lco,ll(A),

c o n t r a d i c t s t h e e x t r e m a l i t y of

[xEIA

i s an extreme p o i n t of L

Accordingly l e t [fJ,

( A ) i s non-empty.

[fl, 4 L f o , l ) ( A ) . I.

E

[hlA

[fJ,.

*

E

E

exLIO,ll(A) and suppose

such t h a t t h e s e t E = { t :

A.

CglA and

[ O , I P

Put g = f

- EX^.

V I A =%([gl,

h =

f +

E S

EX

+ ChlA). This

E'

EXTREME AND FXPOSED POINTS

60

Suppose A

LEMMA 1.

p oi nt s of L

L0,ll

{h : h If h

Let E

then IAl(hfE)

Ihl(x, fE) = IhI(E). while i f h

xE

E

-

x ~ i s- i n~B M ( S ) .

xE

t

- \AIT-+(h)

IAl,(h)

=

11 n L

C0,ll

Consequently H supports L

A-a.e.

/Al(E).

Also,

(A) then -IAlT-+(h) = lXl,(l C0,ll

(A) only a t

xE

-

h),

and so

expLCo,lJ(A). Further, suppose Ih } i s a sequence of elements of L

IXI(hn fE) +

Thus

L1(A), Ihl(hfE) = lhl(S)l is a hyperplane i n L (A). Lco,lJ(A).

that i s h =

xE

S. Then fE =

t

1

t E

Then the s e t s of exposed and strongly exposed

ca(S).

(A) re2ative t o t h e ~ ( h topology ) coincide w i t h S(A).

P r o o f.

X

E

111.7

+

lAIT-E(hn)

+

Consequently hn

0.

For a family of measures A

[O,ll

( A ) such t h a t

IAl(G). Then IXIE(hn) - IX(T-E(hn) + I X l ( E ) , and so I X I E ( l -+

xE

s t r o n g l y exposed p o i n t of L l o , l l ( X ) .

which L

C0,lJ

c

If A

c

SO

xE

hn)

+

is a

The r e s u l t now follows from Theorem 1.

c a ( S ) it i s easy t o c o n s t r u c t examples f o r

( A ) has no exposed p o i n t s .

THEOREM 2 .

i n t h e T ( A ) topology, and

-

However,

c a ( S ) then the strongly extreme points of L L o , l , ( A )

r-elative t o the r(A) topology coincide with S ( A ) . P r o o f.

of ca(S).

F i r s t l y we can suppose by Theorem 2 . 3 t h a t A i s a s u b - l a t t i c e

Suppose E

t

S and CxEIA i s not a s t r o n g l y exposed p o i n t of L C o , l l ( A ) '

Then t h e r e must e x i s t a neighbourhood V o f CxEIA, which we can t a k e of t h e form V = {Cfl, E LLo,ll(A) : p,(Cfl,

p A , such t h a t Cx,lA

{CfalAlacA, V)

t

with CfaJA

- CxE 1A

€1

f o r some continuous semi-norm

o t h)e r G ( L ~ ~ -, v). ~ ~ In (A +

IxEl,,

words t h e r e e x i s t s a n e t

i n t h e T ( A ) topology and [falAE C O ( L [ ~ , ~ - , ( A ) -

. If we d e f i n e VA = V/pi1(0), pA a s above, then t

co(L (A) C0,ll

- VA),

a

E

A.

pA(CfaJh - CxEIA) +

0 and

This means t h a t CxEIA is not a s t r o n g l y

FUNCTION SPACES I

111.8

61

r e l a t i v e t o t h e T(X) norm topology.

extreme p o i n t of L I o , l l ( X l

As t h i s c o n t r a d i c t s

Lemma 1 t h e r e s u l t follows.

8. Vector-valued f u n c t i o n s Let T be a s e t and S a a - a l g e b r a of i t s s u b s e t s .

1 F i r s t l y , t h e v a l u e s of elements of L ( A ) can

3 can be extended i n two ways.

be taken i n any Banach space i n s t e a d o f R

1

.

L e t H be any r e a l Banach space with t h e norm 1 I *!I.

l e t L ( H , A ) be t h e s e t of a l l f u n c t i o n s f : T with r e s p e c t t o every A

E

A.

The r e s u l t s of s e c t i o n

-f

For a set A

c

ca(S),

H Which a r e Bocher i n t e g r a b l e

Two f u n c t i o n s f,g i n L ( H , A ) w i l l be c a l l e d A-

equivalent i f J,+f-gl\a\Xl = 0, f o r every X e A .

Let L ~ ( H , A = )

{c~I,,: f

L(H,A)~,

where [flAis t h e c l a s s of a l l f u n c t i o n s i n L ( H , A ) wLich a r e A-equivalent t o f. On t h i s space we d e f i n e again t h e topology and u n i f o r m i t y T ( A ) t o b e t h e one

determined by t h e family { p , : A

f o r any X

E

E

A ) of semi-norms, where

A , i s t h e Bochner semi-norm.

of elements of H i s c a l l e d a Schauder b a s i s f o r t h e

A sequence

space H, i f every element y e H can be expressed uniquely i n t h e form N

m

... .

where cn ( y ) a r e r e a l numbers, n t h a t i f we put ( y n ‘ , y ) = cn ( y ) , y

H , n = 1 , 2 ,...

E

I t i s well-known (C771 Section 2 . 2 2 ) ,

H , then yh i s a bounded l i n e a r f u n c t i o n a l on

.

THEOREM 1.

every element f

I,”,

Let H be a Banach space with .Schauder basis E

L ( H , A ) can be written uniquely as

!&en

IIl.8

VECTOR-VALUED FUNCTIONS

62

m

where

f is a A-integrable real-valued function f o r each n

f. For any function f

o

.. .

if S ( A ) is r(A)-compZete then so is L 1 ( H , A j .

Moremer,

P r o

= 1,2,.

m

o f real-valued functions If,},=,

: T +

H there exists a unique sequence

such that (1) holds. Furthermore, there exist

constants kn, n = 1,2 ,... , such that If,(t)I

2

knllflt)II, t

T, n =

E

Consequently, if f is 5'-measurable then so is fn, for n = 1.2,.. A-integrable for some X

ca(S), then so is f,,

E

n = 1,2,.

.. .

.

1,2

,... .

, and if f i s

It follows

further that,

The inequality (2) and the uniqueness of the representation (1) implies that if g is another function in L ( H , A ) and g ( t ) = Il,lgn(t)en, t and g are A-equivalent if and only if f n = 1 , 2 ,...

E

T , then f

and gn are A-equivalent for every

.

Assume now that S ( A ) is T(A)-complete. 1

net in L (H,A). Let fact) = ~ ~ = , < ( t ) e n Jt

C C f ~ l A 1 a c A is ~(A)-Cauchyin L

1

(A),

Let {[fal,),,,

k T, a

E

A.

be a r(A)-Cauchy By (2),

for every n = 1,2,.,.

.

the net

Now, by Theorem

1

... .

3.2, this net is r(A)-convergent to an element [f 1 of L ( A ) , n = 1 , 2 , n h We will show that Cfl, E L 1 ( H , A ) and that Define the function f by (1).

Cf1,

is the T(A)-limit of the net {LfalA)aEA. Choose

h

E

A.

1

It is well-known that the space L ( H , h ) is T(h)-complete

("261 Theorem 3.6.1).

x

Consequently, the net { C f a l x ~ a E A must have a limit [f 1,

p(t)

[GI,

lzzlfi(t)en, t E T , the net converges, by 1 h ( 2 ) , to Ifn;, in the topology T ( X ) of L ( A ) , and SO Cf,lA = Cfnlx, n = 1,2,... X 1 Since this holds for every It follows that If], [f I,, or Cfl, E L ( H , X ) .

in this space.

If

2:

.

FUNCTION SPACES I

111.8

63

1

The notation L v ( H , A ) is almost self-explanatory. If V stands for all elements Cfl that f(t)

V for t

E

E

of L'(H,A)

A

c

1

H , then L V ( H , A )

with representatives f

L ( H , A ) such

E

T.

COROLLARY 1. If V

1 H is a closed set then L V ( H , A ) i s a T(A)-cZosed subset

c

of L'(H,A). 1 P r o o f. This is a consequence of the fact that L ( H , A ) is a closed V

subset of L'(H,h)

f o r each h

In fact, by the Chebyshev inequality any

A.

E

r(A)-convergent sequence of elements of L1 ( H , A ) , has a A-a.e. convergent subsequence, and

A

E

so

it easily follows that L1(H,X) is V

~ ( h closed )

in L'(H,A),

A.

COROLLARY 2 .

If V

c

H is a closed set and if S(A) is r(A)-conrplete then

1 L ( H , A ) is T(A)-COmpZete. V

Let us now turn to the second generalization of the results of Section 3 . Let B = B ( l 0 , l l ) be the Bore1 o-algebra on C 0 , l l and let 1 be Lebesgue measure on S

F

1 E

B. In the next Theorem we use the notation T1

T

S o 8 , i.e. S, i s the o-algebra generated by all sets E

B.

Moreover, for any h

measures A and 1.

1

1

c

)

F with E

E

S and

ca(S), we denote X1 = X o 1, the product of the

ca(S), then A1 = {Al

The mapping E

P r o o f.

subset of S (A

If A

E

x

CO,ll, and

x

++

E

x

whose relative T(A')

[0,11,

E

:

E

A

E

A};

so

A1

c

ca(Sl).

S, identifies S ( A ) with a closed

topology is T ( A ) .

Hence the T(A~)-

111.8

VECTOR-VALUED FUNCTIONS

64

completeness of Sl(hl) implies T(h)-COmplet€m3SS of S ( h ) . The interesting part of this Theorem is the converse statement, Assume that S f h ) is r(h)-complete and that {TEaJhl}aeA is a T ( A 1 )-Cauchy net in

S,(A,). We will apply Theorem 1 to the space H = L'!rO,ll), (the classes of) I-integrable functions on [ O , l l ; H is denoted by II*II,

the standard space of

the natural Lebesgue norm on

Tt is known ( r 7 7 1 Example 2,3) that the sequence {?z,);=~

of Haar functions form a Schauder basis for H .

For any set F e S1 and t

Clearly F~

E

E

T we define Ft = {y

:

y e CO.11,

(t,y)E F 1 .

B.

p

H be the function defined by fOL(t) = x t Ea 1 T , where x t represents an element of H . Then rfalA E L V ( H , A ) , where V is Ea

For each a

t

E

A , let

E

: T

+

the set of elements in H whose representatives are characteristic functions of sets in B .

The set V is closed in H .

1 The net {TplA}aEA is T(A~-Cauchy in L ( H . h ) ,

- fB (t)lldlhl4t)

pf%: T

=

J(/lXEt(Y) T

=

t

12(Ea A T

O

t E 1 3 ) d l A l ( t )= Ih,l(E:

Now, by Corollary 2

x

t(y)ldZ(9))dlhI(t)

Ei)

+

0,

a,B e

A. =

A.

Theorem 1, there exists an element rfl,

to

E

El3

a A

Indeed, for any a

From rfl,

which is the T(h)-limit of the net

E

1 Lv(H.h)

we construct an

element of S ( h l ) which is the ~(h~)-limit of the net ICE 1 I a E A . a A1 By Theorem 1, f(t) = ~~=,f,(t)h,, t

f,, n

= 1,2,.,,

.

For every t

E

E

4,with unique A-integrable functions

T, the value of f(t) is an element of V ,

SO

it can be represented by an 2-integrable function on C0.11 taking on values 0 o r 1 at 2-almost every y

E

r0.11.

Denote its value at y

It is known (C771 Example 2 . 3 ) that f,(t)

1

/$(t)(y)

E

C0.11 by f(t)(y).

h n ( y ) d y , n = 1.2,...

.

A

FUNCTION SPACES I

111.8

65

c l a s s i c a l r e s u l t about Haar functions (L771 Example 2 . 3 ) s t a t e s t h a t , given

t

E

T,

m

f(t)(y)

= lim ni-

f o r 1-almost every y

E

E = f(t,y) Clearly E

c

CO.11. :

t

1 fn(t)hn(y)

n=l

Define m

T, y

E

[0,11, I i m

E

1 ~,(t)h,(y)

m ~ n=l o

S1.

The proof w i l l be f i n i s h e d by showing t h a t IXII(Ea A E ) every A1

E

= 1).

A.

Let A 1

E A1,

A1 = X

@

Z with X

E

A.

-+

0, a

E

A , for

Then

I t i s i n t e r e s t i n g t o observe t h a t t h e s e t o f elements [ElA of S,(A,) 1

which have r e p r e s e n t a t i v e s E = {(t,y) function f : T

-+

C 0 , l J i s closed i n

: 0 S

S,(Al).

y s f(t)) f o r some S-measurable F u r t h e r t h e r ( h l ) topology on t h i s

s e t i s t h e same as t h e T ( A ) topology on Lco,ll(A). Hence Theorem 2 g e n e r a i i z e s t h e Corollary t o Theorem 3 . 2 . Theorem 2 i t s e l f can be g e n e r a l i z e d . and t h e family of s e t s o f f i n i t e A Lebesgue measure on t h e whole of A

1

x

2 measure f o r every A

(-m,m).

E

x

(-a,-),

A , where Z i s

Then, i f t h e s e t of t h e s e measures i s

t h e topology T(A ) can be n a t u r a l l y defined and t h e corresponding Theorem 1

stated. T

Q

We could consider t h e space T

Using Theorem 2 and t h e decomposition o f T

[ n , n + 11, n = 0,+1,+2,,..

x

(--,-I

into sets

t h e proof of t h i s extended Theorem can be given.

We do not go i n t o d e t a i l s a s we w i l l have no opportunity t o use t h i s Theorem

I11

REMARKS

66

i n t h e sequel.

Remarks The o r i g i n s of t h e technigue of considering a v e c t o r measure a s a mapping on a s u i t a b l e space is hard t o t r a c e back. i n t h i s d i r e c t i o n d e r i v e s from 1221.

I t was taken over i n C41 and a v a r i a t i o n

I t i s very c o n s i s t e n t l y used i n C21 which w i l l

of t h e approach used i n C173.

be r e f e r r e d t o more i n Chapter V I . there.

Possibly t h e most important stimulus

The proof of Lemma 2 . 1 e s s e n t i a l l y appears

This p o i n t ofview i s a l s o c o n s i s t e n t l y used i n C141.

The topologies ? ( A ) and a(A) were defined i n L391, and t h e Corollary t o Theorem 3 . 2 was proved t h e r e .

The r e l a t i o n of t h i s Corollary t o s p e c t r a l

theory may be worth n o t i c i n g . Section 4 r e l a t e s t h e concept of T(A)-completeness t o t h e concept of a l o c a l i z a b l e measure space C731.

Some i d e a s from t h i s s e c t i o n appear i n

various contexts i n t h e l i t e r a t u r e , i n p a r t i c u l a r 1561.

Theorem 5 . 1 and i t s

Corollary appear i n C391, and Theorem 5 . 2 i n C40l. The t r i c k i n i t s proof was suggested t o us by P . Dodds.

Theorem 6 . 1 i s again from C391.

Theorem 6 . 2 i s

c l o s e l y r e l a t e d t o Theorem 3 . 4 i n S e g a l ' s fundamental paper C731. The method of t h e proof of Theorem 7 . 1 d a t e s back t o C30l. afterwards by s e v e r a l authors.

Lemma 7 . 1 i s from C2l.

I t was used

CLOSED VECTOR MEASURES

IV.

Equipped with t h e information i n Chapter I11 we f i r s t l y r e t u r n t o t h e study of t h e p r o p e r t i e s of t h e i n t e g r a t i o n mapping with r e s p e c t t o a v e c t o r measure.

Then t h e concept of a closed measure i s introduced.

I t is perhaps

t h e c e n t r a l concept of t h e whole t e x t , and w i l l be used i n a l l subsequent Chapters.

Closed v e c t o r measures a r e those f o r which most of t h e c l a s s i c a l

theory of L

1

spaces c a r r i e s over, e s p e c i a l l y r e s u l t s concerning completeness.

The phenomenon of non-closed measures i s observable only i f t h e range space i s not m e t r i z a b l e .

1 . P r o p e r t i e s o f t h e i n t e g r a t i o n mapping

Suppose X i s a 1 . c . t . v . s .

rn)

{(XI,

: X'

E

XI}.

and m :

S + X i s a v e c t o r measure.

Let X1.m

=

T h e n X ' o m c c a ( S ) and t h e following Lemma follows d i r e c t l y

from t h e d e f i n i t i o n s . LEMMA 1. oCY'Om)

The integration mapping m

: L

1

(m)

+

X i s continuous betmeen the

topology on L 1( m ) and the weak topoZogy on X .

By Corollary 1 t o Theorem 11.1.1 t h e r e i s a family of measures, A c c a ( S ) , equivalent t o m. exists a h UCY'om)

E

A with

Q

A.

1

E

c a ( S j f o r which t h e r e

1

Then L ( m ) = L ( A ) as s e t s , and s i n c e X ' a m c n, t h e

1 topology i s weaker than t h e o ( G ) topology on L ( A ) .

THEOREM 1.

the

Let 0 be t h e s e t of a l l measures

o(Qj

The integration mapping m

:

1 L (A)

-+

Consequently,

X is continuous betmeen

1

topology on L ( A ) and the weak topoZogy on X.

This Theorem can be strengthened i f t h e i n t e g r a t i o n mapping i s r e s t r i c t e d 67

11'. 1

PROPERTIES OF INTEGRATION

68

to bounded subsets of its domain. Namely, THEOREM 2. The integration mapping m the T ( A ) topology on L r o , l , ( A )

:

LCO,l,fh)

+

X i s continuous between

and the Mackey topology on X .

P r o o f. Suppose P is a family of semi-norms determining the topology of X and A = {A

: p E P I a corresponding family of equivalent measures to m. P [Corollary 1 to Theorem 11.1.1).

Assume that there exists a net {fa),,,

f,

+

f in the

T ~ A ) topology

on Lr0,,,(m!, but p ( m ( f a

converge to zero, f o r some p n = 1,2,.

..

of members of L

E

P.

such that

C0,ll

- f,)does not

Then there must exist a subsequence {fn :

1 of the net { f a } a E A with p ( m ( f n

- f))

ft

But f n

0.

+

f in the T ( X )

P

topology and so by Lemma 111.2.1 it converges in the uniform T ( A ) topology, where A = { ( z t , m )

and so p ( m ) ( f n

-

: 3:'

E

f ) + 0.

Uoj.

P

By Lemma 11.2.2,

-

Consequently p ( m ( f n

f)

-+

0 and this contradiction

gives the result. As the

o(G)

and ~ ( h ()=

T(m))

topologies coincide on S(A) = S ( m ) (Lemma

111.2.2) we have COROLLARY 1. The integration mapping i s continuous from the o(n) topology on S ( h ) t o the Mackey topoZogy on X.

LEMMA 2. If

p

i s a f i n i t e measure, then for every sequence Cxnl of elements

of GvCS) converging t o

converges t o {f

:

f

E

t,

the sequence o f s e t s {f

Lro,ll(lpl), ~

( f )=

XI

:

f

E

L ~ ~ , ~ , ( I L I I ) ,v ( f ) =

i n the Hacsdorff metric on the

space of (T(Y)) closed subsets of Lco,l,(lvl).

tJ

IV. 1

CLOSED VECTOR MEASURES

P r o o f.

69

L e t d be t h e Hausdorff metric on t h e c l o s e d s u b s e t s of

Lco,l;(lull)and l e t

denote t h e r e s t r i c t i o n of t h e i n t e g r a t i o n mapping

po

L ~ ~ ~ , ~ , ( I ! A } ) . Suppose

T',

t

B = p ( T 1 , and a = u ( T - 1 .

2'-is t h e Hahn decomposition of T r e l a t i v e t o p, set

Then G u ( S ) = ~ o ( L c o , l J ( I p I ) )

s u f f i c e s t o show t h a t , f o r any y

E

E

-1

uo ( { y ) ) such t h a t Iul(6 -

I f y = x, we t a k e 11, = v . similarly.

Ca,BI.

It cle,rly

Ca,B1,

In f a c t (1) w i l l follow i f we can show t h a t f o r any y t h e r e e x i s t s a J,

u to

@)

= 1x

I f not we may suppose

E

Ca,BI and any

-

yI.

z> y

9 E

-1 uo (hl),

as t h e converse follows

Since x > y , x > a and t h e f u n c t i o n J, = P t

is well defined.

s ( x T -

- 9)

Also

The Lemma then follows from our e a r l i e r remarks.

THEOREM 3 .

If m

:

k

S +IR , i s a vector measure and k a p o s i t i v e integer,

then the integration mapping m

:

LCo,ll(m)+ m(LCo,13(m))with the

on i t s domain and t h e usual topotogy on i t s range, i s open.

T(m)

topotogy

CLOSED VECTOR MEASURES

70

P r o o f.

Suppose

IV.2

X i s a f i n i t e , p o s i t i v e measure equivalent t o rn,

(Corollary 2 t o Theorem I I . 1 . 1 ) , and l e t mo denote t h e r e s t r i c t i o n of t h e i n t e g r a t i o n mapping rn t o L Co,ll(rn) each m

= Lco,l,(X).

i is a ( f i n i t e ) real-valued measure, i = 1,...,k .

r n k ( f ) ) f o r each f

6

converges i n T(A)

to y

-1 rno ({y,))

I f some sequence { y f of elements of rn ( L

Lco,ll(X).

n

0, a s

n

,...,k , + m.

=

(Y,,;

+

m.

k

and y

)k

( y i ) i = l , then w e have ynJi -1

and so by Lemma 2 , d(mi tyn,i}

n LLo,ll(A),rni

+

-1

{yij n L[o,ll(h))

c

L

L0,ll

(XI (C461) and so,

Then d(rn,l({yn)),rn,l({y}))

+

0 , as n

+

m.

Now i f rno is not open, t h e r e e x i s t s an open subset 0 of Lco,17(A) such

t h a t rno(0) is not open i n rn ( L 0

C0,ll

Consequently w e can f i n d a sequence

(A)).

) converging t o an element z of elements {a: f i n r n o ( L ~ o , l l ( ~- ~rno(0)

E

rn,,(O).

-1 The s e t rno ( { X I ) n 0 cannot be empty, and i f f belcngs t o i t , t h e r e must be

a closed b a l l with c e n t r e f and r a d i u s

..., d(f, rno-1( { x n f ) 2

192,

(A))

yi f o r each

The operation ( A , B ) + A n B is continuous with r e s p e c t t o d, f o r

any closed s e t s A,B

as n

C0,ll

converges t o rno?{yf) i n t h e Hausdorff m e t r i c , d, on t h e c l o s e d sets of

For, if Y, 1,2

0

then we claim t h a t t h e sequence of s e t s

E rno(LCo,ll(h)),

50 , l $ A ) . i =

,..., mk), where Then m(f) = ( m l ( f ) ,...

Suppose rn = frn,

E,

E,

s a y , contained i n 0.

For each n =

whence

which c o n t r a d i c t s t h e f i r s t s e c t i o n of t h e p r o o f .

Hence rno i s open.

2 . Closed v e c t o r measures Suppose S i s a o-algebra of s u b s e t s o f a s e t T. X a l . c . t . v . s . ,

and

+

IV. 2

rn

:

CLOSED VECTOR MEASURES

S -+ X a v e c t o r measure.

71

In Section 1 1 . 2 , t h e Boolean a - a l g e b r a S ( m ) was 1

S ( m ) is a subset of t h e space L (m), and so we can consider t h e

introduced.

topology and uniform s t r u c t u r e

T h ) .

o r r a t h e r , i t s r e l a t i v i z a t i o n , on S ( r n ) .

I f S ( m ) i s a complete uniform space with r e s p e c t t o t h e uniform s t r u c t u r e

~ ( r n ) ,then t h e measure rn i s c a l l e d a closed v e c t o r measure. Referring t o t h e d e f i n i t i o n of t h e uniformity

T(rn),

a n e t fCEalrnlaEA of

elements of S ( m ) i s T(rn)-Cauchy i f and only i f , f o r every continuous semi-norm

p on X , and every f o r any a t

E

A,

> 0, t h e r e e x i s t s an a.

E

all

E

A such t h a t a.

2 a t , a.

A such t h a t p ( r n ) ( E a l a Eall)
0 for all t

i =

2,3,.

E

S e t El = { t : u ( t )

T.

.. .

2

11,

Ei = { t : l/i 5 u ( t ) < l/(i - 111,

A s b e f o r e n ( r e s p . m) i s t h e d i r e c t sum o f t h e measures nE

( r e s p . mE ) i = 1 , 2 ,

i

...

i

, and t h e r e s u l t f o l l o w s by Lemma 2.

I n t h e g e n e r a l c a s e i f we d e f i n e E

1

= { t : u ( t ) > O } , F2 = {t : u ( t ) < O } ,

and E as g i v en , t h e r e s u l t f o l l o w s from Theorem 2 as n ( r e s p . m ) i s t h e d i r e c t sum o f t h e measures nE, nE1, nE2 ( r e s p . mEJ mEl, m E 2 ) . COROLLARY 1.

If {t

: u(t)

0)

i s m-negligible i n Theorem 3, then m i s

Liapunov if and only if n i s Liapunov.

88

LIAPUNOV DECOMPOSITION

v.3

3 . Liapunov decomposition

Clearly t h e extreme case of a non-Liapunov measure i s a measure rn : S such t h a t , f o r every E

S with LEI,

E

0. t h e r e i s a F

;z

and mF : L C O , l , ( m F ) + X i s i n j e c t i v e .

E

-+

X

SE such t h a t Elm f 0

Such measures w i l l be c a l l e d anti-Liapunov.

I t w i l l be shown t h a t any closed v e c t o r measure i s a d i r e c t sum o f a Liapunov measure and an anti-Liapunov measure.

If rn

THEOREM 1.

:

S

-+

X i s a closed vector measure there e x i s t s an m-

essentiaZly unique s e t E i n S such that the measure mE and mT-E

:

sT-E

SE

-+

X is Liapunov

[El,,, and [T-El are the maximal elements rn

X i s anti-Liapunov.

+

:

of S ( m ) such that mE i s Liapunov and mr-E i s anti-Liapunov. P r o o f.

Let

mG i s anti-Liapunov. elements [GI, of G .

G be t h e family of

a l l elements [GI, of S ( m ) such t h a t

Let P be a s e t i n S such t h a t CFIm i s t h e union of a l l I t s e x i s t e n c e i s guaranteed by Theorem IV.5.1.

The v e c t o r measure mF i s anti-Liapunov.

I f CFIm *,O, choose an a r b i t a r y s e t H

then it i s obvious. t h e r e is [GI,

E

In f a c t , i f G contains only r01,

G such t h a t CG n H I m

f

0.

c

F , CHI,

*

0.

Then

Since mG i s anti-Liapunov, G n H

contains a s e t on which t h e i n t e g r a t i o n mapping i s i n j e c t i v e .

Consequently H

contains a set on which t h e i n t e g r a t i o n mapping i s i n j e c t i v e . Let E

The maximality o f CFl, and Theorem 1.1 imply t h a t mE i s

T - F.

Liapunov. The m-uniqueness of E, i . e . t h e uniqueness of r E l m , f o l l o w s a l s o from t h e maximality of CT

-

Elm = [Fl,.

A vector measure m :

mapping m : Z m ( m ) Liapunov. measure.

-+

S

-+

X w i l l be c a l l e d i n j e c t i v e i f t h e i n t e g r a t i o n

X is injective.

An i n j e c t i v e measure i s obviously a n t i -

The vector measure i n Example IV.6.1 i s a c a s e of an i n j e c t i v e The following Theorem s a y s t h a t it i s , i n a sence, a t y p i c a l case.

LIAPUNOV VECTOR MEASIJRES

v.4

89

Every anti-Liapunov measure can be b u i l t up as a d i r e c t sum of i n j e c t i v e measures.

If m

THEOREM 2.

:

S

+

X i s a closed, anti-Liapunov vector measure then

there e x i s t s a family F of pairwise m-essentia22y d i s j o i n t s e t s i n S such t h a t , f o r every F i n F, the measure mF i s i n j e c t i v e , and the union in S ( m ) of a l l !FIm

f o r F i n F i s [TIm. If m : S

P r o o f.

+

X i s a n o n - t r i v i a l , anti-Liapunov measure, then

t h e r e e x i s t s a non-m-negligible s e t G i n S such t h a t mG i s i n j e c t i v e .

The

r e s u l t follows by exhaustion based on t h e Theorem IV.5.1. The family

F

i n Theorem 2 need not be unique, as it can e a s i l y be shown by

examples. We say, a s i n C351, t h a t t h e space X has t h e p r o p e r t y ( C ) i f any family of

i t s elements summable, by t h e n e t o f a l l f i n i t e subfamilies ordered by i n c l u s i o n , contains a t most countably many non-zero terms.

The c l a s s of spaces with

property ( C ) i s e f f e c t i v e l y l a r g e r than t h e c l a s s of m e t r i z a b l e spaces. I f t h e space has property ( C ) then t h e family f of Theorem 2 is a t most countable.

I f F i s countable t h e elements of F can be made a c t u a l l y d i s j o i n t .

Theorems 1 and 2 combine t o g i v e t h e following d e s c r i p t i o n of t h e s t r u c t u r e

of closed measures. TBEOREM 3 .

I f m : S + X i s a closed vector measure then there e x i s t s an

m-essentially unique s e t E i n S and a f a m i l y F of p a i m i s e m-essentia2ly d i s j o i n t s e t s i n S such that mE i s Liapunov, E n F = 0, mF i s i n j e c t i v e for every F

E

and the union of F i n S h ) is [T-Elm.

4 . Moment sequences

The aim of t h i s s e c t i o n is t o p o i n t t o an i n t e r e s t i n g source of Liapunov

F,

MOMENT SFQUENCFS

90

v.4

measures by showing t h e r e l a t i o n between t h i s concept and t h e moments s f an incomplete system of f u n c t i o n s . I f ( T , S , A ) i s a measure space we c a l l a sequence valued A-integrable f u n c t i o n s on T complete on a s e t E

I, fqR dA = 0 ,

...

f o r each n = l , Z ,

E

= 1.2.

S if f

E

... 1

of r e a l -

BM(SE) and

, implies t h a t f i s A-equivalent t o 0.

sequence i s not complete on E i f t h e r e e x i s t s f f q n dA = 0. f o r n = 1,2,.

on E with

{vn: n

E

This

BM(SE) not A-equivalent t o

0

.. .

I f t h e f u n c t i o n s of t h e sequence C'P,

:

n = 1,2,

... 1

2

belong t o L (1) then

it can be e a s i l y shown t h a t t h e sequence i s complete on a s e t E

E

S of non-zero

2

measure i f and only i f t h e L -closed l i n e a r span of t h e f u n c t i o n s {vnl equals

Let X = Rm be t h e product of countably many copies of t h e r e a l l i n e . t h e product topology X i s a complete 1 . c . t . v . s . . n o t ) {qn : n = I , ? ,

m

:

... 1

In

Now any sequence (complete o r

of A-integrable f u n c t i o n s on T d e f i n e s a v e c t o r measure

S + X by m(E) =

(1)

(

I qldA, I Ip:,dh ,...

),

E

f o r every E

E

S.

In many s i t u a t i o n s , including p r a c t i c a l l y a l l c l a s s i c a l ones, we can reduce m

t h e space X t o a proper subspace o f R For example, i f t h e system

{vn

:

and considerably s t r e n g t h e n t h e topology. 2

n = l,Z,.., 1 i s orthonormal i n L (A),

it is

n a t u r a l t o t a k e X = 12. For any function f

E

BM(S). t h e i n t e g r a l m ( f ) with r e s p e c t t o t h i s measure

i s t h e moment sequence of f with r e s p e c t t o t h e sequence of f u n c t i o n s {'P,

n

1,2.

... I .

THEOREM 1. 12,

...

:

Suppose ( T , S , A ) i s a localizable measure space and

1 a sequence of A-integrable functions.

{Ipn

:n =

Then the vector measure

m

91

LIAPUNOV VECTOR MEASURES

v.4

: S +

X defined by (1) i s closed.

incomplete on every s e t E

I f the sequence I'P,

:

..

n = 1.2,.

1 is

S of non-zero A-measure then t h i s vector measure i s

E

Liapunov.

P r o o f.

The v e c t o r measure is closed by Theorem IV.7.3 s i n c e it Then t h e proof follows from Theorem 1.1.

obviously has a d e n s i t y .

I t i s not d i f f i c u l t t o c o n s t r u c t t h e s i t u a t i o n modelled i n Theorem 1, e . g . Example 2 . 1 .

However, it is more i n t e r e s t i n g t h a t t h i s Theorem a l s o a p p l i e s t o

some c l a s s i c a l systems of orthogonal f u n c t i o n s on t h e i n t e r v a l C 0 , l I with r e s p e c t t o Lebesgue measure. A sequence

on [ O , l l

{vn

:

...

n = 1,2,

1 of real-valued Lebesgue i n t e g r a b l e f u n c t i o n s

i s c a l l e d a Riesz system i f t h e r e e x i s t s c o n s t a n t s

f o r any numbers el, e2,

... , eN and N

A1

and A1 such t h a t

.. .

= 1,2,,

The following Theorem d e s c r i b e s s i t u a t i o n s where Theorem 1 i s a p p l i c a b l e . The proof i s given i n [643 and 1 7 4 1 . THEOREM 2.

Let

{'P,

:

...

n = 1,2,

be an orthonorma2 sequence of functions

1

2

i n L ( [ O , l l ) and l e t Y be the L -cZosed linear span of the functions

{'P~;.

Y', regarded as a subspace of Lm, i s separable, and i f f o r every E

C0,ll with

non-zero Lebesgue measure

{"IE

:

n = 1,2,.

..

c

If

1 i s a Riesz system, then {vn1 i s

incomp2ete on E. Let {wnl be any subsystem of the WaZsh functions (C771 p. 398) i n L2(C0,11) with the property t h a t , f o r any natural numbers bl,. ..,bk,

0 "1

22

"k

il ,..., ik,

MOMENT SEQIJENCFS

92

f o r my k = 1.2,

... .

v.4

Iwn 1 i s incomplete on any s e t of positive measure.

Then

I f 19n1 i s a lacunary subset of e i t h e r the Haar functions, or the Sckauder 2

functions i n I; ( L O , l l ) ,

then {qn1 is incomplete on every

(C771 Example 2 . 3 ) .

set of positive measure. Let T = [0,11,

EXAMPLE 1.

1,2,.., 1 be t h e sequence o f Rademacher f u n c t i o n s .

Let {rn : n

( jErl dZ, hr2dZ,...

m(E) =

m

:

S

+

Z2.

S = K ( C 0 , l l ) and l e t l be Lebesgue measure.

) , f o r every E

E

Then s e t t i n g

S , d e f i n e s a v e c t o r measure By Theorem 2 and Theorem 1

This measure i s closed by Theorem I V . 7 . 3 .

t h i s measure i s Liapunov, s i n c e t h e Rademacher f u n c t i o n s a r e a lacunary subsystem of t h e Walsh system s a t i s f y i n g Theorem 2 . E

E

I f follows t h a t m ( S ) = { m ( E ) :

K ( C O , l l ) } i s a weakly compact, convex subset of 2,. I t i s , perhaps, of i n t e r e s t t o n o t i c e t h a t t h e s e t m ( S ) has a non-empty

i n t e r i o r i n 2,.

Indeed, i t is a c l a s s i c a l r e s u l t of Banach, s e e C291, p. 250,

t h a t f o r every element x that m(f)

= z.

E

Z 2 , t h e r e i s a continuous f u n c t i o n f on 1 0 , l l such

I t follows t h a t l 2 = U i - l m ( M (K([o,ll))). [-n,nl

As Z2 i s a

complete metric space, t h e Baire Category Theorem g i v e s t h a t t h e r e e x i s t s an

n such t h a t t h e s e t m(MC-n,n, ( B ( C 0 , l l ) ) ) has a non-empty i n t e r i o r . and t r a n s l a t i o n m(M

C0,ll

( K ( [ o , l l ) ) ) has non-empty i n t e r i o r .

i s closed and Liapunov implies t h a t m ( S ) = m(M

C0,ll

By c o n t r a c t i o n

The f a c t t h a t m

~K(C0,lI))).

Theorem 3 . 3 applied t o t h e s i t u a t i o n considered i n t h i s S e c t i o n g i v e s t h e following r e s u l t . THEOREM 3 .

I f ( T , S , A ) i s a Localizable measure space and {vn

:

n = 1.2,

... 1

is a sequence of integrable functions, then there e x i s t s a countable f a m i l y of pairwise disjoint sets iEi jq

n

:

i =

0,1,2,.

.. 1 from S ,

with Uiz0Ei

= T , such that

1 i s complete on each Ei, i = 1,2 ,... , but not complete on any F m

sets E o and UizlEi

are A-unique.

E

SE

0

.

fie

93

LIAPUNOV VECTOR MEASURES

v.5

5. Liapunov Extension THEOREM 1. Suppose T i s a s e t , S a a-algebra of subsets of T , and m : S

v'

and a closed Liapunov vector measure rn : S1 1 co r n ( S ) .

&l'

+

-

f

E

E 1,

S

,where

Define T

1

Et = Iy

1

C0,ll

X such t h a t rn1 ( S1 1 =

ml(Sl) =

Let 8 be t h e i3orel s e t s on t h e i n t e r v a l C0,ll and 2 : 8 +IR

Lebesgue measure.

for E

X

Then there e x i s t s a s e t T a a-algebra S of subsets of 1 1

a closed bector measure.

P r o o f.

+

:

= T x CO,ll,

(t,y, E E l , t

S1 =

vn)) = z r n ( S ) , as m i s closed.

Then E = { ( t , y )

T1

E

: 0 5

y

S

Clearly rn ( S

T.

E

+ X by

S o 8 , and m1 : S

1

1

j

c

I

IT fdm

For t h e converse suppose f

f c t ) } i s S1-llieasurable and ml ( B ) =

IT

E

:

Llo,l,(m).

fdm.

By

Theorem 111.8.2, m i s closed and we show i t i s Liapunov using Theorem 1.1. 1 Suppose f

E

BM(S,).

f

Then s i n c e both

e x i s t , by F u b i n i ' s Theorem we have f o r any

Now suppose E

E

S

1

f(t,y)dm,(t,y)

T1

is not rn - n u l l . 1

X I

E

and & , l i f ( t , y ) d y d m ( t )

XI,

By analogy with Example 1.1 we can

f i n d a bounded, S -measurable f u n c t i o n f such t h a t

1

1

f f(t,y)x,(t.y)dy

= 0

0

for a l l t

E

T.

I t can e a s i l y be shown t h a t

[fl,

f

1

Theorem 1.1 then g i v e s t h e r e s u l t .

0 , however

j E fh1=

0.

94

NON-ATOMIC VECTOR MEASIIRES

V.6

I t may be worth a remark t h a t by Theorem 11.6.1 [and its converse, which can be proved s i m i l a r l y ) , t h a t i f S i s m - e s s e n t i a l l y countably generated, and

m non-atomic, we can f i n d a closed, Liapunov measure m

-

1

:

S

-+

X, with ml(S) =

co m(S).

6. Non-atomic v e c t o r measures In t h i s s e c t i o n we w i l l i n v e s t i g a t e t h e compactness and convexity of t h e range under weaker assumptions than Section 1. LEMMA 1.

I f S i s countably generated and

then there e x i s t s an atom B of S such t h a t B For any s e t E

P r o o f.

n = 1,2,

... 1 be

c

c

if A i s an atom ofm, A and m ( B ) = m ( A ) .

T , put E1 = E , and

a countable family generating S.

A belongs t o i t , and t h a t i n f a c t El

Put

= A.

(Section 11.6)

E

1

EV1 = T - E .

Let { E n :

Obviously we can assume t h a t = 1 and determine i n d u c t i v e l y

n t o be 1 o r -1 i n such a manner t h a t

E

This i s p o s s i b l e s i n c e A is an atom of m.

then B

E

S and m ( B ) = m ( A )

f

0.

Hence B

Moreover, i f we put

*

0.

I t follows from t h e construction

t h a t B i s an atom of m. Let Y a 1 . c . t . v . s . and l e t

LEMMA 2 .

:

X

-f

Y be a continuous l i n e a r map

If m i s a non-atomic vector measure and S i s m-essentially

countably generated (Section 11.6), then also

@om

i s a non-atomic vector measure.

V.6

LIAPUNOV VECTOR MEASURES

95

P r o o f. Let SO c S be a countably generated a-algebra as required by

the definition that S is m-essentially countably generated. If m has no atoms then neither has the restriction mo of m to S O , But, if A were an atom o f @Om0

then, by Lemma 1, A would contain an atom B of S O . Since B is also an

atom o f

B must be an atom of m o .

@ O m o ,

The last observation, completeing the

proof, is that if @om0 is non-atomic then LEMMA 3 . E

P r

is non-atomic.

I f X i s a non-negative, non-atomic measure and i f X ( E )

S, * l i e s

E

@om

mCE)

o o f.

-+

0, then m

Assume

+

i s non-atomic,

that A is an atom of m, Using the classical result

that the range of A is an interval, we can construct inductively sets E such that A have X ( E n ) We

3

-+

En

3

En+l, X ( E

0 but not " ( E n )

A vector measure m : S

X', the measure

coincides with o o

We will

+ m,

@om

to be non-atomic

@om

i s non-atomic.

( X I ,

@om

-+

instead of m gives the result.

X is called scalary non-atomic if, for every

m ) E ca(S) is non-atomic.

I f m is scalarZy non-atomic then t h e weak closure of m ( S )

LEMM4 5 .

P r

S

The non-atomicity of m implies the non-atomicity of

Then Lemma 3 applies to

E

0, n

.....

E

By Corollary 2 to Theorem 111.1.1, there exists a non-negative

measure X equivalent to m.

x'

+

= m ( A ) , n = 1,2

I f X is metrizable a d m non-atomic, then

P r o o f.

X.

) = 2 c n ( ~ ) ,m ( E n )

will now give another sufficient condition for

LEMMA 4 .

0,

m(S).

f. Suppose 3:

E

m ( S ) , and we are given a natural number n , and

NON-ATOMIC VECTOR MEASURES

96

elements xi E *

( ( ~ 1 ,

,..., x'

rn)(E),

E

X'.

S i n c e t h e n-dimensional s p a c e valued measure

..., (x;,

m)(E)), E

(C o r o l l ar y 1 t o Theorem 2 . 1 ) exists a set E

E

V.6

E

S, i s non-atomic, by Liapunov's Theorem

i t s r a n g e i s compact and convex.

S w i th (xi, r n ) ( E ) =

("1, x), i =

1,2,.

. . , n.

Thus t h e r e

Thus ev er y weak

neighbourhood of x c o n t a i n s a n element o f m ( S ) . Applying Lemmas 2,4,5, THEOREM 1.

Let m : S

we th e n have

+

X be a non-atomic measure.

If

S is m-essentially

countably generated o r i f X i s metrizable, then the veak closure of m ( S ) coincides w i t h

m(S) .

N e i t h e r of t h e s e c o n d i t i o n s can b e o m it ted a s t h i s example shows. EXAMPLE 1.

Choose a s e t T , and a - a l g e b r a S such t h a t S h as no atoms

( e . g . l e t S b e t h e product o f a n uncountable f am i l y of c o p i e s o f a n o n - t r i v i a l

a-algebra).

For every t

E

T , l e t X t s t a n d f o r t h e sp ace o f r e a l numbers w i t h

i t s u s u a l topology, and l e t X.be t h e t o p o l o g i c a l p r o d u ct of t h e sp aces X t , Hence X i s a Monte1 s p a c e .

w el l d ef i n ed X-valued measure on

E

S.

x ( t ) , f o r each x

a p u r e l y atomic measure. S i n Lemma 1 i s e s s e n t i a l .

2'.

I t follows t h a t m w i l l b e a

S.

A s S h as no atoms, m must b e non-atomic. =

E

Define m ( E ) t o b e t h e c h a r a c t e r i s t i c f u n c t i o n of E

c o n s i d er ed a s an element o f X, f o r each E

X d e f i n e d by ( x i , x )

t

E

But i f z' i s t h e f u n c t i o n a l on

t

X and some t

E

T , t h en x i o m w i l l b e

T h i s measure can b e used t o show t h a t t h e assumption on The measures m , and x'om show a l s o t h a t t h e assump-

t

t i o n s on S i n Lemma 2 and on X i n Lemma 4 a r e e s s e n t i a l . Lastly the cl o s u re of m ( S ) i n

X i s , a t t h e same time, t h e weak c l o s u r e o f

m ( S ) and c o n s i s t s of f u n c t i o n s t a k i n g v a l u e s 0 and 1 o n l y .

97

LIAPUNOV VECTOR MEASURES

V.6

Under c e r t a i n assumptions t h e r e s u l t of Theorem 1 can b e s t r e n g t h e n e d .

Let X be a Banach space which i s e i t h e r a r e f l e x i v e space or

THEOREM 2 .

a separabZe dual space.

If m

: S

+

X is a vector measure w i t h bounded v a r i a t i o n ,

then t h e strong closure of the range o f m i s norm compact.

Further, if m is

non-atomic then the cZosure of t h e range of m is compact and convex.

P r o o f.

Then m is a b s o l u t e l y co n t i n u o u s

Suppose h i s t h e v a r i a t i o n o f m.

with r e s p e c t t o h s o , by C631 p . 3 0 and C161 Theorem 2 . 1 . 4 , t h e assumptions on

X

guar an t ee t h e e x i s t e n c e o f an X-valued f u n c t i o n f which i s Rochner i n t e g r a b l e w it h r e s p e c t t o X and f o r which m ( E ) =

1,

fdh, E

S.

E

1 S e l e c t a sequence { f n } o f s i m p l e f u n c t i o n s i n L ( X , A ) converging t o f i n t h e Bocher norm, i . e . j,lf and I n ( g ) =

-

fnlldh

gf A , n = 1 , 2 , .

T n l&lder's i n e q u a l i t y

t he y a r e bounded o p e r a t o r s .

..., each

.. .

0,

n

For g

-+ a.

E

Lm(A) s e t I(g) =

-+

m.

j T gfdh,

Then I , I n : Lm(h) + X are l i n e a r and, by

I n a d d i t i o n (1) shows t h a t In converges t o

uniform o p e r a t o r topology as n

n = 1,2,

+

I in the

S i n c e each f i s a si m p l e f u n c t i o n ,

In h a s f i n i t e dimensional r an g e, and so i s a compact o p e r a t o r .

Consequently I i s compact, and as t h e set

Ix,

:

B

E

Sl i s co n t ai n ed i n t h e u n i t

b a l l of Lm(A),

i s a norm precompact s e t i n X.

T h i s proves t h e f i r s t a s s e r t i o n .

If m i s non-atomic, by Lemma 4 it i s s c a l a r l y non-atomic, and s o t h e weak c l o s u r e o f m ( S ) i s weakly compact and convex.

However t h e norm c l o s u r e of m ( S )

i s norm compact, hence t h e norm c l o s u r e of m ( S ) e q u a l s t h e weak c l o s u r e of m ( S ) ,

EXAMPLES

98

and t h e r e s u l t follows.

COROLLARY 1. Suppose (T,S.A) is a measure space, X is a Banach space, and

f

: T

+

X i s Bocher integrable with respect t o A,

m :S

-+

X defined by m(E) = ,fE f d h , E

E

then the vector measure

S , has precompact range, and if h i s

non-atomic the norm closure of the range of m i s norm compact and convex.

7. Examples of bang-bang c o n t r o l A very important f e a t u r e of f i n i t e dimensional l i n e a r c o n t r o l systems i s

t h e "bang-bang" p r i n c i p l e [ 2 4 l .

Namely, any p o i n t , , which is reachable by a

n

c o n t r o l taking values i n some compact,convex s e t U ofIR c o n t r o l taking values on t h e extreme p o i n t s of U.

, is

reachable by a

H e r e a f t e r we r e s e r v e t h e I t i s well

use of t h e term "bang-bang" t o d e s c r i b e p r i n c i p l e s of t h i s type.

known t h i s r e s u l t is a consequence of Liapunov's theorem s t a t i n g t h a t t h e range of any f i n i t e dimensional non-atomic v e c t o r measure i s compact and convex. However i n need not hold.

i n f i n i t e dimensional c o n t r o l systems t h e bang-bang p r i n c i p l e This i s an exact p a r a l l e l t o t h e f a c t t h a t Liapunov's theorem

need not hold f o r i n f i n i t e dimensional non-atomic v e c t o r measures. Most i n f i n i t e dimensional extensions of t h i s p r i n c i p l e t o d a t e consider systems whose c o n t r o l s t a k e values i n some compact, convex s e t U i n a Banach o r H i l b e r t space, e.g. CSSl, Chapter 3 , 817.3.

I n t h i s s e t t i n g a l l t h a t can be

s a i d i s t h a t any p o i n t reachable by a c o n t r o l t a k i n g values i n U i s reachable by a c o n t r o l t a k i n g values on t h e boundary of U, which may be a very l a r g e s e t In p a r t i c u l a r , suppose t h e c o n t r o l i s a f u n c t i o n ( x , y ) * f ( x , y ) of two v a r i a b l e s , which for f i x e d x i s regarded as an element of a H i l b e r t space.

I f U is t h e

u n i t b a l l i n t h e H i l b e r t space and i f t h e c o n t r o l i s r e s t r i c t e d t o t a k e values on U then t h e known Theorems s t a t e t h a t any p o i n t reachable by such a c o n t r o l

LIAPUNOV VECTOR MEASURES

v.7

99

is reachable by a control having values with norm exactly one. Since many elements of norm one in the Hilbert space correspond to unbounded funtions, such Theorems do not apply if the controls f are restricted to take their values in the interval C-l,lI, say. The aim in this section is to consider the bang-bang principle for systems in infinite dimensions with bounded controls, Analagous to the finite dimensional case such systems will only be "bang-bang" if the vector measure determining them has convex range which is compact in a suitable topology, in particular if the vector measure is Liapunov. Perhaps it is worth noticing that if the vector measure determining the system is just non-atomic, in general all we can say is that optimal controls can be approximated by bang-bang controls, e.g. Theorem 6 . 1 o r 6 . 2 . If m is Liapunov then for every f E MC-1,lI(3)there exists an f o such that

/$chi

= /2focbn.

E

M

t -l,dS)

Clearly, any control system which is given by a

Liapunov vector measure will satisfy the "banE-banE': principle, !Vith the aid of Theorem 1.1 we can identify a number of control systems with distributed parameters for which the "hang-bang" principle holds, It is stated in sufficient generality to be flexible in applications, While the range space is not necessarily a Banach space, situations when it is not metrizable occur rarely, hence the problem of proving that the measure is closed does not occur. It is important, as can be seen in several of the applications given here, that the measure in the Theorem is not assumed to possess a density, There are cases where the Theorem 1.1 is immediately applicahle. EXAMPLE 1. If in a problem of the linear Theory o f rlasticity, the validity of St. Venant's Principle is accepted, then the relation between

v.7

EXAMPLES

100

deformations ( o r s t r e s s e s ) and t h e f o r c e s c a u si n g them i s a t r a n s f o r m a t i o n e x p r e s s i b l e as i n t e g r a t i o n w i t h r e s p e c t t o a Liapunoy v e c t o r measure. Assume t h a t a s e t T r e p r e s e n t s a p a r t o f an e l a s t i c body where some f o r c e s

For s i m p l i c i t y assume f i r s t t h a t such a f o r c e i s g i v en by a

are applied.

s i n g l e bounded B(T)-measurable f u n c t i o n f, which we i n t e r p r e t as t h e d e n s i t y o f The corresponding s t r e s s e s i n a p a r t A o f t h e body a r e g i v en

the applied force. by a k - t u p l e

ip

= (v 1’””

P k ) o f continuous f u n c t i o n s .

Let X b e t h e sp ace o f

S in c e t h e r e l a t i o n between f and t h e corresponding

a l l such k - t u p l e s .

v

is

assumed l i n e a r and continuous i n a c e r t a i n s en se, t h e r e e x i s t s a measure

rn

:

B(T)

+

X such t h a t

f o r any f

/+%n,

9

E

M ( B ( T ) ) and co r r esp o n d i n g stress

9.

I n some s i t u a t i o n s , e s p e c i a l l y when t h e s i z e o f T i s r e l a t i v e l y small compa r ed with t h e d i s t a n c e from T t o A , S t . Venant’s P r i n c i p l e ( s e e 1201) i s assumed t o hold.

That is, i t i s assumed t h a t t h e s t r e s s caused by f o r c e s s t a t i c a l l y

e q u i v al en t t o ze r o a r e none, o r a t l e a s t , a r e n e g l e c t e d . measurable set E

I t i s c l e a r t h a t any

T o f non-vanishing measure s u p p o r t s a f u n c t i o n f which is t h e

c

d e n s i t y of a non-vanishing f o r c e s t a t i c a l l y e q u i v a l e n t t o z e r o , i . e . with v aq i sh i n g z e r o t h , and f i r s t moments. t h e measure m i s Liapunov. d e n s i t y f wi t h 0

fo(t) E

t 0, 1} ,

5

Under t h e s e assumptions, t h e Theorem g i v e s t h a t

Hence t h e same s t r e s s e s a s a r e caused by a f o r c e with

f(t) 5 1, t

f o r every t

E

E

T, a r e caused by a f o r c e with d e n s i t y fo such t h a t

z‘.

In a more g e n e r a l (and more r e a l i s t i c ) s i t u a t i o n t h e f o r c e s a r e g i v en by an n - t u p l e f = (fi,

...,G of measurable

functions representing d e n s i t i e s of

d i f f e r e n t components o f f o r c e s and t o r q u e s .

If fi,

....f‘ n a r e

interpreted as

components o f a c o n t r o l f, we have a s i t u a t i o n co r r esp o n d i n g t o a c o n t r o l system w it h n - d i m en s i o n a l c o n t r o l s .

If U i s a compact, convex set i n I R

n

and S t . Venant’s

P r i n c i p l e i s assumed t o h o l d , t h e n any system o f s t r e s s e s caused by f o r c e s f w i t h f(t) E 0 f o r every t

E

T, w i l l a l s o b e caused by f o r c e s f, such t h a t fo(t)

LIAPUNOV VECTOR MEASURES

v.7

is an extremal point of U , for every t

E

101

T.

In this example, it is an immediate consequence of St. Venant's Principle that the vector measure o f the.contro1 system mediating between forces and stresses is Liapunov. Of course, this is no way contributes to the discussion on the validity o r the extent of validity of St. Venant's Principle itself. The solution of many problems in Mathematical Physics is given in the form of an integral transform.

If interpreted as integration with respect to a

vector measure it is interesting to give conditions for such a measure to be Liapunov. Many examples can be shown to be variations of the measure described in the following LEMMA 1. Let I and J be i n t e r v a l s , T = I

x

suppose K i s a Lebesgue integrable function on T .

1

J . S = B ( T ) , X = L ( J ) , and

The mapping which associates

with every bounded measurable function f on T the element of X defined by

f o r almost a l l y

E

which i s Liapunov.

J , i s integration with respect t o an X-valued measure m,

For any s e t E

given by the formula (1) f o r f = P r o o f.

E

S the value m(E) of m i s the element

IP

\o

of X

-+

X

xE.

Using Fubin's Theorem, it is easy to show that if m

is defined by putting m ( E ) = E

E

:

S

to be the element of X obtained in (1) for f =

S, then m is a vector measure and (1) holds if and only if

=

xE,

I f h a

We show that rn is Liapunov by using Theorem 1.1. For siniplicity we assume that I is one-dimensional. Let E be a set in S which is not m-negligible. For y smallest number o r

--

satisfying the condition

E

J , let a ( $ ) be the

v.7

EXAMPLES

102

j K ( z , y ) xE(z'y)cI;c -

Now we define f ( z , y ) z

E

I, y

E

J, z

EXAMPLE 2 .

E

1 if

z E I, y

J. z

E

=

K(x,y) xz(z,y)& In(a(y)rm)

I n ( --,a (Y ) )

E

(-m,a(y)),

and f ( z , y )

0.

= -1 i f

Then f i s a bounded measurable f u n c t i o n and

(a(y),m).

/Gfd"

= 0.

Consider t h e problem of c o n t r o l l i n g a q u a n t i t y obeying t h e

three-dimensional wave equation, by varying t h e i n i t i a l c o n d i t i o n .

W e a r e given

t h e i n i t i a l value problem n

u

tt

u(p.0)

cLAu, 0,

p

3

JR ,

u t f p , O ) = f(p), p

3I

where t h e c o n t r o l f i s a measurable f u n c t i o n taking v a l u e s i n C-1.11. point p o

E

lR3 and a l o c a l l y i n t e g r a b l e f u n c t i o n q on

CO,m),

Given a

we wish t o choose f

s o t h a t t h e s o l u t i o n u of ( 2 ) s a t i s f i e s u ( p o , t ) = q ( t ) , f o r almost a l l t

E

IO,m).

By symmetry we can assume t h a t p o i s t h e o r i g i n of our coordinate systems, p o Then i f we write f ( p ) = f ( p . e , A )

using s p h e r i c a l coordinates f o r p , t h e Poisson-

Kirchoff formula gives t h e value 2n n

u(0.t) =

//

f ( c t . 8 , A ) s i n ti de d h ,

0 0

of t h e s o l u t i o n u at t h e o r i g i n f o r almost every t i m e - i n s t a n t t 5: 0.

Hence we wish t o have

f o r almost every t

5

0.

I f t i s confined t o a bounded time i n t e r v a l J then, by Lemma 1, t h e r e l a t i o n between f and IP i s given by i n t e g r a t i o n with r e s p e c t t o a Liapunov

0.

LIAPUNOV VECTOR MEASURES

v.7

103

vector measure with values in L I L T ) . The same argument as in Lemma 1 shows that, for every non-negligible measurable set E there is a non-vanishing function

f such that

/I f(ct,e,h)sine de dh

= 0,

E

for every t

0. Hence the solution ( 3 ) is given by integration with respect to

2

a Liapunov measure with values in L1loc (CO,=l). It follows that if (3) is satisfied by a function f with values C-1.13 then it can be satisfied by a function f o with values in

f-l,lj.

Let 11, 12, J be intervals, T = II x 12,S = B(T).

LEMMA 2 .

Let X be the

space of continuacs functions on J with the topology of Zocally uniform convergence. Let L be a bounded continuous function on I2

x

J and 2et M be

a Lebesgue integrabte

function on Il x 12. The m a p p i n g which associates with every bounded measurable function f on T the element IP of X defined by (4 1

IP(Z)

=

/I L(y,z) M(x,y)

f ( Z r 3 ) dk dYJ

T

f o r every z

E

J J is integration with respect t o a Liapunov vector measure

m:S+X. For any E

E

S, the value m ( E ) of the measure m i s the element IP of X

obtained i n (4) f o r f =

P r

o o

xE.

f. If, for any E

E

S, we define n ( E ) = g where

g(y) =

/

M(x,y) XE(".Y)

h.

12

for almost all y

E

I2' then g

6

1 1 L ( I ) , and the application n : S + L (I2)is, 2

by Lemma 1, a Liapunov vector measure. If, further, for m y g

E

L ' ( 1 2 ) , we put

104

EXAMPLES

@ ( g ) , where

9

E

J , then 9

E

I

L ( y , z ) g ( y ) dy , I2 X and the mapping @ : L 1 ( I 2 )-t X is linear and continuous. q(z) =

z

v.7

, E It follows from Fubini’s Theorem that m(E) = @ ( n ( E ) ) for

E

s. Lemma

2.1

/$dm

gives that m is a Liapunov measure. Since (4) holds if and only if 9 the Lemma is proved. EXAMPLE 3 .

This is an example of a process governed by the diffusion

equation. Consider the problem,

(51

of finding the distribution of temperature u in the half-plane LO,-)

x

(--,m)

dependent on time, if the initial temperature was zero everywhere and the edge x

0 at the point y and at time

Having fixed a line y for every y

E

(-m,-)

and

tion along the line y

=

= c

T E

t is kept at temperature f ( y , t ) .

and an instant of time t we wish to choose f ( y , ~ ) ,

(O,t),in such a way that the temperature distribu-

c at time t is prescribed, That is, given a continuous

function 9 of one variable we wish to choose f in the interval (--,-I so

x

(0,t)

that if u is the corresponding solution of (5) then u ( z , c , t ) = IP(X)for

every z

E

(a,-).

In this setting,

105

LIAPUNOV VECTOR MEASURES

v.7

for x

> 0,

y

E

> 0.

t

(-m,-),

Hence we wish to determine f so that

1I

@ ( 2 )=

L(S,T)

M(V,T)

dr,

f(V,T)

dT,

C0,tlx(--,m) 3: >

0, where L ( ~ , T ) =

( l / ( t - T ) )

2

exp(-x / b k ( t - r ) ) .

From Lemma 2 the relation between f and where m

:

IP

M(q,T) =

is given as

ip

=

2

exp(-(c-s) / b k ( t - r ) ) .

jfl&,

T = [O,t3

x

(--,m),

B(T) + X is a Liapunov measure with values in the space X of continuous

functions on

CO,m)

with locally uniform convergence. If the control function

f is constrained to belong to Ml - l , i l ( B ( T ) ) , say, and if a result IP

is a temperature distribution @ along the line y such a function, then

IP

= c

E

X, that

at time t , is reachable by

is reachable by a bang-bang control f o

E

h41-1,11 ( B ( T ) )

in the same time. The kernels of many problems are, of course, not necessarily factorized in the form which permits direct application o f Lemma 2.

It can happen, however,

that by a suitable transformation they can be reduced to a form where Lemma 2 is applicable. A sufficient condition for this is given in the following. LEMMA 3 .

Let T

c

1~~ be an open s e t , J an internal.

Let K

:

T

x

J +R

1

be a continuous function such that (x,y) * K(x,y,z) i s continuously differentiable

i n T for every z

E

J.

Assume that there are continuous functions a and b i n T

s a t i s f y i q the Zinear homogeneous equation,

for every ( x , y ) ( 6 ) has

E

T and z

E

J.

Assume further1 that the space of solutions

0.f

dimension 1.

Let X be the spuce of continuous ftmctioris on J equipped iJith the topozory of locaZ1y uniform convergance, and l e t M be a Lebssgue iztegrcble functior. on T.

106

v.7

EXAMPLES

The mapping which associates to every function f E M ( B ( T ) ) the element q of X defined by (7)

z

E

J , is integration with respect to a Liapunov measure m : B ( T )

for any E

f

B(T), the value m(E) of m is the element

E

v

-+

X, where,

of X obtained i n (7) for

= XE.

P r o o f. We shall only prove that the measure in question is Liapunov. Let E be a non m-negligible set in B ( T ) . Noticing that E is non m-negligible if and only if dy for some z

E

f

0,

J, it is easy to see that a set which is non rn-negligible contains

arbitrarily small sets which are not m-negligible. Let z o be a fixed point in J .

Assume that E is a subset of an open disc

D in which the partial derivative K (x,y,zo),

Y

say, is bounded away from zero.

Then the manifolds

for 5

E

(--,-),are non-intersecting, and by condition ( 6 ) , the family of

manifolds (8) is identical to

for 5

E

(--,-),

any z

c J.

(Of course, a member of this family can be obtained

for a different value of 6 in the presentation (8) then in (S), the set

(8)

can be empty.)

and for some 5

LIAPUNOV VECTOR MEASURES

v.7

107

Let

1) E

be a family bf orthogonal trajectories to (8).

(-m,m),

assumptions made that the mapping (z,y) *

is invertible in D.

for (5.9

E

It follows from the

defined by

(g,q),

Let the inverse transformation be

C,an open set which is the image of D under (11).

are identical, there exists a continuous function L

K(a(S,o),B(S,o),z) = L ( 5 . v ) . for

(5.1)) E C,

z

E

: C x J

Since

(8)

and (9)

+R2 such that

J.

Hence using the substitution (12) we obtain

11 K ( t , y , z )

(13)

M(z,y) f(x,y) dx dy

E =

L ( S . 2 ) M a ( C S 1 ) ) . B ( 5 , v ) ) g(5.1))

J ( 5 . v ) dS &,

F

where F is the image of E under (ll), g ( 5 , s )

f(a(5,1)).@(5,1))), and J ( S . 1 1 )

the absolute value of the Jacobian of (12).

For the right hand side of (13)

Lemma 2 is applicable. Hence g , and consequently f , can be chosen

SO

is

that (13)

vanishes. It is clear that a similar statement to this Lemma can be made in higher dimensions. In practice the transformation is often seem directly, and the Lemma may not have to be used.

EXAMPLES

108

EXAMPLE 4 .

v.7

Consider the following control problem governed by the heat

equation in the plane. We wish to choose the initial temperature f(3:,y) (r,y)

E

x

(-m,m)

(-m,m),

at the origin, f o r t

E

to give a desired time distribution of temperature q(t) C 0 , t o l a given time interval. That is we are given the

initial-value problem,

where f is a measurable function taking values in C-1.11, and we are given a function q continuous in C0,tol. The aim is to choose f so that for the corresponding solution u of (14) the relation u ( O , O , t )

q ( t ) , for t

E

CO,tol,

will hold, From Fourier's solution to this problem we have

51 / m

u(O,O,t)=

-m

t

E

2

m

f(Z&t3:,

2&ty)P

2 -y clx dy,

-m

C0,tol. The relation between f and

9 is given by integration with respect

to a vector measure m on B ( I R 2 ) with values in C(C0,tol). We show this measure

rn is Liapunov. Any set E

2

E

B(1R ) is not rn-negligible if and only if E is not negligible

with respect to Lebesgue measure on IR B ( I R L ) , and a function f

E

E

3:

= p cos

g

E

e, y =

M(B(IR2))

E

2

.

M ( B ( I R L )),

Consequently given a non-negligible set by transforming to polar coordinates

p sin 8 . we can find a non Lebesgue null set F

with

E

B(IR2) and

v. 7

t

E

LIAPIINOV VECTOR MEASURES

CO,tol.

109

The r e s u l t follows by applying Lemma 2 , and then transforming

back t o Cartesian coordinates. A similar r e s u l t holds i f t h e time i n t e r v a l i s not bounded.

t h e temperature

q(t),

f o r every t > 0 .

Then P i s considered as an element of

t h e space X of continuous f u n c t i o n s on C0,m) uniform convergence.

We can p r e s c r i b e

w i t h t h e topology of l o c a l l y

Again t h e r e l a t i o n between f and q i s given by an X-valued

Liapunov vector measure. EXAMPLE 5.

As an example of a c o n t r o l system governed by an e l l i p t i c

p a r t i a l d i f f e r e n t i a l equation, consider t h e following problem of s t e a d y s t a t e heat conduction i n a s e m i - i n f i n i t e s o l i d M = I(x,y,z)

,IR3 : z 2 0) whose

s u r f a c e temperature i s c o n t r o l l e d t o be f(x,y),

E

(z,y)

(-m,m)

x

(-m,m),

for

some measurable f u n c t i o n f with values i n C--1,11. The temperature i n t h e body s a t i s f i e s t h e boundary value problem

C W

Au = 0 i n M ,

We wish t o c o n t r o l t h e s u r f a c e temperature t o o b t a i n a d e s i r e d temperature ~ ( z ) a t p o i n t s ( O , O , z ) , z > 0 , i n s i d e t h e body.

I f u i s t h e s o l u t i o n of ( 1 5 ) , then

Hence, given t h e continuous f u n c t i o n IP i n ( 0 , m ) we have t o determine t h e f u n c t i o n

f so t h a t

f o r every z > 0.

Lemma 3 or transformation t o p o l a r c o o r d i n a t e s shows t h a t t h e

r e l a t i o n between f and q i s given by i n t e g r a t i o n with r e s p e c t t o a Liapunov

110

V

REMARKS

vector measure on B(IR')

with values in the space of continuous functions on

(0,m) under the topology of locally uniform convergence.

Consequently, the

system is bang-bang.

Remarks The subject of this Chapter possibly starts with the work of Sierpiiski C751

who proved that the set of values of a non-atomic real-valued measure is

an interval (see also CIS1 and C19l).

Neymann and Pearson C591 is probably the

first instance of an application of this fact. Buch 191 reproved the result of C751

and also showed that an IR2-valued non-atomic measure has a compact and

convex range, The name Liapunov's Theorem derives from the fact that Liapunov proved in C511 that any non-atomic IRn -valued measure has a compact and convex range. He gave a counter-example in C521 to show this is not necessarily the case for non-atomic infinite dimensional measures. A simpler example to this effect is in C611. There are several proofs of Liapunov's Theorem now available, C231. C331. [71.

An outstanding new proof was given by Lindenstrauss C531. whose paper

revived interest in the subject. A condition for an infinite dimensional measure with density to be

Liapunov was given by Kingman and Robertson C331 (see also C861). is from

C441.

Theorem 1.1

It is based on the idea of Kingman and Robertson. The idea

contained in the Corollary to Theorem 1.1 goes back to Karlin C30l.

In fact,

the ideas of Karlin are reflected in practically all proofs of Liapunov's Theorem and its generalizations that came after. They are inherent in C331, C531

and in Theorem 2 . 1 and its Corollaries. The condition (i), (ii) and (iii) in Theorem 1.1 need not be sufficient

LIAPIINOV VECTOR MEASURES

V

for

61

111

t o be Liapunov i f they a r e s a t i s f i e d not f o r a l l s e t s E which a r e not

rn-null but only f o r s e t s i n a smaller family, even i f t h i s family i s T(rn)-dense in S(rn).

1

For example, l e t T = (O,l), S = B ( T ) , X = L ( 0 , l ) based on t h e

Lebesgue measure 2.

Let r l , r 2 ,

be numbers i n T such t h a t F = UE=l(sn,tn),

8,


4.

$(s,+t,)

and

l;=l(tn-sn) =

4.

Let xo be a f i x e d element of X.

X by m(E) = Z(EnP)zot

xEnG,

E c S.

Let Define

Then rn

E i s not

i n j e c t i v e whenever E i s a union of i n t e r v a l s b u t rnG i s i n j e c t i v e . Section 3 i s based on C421.

Tweddle C791 be obtained t h e r e s u l t contained i n

Theorem 3 . 3 f o r v e c t o r measures having a d e n s i t y with r e s p e c t t o a a - f i n i t e measure. A s p e c i a l case of Theorem 5 . 1 on Liapunov extension i s Theorem 1 . 6 i n [ E l .

There a r e many authors who proved Theorem 6 . 1 , o r r a t h e r Lemma 6 . 5 , i n special cases.

F o r i n s t a n c e 1 4 8 1 , C711, [ E l l ,

C271, C391.

Theorem 6 . 2 and

i t s Corollary i s due t o Uhl [ 8 2 1 . Section 7 was i n s p i r e d by an attempt t o extend t h e approach and r e s u l t s concerning t h e c o n t r o l of systems with a f i n i t e number of degrees of freedom, t o systems governed by p a r t i a l d i f f e r e n t i a l equations. I t seems t h e f i r s t mathematically f e a s i b l e formulation of t h e "bang-bang" p r i n c i p l e i s i n C51 and C471.

Of course, i n f i n i t e dimensions t h e r e i s much

more l i t e r a t u r e concerning t h e s u b j e c t .

In p a r t i c u l a r , we r e f e r t o C241 where

t h i s s i t u a t i o n i s well summed-up and t h e r o l e o f Liapunov's Theorem i s c l e a r l y shown.

EXTREME AND EXPOSED POINTS OF THE RANGE

VI.

In t h i s chapter t h e p r o p e r t i e s of t h e closed convex h u l l of t h e range of a v e c t o r measure a r e examined f u r t h e r , e s p e c i a l l y from t h e p o i n t of view of t h e extremal s t r u c t u r e .

The r e s u l t s i n t h i s d i r e c t i o n have i n t e r e s t i n g measure-

t h e o r e t i c a l consequences.

There a r e a l s o a p p l i c a t i o n s t o c o n t r o l theory, as

t h e uniqueness of c o n t r o l s i s r e l a t e d t o t h e extreme p o i n t s of t h e a t t a i n a b l e set.

1. Extreme p o i n t s

We s t a r t with a c h a r a c t e r i z a t i o n of t h e extreme p o i n t s of t h e closed convex h u l l of t h e range of a v e c t o r measure m : S

X i n terms of t h e i n t e g r a t i o n

i .

mapping. I f x is an extreme point o f the s e t m ( L

THEOREM 1.

e x i s t s a unique element belongs t o LIo,ll(m)

Cfl,

of L

C0,lJ

( m ) ) then there co,13 ( m ) such that x = m(f) and this eZement

Sh).

I f x belongs t o m(LiO,ll(m))and if x i s reached by m by a unique element

of LCo,l,(m)and if this element belongs t o L { o , . i ) ( m ) = S ( m ) , then x is an extreme point of m ( L C o , 1 3 ( m ) ) . P r o o f.

such t h a t E

6

CfIm

Suppose t h a t x E

E

(m)) and t h a t x = m(f) f o r some f em(L C0,ll

Lco,l,(m) - L{o,ll(m). Then t h e r e e x i s t s an

S which i s not m-negligible such t h a t

assume t h a t m(E)

f

0;

E,

s f(t)5 1 -

E,

> 0

for t

E

and a s e t E.

We can

i f n o t , we can choose a subset o f E with non-zero measure.

Define functions g,h by g ( t ) and h i t ) = f ( t ) t

E

E

for t

E

h(t) E.

f ( t ) ,f o r t

E

T - E , while g ( t ) = f ( t ) -

Then t h e functions g,h Erenot m-equivalent and 112

E

113

EXTREME AND EXPOSED POINTS

V I .1

both belong t o Llo,l,(m). Further, m ( h ) = mT-E ( f ) t mE(f

-

= x t

E)

E

m(E).

Similarly, m ( g ) = x -

E

and, t h e r e f o r e , m ( g )

m ( h ) , t h i s c o n t r a d i c t s t h e extremal c h a r a c t e r o f x .

f

Consequently x = % ( m ( g ) t m ( h ) ) . Since m ( E )

m(E).

Moreover, i f m ( E ) = m(F), with E , F then x =

m(4(xE

xF ) ) ,

t

E

*

0,

S , and i f E,F a r e n o t m-equivalent,

which i s not p o s s i b l e s i n c e

C4(xE t

y.

F

) Im does not

belong t o S ( m ) = LIo,ll(m). Suppose now t h a t x = m ( E ) , f o r some E

f

E

L E O l j ( m ) implies t h a t f

LCo,l,(m),then 4(g + h )

m(g) = m(h) =

2,

CxElm.

S, and t h a t i f x = m(f) with

Then, i f x = %(m(g) t m ( h ) ) , with g,h

Consequently g E

E

E

C0,ll

E

CxElm.

Hence

( m ) ) are contained i n the range

w z m ( S ) i f and onZy i f {Cflm : m ( f ) = x,

LCo,l,(m)l i s a singleton belonging t o S ( m )

P r o o f.

CxEIm, h

E

e m ( L r 0.1 1(m)

The extreme points of m ( L

I f m i s closed then x

m ( S ) of m. E

[xElm.

which means t h a t x

COROLLARY 1.

Cfl,

E

E

E

L{o,ll(m).

The only e x t r a information needed i s given i n Theorem IV.6.1.

The assumption t h a t m i s closed i s needed f o r t h e extreme p o i n t s of

co m ( S )

t c belong t o m(S). The Example IV.6.1 e x h i b i t s a v e c t o r measure m such

that

m ( S ) has many extreme p o i n t s not belonging t o m ( S ) .

The v e c t o r measure

m is not closed, of course. COROLLARY 2 .

co m ( S ) are

:

S + X , the extreme points of

contained i n the closure ( i n the topology of X ) of m ( S ) .

P r o o f. p o i n t s of

For any vector measure m

Let

h

:

S

-t

X be t h e c l o s u r e of m.

By Corollary 1 a l l extreme

G h ( S ) belong t o h ( S ) , b u t , by t h e Corollary t o Theorem IV.3.1, G(S)

i s contained i n t h e closure of m ( S ) . COROLLARY 3 .

I f m i s a closed measure which i s e i t h e r Liapunov or i n j e c t i v e ,

VI.1

EXTREME POINTS

114

then x i s an extreme point of -& m ( S ) i f and onZy if there e x i s t s a unique eZement LEIm of S ( m ) with m(E)

P r

o o

x.

f. The necessity of the condition follows from Theorem 1. Conversely,

suppose there is just one element [ E l , of S ( m ) with x family of equivalent measures for m.

Let n = {p

E

m ( E ) , and let A be a

ca(S)

: P

4 A for some A

E

A}.

By the Corollary 1 to Theorem V.l.l all extreme points of the set {Cfl m : Cfl, E LCo,ll(m)and m(f) = slbelong to S ( m ) , so the only extreme point o f this set is Since this set is convex and o(n)-compact, the Krein-Milman Theorem

[El,.

implies that it consists of the single element [ E l that x

E

Then Theorem 1 implies

exm(lC0,13(m)).

LEMMA 1. For any vector measure m

co m t S )

m'

:

S

+

X , the s e t s m ( S ) , % ( S ) , and

have the same supporting hyperplanes.

P r o o f. Given any

T with respect to

( X I ,

Y'E

m).

XI, let Tt and T- be the Hahn decomposition of

Then

and sinilarly for the inf. It is known (1681 p. 753 that the extreme points of a weakly compact convex set in X need not be strongly extreme. However, if the set is the closed convex hull of the range of a vector measure, the situation is more favourab1e . THEOREM 2 .

co m ( S ) i s

If m

:

s -+

strongZy extreme.

X i s a vector measure, then every extreme point o f

115

EXTREME AND EXPOSED POINTS

VI.2

P r o o f.

Since the ranges of a vector measure and o f its closure have

the same closed convex hull, from the outset we will assume that rn is closed. Suppose x

Then there exists a neighbour-

exco m ( S 1 and x 4 st.exzm(S).

E

hood Y of z in G m ( S ) , in the relative topology of G m ( S ) as subset of X, and such that x

_ co(co

E

In other words, there exists a net { x a I a E A

m(S) - V).

k"

a

a

of elements of co(Z m ( ~ )- V ) converging to z. Let za = lj=lyj y j , lj=lyj k a a = 1, y g

m ( S ) - V , for all j and

E

a E

A.

yg 2 0,

Since m is closed, by

Theorem IV.6.1, there exists f? E Lco,ll(m) such that m(f?) = y" 3 3 i* For every ka y a fa a E A, define f a = j; and so f a E LCo,ll(m).

ljzl

Suppose A is a family of measures equivalent to m, and p

< A for some X

E

E

LIO,lI(A) in the

{p

ca(S)

E

o(Q)

of the net {CfalA!aEA,

:

( A ) and the weak topology

[O,ll 0 weakly in X, and so m(f) = x. As x

Theorem 1, there exists a set E The set WE = {Cfl,

E

S with

E

L[o,l,(A)

converging

topology. Further, since the integration

mapping is continuous with the a(a) topology on L on X, m(fg - f) -+

=

Then LEo,l,(A) is a(Q)-compact (Corollary 1 to Theorem

A).

III.S.l), and so there exists a subnet {CfslAl

to some Cfl,

Ci

: m(f)

[fl, E

=

E

ex=

m ( S ) , by

CxEIA and also m(E)

= x.

V } is a T(A)-neighbourhood o f

[xEIA in L[o,ll(A) by the continuity of integration mapping, (Theorem

IV.l.Z)..

On the other hand we have just proved that CxEIA belongs to the o(Q)-closure of the set co{CfI,

convex, Cx,lA

E

LLo,ll(A)

:

m(f) E z m ( S ) - V 3 .

Since this set is

-

is in its T(A)-closure, i.e. CxElAe C O ( L ~ ~ , ~ ~-(WE'. A)

This

contradicts Theorem 111.7.2.

2. Properties of the set of extreme points

THEOREM 1. ~f m

:

s

-f

x

i s a vector measure then, on ex

CO m ( S ) , a t 1

topologies consistent with the duality between X and X' coincide.

PROPERTIES OF EXTREME POINTS

116

VI.2

-..

P r o o f,

If $ is t h e c l o s u r e o f m, t h e n , by Theorem IV.3.1, co m ( S ) =

-

co m ( S ) , hence we can assume without l o s s of g e n e r a l i t y t h a t m i s c l o s e d . . C l e a r l y , i t s u f f i c e s t o show t h a t t h e ( r e l a t i v e ) Mackey t o p o l o g y on

m(S) is n o t s t r o n g e r t h a n t h e ( r e l a t i v e ) weak t o p o l o g y u ( X , X ’ ) .

ex

n e t o f elements o f

As ev er y

X which does n o t converge i n t h e Mackey topology t o an

element x, has a s u b n e t , no s u b n e t o f which converges t o x , it s u f f i c e s t o show

corn($)converging weakly

t h a t ev er y n e t IxolIaEAof elements o f ex

x

E

t o an element

m(S), has a subnet Mackey convergent t o x.

ex

Let E

E

S be t h e m-unique s e t such t h a t x

= m(Ea), a

be such t h a t x = m(E). These s e t s e x i s t by Theorem 1.1.

E

A , and l e t E

E A

LLo,ll(A) i s u(R)-compact.

w it h p

* A.

E

ca(S) f o r

Then, by C o r o l l a r y 1 t o Theorem 1 1 1 .5 .1 ,

Hence t h e n e t

{CxE I A l a E A has

a s u b n e t , which we

a

can suppose i s t h e n e t i t s e l f , which converges i n o ( Q ) t o an element LCo,ll(A)).

S i n ce t h e mapping

w i t h i t s weak and L

=

: LCo,l,(A)

CxE 1A

sequently

-+

rfl, o f

X i s co n t i n u o u s i f X i s equipped

( A ) with i t s o ( Q ) topologies, x

m(E) i n t h e weak topology o f X.

m(f) = x

IxE 1m

C0,Il

m

= m(Ea) = m ( x E )

=

Cx,l,

-+

a

On t h e o t h e r hand, by Theorem 1.1,

i s t h e unique element o f L r o , l , ( A ) w i t h x = m(XE) = m(E).

Cfl,

S

Let A be a f am i l y o f

measures e q u i v a l e n t t o m, and l e t fi be t h e s e t o f a l l measures p which t h e r e e x i s t s a I

E

Con-

and t h e n e t ICEalAIaEA converges i n u ( 0 ) t o CEI,,. Now

C o r o l l a r y t o Theorem IV.1.2 i m p l i e s t h a t Im(Ea)laEA converges i n t h e Mackey topology t o m(E). THEOREM 2 .

I f X i s a Banach space and m : S + X a vector measure then

through every extreme point of

CO m ( S ) passes a supporting hyperplane.

P r o o f , The c l o s e d l i n e a r span o f we may assume t o be X i t s e l f .

-

Theorem 4 , co m ( S ) =

As

exp z m ( S ) .

m(S) i s a Banach sp ace, which

m ( S ) i s weakly compact and convex, by C11 Then by Milman’s Theorem (C321 p . 132)

117

EXTREME AND EXPOSED POINTS

VI.2

the s e t of exposed p o i n t s o f G r n ( S ) must be weakly dense i n t h e s e t of extreme p o i n t s of G r n ( S ) . Hence, by Theorem 1 , t h e exposed p o i n t s of G r n ( S ) a r e norm dense i n t h e extreme p o i n t s . Let a:

-

such t h a t IIzn

Choose a sequence la: 3 of exposed p o i n t s of

miS).

ex a)

IR. Since U is clearly

the result follows by the hypothesis. Let F,G

LEMMA 2.

t

E

=

S. Because of the compactness of F(t), the supremum in

the definition can be replaced by maximum and so I t :

* s(z',F(t))

F(t)I, is measurable if and only if, for each a

{t : s(z',F(t))> a}

{t

S.

:

T

-+

CCIRm be measurabk.

I f we define H ( t ) = F(t) n G(t),

is "on-empty, then it is measurable.

T , and H

P r o o f. By Lemma 1 we need only show that for any open set U c R

{t

:

F(t) n G(t) n U # 01

If g

:

E

5'.

:

F(t) n (g(t) + V ) n U # 0)

T

choose measurable functions f

:

dense in F(t) for each t

Then

3

E

T.

,

The proof falls into two parts.

T + IRm is a measurable function and U,V are open sets in IR

show that {t

m

-t

IR

m

E

m

we

S. The fact, by Theorem 2 , we can

such that

If 3.(t) : j

= l,Z,..

.

is

SEQUENCES OF MEASURES

148

It

:

F(t) n (g(t) t V ) n U

* 01

VIII.3

iii(fjl(~) n (f. 3

=

-

Let Vn be open s e t s i n Rmhaving only 0 i n c m o n and Vn Suppose g

i

i = 1,2,.

.. 3

It

:

T

:

-+

IRm a r e measurable f u n c t i o n s i

i s dense i n G ( t ) , f o r each t

*

F(t) n G(t) n U

nzl

01 =

E

iclIt

T. :

1,2,.

g)-l(V))

3

S.

E

Vn+l, n = 1,2,.

.. such t h a t

{gi(t)

. .

:

Then

F(t) n (gi(t) + V n ) n U

*

01

which belongs t o S by t h e f i r s t p a r t of t h e proof.

3 . Sequences of measures

Let cca(S) be t h e s e t of a l l sequences p = (pi) o f measures

i=

1,2,.

..

m

, with ,&=ll

For a s e t A

pil

c cca(S).

(TI
0 be a f i x e d p o s i t i v e number, and suppose a

X i s g iv e n .

The f u n c t i o n z can b e i n t e r p r e t e d as t h e

t r a j e c t o r y o f t h e t a r g e t t h a t t h e c o n t r o l system i s t o r each .

For ev er y t

E

1 l e t m t be a c o n t r o l system on t h e Bore1 s e t s B ( l 0 , t J ) o f t h e i n t e r v a l C0,tJ. If F : i O , t o l + CCR”is a given bounded, measurable s e t - v a l u e d f u n c t i o n , C0,t

0

t h en t o s h o r t e n n o t a t i o n we a b b r e v i a t e A F ( m t ) t o A ( t ) . If t h e r e e x i s t s a minimum time t* f o r which t h e t a r g e t z ( t * ) belongs t o A(t*),

t h en t* i s c a l l e d t h e optimal time, and c o n t r o l s r each i n g z ( t * ) i n time

t * a r e c a l l e d optima? c o n t r o l s .

In t h i s s e c t i o n we g i v e some c o n d i t i o n s f o r

t h e e x i s t e n c e of t h e o p t im a l ti m e . Suppose a c o n t r o l system m = (mi) is given on B ( r O , t o l ) ,

i s a f i x e d measurable, bounded s e t - v a l u e d f u n c t i o n .

rot, f o r ev er y t

mt =

E

[O,tol,

and F : T

CCR

co

Define t h e c o n t r o l system

as t h e r e s t r i c t i o n o f m t o t h e i n t e r v a l [ O , t l ,

The a t t a i n a b l e s e t f o r t h i s system i s o f t h e form

( ( V I ~ ) ~ ~ , ~ , ) .

-f

i.e.

IX.5

CONTROL SYSTEMS

163

L e t m be a contro2 system on B(Co,t,l) and F : T

THEOREI! 1.

bounded measurab2e set-valued f u n c t i o n . d e f i m d above, f o r each t

t

Co, t , l .

continuous, and i f there e x i s t s a t'

Suppose the contro2 system mt i s as

If the target E

* CCRma

CO,t,l

z : Co,t,l

f o r which z ( t t )

+

X is ueakZy

E

A ( t ' ) , then

the optima2 time e x i s t s . S e t t* = i n f I t '

P r o o f.

show t h a t z ( t * )

t

: z(t') E

A(t')I.

There e x i s t s a n o n - i n cr easi n g sequence t

A(t*).

LO,t,l, and an a s s o c i a t e d sequence of c o n t r o l s

For each x'

t

The a i m o f t h e proof i s t o

=

+

(c)

with

X' we have, m

+*

F i r s t l y we s h a l l show t h a t t h e t h i r d term i n (1) t e n d s t o z e r o as n Let L O , t , l

LO,t,l,

x

( S ect i o n 1 1 . 7 ) .

+ m.

LN be t h e d i s j o i n t union o f co u n t ab l y many c o p i e s o f t h e s e t

and mIN : Sm

-+

X t h e d i r e c t sum of t h e measures mi, i = 1,2,. . .

I f En i s t h e s e t i n Sm whose p r o j e c t i o n o n t o C O , t , l

f o r ev er y component, th e n c l e a r l y I ( x ' , mm) ((E,)

+ 0 as n -+

as n

(4)

+ m,

t*, tn E

f o r some c o n s t a n t c, as t h e f u n c t i o n s

f o r each n = 1,2,...

m,

i s C t ,t*l

and s o

are u n i f o r m l y hounded

.

As z i s weakly continuous t h e second t e r m i n (1) t en d s t o zer o a s

so z(t*) must be t h e weak l i m i t o f a sequence belonging t o A ( t * ) . 1.1 t h i s s e t i s weakly c l o s e d , hence z ( t * )

E

A(t*).

n * -, and

By Theorem

164

TIMF-OPTIMAL CONTROL

IX. 5

I t would perhaps be worth n o t i c i n g t h a t t h i s r e s u l t can be extended t o

s i m i l a r systems d e f in e d on n-dimensional i n t e r v a l s . I n t n e n ex t Theorem we c o n s i d e r a c o n t r o l system o f t h e co n v o l u t i o n t y p e . Such systems o ccu r f r e q u e n t l y .

Let I

c

ni",J

c

E n be Bore1 measurable s e t s and l e t

Suppose t h a t F : k O , t o l and t h a t

K

:

LO,t,]

x

I

i n the f i r s t variable.

s

= 8 ( L0 ,t o :

X

I).

+

CCIR i s a bounded, measurable s e t - v a l u e d f u n c t i o n ,

x

J + IR i s a bounded i n t e g r a b l e f u n c t i o n , co n t i n u o u s

1 Suppose X = L ( J ) . and t h e c o n t r o l system mt

: S

-L

L1(J)

i s of t h e form

t m t ( E) ( y )

1 /K(t or

T , X , ~ ) X ~ ( T , Z ) ~d ZT ,

y

E

J,

t

E

iO,t,l.

In o t h e r words, t h e a t t a i n a b l e s e t i s

t A ( t ) = {g

f o r some f L e t mt,

THEOREM 2 .

If t h e t a r g e t z t'

E

:

[O,t,l

t -r

: g(y) =

L'iJ)

E

E

4IF\S); y

t

J),

~,x,y:f(x,~)dXd

E

T ,

Io,t,l.

1

L ( J ) is (norm) continuous, rmd t h e r e exists a time

t h e n t h e r e exists an o p t i m a l time t * .

E

A(t'),

For each t

E

LO,t,J,

able s e t , A ( t )

-

IK(t

or

LO,t,l, be t h e c o n t r o l spstem d escp i b ed above.

E

t o , t , l f o r which z ( t 1 ) P r o o f.

E

I

{IL,,t,xIfdmt

:

f

E

m t is a cl o sed measure and so t h e a t t a i n 1

MF(S)) i s weakly compact i n L ( J ) .

(Theorem I V . 6 . 1 . ) From t h e d e f i n i t i o n t * = i n f { t ' : z ( t ' ) no n - i n cr eas i n g sequence t lvj

+

F i s ) wi t h z ( t n ) ( y ) = / $ & K ( t n

E

A(t')},

and so t h e r e e x i s t s a

t*, and an a s s o c i a t e d sequence of c o n t r o l s fn

-

F i r s t l y consider t h e integral

-r,x,y)fn(T,x)dX

dT, n = 1,2,

...

,y

E

J.

E

IX

165

CONTROL SYSTEMS

tn

jjj

(2)

J I O

t

-

Now, j,*IK(tn

I f n ( ~ ) l IK(~,

T,z,~)

-

-

K(t*

- ~ ( t -* T,z,Y)~~T

T,z,~)

T,x,~)I~T

+.

0

c ~ y .

f o r a l l z,y as n

+.

m

by t h e

Dominated convergence Theorem, s i n c e K i s bounded, and co n t i n u o u s i n t h e f i r s t component.

But t h e i n t e g r a l (2) i s l e s s t h a n o r eq u al t o c

r,y) - K(t*

-

T,Z,LJ)~~T

0 a s n -+

-+

t /OOIK(tn -

T

f o r some c o n s t a n t c as F i s bounded, and so by

&dy,

Dominated convergence (2)

,/ I,

m.

However,

t

(3)

IlZ(t*)

- /

5 IlZ(t*)

5 IIz(t*)

-

-+ I l Z ( t , )

Z(t,)lll

-

jfn(T,Z)K(t*

O I

-

t*

/

/fn(T,Z)K(t*

or

tn - z(tn)II + II( j ’ f , ( ~ , ~ ) ~ ( t-,

O I

t* + II/

1 f,(T,Z)K(t*

-

&lll

T,X,y)dT

T,z,~)

-

T,Z,y)dT

-

T,Z,y)dT

&Ill

K ( ~ * - T , z , ~ ) ) ~ T& I l l

&Ill.

tn I Since the as n

-+

m,

Ifn}

a r e u n if o r m ly bounded, t h e l a s t term o f ( 3 ) t e n d s t o z e r o

and t h e second term t e n d s t o zero because z i s norm co n t i n u o u s.

F i n a l l y t h e t h i r d term o f (3) i s dominated by t h e i n t e g r a l ( 2 ) and so it must converge t o 0.

1 I n o t h e r words z(t*) i s t h e l i m i t ( i n L -norm) o f a sequence

of p o i n t s helonging t o A ( t * ) .

S in c e A ( t * ) is weakly compact and so L’-closed,

z(t*) E A ( t * ) .

Remarks The r e l e v a n c e of v e c to r - v a lu e d measures t o t h e problems o f t h e time-optimal c o n t r o l t h e o r y i s s a l i e n t l y e x h i b i t e d i n t h e monograph of Hermes and LaSel l e C241.

They c o n s i d e r systems with an a r b i t r a r y f i n i t e number o f d eg r ees of

freedom and s t e e r e d by a f i n i t e number of c o n t r o l s .

Consequently, t h e t h e o r y

IX

REMARKS

166

n

inv o l v es IR -valued measures.

The r o l e o f Liapunov's Theorem and i t s v a r i o u s

g e n e r a l i z a t i o n s i s a l s o c l e a r l y shown. dimensions i s well summed u p .

In [ 2 4 1 much o f t h e s i t u a t i o n i n f i n i t e

We r e f e r t o 1241 f o r b a s i c r e f e r e n c e s on t h e

s u b j e c t t u r n i n g s p e c i a l a t t e n t i o n t o O le c h ' s work C601. The p r e s e n t Chapter i s a c o n t r i b u t i o n t o t h e programme o f e x t e n d i n e I 2 4 1 t o the infinite-dimensional s i t u a t i o n .

Such an e x t e n s i o n i s m o t i v at ed by t h e

d e s i r e t o have t h e methods and r e s u l t s d e s c r i b e d t h e r e f o r systems o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , a v a i l a b l e f o r systems governed by p a r t i a l d i f f e r e n t i a l equations.

Admitting i n f i n i t e l y many c o n t r o l s ( i . e . t a k i n g c o n t r o l s i n ?!I

m

rn

i n s t e a d of IR ) i s a n a t u r a l g e n e r a l i z a t i o n which could b e o f i n t e r e s t s i n c e t h e space IR- is " f a i r l y u n i v e r s a l " . Theorem 1.1 h a s i t s o r i g i n i n K a r l i n ' s p ap er 1301, whose r e s u l t i s covered by ours i f X = Rn, Rm i s r e p l a c e d by If?,

and F i s a c o n s t a n t s e t - v a l u e d f u n c t i o n .

There a r e s e v e r a l a u t h o r s e x te n d in g K a r l i n ' s r e s u l t , 1601, L l O l , and o t h e r s . The o r i g i n o f Theorem 2 . 1 a l s o d a t e s back t o L241.

It is, clearly, related

t o Liapunov's r e s u l t i n h i s famous paper 1511, as p o i n t e d o u t i n t h e remarks t o Chapter V I .

The r o l e o f t h i s Theorem i n Co n t r o l Theory i s shown i n 1241.

The importance o f t h e e x i s t e n c e o f measurable s e l e c t i o n s i n t h e p r o o f o f Theorem 2 . 1 should be a p p r e c i a t e d .

I t permits extension of the r e s u l t t o the

c a s e where F i s n o t c o n s t a n t . The r e s u l t s of S e c t i o n s 3 and 4 a r e a d i r e c t g e n e r a l i z a t i o n o f t h e corresponding f i n i t e - d i m e n s i o n a l r e s u l t s e . g . 1101. The p r o o f o f Theorem 4.1 fol l o ws a t r i c k i n V a l a d ie r L 8 4 1 .

I t i s c l e a r l y r e l a t e d t o t h e r e s u l t s of

S e c t i o n V.6. One aim of S e c t io n 5 i s t o show how t h e geometric p r o p e r t i e s o f t h e a t t a i n a b l e s e t ( i t s compactness, convexity e t c . ) can be used i n Co n t r o l Theory. The theorems t h e r e c o n s i d e r o n l y two of t h e p o s s i b l e forms t h e c o n t r o l system

IX

can take.

CONTROL SYSTEMS

167

The relevance of these theorems to control of distributed systems

can be seen from the examples of Section V.7.

This Page Intentionally Left Blank

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NOTATION

INDEX

4

9

162

9

155

23 38

10

11 7

9 22 148 27

143 39

coA

4

-

61

coA

c

=

U

C(X)

24.26

11 38

d K ( x ') A(m)

A

P

7 131 71

ex A

4

ex F

152

exgA

5

exp A

4

iEJA

9

[ElA

38

iEJ,

25

177

NOTATION INDEX

178

9

st . e x p A

5

142

4

11

143

16

41

22

2

15 8 72 9

25 24

IN

149

40

17

2

23

v (rn)

P

39 131

Q"

X

P X'.

41

2

x*

X'* 1,8.11 142

1

1,134

x', x*

1

x;

8

39 40

16

8

X'vn

67

(x*, r n )

15

8

llX'llK 39 25 149

s @-a

53

137

INDEX

Anti-Liapunov measure, 88

Family o f e q u i v a l e n t measures, 2 1

Atom

F i n i t e measure sp ace, 10

o f a o - al g eb r a , 32

Function

o f a v e c t o r measure, 32

bounded measurable, 9 A - eq u i v al en t , 38

Banach s p ace, 1

A - i n t eg r ab l e,

38

Bang-Bang p r i n c i p l e , 98

m easu r ab l e, 9

Beppo-Levi's theorem, 27

m - eq u i v al en t , 23

B-P p r o p e r t y , 31

r n - i n t eg r ab l e, 2 1 rn-null, 2 2

Closed v e c t o r measure, 71 Cl o s u r e o f a v e c t o r measure, 72

I n j e c t i v e vector. measure, 88

Complete weak s p a c e , 11

I n tegra1

Conical measure, 10

i n d e f i n i t e , 22

l o c a l i z e d on a compact

on a l a t t i c e , 39

s e t , 135

P e t t i s , 10

r e s u l t a n t o f , 11

p-upper, 23

Co n t r o l system, 154 F-Liapunov, 156

Lexicographic o r d e r , 145

non-atomic, 160

Liapunov v e c t o r measure, 82 L o c a l i z a b l e measure sp ace, 9

Daniel1 i n t e g r a l , 1 2

L o c a l l y convex t o p o l o g i c a l

Denting p o i n t , 14

v e c t o r sp ace ( l . c . t . v . s . ) ,

D i s j o i n t union of s e t s , 35

Mackey t o p o l o g y , 3

Dominated convergence

property (Z),

theorem, 30 Dual o f a l . c . t . v . s . ,

89

series i n , 3 1

weak t o p o l o g y , 2

Equicontinuous f a m i l y o f l i n e a r

Measurable c a r d i n a l , 49

functionals, 2

Measure sp ace, 9

Equivalent measures, 9 Exposed p o i n t , 4

Optimal control, 162

Extreme p o i n t , 4

Optimal time, 162 179

1

180

Orlicz-Pettis lemma, 4 p-measure, 16 p-semi-variation, 17 p-variation, 16 Rybakov's theorem, 121 Scalarly non-atomic vector measure, 95 Schauder basis, 61 Set-valued function, 143 St. Venant's principle, 100 Strongly exposed point, 4 Strongly extreme point, 4 Supporting hyperplane, 5 Uniformly absolutely continuous family of measures, 9 Vector measure, 16 anti-Liapunov, 88 closed, 71 direct sum of a family of, 35 injective, 88 isomorphic, 32 Liapunov, 8 2 non-atomic, 32 scalarly non-atomic, 95 Zonoform, 130 Zonohedron, 129

INDEX