Convex Cones

CONVEX CONES

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NORTH-HOLLAND MATHEMATICS STUDIES

56

Notas de Matematica (82) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Convex Cones BENNO FUCHSSTEINER and WOLFGANG LUSKY University of Paderborn Germany

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

North-Holland Publishing Company, I981

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 86290 0

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors for the U.S.A. and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

F u c h s s t e i n e r , Benno. Convex cones. (North-Holland mathematics s t u d i e s ; 56) Bibliography: p . 1. Functions of r e a l v a r i a b l e s . 2 . Convex bodies. 3 . Cone. I. Lusky, Wolfgang, 194811. T i t l e . 111. S e r i e s .

PRINTED IN THE NETHERLANDS

PREFACE

The aim of t h i s book i s t o o u t l i n e an elementary theory o f l i n e a r funct i o n a l s on convex cones, b u t convex cones a r e here taken i n a s l i g h t l y more general way than usual, they need not be imbedded i n a vector space. In consequence, we do not have a general c a n c e l l a t i o n law f o r t h e a d d i t i o n . Typical examples f o r t h e cones we have in mind a r e 6 = R u I - -1 o r the upper semicontinuous 6- valued functions on some topological space. Accordingly, l i n e a r functionals on such cones a r e allowed t o a t t a i n values i n 6 instead of R . This g e n e r a l i t y has advantages w i t h respect t o extensions of l i n e a r f u n c t i o n a l s .

We believe t h a t , t o a l a r g e e x t e n t , t h e work of an a n a l y s t c o n s i s t s of a s k i l l f u l handling of i n e q u a l i t i e s . I n order t o t r e a t i n e q u a l i t i e s in their proper context, we combine t h e theory of l i n e a r f u n c t i o n a l s w i t h order t h e o r e t i c aspects i n s o f a r as we mostly deal with f u n c t i o n a l s being monotone w i t h respect t o s u i t a b l e order r e l a t i o n s . For those a p p l i c a t i o n s we have i n mind, monotonicity usually implies c o n t i n u i t y . Therefore the material of t h i s book i s presented i n an order t h e o r e t i c r a t h e r than topological s e t t i n g . Very few topological arguments a r e used in this book, b u t this i s n e i t h e r done t o c a r r y our viewpoint t o t h e extreme nor t o show disregard f o r t h e many beautiful arguments and techniques topology has contributed to analysis. We have adapted, and sometimes extended, many basic techniques which were developed in the study of compact convex s e t s . Accordingly, measure theory takes an important place throughout this book. For those readers who a r e not well acquainted w i t h t h a t f i e l d , we have gathered some elementary r e s u l t s and techniques i n t h e A p p e n d i x . V

vi

Preface

We want t o emphasize t h a t these notes a r e w r i t t e n f r o m a h i g h l y personal p o i n t o f view. The c h o i c e o f what t o i n c l u d e has been q u i t e s u b j e c t i v e , we c o n c e n t r a t e h e a v i l y on work done b y t h e f i r s t a u t h o r d u r i n g t h e l a s t years. While t h e i n t e r e s t o f b o t h authors l i e s more on t h e c o n c r e t e s i d e of a n a l y s i s , nevertheless we a r e aware o f t h e d e p l o r a b l e f a c t t h a t t h i s w i l l n o t always b e t r a n s p a r e n t t o t h e r e a d e r . To h e l p compensate f o r t h i s disadvantage, we have added many examples and a p p l i c a t i o n s from d i f f e r e n t areas. But, due t h e l a c k o f space and time,we have had t o o m i t many i m p o r t a n t examples, e s p e c i a l l y from e q u i l i b r i u m t h e o r y i n Mathematical Economics and from S t a t i s t i c a l Mechanics. The book i s m a i n l y addressed t o graduate s t u d e n t s . S i n c e no s p e c i f i c p r e r e q u i s i t e s a r e necessary, however, and no deeper knowledge o f any s p e c i f i c p a r t o f mathematics i s r e q u i r e d , these notes can as w e l l be read by every s t u d e n t . We have t r i e d t o p r e s e n t a l l arguments i n d e t a i l , although sometimes t h i s i s done r a t h e r b r i e f l y and t h i s i s e s p e c i a l l y t r u e i n the a p p l i c a t i o n sections. The book i s divided i n t o two chapters, which a r e t h e n s u b d i v i d e d i n t o two and s i x s e c t i o n s , r e s p e c t i v e l y . I n t h e f i r s t chapter, we p r e s e n t t h e b a s i c m a t e r i a l about l i n e a r f u n c t i o n a l s . We g i v e o u r f a v o u r i t e v e r s i o n o f t h e Hahn-Banach Theorem, and we o b t a i n i m p o r t a n t i n f o r m a t i o n about extens i o n and decomposition o f l i n e a r f u n c t i o n a l s . The second p a r t i s m a i n l y devoted t o t h e r e p r e s e n t a t i o n o f 1 i n e a r f u n c t i o n a l s by i n t e g r a l s . When r e f e r r i n g back t o items w i t h i n a chapter, t h e c h a p t e r number i s omi t t e d . We owe a g r e a t deal t o Mrs. Waltraud Bohmer f o r h e r e x c e l l e n t t y p i n g o f t h e manuscript. F i n a l l y , we thank t h e e d i t o r , Leopoldo Nachbin, and t h e p u b l i s h e r , E. F r e d r i k s s o n , f o r i n c l u d i n g t h i s volume i n t h e i r s e r i e s .

TABLE OF CONTENTS

PREFACE CHAPTER I

V

LINEAR FUNCTIONALS

Chapter I . 1.

1.1

1

The Sandwich Theorem

Semi groups

2

1.2

Cones

6

1.3

Some Consequences o f t h e Sandwich Theorem

11

1.4

Sum Theorem and F i n i t e Decomposition Theorem

14

1.5

Some elementary Appl ic a t i o n s

20

1.5.1

The Phragrnen-Lindelof

1.5.2

Bishop's General V e r s i o n o f t h e "Three

1.5.3

Seevers' P r o o f f o r t h e E x i s t e n c e o f

Principle

21

C i r c l es Theorem"

22

Jensen Measures 1.5.4

When i s a F u n c t i o n Dominated by an A r i t h m e t i c Mean o f o t h e r F u n c t i o n s ?

1.5.5

20

24

Decomposition w i t h r e s p e c t t o P o s i t i v e l y Independent F u n c t i o n a l s

26

1.5.6

The Jordan Decomposition

28

1.5.7

The Lemma o f Farkas

29

1.5.8

The S e p a r a t i o n Theorem

33

1.5.9

A f f i n e I n t e r p o s i t i o n and t h e M a z u r - O r l i c z Theorem

34

The Riesz-Konig Theorem

37

1.7

A S t r a s s e n - t y p e D i s i n t e g r a t i o n Theorem

42

1.8

Some Appl i c a t i o n s

49

1.6

1.9

1.8.1

An a b s t r a c t Flow Theorem

49

1.8.2

Flows i n networks

54

1.8.3

Supply-Demand Problems

58

A d d i t i o n a l Remarks and Comments vii

72

Table of Contents

viii

Chapter 1.2

Order Units and L a t t i c e Cones

Order U n i t Cones 2.2 The Kakutani-Krein-Stone-Yosida Theorem Order Complete Vector L a t t i c e s with Order Unit 2.3 L a t t i c e Cones 2.4 2.5 Riesz Property and F i n i t e Sum Property The Positive Dual Cone 2.6 Dual Orders and t h e Cartier-Fell -Meyer Theorem 2.7 Free L a t t i c e Cones 2.8 2.9 Simplicia1 Cones 2.10 Characters 2.11 Some Examples

2.1

87 89 93 96 101 106 113 118 122 130 139 149

2.11.1 Absorbing Functionals and Banach L a t t i c e s

149

2.11.2 AM- and AL- Cones

154

2.11.3 Characters o f AM-Cones

158

Kakutani's Characterization o f S u b l a t t i c e s o f C ( K ) - Spaces 2.11.5 Korovkin's Theorem

162

2.11.4

2.12

Remarks and Comments

CHAPTER I1

1.2 1.3 1.4 1.5

Countable Decomposition

177

Preliminaries The Main Decomposition Theorems Dini Cones Weak Dini Cones Remarks and Comments

178

Chapter 11.2 2.1 2.2 2.3 2.4 2.5 2.6

170

REPRESENTING MEASURES

Chapter I I . 1 1.1

167

Representing Measures

Decomposition Properties and Measure Theory Dini Cones and Representing Measures Weak Dini Cones and Signed Representing Measures Representing Measures on Weighted Cones Dirichlet States Elementary Examples and Applications

182 189 196 198 201 202 210 213 218 220 227

Table of Contents

2.6.1

The Riesz R e p r e s e n t a t i o n Theorem f o r non-Hausd o r f f Sets

2.6.2 2.6.3 2.6.4 2.6.5 2.7

Regul a r i t y and Support R e p r e s e n t a t i o n by Unbounded Measures Adapted Cones R e p r e s e n t a t i o n o f F u n c t i o n a l s w i t h Zero Mass

Remarks and Comments

Chapter II . 3

Boundaries

3.1

F i x p o i n t Boundaries, B a u e r ' s Maximum P r i n c i p l e and

3.2 3.3 3.4 3.5 3.6 3.7

More Boundaries

t h e K r e i n-Mi 1man Theorem Choquet's Theorem Maximal Measures The Choquet-Meyer Theorem D i n i Boundaries Remarks and Comments

Chapter 11.4

ix

I n t e g r a l R e p r e s e n t a t i o n o f Operators t a k i n g

227 2 28 232 237 241 245 251 252 258 26 7 274 281 287 303 305

Values i n an Order Complete Vector L a t t i c e Chapter 11.5

Generalized Hewitt-Nachbin Spaces

5.1

B a s i c D e f i n i t i o n s and t h e i r Meaning i n t h e

5.2 5.3 5.4 5.5 5.6

The F- C o m p a c t i f i c a t i o n

Classical S i t u a t i o n The F- R e a l c o m p a c t i f i c a t i o n F- Pseudocompactness Some Consequences Remarks and Comments

Chapter 11.6

6.1 6.2

Examples and A p p l i c a t i o n s

Completely Monotonic Functions

314 318 324 328 331 337 340 341

Kendal 1 ' s Theorem on I n f i n i t e l y D i v i s i b l e Completely Monotonic F u n c t i o n

6.3 6.4 6.5

313

M u l t i p l i c a t i v e Cones Banach Algebras and S p e c t r a l Theory The Bochner-Weil Theorem

345 349 352 362

Table of Contents

X

The L’evy-Khi n t c h i ne

6.7

Remarks and Comments

378

Measures and t h e Riesz R e p r e s e n t a t i o n Theorem

380

APPENDIX:

A 1

A A A A

(I-

Formul a

37 1

6.6

Algebras

380

2

Measures

383

3

The Riesz Representation Theorem

386

4

The Radon-Nikodym Theorem

393

5

Signed and L a t t i c e - v a l u e d Measures

396

REFERENCES

403

AUTHOR INDEX

423

SUBJECT INDEX

425

C H A P T E R

I

LINEAR FUNCTIONALS SECTION 1.1 THE SANDWICH THEOREM

E x t e n s i o n theorems f o r 1 i n e a r f u n c t i o n a l s b e l o n g t o t h e most fundamental t o o l s f o r l a r g e p a r t s o f modern a n a l y s i s , e s p e c i a l l y o f

functional

a n a l y s i s . The same i s t r u e f o r convex cones, where l i n e a r f u n c t i o n a l s p l a y an i m p o r t a n t r o l e . A s k i l l f u l 1 h a n d l i n g o f l i n e a r f u n c t i o n a l s t u r n s o u t t o be i n d i s p e n s a b l e i n t h i s area o f a n a l y s i s . Consequently we s t a r t t h i s book w i t h a s u i t a b l e v e r s i o n o f t h e Hahn-Banach theorem f o r semigroups and convex cones. We show t h a t between a s u b a d d i t i v e and a s u p e r a d d i t i v e f u n c t i o n a l t h e r e i s always an a d d i t i v e f u n c t i o n a l h a v i n g t h e same m o n o t o n i c i t y p r o p e r t i e s as t h e s u b a d d i t i v e f u n c t i o n a l (Sandwich Theorem). Since no a d d i t i o n a l work i s r e q u i r e d we immediately prove t h i s r e s u l t f o r maps t a k i n g values i n an o r d e r complete v e c t o r l a t t i c e . S i n c e we a r e d e a l i n g w i t h semigroups and cones i n s t e a d o f groups and v e c t o r spaces we have t o a d j o i n a smallest element t o t h e space where t h e f u n c t i o n a l s t a k e t h e i r values. We proceed g i v i n g two i m p o r t a n t consequences o f t h e Sandwich Theorem, namely t h e F i n i t e Decomposition Theorem and t h e Sum Theorem. The power o f these r e s u l t s i s i l l u s t r a t e d

a t a few s i m p l e a p p l i c a t i o n s ( s e c t i o n 1.5)

and l a t e r on we extend these theorems t o a more s o p h i s t i c a t e d s i t u a t i o n , where measures and i n t e g r a l s t a k e t h e p l a c e o f f i n i t e sums. The Sum Theorem leads t o a Strassen-type d i s i n t e g r a t i o n theorem and, f o r measures on compact spaces, t h e F i n i t e Decomposition Theorem

leads t o t h e Riesz-Konig Theorem

( a general and e f f e c t i v e v e r s i o n o f t h e Riesz R e p r e s e n t a t i o n theorem). Since t h e Riesz-Konig Theorem i s g e n e r a l i z e d l a t e r on t o a non-compact s i t u a t i o n we have postponed a p p l i c a t i o n s f o r t h i s r e s u l t u n t i l c h a p t e r 11. A l l t h e examples i n s e c t i o n 1.9 deal w i t h a p p l i c a t i o n s o f t h e D i s i n t e g r a t i o n Theorem t o networks and p r o b l ems i n Mathemati c a l Economi cs

. Finall y , i n

s e c t i o n 1.9 we mention a few f u r t h e r r e s u l t s and some r e l a t e d problems. Furthermore, we g i v e ( f r o m a h i g h l y personal v i e w p o i n t ) a s h o r t survey on 1

2

Linear Functionals

t h e h i s t o r y o f t h e Hahn-Banach theorem.

1.1 SEMIGROUPS

L e t S = ( S , t , < )be a preordered a b e l i a n semigroup always w i t h n e u t r a l element 0. By p r e o r d e r we mean a r e f l e x i v e and t r a n s i t i v e ( n o t n e c e s s a r i S . Furthermore we r e q u i r e c o m p a t i b i l i t y ~ ~ , s ~ ,t2 t E~ S, such t h a t

l y a n t i s y m m e t r i c ) r e l a t i o n < on

w i t h t h e semigroup s t r u c t u r e , i . e . whenever s1 < s2 and

t2 t h e n

tl<

We c o n s i d e r l i = IR

u I- -1

r e l a t i o n o f IR t o lii 0

*

s1

tl <

t

s2

t

t2 .

and extend t h e a l g e b r a i c o p e r a t i o n s and o r d e r

i n t h e usual way, i . e . l e t (-m)

-

m

= O = i n f IR

A > O

~ . ( - m ) = - m

if

- m + r

for all

= - m

r E

IE

.

The p r o o f s o f t h e f o l l o w i n g a s s e r t i o n s a r e m o s t l y based upon elementary computations w i t h i n f i m a and suprema. Therefore, t h e r e a d e r s h o u l d r e c a l l i d e n t i t i e s such as

I

a E A3 =

-

i)

supIa

ii)

supCxa l a E A 1 = X supIa

iii) sup{a

t

b

1

inf{- a

a E A 1 = supfa

I

I

a E A3

a E Al,

1

X 2 0

a E A3 + b

3

b E IR

where A i s a bounded subset o f IR .These i d e n t i t i e s remain t r u e i f we r e p l a c e IR by an a r b i t r a r y v e c t o r l a t t i c e R. R e c a l l , a v e c t o r l a t t i c e i s o r d e r complete i f f o r any subset A c R which i s o r d e r bounded from

R

r E R w i t h r 2 a f o r a1 1 a E A) t h e supremum A e x i s t s i n R. I n view o f i)t h e n t h e infimum ( i n f B) e x i s t s f o r any subset B c R which i s o r d e r bounded from below. S i m i l a r l y

above (i.e. t h e r e i s (sup A) o f

3

The Sandwich Theorem

as lk we define R

=

R u I - -1

with

0 . (--)=O

-

= inf R

m

for a l l for a l l

~ . ( - m ) = - m

- w + r = - m

x > 0 r E R

In the sequel we want t o deal with functions whose values are elements of I t turns o u t t h a t in a l l our definitions and proofs of the assertions in the f i r s t sections we need not change a single word i f we consider a t once the more general situation where a l l functions involved are allowed t o take values in R with order complete vector l a t t i c e R.

R .

So, from now on, l e t R be an order complete vector l a t t i c e . If the reader feels uncomfortable with the notion of general vector l a t t i c e s , he i s invited t o take R = IR

.

1.1.1.

Def i ni t i on :

i)

p : S fi i s subadditive, i f p ( 0 ) = 0 and p ( s + t ) I p(s) t p ( t ) for a l l s , t E S

ii)

q : S

iii)

p

: S

+

6 i s additive, i f

iv)

f :S

+

R i s monotone i f f ( s ) s f ( t ) whenever s

-.

+ fi i s superadditive i f q ( s + t ) 2 q ( s ) + q ( t ) for a l l

p

q(0) = 0 and s,t E S i s sub- and superadditive < t, s , t E S

Note, that the function which i s -- on S can neither be sub- nor superadditive since we required p(0) = 0 and q(0) = 0 in the preceding definition. On the other hand, define q : S + R by q ( s ) = if s E S 'I01 and q(0) = 0. Then q i s superadditive and even additive there i s no t E S with s + t = 0. Usually, we i f for every s E S'COI denote subadditive maps by p and superadditive maps by q . I t i s convenient t o use the symbol s for the pointwise order relation between functions on S , i . e . f 5 g i f f f ( s ) s g ( s ) for a l l s E S , where f,g: s + R .

-

Linear Functionak

4

1.1.2.

Sandwich Theorem:

Let p : -

S

be monotone and s u b a d d i t i v e -and l e t q : S + 6 -be superq s p. ---Then t h e r e i s a monotone a d d i t i v e p : S + R with

+

additive with

Remark

. q

, only

s u p e r a d d i t i v e and

q Ip

Note, t h a t we do n o t need monotony f o r

for

p.

Proof: In Q =

{p

pointwise

: S

R

+

I

o r d e r on

p

5

-

Q

S . C e r t a i n l y , q E Q, hence

pl

we c o n s i d e r t h e

* +. By Z o r n ' s

lemma

t h e r e i s a maximal c h a i n Q, i . e . a l i n e a r l y ordered subset o f Q, which .. D e f i n e ~ ( s )= s u p I p ( s ) l p E Q

i s maximal w i t h r e s p e c t t o i n c l u s i o n .

(1)

i s superadditive

p

We show t h a t

p

E Q with

p

p,

5

.

i s a maximal element o f p,

. Hence

would be a c h a i n c o n t a i n i n g

of

tj .

so E S

F i x an a r b i t r a r y a

= inf{

for

We c l a i m (1) - ( 5 ) and postpone t h e p r o o f s t o t h e n e x t page.

a l l s E S.

p 5 p

6

Q. Indeed, o t h e r w i s e t h e r e i s

for all

p

E

6

and thus

Q

U

{pol

properly. This contradicts the maximality

and d e f i n e

1 (p(n s o t t ) - p ( t ) ) l n €IN, t E S with u ( t ) + - - I

and

L(s) = sup{p(t)

t

0

s E S , where No

for all

(2)

u s u s p .

(3)

p

Thus,

= N U

with

t t m so < sl

IOl. We o b t a i n

i s monotone and s u p e r a d d i t i v e . p = p

be monotone.

since

m as ( m E No, t E S

p

,

i n view o f t h e m a x i m a l i t y of

We have

i s superadditive.

p

. I n particular,

p

must

5

The Sandwich Theorem

, we

Furthermore, u s i n g t h e d e f i n i t i o n o f Hence,

. Since

p ( s o ) = as

1

0

so E S

-

~ ( 5 )= as = i n f I T i ( p ( n s + t )

(5)

s

The map

+

conclude

s ;(so).

a

was a r b i t r a r y , we have

I

p(t))

s E S

for all

n E W, t E S l

.

i s subadditive.

as

Hence t h e theorem i s proved.

0

For r e a d e r s who a r e n o t t o o f a m i l i a r w i t h sub- and s u p e r a d d i t i v i t y we f i l l i n t h e d e t a i l s o f (1)

-

(5):

(1):

p ( 0 ) = 0 f o l l o w s from t h e d e f i n i t i o n . L e t

pl,p2

E Q

. Since

Because p

i s a chain, t h e r e i s

Q

s1,s2 E S . Consider with

E Q

p

pi

Ip ,

i = 1,2.

i s s u p e r a d d i t i v e , we have

P1(S1) + PZ(S2) 5 P(S1) + P ( S 2 ) I P(S1+S2) I !J(Sl+S2)

Hence, t a k i n g t h e suprema, we o b t a i n

+

p(sl)

.

p ( s 2 ) 5 p(sl+s2)

T h i s proves (1).

(2):

m = 0

Taking

We have m as 5 p(m so

+ t)

0

p(t)

I supIp(m

Ip ( s )

Take

+ so

m as

p(t)

for all

I m E INo, t E S

0

+

t)

for all

t sup{p(tl+t2)

hS1) +

u

+

I

(ml+m

with

m E INo, t E S w i t h

s E S

The monotony o f s1,s2 E S. Then

because

-

yields

m E INo

.

u 1.

and

t E S

with

. Hence,

P --m

i ( s ) = supIp(t)

(3):

;

i n the definition o f

(Recall, p

t

+

m so<

s)

t + m so< s l

i s monotone)

. .

i s an easy consequence o f t h e d e f i n i t i o n of ;(s1+s2) = s u p I u ( t ) + m as 1 m E H o , t E S , t + m S o ~ S 1 + S 2 } 0

)a

so

Imi

EN^,

ti E S

such t h a t

ti +mi so<si,i=1,21

ds2) i s superadditive.

( 4 ) : Using t h e s u p e r a d d i t i v i t y o f

u

s u c c e s s i v e l y we o b t a i n

Linear Functionals

6

n E

for a l l

n

p(so) t

(5):

IN

. Hence,

p(t) I

We have

p(n

so

t ) I p ( n so t t), n E N , t E S

t

p(m s) I m p ( s )

for all

s E S

s u b a d d i t i v e . Hence w i t h (*) we o b t a i n f o r f i x e d p(s

0

) = infI

n.m ( p ( n m

so t m t ) -

NOW, from t h e s u b a d d i t i v i t y o f

s1.s2

infer with

p

p(mt))

and t h e

and

. This

y i e l d s (4).

m E No s i n c e

p

is

m E IN

I n E H,t E S w i t h

p(t)+ -a)

superadditivity o f

u we

-

E S

d S 1 ) + u'(S2) =

inf{T1 (p(nmsltmtl)

-

p(mtl)+p(nm

s2+nt2)-p(nt2))lny m E N

tl,t2 E S

with

u(ti)

* - m y i=1,2}

The assumption on monotony o f p i n t h e a s s e r t i o n o f t h e Sandwich Theorem i s b y no means a r e s t r i c t i o n . I f an a r b i t r a r y s u b a d d i t i v e map p on S i s g i v e n w i t h q I p , t h e n we c o n s i d e r " e q u a l i t y " i n S, i . e . t h e pre-

.

o r d e r on S d e f i n e d b y s < t' i f f s = t, s, t E S Trivially, p monotone w i t h r e s p e c t t o < and t h u s t h e Sandwich Theorem y i e l d s an additive

1.2.

p

: S

+

with

q 5

p I

is

p.

CONES

F o r many a p p l i c a t i o n s o f t h e Sandwich Theorem 1.1.1. we need t h e n o t i o n o f cones. These a r e semigroups endowed w i t h mu1 t i p 1 i c a t i o n w i t h nonnegative s c a l a r s , i . e . elements i n RI, = {x E IR 1 x 2 01 . 1.2.1

Definition:

i) L e t (F,t) be an a b e l i a n semigroup. or ( f o r s h o r t ) a cone i f

F

--

i s d e f i n e d t o be a convex cone

7

The Sandwich Theorem

there is a map IR+ x F + F ((x,f) + x f , x E IR+,f E F) usual laws of multiplication with scalars, i.e. (Al

(Cl)

+

h2)

satisfying the

f = h l f + x2 f

(C2)

h(fl + f2) = xfl + xf2

(C 3 )

q$f) = (59) f

(C 4)

If = f

(C 5)

0.f = 0

for all x,hl,h2

E IR+

and f, fly f2

E

F.

cone F is called a preordered cone with preorder < , if (F, +, < ) is a preordered semigroup and < additionally satisfies h f < x g whenever f < g, f, g E F, x E IR+ . A

ii)

Note, that (C 5) is independent from (C 1) - (C4) . For example, take the semigroup I0,lI with 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 1 and define a multiplication by xf = f for all x 1 0 , f = 0,l. Then obviously ( C l ) - (C4) hold but (C5) is false since 0 * 1 = 1. On the other hand, (C5) is needed to prove the following trivial but indispensable Remark : Let F be a cone. Then A 0 = 0 for all x EIR, .

-

Indeed, (C5) implies 0

1.2.2 i) X

R+

.0=

0. Then with (C3)

x

n

= ( x O ) O = O . 0 = 0.

Examples Let X be a nonempty set. Then RX (E- valued functions on X ) and

(R+- valued functions on X )

are cones under the pointwise operations.

Here, R+ = { r E R I r 2 03 . If these cones are equipped with the pointwise order (i.e. f > g iff f(x) 2 g(x) for all x E X ) or with the pointwise order on some subset Y c X (i.e. f > g iff f(Y) 2 g(Y) for all y E Y ) then they are preordered cones.

8

Linear Functionals

ii) L e t T be a t o p o l o g i c a l space. Then t h e s e t s o f a l l I i - valued upper semicontinuous as w e l l as t h e l o w e r semicontinuous f u n c t i o n s a r e cones. iii) Vectorspaces iv)

( o v e r IR) w i t h t h e usual o p e r a t i o n s a r e cones.

< ) be an a b e l i a n semigroup and denote by (S, < ) * t h e monotone R- valued a d d i t i v e maps and t h e s u b a d d i t i v e

L e t S = (S,

t,

and (S, < )# maps r e s p e c t i v e l y . Then these s e t s a r e cones w i t h r e s p e c t t o t h e p o i n t w i s e

Rs

o p e r a t i o n s . I n f a c t t h e y a r e subcones o f

F a subset

G

c

F

i s s a i d t o be a subcone

. Here,

if

o f course, f o r a cone i s c l o s e d under a d d i -

G

t i o n and m u l t i p l i c a t i o n w i t h nonnegative r e a l numbers, i . e . t X2g2 E G

Xlgl

set T

x

Let al

t

tl,

g1,g2

S

o f a semigroup

whenever v)

whenever

. (Similarly,

if

tl t t2 E

whenever

al,a2

c

E A

F

x for

and

0 Ix 5 1

.

F. We endow

EatblaEA,bEBI

A = I x a l a E A3

A, B E Conv(F),

Conv(F)

=

T

i s s a i d t o be convex i f

L e t Conv(F) be t h e f a m i l y o f a l l nonempty convex subsets o f t h i s f a m i l y w i t h the f o l l o w i n g operations

A t B

a sub-

0 E T).

F be a cone. Then a subset A (l-x)a2 E A

t 0

i s c a l l e d a subsemigroup

and

t2 E T

x1 , x2

and

E G

x

L 0.

Then ( C l ) t o ( C 5 ) a r e e a s i l y v e r i f i e d , and

i s a cone under these o p e r a t i o n s . The n e u t r a l element o f t h i s

cone i s g i v e n by t h e s p e c i a l convex subset preordered cone i f we d e f i n e

A< B

iff

I01

c

F. Conv(F)

becomes a

A cB

Remark: The c o n v e x i t y o f t h e subsets i s necessary, o t h e r w i s e ( C 2 ) would n o t h o l d . ( v i ) A n i n t e r e s t i n g subcone o f Conv(F)

S(F) = I G G F 1 G

F I . For t h i s cone t h e c a n c e l l a t i o n l a w

i s a subcone o f

G t G 1 = G t G 2

*

i s g i v e n by

G1=G2

i s o b v i o u s l y n o t v a l i d . F o r example t a k e F = IRL and c o n s i d e r t h e f o l l o w i n g elements o f S(F) g i v e n by G1 = C(0,x) x E I R I ,

I

G2 = I(x,O)

Ix

E IRI

and

G = C(x,x)

1

x E I R I . Then

9

The Sandwich Theorem

G + G

3ut

+

G1

-IR 2 = G + G 2

1 -

G2. T h i s shows t h a t t h e r e a r e cones which a r e n o t sub-

cones o f vectorspaces s i n c e e v e r y subcone o f a v e c t o r space c e r t a i n l y satisfies (CL) (vi)

.

A r a t h e r odd example of

a cone i s t h e f o l l o w i n g . Consider

( t h e nonnegative f u n c t i o n s on some nonempty s e t X) multiplication and a d d i t i o n + by:

IR+ X

and d e f i n e s c a l a r -

0

for a l l fitg

A * f

=

fX

fu-g

=

f - g

and

h t 0

I: f,g E R

instead o f

hf

and

f

(Here we have chosen t h e symbols

x

+ g i n order t o prevent confusion w i t h the

usual m u l t i p l i c a t i o n and a d d i t i o n o f f u n c t i o n s ) . Then, i f we d e f i n e R:

f,

o

0' = 1,

becomes a cone under these o p e r a t i o n s . T h i s i s another example f o r a

cone which does n o t s a t i s f y t h e c a n c e l l a t i o n law.

1.2.3. Let

Definition: F

be a cone. A map

p : F

i s called functional.

+

A functional A E IR+, f E F.

p i s s a i d t o be homogeneous i f p ( x f ) = x p ( f ) f o r a l l A f u n c t i o n a l i s c a l l e d s u b l i n e a r ( s u p e r l i n e a r ) i f i t i s homogeneous and s u b a d d i t i v e ( s u p e r a d d i t i v e ) . A f u n c t i o n a l i s l i n e a r i f i t i s sub- and super1 inear.

1.2.4. Let -

Proposition: p be

(*) for a l l --

l i m sup

EJ.0

p ( ~ f )= 0

f E F. -Then e v e r y a d d i t i v e

(Recall t h a t

F and assume ~ that

s u b a d d i t i v e f u n c t i o n a l --on t h e cone

l i m sup

E J . 0

p I

p

i s linear. --

p ( ~ f )= i n f sup I p ( 6 f ) &

>O

I

E

t 61)

-

Linear Functionals

10

Proof: We have

mp

(1 m f)

=

n m

p(

since i s additive. Hence Consider some X 0 , x E IR,

*

rn

2 0

p ( f~)

such t h a t rn --t = p((x

- r n )f

1

f)

=

n p ( f ) for a l l

n,m E

IN

,f E

F

,

r ( f ) for a l l r E 4' r 2 0 . , and take an increasing sequence rn E Q ,

p(rf)

=

x ( r n + x in short). We have for f

) + u(rn f ) 5 P ( ( X - r n ) f

1

+

E F

rn ( f ) .

Taking the lim sup on b o t h sides we obtain by ( * ) t h a t

v ( X f ) s X p(f).

To prove the converse inequality we take a sequence rn E Q , rn

4 A.

Then

Certainly, every homogeneous functional p s a t i s f i e s (*). This enables us t o s t a t e the Sandwich Theorem now in terms of cones and functionals as a direct consequence of 1.1.2. and 1.2.4. This version of the Sandwich Theorem plays a central role in our theory. 1.2.5.

Let

Theorem:

g preordered cone and l e t p : F -, be monotone and sub1 inear, q : F + k superlinear with q s p. Then there i s 5 monotone linear p : F + R with q s p s p . F

Again, we emphasize t h a t the monotony of p does n o t mean a restriction. If no explicit preorder on F i s given, consider "equality". Then p becomes a monotone functional and 1.2.5. i s applicable. The notion of linearity we have adopted here indeed generalizes the usual definition of a linear map p : F + R , where F i s a vector space. To see this, l e t F be a vector space and take a functional p : F + R which i s linear according t o Definition 1.2.3. Since f E F, we obtain p((-x) f ) t p ( x f ) = ~ ( 0 = ) 0, x E IR, In particular p ( f ) > - m Thus our p ( ( - A ) f ) = - p ( f~ ) = ( - A ) p ( f ) . notion o f linearity coincides with the usual terminology in the vector space case. Let S be a group, i.e. an abelian semigroup with neutral element 0 such

.

11

The Sandwich Theorem

t h a t for every s E S there i s a n element -s E S vrith s t ( - s ) = 0. If U,V : S + R are additive maps with p I v on S then we can conclude p = v . Indeed, we have ~ ( s I ) ~ ( s ) and p ( - s ) s v ( - s ) , for a l l s E S. As before, we have p ( s ) t p ( - s ) = p ( s t ( - s ) ) = p ( 0 ) = 0, i . e . L I ( S ) = - u ( - s ) , and, similarly, v ( s ) = - v ( - s ) . Hence - ~ ( s )I - V ( S ) for a l l s E S. This yields p ( s ) = v ( s ) for a l l s E S . Thus, in the case of groups and additive maps, "domination means equality". Of course, here, the case of vector spaces and linear maps i s included.

SOME CONSEQUENCES OF THE SANDWICH THEOREM

1.3

Here we present some fundamental applications of the Sandwich Theorem. The theorems in t h i s and the next section are formulated in the context of semigroups. The corresponding assertions f o r cones are presented as corollaries (via Proposition 1 . 2 . 4 . ) However, we emphasize that the proofs for the semigroup case can be adapted immediately t o the case of cones. The reader merely has t o replace the words "sub- and superadditivity" by "sub- and super1 inearity", respectively. 1.3.1 Dominatina Extension Theorem: Let T -

be-a subsemigroup of the preordered abelian semigroup (S, < ) . Consider an additve map p : T + R and - a monotone subadditive p : S +

with l?t)

S

P(t)

for all t --

Then there i s a monotone additive ---L I S P 0" s. (by

'I

I LI

on T

I'

p

: S

+

+

R

with F(

we mean o f course ; ( t ) I

Proof: The map q : S

.

E T

R defined by

q(s) =

-

I LI

for a l l

p(t)

T

and -

t E T).

s E T

i(s)

if

-

otherwise

i s superadditive. The Sandwich Theorem 1.1.2. yields then a monotone additive p on S with q I LI I p , which means ;IU on T . o

12

Linear Functionals

The same argument as i n t h e preceding p r o o f , now w i t h an a p p l i c a t i o n o f Theorem 1.2.5 i n s t e a d o f t h e Sandwich Theorem 1.1.2.,

y i e l d s t h e cone v e r -

s i o n o f t h e Dominating Extension Theorem. On t h e o t h e r hand we can i n f e r t h i s v e r s i o n d i r e c t l y from t h e Dominating Extension Theorem by u s i n g P r o p o s i t i o n 1.2.4. Corollary: Let G -

(F, < ) . Consider -a linear G -and a monotone s u b l i n e a r p on F with

a subcone o f t h e p r e o r d e r e d cone

functional

i 2

Then t h e r e i s a monotone l i n e a r ----

p

p s p 0" F.

on F with -

I

on -

p

G and

The q u e s t i o n seems n a t u r a l whether i t i s always p o s s i b l e t o f i n d a monotone a d d i t i v e that is,

i n t h e p r e c e d i n g theorem which extends

p

G(t) = p ( t )

for all

i s a group. I n f a c t , i f T and hence ;(t) equivalent t o T

5

p(t)

and

;(t) = p ( t )

t E T.

t) I

for a l l

,

properly

T h i s i s c e r t a i n l y t h e case, i f

i s a group, t h e n

;(-

;

-t E T

for a l l

p(-t)

t E T.

f o r every t E T

t E T

which i s

I n p a r t i c u l a r , i n case t h a t

i s a vectorspace, t h e Dominating E x t e n s i o n Theorem i n c l u d e s t h e usual

v e r s i o n s o f t h e Hahn-Banach e x t e n s i o n theorem. The n e x t theorem p r o v i d e s us w i t h a necessary and s u f f i c i e n t c o n d i t i o n for

i

1.3.2.

t o admit a proper e x t e n s i o n . E x t e n s i o n Theorem:

canbe extended t o-a Then T, S, G , p be g i v e n as i n 1.3.1. on S ( i . e . 11 = u on T) with p 5 p on monotone a d d i t i v e p -

Let -

i f and o n l y i f ----

whenever

tl,t2 E T, s E S

and

tl < tp + s

.

S

'r

13

The Sandwich Theorem

Proof: (only i f part). then t ,t

1

;(tl) 2

= u(tl)

E T

and

(if part): Let for a l l

If

r E S.

p

5 p

u = u

i s a d d i t i v e and monotone and

I u ( t 2 + s ) = u ( t 2 ) + U(S) 5 ;(t2) + p ( s )

T

on

s E S,

for

tl< t2 + s .

+

$r) = i n f { i ( t )

p

p(s)

I

t E T, s E S

with

i s monotone, s u b a d d i t i v e and we have

From o u r assumption we o b t a i n

c 5 i;

on

r< t

p .

Hence

and t h e Sandwich Theorem y i e l d s a monotone a d d i t i v e u I U 5 u I = i on T. Now, n e c e s s a r i l y , u = ; on T

s)

i; I p on S .

T. On t h e o t h e r hand,

which f o l l o w s d i r e c t l y f r o m t h e d e f i n i t i o n o f

+ I on

=

i;

;, T

with

Again, these arguments c a r r y o v e r t o t h e case o f cones. A1 t e r n a t i v e l y , we can prove t h e cone v e r s i o n u s i n g P r o p o s i t i o n l . 2 . 4 . Corol 1a r y : Let -

G, F,

G,

Theorem 1.3.1. (i.e.

=

whenever

p

be g i v e n --as i n t h e c o r o l l a r y o f t h e Dominating p -can be extended t o-a monotone l i n e a r p

Then

u 0" G)

g1,g2 E G

with

,f

p

E F

I p ')" F ---i f and o n l y i f

and

_ .

91 < 92

Extension on F -

+ f -

As a consequence o f t h e E x t e n s i o n Theorem we o b t a i n a g e n e r a l i z a t i o n o f t h e w e l l known f a c t , t h a t t h e norm o f an element i s equal t o t h e maximum o f

1.3.3

x

x

o f a normed space

on t h e dual u n i t b a l l .

Norm Theorem:

Let S be-a preordered a b e l i a n semigroup. Consider a monotone s u b a d d i t i v e y p : S + R w i t h p ( n s ) = n p ( s j -f o r a l l s E 5 and n = 0,1,2, ... . Then p i s t h e p o i n t w i s e supremum o f all monotone a d d i t v e p 5 p and i s a t t a i n e d -a t some u , i . e . -for all f-o r e v e r y s E S t h i s supremum s E S -we have p ( s ) = maxIp(s) I p : S + R i s monotone, a d d i t i v e w i t h - p 5 p 0" S ) _ .

14

Linear Functionals

Proof: Let

s E S

and d e f i n e t h e subsemigroup

ii : T

L e t t h e a d d i t v e map n = 0,1,2,...

.

+

R

T = {ns

be d e f i n e d by

I

n= 0,1,2,

...I .

;(n s ) = p ( n s )

for all

Then t h e Dominating E x t e n s i o n Theorem p r o v i d e s us w i t h an

a d d i t i v e , monotone

!.I

:S

-,

R

such t h a t

!.I I p

T. Here t h e l a t t e r f a c t i m p l i e s e q u a l i t y on T

on

S

and

I !.I I

p

on

by d e f i n i t i o n o f

Corol 1a r y : Let -

p Then -p

the

&

p r e o r d e r e d c ~ n e F. monotone sub1 i n e a r f u n c t i o n a l fi i-s the pointwise ~ supremum o f a ll monotone li n e a r !.I I p

for every f E F

_.-

p ( f ) = max{U(f)

this supremum 3 a t t a i n e d

I 11

monotone, l i n e a r and

!.I

a t some

!.I

Ip

FI

,

and

i.e.

.

1 . 4 . SUM THEOREM AND FINITE DECOMPOSITION THEOREM As before, we f o r m u l a t e t h e theorems f o r b o t h cases, a b e l i a n semigroups and cones.

1.4.1.

Sum Theorem:

Let

be an a d d i t i v e 9 -on t h e

_.

!.I

-and l e t

pl,.

. .,p,

: S

+

!.I I p1 t

p r e o r d e r e d a b e l i a n semigroup

whenever

p2 t

*.. t

Pn

*

R with

uk

If, i n addition, -

!.I

...,un : S !.I,,

.

-,

I

that

r, t, sl,...,sk

< )

be monotone and s u b a d d i t i v e -such t h a t -

Then t h e r e a r e monotone, a d d i t i v e ul, --k = l,...,n, and !.I I !.I t !.I~t... t tone and i f we have ------

(S,

E S

such t h a t --

r < t t sk

, k = l , ...,n,

pk

,

i s mono--

The Sandwich Theorem

15

then we can assume

(**I

lJ =

+ u2 +...+ lJn

Proof: We consider

@

= S { l y * * * y n=l { ( s l ,

... , s n ) I

sk E S , k = l ,

..., n }

and denote

the diagonal by A @ ={(S,S,

...,S) I

s E

s}

@ i s a preordered abelian semigroup i f the preorder relation and the operations are defined pointwise, i .e.

+ tl’ ..., S n + t n ) n (s1,...,sn) < ( tl , . . . , t n ) i f f sk < t k for a l l k=l, ...,n ( s y . ., s n ) + (tl’“ . ’ t ) = (sl

.

i s a subsemigroup of 0 . We define an additive map : A @ -, R by (s,s,...,s) = ~ ( s ) , s E S . On @ we consider the monotone subadditive map p given by Obviously,

P(S”’..”Sn)

A @

=

Pl(S1) +

Pz(S2)

+...+

Pn(Sn),

S1’S2’“ ” S n

E

s.

We have I p On A @ in view of our assumptions. Hence we can apply the Dominating Extension Theorem 1.3.1. t o obtain a monotone additive map v : @ -, f? with 5 v on A @ and v 5 p on @ .I f , in addition, (*) holds, then the assumptions of the Extension Theorem 1.3.2. are s a t i s fied and we may even assume v = on A Q . Define p k ( s ) = V ( A s~ ) for a l l s E S , k = 1, ..., n , where s i s the k - t h component of A

i.e. a l l other components of A

A ~ S =(O,...,O,s,O,...,O), ~

Sand

~

S

are zero. The p k are monotone, additive and satisfy the required condi+ A,,S for a l l s E S. 0 tions since (s,s,...,s) = A ~ S ...+ Similarly as in the preceding section the cone case can be proven with the analogous arguments. On the other hand, using 1.4.1 and Proposition 1.2.4., we can derive the following corollary directly from the Sum Theorem.

16

Linear Functionals

Corol 1a r y : Let -

P

.,p,

ply..

&a

l i n e a r f u n c t i o n a l -on t h e p r e o r d e r e d cone (F, monotone sub1 i n e a r on F -such t h a t

and

P I

p1 t p2 t... t Pn

--Then t h e r e a r e monotone and

k = l,...,n,

linear

P I p

<) and let

*

pl,...,un

t p2 t.. .t pn

.

on F w i t h p k 5 pk,

If i n addition,

p

2 monotone

and i f we have ----

whenever

h,g,

E F

fl,...,fn

-such t h a t

h < g t f k ’ k = l,...,n,

then we -can assume --

(**I

P =

PI

-I.

v2

t... t pn

Note, t h a t i n t h e p r e c e d i n g a s s e r t i o n s , (*) i s a l s o necessary f o r ( * * ) . C o n d i t i o n ( * ) i s o f t e n s a t i s f i e d , f o r example, i f S has t h e s e m i i n t e r p o l a t i o n property:

A preordered a b e l i a n semigroup (S, < ) has t h e s e m i i n t e r p o l a t i o n p r o p e r t y . (SIP i n s h o r t ) i f f t h e r e i s s E S w i t h r < t + s and s < s k y k=1,2 whenever

1.4.2.

r,t,sk

E S

and

r < t t sk, k=1,2.

Lemma

semigroup (S, < ) -have t h e S I P . Then c o n d i t i o n ( * ) i---n t h e Sum Theorem 1.4.1 holds a u t o m a t i c a l l y f o r monotone a d d i t i v e Let the abelian ---p :

s+R.

Proof: Let

r, t, sk E S

S I P y i e l d s some

and

r< t

s E S

with

t sky

k = 1,2,...,n.

s < s k y k = 1,2,...,n,

Then, b y i n d u c t i o n , t h e and

r < t t s.

17

The Sandwich Theorem

Here t h e l a s t i n e q u a l i t y f o l l o w s from t h e monotony o f

Examples o f preordered semigroups w i t h t h e SIP i)

Let

Let

s

s k y r, t, s

t

k=1,2

r < t

and

E S, k= 1,2. Then w i t h

k t

e v e r y v e c t o r space has t h e

s . Hence e v e r y group

s1,s2

t < s k y k=1,2,

+

( - t ) we have

< ) such t h a t e v e r y two s, i . e . whenever t E S

(S,

have a g r e a t e s t l o w e r bound

E S

then

s = r

.

and, i n p a r t i c u l a r ,

.

SIP

ii) Consider a preordered a b e l i a n semigroup elements

.

be a group endowed w i t h an a r b i t r a r y p r e o r d e r r e l a t i o n

S

r < t

< s k y

.

11

t < s . I n t h i s case, o b v i o u s l y ,

S

has t h e

SIP.

We observe t h a t a g r e a t e s t l o w e r bound need n o t be unique, i f < i s n o t antisymnetric. Note, t h a t i n t h e r e m a i n i n g theorems o f t h i s s e c t i o n we have t o r e s t r i c t o u r s e l v e s t o t h e case

R = IR

.

1.4.3. F i n i t e Decomposition Theorem ( f i r s t p a r t ) : Assume t h a t -l e t 9, ...,pn -

S

: S

f o r a l l s E S. -xl t +...+ x n

Define

CI =

+6 be s u b a d d i t i v e -such t h a t

Then t h e r e a r e

X k 2 0,

k=l,

be a d d i t i v e and

...,n , with

= 1 -such t h a t

p I

Proof:

a b e l ia n semigroup. L e t 1-1 : S + l i ~

i s an -

IP1’.

Alp1

..

Y

t

Pnl

x2p2

+...+ x n D. n

= {q

I CP : { pl,...,pn}

+lil

.

@

t h e p o i n t w i s e o p e r a t i o n s . We c o n s i d e r t h e p o i n t w i s e o r d e r :

i s a cone under

18

Linear Functionals

iff

( p k ) 2 cp2 ( pk ) for a l l k = l , . . . , n . i s then a preordered cone. p : ID + I i defined by

'pl "cp2

(ID,i)

cp

1

p(cp) = max((p(pl), . . . , t p ( p n ) ) , q : ID

+ k , with

, i s a monotone sublinear functional.

cp E ID

q(Q) = S U p { p ( S )

IS

E

s with pk(s)<(p(pk), k = l ,... ,nl

for a l l cp E I D ,i s superadditive. (Recall sup(0) = - a ) . In view of our assumptions we have q Ip . Hence, by the Sandwich Theorem 1.1.2 there i s a monotone additive v on Q with q I v I p. From Proposition 1 . 2 . 4 and the homogeneity o f p we infer t h a t v i s indeed a linear functional on ID.Let s E S and define S E ID t o be the function with c ( p k ) = p k ( s ) , k = ~ , . . . , n . Let t i E ID be the element with Put

Xk =

&k 2 0

v(tk)y

and

v

&i(Pk)= 0 i f

k

#

k=l

k=l

n

- k=l Now ,

Ik

=

since

k

1 since

k=l

n

r

=

+Gk)

k=l

Indeed, t h i s i s clear, i f

-

l,...,n.

n

v ( - 1 ) Ip(-1) =

pk(s) > -

such t h a t

& k for a l l

00

pk(s) =

n

E N

-

1

n

(s)

Assume there i s k

=

o for a l l

=

1 w ( t k ) = ~ ( 1I ) p ( 1 ) = 1 and

u ( s ) Iq ( c ) I v

I

lk2

n i s monotone. We conclude t X k n

t .1( p 1. ) = 1, i,k

and

Then we have

k = l,...,n.

1 Xk =

Pk(s) &k

i

-

n

.

for a l l

s E S

for a l l

k = l,,..,n

m.

. Hence,

.

Then we have by the monotony of

v

,

Thus the preceding estimate holds in t h i s case, t o o , and the theorem i s proved

19

The Sandwich Theorem

Note t h a t the Finite Decomposition Theorem 1.4.3 includes already the case t h a t S i s a cone and a l l functionals involved are homoqeneous. So, in contrast t o the preceding theorems i t i s n o t necessary t o s t a t e the cone version of 1.4.3 separately. As an immediate corollary of 1.4.1., 1.4.2., 1.4.3 we obtain the second p a r t o f the Finite Decomposition Theorem. 1.4.4

Finite Decomposition Theorem ( p a r t two):

Let IJ:S -, 6 be additive -on the preordered abelian semiqroup : S +IR with and consider subadditive, monotone p l , . . . , p n ~ ( s )I max(pl(s), . . . , p n ( s ) ) 2 0,...,An

A1

n

1

k=l

Ak =

2 0

-for _ _a l l

s E S.

a n d monotone additive -

(S,<

)

Then there are ---

pk 5

pk, k

=

1,

...,n , -such t h a t

1 and

n

I n case t h a t ---

( S , <) -has the S I P we can assume t h a t

Corollarv (cone version o f 1.4.4.1

:

Let -

IJ : F -, t i be l ine a r on the preordered cone ( F , 4 ) and consider sublin ea r , monotone p l , . . . , p n : F +I6 with v ( f ) 5 max(pl(f), . . . , p n ( f ) )

-for a l l k

=

f E F.

l,...,n,

In case t h a t ---

--Then there are

n such t h a t z --

k=l

(F, < )

A1 2 0

and monotone linear

hk = 1 and

has the SIP we can assume~ that --

n

-

IJ k

5 P k Y

20

Linear Functionals

1.5 SOME ELEMENTARY APPLICATIONS

In this book we do not intend to go into details of semigroups. But since we stated the Sandwich Theorem in the context of abelian semigroups we like to start with some applications in this field. 1.5.1 The Phragmen

-

Lindelof Principle

An important tool in many areas of classical analysis - for example interpolation theory [331 - is the Phragmen - Lindelof principle. We demonstrate that this principle is a corollary of the Finite Decomposition Theorem. Theorem: L e t F(z) &a bounded analytic function on the strip 0 c Re(z) c 1 and assume that F ( z ) fi continuous on the closed strip 0 I Re(z) I 1. Then for 0 I o I 1 we have for all t E 1R

Proof: Let H be the multiplicative semigroup of all bounded analytic functions on 0 < Re(z) c 1, which are continuous on the closed strip 0 I Re(z) I 1 and fix some zo = xo t i yo in this closed strip. By "multiplicative" we mean that H is considered as a subsemigrobp of the cone given in example 1.2.2 (vi), i.e. that the semigroup operation in H is given by f fig = f g. Then by

an additive function is given on the mu1 iplicative semigroup H. The maximum principle for analytic functions imp1 ies

where po,pl,p are the following subadd tive functions:

21

The Sandwich Theorem

p ( h ) = l o g ( 1 i m sup yaR, 1-

sup o<x
I h(x

:rom t h e F i n i t e Decomposition Theorem we g e t

, + x1 + x

xo

= 1

I)

,

xo,xl,X 2 0 w i t h

= 1 such t h a t

It remains t o c a l c u l a t e t h e numbers

functions

+ i y)

i n the inequality (*).

h E H

.

- xo

Thus

-

by introducing special

xo,Xl,h

We c l a i m

by t a k i n g e x p o n e n t i a l s i n ( * )

1 = 0,

x1 = xo and

- t h e theorem i s proved

2 P r o o f o f t h e c l a i m : Consider t h e s p e c i a l f u n c t i o n h,(z) = ez , t h e n 2 2 p(ho) = l o g ( o ) = and u ( h o ) = xo - yo > - a3 . Hence A = 0.

-

Now, we i n s e r t

hl,2(z) i

Hence

hl = x

0

.

= e

i n (*)

xo = v ( h l , ? )

We c o n s i d e r a nonempty s e t

,t h e

X

0

=

i

xl l o g ( e )

1 we o b t a i n

= i

x0

x1 = 1

- x0 .

and a m u l t i p l i c a t i v e subsemigroup

bounded complex-valued f u n c t i o n s on

semigroup we mean, o f course, t h a t If

I xo.O

xo+xl+x

Because o f

and o b t a i n :

0

B i s h o p ' s (;enera1 V e r s i o n o f t h e "Three C i r c l e s Theorem"

1.5.2

BC(X)

iz

#

YcX

then

I( I I y

hl

denotes t h e

-

X

h2 E H(X)

H(X)

. By a m u l t i p l i c a t i v e whenever

Y-sup-norm g i v e n by

hl,h2

of

E H(X).

22

Linear Functionals

Theorem: - Let m : H ( X ) m(hl

*

h2) = m(hl)m(h2)

+

C be multiplicative, i . e .

for a l l

hl,h2 E H ( X ) .

m ( h ) I 11 h l l y

(*)

for a l l

h E H(X)

,

...,Y n

subsets such t h a t _ _ _ -of Y -- are nonempty _ n Y = Y1u Y2 u... u Y n then there are h i 2 0 & y n i = l i =1 such that -'n

and i f --

Y1,

Proof:

We may assume that H ( X )

contains the neutral element

lX

(constant function equal t o 1 on X ) , because otherwise we extend m without any difficulty to a multiplicative function on H ( X ) u Ilx1 by putting m ( l x )

=

1

.

Now, we consider the additive li- valued function given by p ( h ) = log(m(h)l and the subadditive functions defined by p i ( h ) = log11 h ( l , i = 1,..., n Then from ( * ) weget v 5 max(pl 'i n Hence by the Finite Decomposition Theorem p I z Xipi , where i =1 n A . 2 0 with 1 hi = 1 1 i =1

.

,...,D n ) .

.

Going back, by taking exponentials, immediately gives the theorem. Usually this theorem i s proved by means of Jensen measures for the rather special case of mu1 tip1 icative 1 inear functionals on function algebras bg]. For applications see [821.

1.5.3 Seevers' Proof for the Existence o f Jensen Measures The following proof i s modelled a f t e r a beautiful argument appearing in an unpublished paper of Galen L . Seever. Actually t h i s paper has drawn the

23

The Sandwich Theorem

a t t e n t i o n o f t h e f i r s t a u t h o r t o semigroup v e r s i o n s o f t h e Hahn-Banach theorem,

I n o r d e r t o a v o i d c o n f u s i n g o u r arguments w i t h measure t h e o r e t i c c o n s i -

d e r a t i o n s we p r e s e n t t h e theorem about t h e e x i s t e n c e o f Jensen measures i n a somewhat m o d i f i e d X

Again, l e t

be a nonempty s e t and l e t

semigroup o f

BC(X). By

f u n c t i o n s on

X. Note t h a t

u :

v : B,i(X)

a linear --

H(X)

# 0 and +

(*)

+

H(X)

form.

be a m u l t i p l i c a t i v e sub-

we denote t h e upper bounded li- valued

Bli(X)

Blk(X) = I l o g l f l I f E BIR(X)1, where

the real-valued functions i n Theorem: - Let assume that a --

-

- and by t h e same t i m e g e n e r a l i z e d

BIR(X)

are

Bg(X).

C be-a m u l t i p l i c a t i v e f u n c t i o n on H(X) and

I u ( h ) l I I( h l I X

li with

-f o r a l l h E H(X).

Then t h e r e i s

such t h a t : p ( f ) 5 sup f ( x ) -f o r a l l f E B,i(X) -XEX

lu(h)I 5 ev(loglh1)

-f o r a l l h E H(X).

I n ( * ) we assume i f and o n l y i f : _ -can __ _ e q u a l i t y ----

(**I such t h a t -Proof:

Consider on t h e mu1 t i p 1 i c a t ve semi group

BIR( X )

t h e f o l 1owing

s u p e r a d d i t i v e f u n c t on I h E H ( X ) w i t h I h l = I'fll, f E BIR(X)

= sup{loglu(h)

q(f

i s d e f i n e d t o be

where, o f course, t..e supremum o f Then

q 5 p

where

p

.

-

i s t h e s u b a d d i t i v e f u n c t i o n g i v e n by

P ( f ) = log11

fllx

I

f

E BIR(X).

From t h e Sandwich Theorem we o b t a i n an a d d i t i v e

i

with

q 5

iI p.

Linear Functionals

24

Hence,

(***) Consider

E3,jj(X).

logla(h)( 2 ;(\hi) O f course,

BG(X)

a d d i t i o n o f t h e f u n c t i o n s i n Bli(X) elements i n IR,

Thus

i

qE(X)

h E H(X)

.

i s a cone i f we t a k e now t h e F o i n t w i s e and p o i n t w i s e m u l t i p l i c a t i o n w i t h t h e

as t h e cone o p e r a t i o n s . Then

a d d i t i v e f u n c t i o n on

(since

for a l l

p ( f ) = ;(ef)

d e f i n e s an

with

was a d d i t i v e on t h e m u l t i p l i c a t i v e subgroup).

u must be l i n e a r b y P r o p o s i t i o n 1.2.4 and (*) f o l l o w s w i t h (***)

from: l o g ( a ( h ) l 5 ;(lhl)

= i(e’oglhl)

= p(loglh1)

I n case t h a t (**) holds we c o n s t r u c t o u r

.

w i t h t h e Extension Theorem

i n s t e a d of t h e Sandwich Theorem. We do t h i s i n such a way t h a t we have e q u a l i t y i n (***) which c l e a r l y i m p l i e s t h e a s s e r t i o n . I n t h i s case we cons i d e r t h e m u l t i p l i c a t i v e semigroup ( H I = { ( h ( \ h € H ( X ) I and d e f i n e an a d d i t i v e u on H by

T h i s d e f i n i t i o n i s n o t ambiguous because o f ( * * ) . Then by t h e E x t e n s i o n Theorem we can extend T h i s i s p o s s i b l e because (**)

u

t o an a d d i t i v e

5 p.

i s j u s t t h e e x p o n e n t i a l form o f t h e necessary

c o n d i t i o n i n 1.3.2.

T h i s shows t h a t (**) i s necessary and s u f f i c i e n t f o r

e q u a l i t y i n (*).

o

1.5.4

When i s a F u n c t i o n Dominated by an A r i t h m e t i c Mean o f o t h e r Functions?

I n t h i s subsection we prove a r e s u l t o f a somewhat c o m b i n a t o r i c a l nature, showing t h a t t h e F i n i t e Decomposition Theorem can be extended t o t h e most general s i t u a t i o n .

25

The Sandwich Theorem

Given a nonempty s e t X and

then our question i s : When i s of the functions f l , ...,f n ? Theorem: (i)

The following __ are

There are

n

cp(x) I

(ii)

cp

i =1

cp

.

,

and f l y . . , f n on X

pointwise dominated by an arithmetic mean

equivalent:

. ,An 2

hl,..

valued functions

li-

0

with z -

hi =

1 -such that

h i f i ( x ) -for a l l x E X

.

-

For every f i n i t e nonempty subset X c X w e have --

x

x

E Proof: ( i ) * ( i i ) : By adding u p the inequality in ( i ) for the we see that t- ~ ( ii s) dominated by an arithmetic mean of the corres-

;EX

ponding sums for the f l metic mean of numbers.

n

to fn

. Now,

( i i ) i s t r i v i a l , since an arith-

numbers i s less t h a n or equal t o the maximum of these

( i i ) ( i ) : From elementary algebra [ 65 , p.211 (construction below) we know that there exists a free abelian semigroup X* 3 X with neutral element which has the property t h a t any map T from X into an abelian semigroup S with neutral element can be uniquely extended t o an additive function from X* into S . Furthermore we know that X i s a s e t of generators of X* NOW, take lE f o r S and extend cp,fl, ,f n uniquely

.

...

t o additive 6- valued maps on X*. Then ( i i ) implies t h a t (p i s dominated by the maximum of the f l , ...,f n (since X generates X * ) . Application of the Finite Decomposition Theorem immediately yields the assertion of ( i ) . A

h

0

Problem: As an exercise the reader may prove a corresponding statement for the geometric mean o f positive functions. Construction o f the free abelian semigroup: a subset of

We may consider X

t o be

26

X*

Linear Functionals

= If

E :N

I

+

f(x)

f o r a t most f i n i t e l y many

0

x E X with t h e function zero.

X*

: X

-,S

: X*

+

‘I

which maps

6,

be a map i n t o an a b e l i a n semigroup S

1 and a l l

x+x

to

S, then o b v i o u s l y

g i v e n by 1 f ( x ) T(x),

=

f E

X€X

i s t h e unique e x t e n s i o n o f

Let

to

by i d e n t i f y i n g

i s an a b e l i a n semigroup under p o i n t w i s e a d d i t i o n . Now, l e t

;(f)

1.5.5

x

x E XI

T

x*

t o an a d d i t i v e f u n c t i o n from

X*

into S

.

Decomposition w i t h r e s p e c t t o P o s i t i v e l y Independent F u n c t i o n a l s F

be an a r b i t r a r y cone. A s e t

p1

,. .. ,p,

:F

-,li

o f sub1 i n e a r

f u n c t i o n a l s i s s a i d t o be p o s i t i v e l y independent i f , whenever n z xi pi 2 o f o r xi 2 0, i = l,...,n, then we must have i=1

x1

=

x2

... = xn

=

i f t h e r e i s some

= 0. T h i s cond t i o n i s q u i t e o f t e n f u l f i l l e d ,

f E F with

Theorem: L e t pl,...,pn ~f u n c t i o n a l s -and l e t pl(f)

5 0,

p2(f) I 0

p :

Proof:

F

= If

I

F +IF

,... , p n ( f ) n

u

pk f ) < 0, k = l,...,n.

be a s e t o f p o s i t i v e l y independent s u b l i n e a r

such t h a t -(*)

especially

k=l

XkPk

be l i n e a r such t h a t 5 0.

Then t h e r e a r e

p ( f ) I 0 whenever

xk

2 0, k = l,...,n

*

I t s u f f i c e s t o p r o v e t h e d e s i r e d i n e q u a l i t y f o r t h e subcone

E F I u(f)

choice o f

xl,...,hn.

+

-

=I

F \ F (*) i s t r i v i a l l y t r u e f o r every So we r e s t r i c t o u r c o n s i d e r a t i o n s t o f . The because on

assumption i n t h e theorem i m p l i e s

27

The Sandwich Theorem

Hence from t h e F i n i t e Decomposition Theorem we n e t

Xk 2 0, k =

1, ...,n,

with

n

and

n

X = I:

k=l

We know t h a t

n

X k 5 1. T h i s can be r e w r i t t e n as

1-

x

f

s i n c e we assumed t h e

0

pl,.

. . ,pn

t o be p o s i t i v e l y

independent. Hence

p s t - 'k

and t h e decomposition ( * )

'

'k

k=l 1-X

i s proved.

0

Now, l e t us a p p l y t h i s theorem t o l i n e a r f u n c t i o n a l s F(X) (i.e.

-

i s a subcone o f sup(f(x)) < XEX

the constant function subset

Y c X

Corollary:

Let -

+

and assume ~ that p(f) A

1> - 0,

Proof:

-

-

lX (beinq supy

sup f ( y ) YEY

for -

5 0

...,A n 2 0 -such t h a t p

I

+ f i , where

f o r a l l f E F ( X ) ) and where we assume t h a t F(X) c o n t a i n s

1 a t every x E X).

.

all

n t X k supx k=l k

We can assume t h a t t h e

F o r a nonempty

the f o l l o w i n q sublinear f u n c t i o n a l :

...,Xn _be_ a_ _f i _n i t e

X1,

: F(X)

lix c o n s i s t i n g of upper bounded f u n c t i o n s

we denote by f

p

covering o f

L

f E F(X) with f

X

(i.e.

I 0.

n U Xk = X ) k=l

Then t h e r e a r e ---

.

X1,.,.,X

n

a r e nonempty ( o t h e r w i s e we drop

28

Linear Functionals

those s e t s which a r e empty). NOW, we observe t h a t t h e f u n c t i o n a l s

, k = l,...,n, a r e p o s i t i v e l y independent s i n c e - lX E F(X). Hence k t h e theorem can be a p p l i e d imnediately t o t h e s i t u a t i o n under c o n s i d e r a t i o n .

supx

1.5.6

The Jordan Decomposition

For t h e n e x t a p p l i c a t i o n (due t o H. Konig [1991),let

be a q a i n a nonempty s e t and denote by B(X) t h e vectorspace o f a l l bounded, r e a l v a l u e d functions. A l i n e a r f u n c t i o n a l p : B(X) - r R i s s a i d t o be bounded i f

where

11 fll

(1 f (1

denotes t h e usual -sup-norm

X

.

= sup ( f ( x ) l XEX

We consider t h e p o i n t w i s e o r d e r r e l a t i o n ( f < g iff f ( x ) I g ( x ) f o r a l l x E X ) on B(X). Note, t h a t a l i n e a r f u n c t i o n a l v i s monotone w i t h respect t o

i f and o n l y i f

5

u ( f ) 2 0 whenever

f 2 0.

Jordan Decomposition Theorem For every -bounded l i n e a r f u n c t i o n a l -

on B(X) 1-1 = 1-1, Up

0" B(X) w i t h

functionals

there are --

-

and 11 1-111

monotone l i n e a r =

I( l-Itl(

t

11 p - l ( .

Proof: Define

p,(f)

for a l l

=

11 1-11)

f E B(X).

s u p I f ( x ) \ x E X I , f E B(X),

p,,

p-

u ( f ) I max(p+(f), p - ( f ) ) = Then 1.4.4 y i e l d s l i n e a r p =

A;,+

(1-k)ii

-

We o b t a i n

-

p,(-f)

f 2 O .

I

-

p,(-f)

I

-

(1

1-111

( 1 f 11

f o r a l l f E B(X). B(X) and 0 I A I 1 w i t h

iJ+(-f) = c,(f)

i - ( f ) I p,(f).

p_(f) = p t ( - f )

a r e s u b l i n e a r and we have

it,;-on G+ I p+, is p-

and

and p u t

Hence

;,(f)

.

I p,(f)

2 0

and and

similarly

( - i - ) ( f )2 0

whenever

29

The Sandwich Theorem

lG+(f)l, conclude 11

Since we

Note, t h a t t h e same p r o o f works f o r bounded l i n e a r f u n c t i o n a l s on subspaces of

E

B(X). On t h e o t h e r hand g i v e n a subspace

11

functional

on

c B(X)

and a l i n e a r

t h e n we may as w e l l a p p l y a t f i r s t 1.3.1 i n o r d e r t o

E

obtain a l i n e a r extension

: B(X) + R

p

of

with

11; I( = 11

.

pll

Then we can use d i r e c t l y t h e theorem above. I t i n c l u d e s t h e Jordan decomposition f o r measures. Consider a bounded s i g n e d measure T on X w i t h r e s p e c t t o some U- a l g e b r a ,I. Define E t o be t h e l i n e a r span of t h e c h a r a c t e r s t i c f u n c t i o n s of a l l a linear functional

p

E by p ( e )

on

=

X

A p p l i c a t i o n o f o u r theorem y i e l d s monotone Let

A E 1 with

IT(A)I

< m

and d e f i n e

edT

X

for all with

p + , 11-

r+(A) =

( l A denotes t h e c h a r a c t e r i s t i c f u n c t i o n o f p o s i t i v e bounded measures on

X

Z- measurable subsets of

T-(A)

p+(lA),

, I

E.

-

p = p+

A ) . Then t h e

w i t h respect t o

e E

and d e f i n e

T+,T-

=

11-

.

p-(lA)

are

and we have

T = T + - T - .

1.5.7

The Lemma o f Farkas

Next we proceed t o Farkas' Lemma which has become a useful t o o l i n mathem a t i c a l economy [104].0ur proof i s b a s i c a l l y t h e same as t h e one g i v e n b y Heinz Koniq i n h i s b e a u t i f u l l e c t u r e notes on Mathematische W i r t s c h a f t s t h e o r i e [ 201, p.321. Farkas' L e m w: space

E

tionals.

consider

a realvalued

l i n e a r functional

of r e a l v a l u e d

and a f i n i t e s e t

A = I~~,...,p~l

v

a vector

l i n e a r func-

If (*)

--then there are (**)

v(f) 5 0 hl

2 0,.

v =

n , I

i=1

whenever

. .,xn xi

pi

2 0

pi(f)

5 0

such t h a t

forall

i = lY...¶n

30

Linear Functionals

Remark: Of course, ( * ) i s also necessary for ( w ) .

.

Proof: We proceed via induction over the number of elements in A Recall t h a t v1 5 v 2 on a vectorspace E implies v1 = v 2 for a l l linear functionals v l y v 2

.

If A has only one element then this i s either equal t o zero or A i s a s e t of positively independent sublinear functionals. In the f i r s t case ( * ) implies v = 0 and ( * * ) t r i v i a l l y holds. I n the second case the assertion (**) i s a consequence of Theorem 1.5.5. NOW, we assume that the theorem i s true for arbitrary E and for a l l havinq 5 m elements.

A

We consider A = ~ U ~ , . . . , U ~If , ~A ~i s+ a~ s~e t. of positively independent functionals then ( * * ) i s again an immediate consequence of Theorem 1.5.5. So we are l e f t with the case t h a t A i s n o t positively independent. Then i t contains some element, say pm+l, such t h a t

-

=

m

1 akpk

k=l

for suitable a k 2 0, k = l y . . . , m

.

NOW, we r e s t r i c t a l l our linear functionals t o the subspace =

Ie E E\y,,+l(e) = 01. pl(e)

Then, for e E

I 0, ...,LI( e ) 5 0

m

By our induction hypothesis we get

u For

= 0

= (v

(i.e.

E =

m

1

k=l

Xkpk)

i) the

x1

E,

by ( * )

ensures

t 0,

...,x,

,

.

o

v(e)

I

1 0

such t h a t

i s equal t o zero on

-

E

.

proof i s finished. Otherwise the rest i s

linear algebra. Because then we can find f E E \ E

and p u t

31

The Sandwich Theorem

n i ng e)f E

i

for a l l

e E E.

v =

For

611 ( f ) 2 0

formula

um+l

t h e a s s e r t i o n i s proved. Otherwise we r e p l a c e i n t h e l a s t

m

- I

by

v =

As i n [201, p.331

ak

k=l

m 1

i=1

and g e t t h e d e s i r e d r e s u l t :

pk

-

(xi

611 ( f ) a i )

ui

I

0

one can use F a r k a s ' Lemma t o p r o v e t h e i m p o r t a n t f a c t

t h a t finite-dimensional

p o l y h e d r i c cones i n H a u s d o r f f l o c a l l y convex

v e c t o r spaces a r e c l o s e d . For t h e sake o f completeness we i n c l u d e t h i s r e s u l t here. C o r o l l a r y 1:

L e t xl,. -

. . ,x,

be elements _ o f -a

Hausdorff l o c a l l yconvex

t o p o l o g i c a l ~v e c t o r space. -Then t h e p o l y h e d r i c cone

K =

n {

1

i=1

XixiIX.20l 1

i s closed. -~ Proof:

Let

E'

, the

E

E t o be E = Cx E E f y ( x ) 5 01. Y Y i s o b v i o u s l y closed. Now, c o n s i d e r t h e elements of E as l i n e a r

and d e f i n e , f o r

E Y f u n c t i o n a l s on

Then

Xl,*..,X

be t h e vectorspace o f c o n t i n u o u s l i n e a r f u n c t i o n a l s on

n'

y E E'

E'

set

and a p p l y F a r k a s ' Lemma t o t h e f u n c t i o n a l s q i v e n by

Linear Functionals

32

This yields K =

n { E Iy

Hence K must be closed

Y

.

with y ( x i )

E E'

5 0,

i

=

l,...,n}.

0

Another useful corollary gives information a b o u t the support of certain linear functionals: Corol 1ary 2: F(X) be a cone of functions f : X + R . Let p : F ( X ) -,6 be 1 inear and consider x1 ,.. . ,xm E X -such t h a t p ( f ) = p ( g ) whenever --

Let -

f,g E F(X)

_ .

and -

Then there are real numbers

f ( x k ) = g ( X k ) ' k = I , . . .,m.

al, . . .,a m - with

p(f) I

Equality --holds for a l l

m

t a k f ( x k ) -for all f E F(X). k=l

f E F(X) with p ( f ) > -

co

.

Proof:

.

We only need t o consider the subcone p(X) = Cf E F ( X ) ( u ( f ) > - -1 Consider the vectorspace E = F ( X ) - ? ( X ) . We may extend p t o a linear functional on E by defininq ;(f-q) = p ( f ) v(q) for a l l f,g E F ( X ) . This definition makes sense since i f f-g = f-i, f , g , f , i E f ( X ) , we obtain p ( f ) t ~ ( 4 )= ~ ( 7 t) p ( q ) . P u t 6,(f) = f ( x ) , x E X , f E E. Then the assumptions o f Farkas' Lemma are s l t i s f i e d for i 6 ,..., * bX and i Indeed, l e t x1 m i 6 (f-g) I 0,i = 1,...,my for some f,q E E. Then f ( x i ) = q ( x i ) , i y...,m. xi Hence p ( f ) = p(g), t h a t i s ;(f-q) 5 0. By Farkas' Lemma n p = 1 (ai , for some a.,B. 2 0, i = l,...,m. This proves i =1 bXi - Bi b x1. ) 1 1

-

.

-

the assertion of Corollary 2.

0

33

Linear Functionals

The S e p a r a t i o n Theorem

1.5.8

Of course, t h e Sandwich Theorem i n c l u d e s t h e usual v e r s i o n s o f t h e Hahn

Banach Theorem. F o r completeness we demonstrate t h i s a t t h e example o f t h e S epara t ion Theorem.

Hahn Banach S e p a r a t i o n Theorem v e c t o r space. L e t E be a r e a l l o c a l l y convex Hausdorff t o p o l o g i c a l ___Consider _ convex_subsets and B of E with A n B = 0 such _ A ~0 -

that A

i s open. Then t h e r e i s a continuous l i n e a r f u n c t i o n a l --

w i t h u(A) n u(B) -

.

0

=

p

on E -

Proof: a.

F i x some A,,

Then of

E A

and d e f i n e

Bo

p ( e ) = i n f I x > Ole E for a l l

= l a - a o l a E A l , Bo = l b - a o l b E B I .

A,

Bo a r e d i s j o i n t convex s e t s and We assume

0.

A.

e E E.

Because Ao,Bo

0 , otherwise

P

x

0

p(e) <

for all

m

a r e convex we o b t a i n t h a t

s u p e r l i n e a r . Furthermore, s i n c e tains

0 ,we have By 1.2.5

e @ p A,

A.

whenever

u(a) < 1 f o r a l l

Hence

n p(B)

0

=

.

a E A.

e E E

since

A.

i s s u b l i n e a r and

=

0

and A.

e E

x

LI : E

and

Bo

+IR

0

=

- m)

i s absorbinq.

p

n Bo

we o b t a i n a l i n e a r

T h i s means p(A)

the assertion i s t r i v i a l . Define

A o I , q ( e ) = s u p l x 2 01 e E AB l ( w i t h sup

We have

q I p.

i s an open neighbourhood

q

is

i s convex and con-

. Thus

0 I I.I I x

and with

q I p Ip .

p(b) 2 1 f o r a l l

b E Bo.

0

Since we have spent so much e f f o r t on D r e s e n t i n g d i f f e r e n t v e r s i o n s o f t h e Hahn Banach Theorem t h i s book would c e r t a i n l y be i n c o m p l e t e i f t h e usual complex v e r s i o n cannot be found. So we do n o t r e s i s t t h e t e m p t a t i o n t o i n c l u d e t h i s theorem. O f course t h e p r o o f goes a l o n g t h e same l i n e s as t h e proof i n

-

a t least

Let

E

map

p : E +IR+

-

t e n thousand o t h e r books.

8 . A seminorm p p(xe) = I x [ p ( e ) f o r a l l

be a vectorspace o v e r with

on

E

i s a subadditive

e E

E, x

E

8

.

34

A map v(a el

Linear Functionals

-,C i s called complex linear i f e2) = a .(el) + B v ( e 2 ) for a l l el,e2 E E and a , B E C

v : E

t

~~

The complex version of the Hahn Banach Theorem: E a complex vectorspace consider a seminorm p Furthermore, -assume t h a t G i s-a subspace of E -and t h a t i s complex linear with [; I 5 p o n G ( i . e . 1; ( g ) l 5 p ( g ) ---Then there i s a complex linear I, : E + 6 with v = G 0" -.-_---

on E. -

Let -

: G

L

-

IvI 2 p

0" E.

+

C

for a l l g E G). -G and

Proof: Let course then a clude Define

and regard E and G a s vectorspaces over IR , then of the elements e 0 and i e are linearly independent. 1.3.1 yields linear u : E + I R with u I p on E and 5 I u on G. We conu = ; on G since G i s a vector space. v(e) = p(e) - i p ( i e ) for a l l e E E Then v i s complex linear

= Re

.

.

and extends Then since

1.5.9

For v ( e )

0

, hence p(i a e ) = 0 , Iv(e)l = p ( a e ) I p ( a e ) = p ( e ) .

= v ( a e ) = u ( a e ) - i p(i a e ) =

v(ae)

i s real

Thus

Affine Interposition and the Mazur-Orlicz Theorem

Let X c F be a convex subset o f some cone F i s called convex, i f

.

A function

f : X

4

fi

35

The Sandwich Theorem

f ( x x t (1-X)y) I A f ( x )

t

( 1 - x ) f ( y ) f o r a l l x,y E X, 0 5 A 5 1.

f

I f always t h e converse i n e q u a l i t y h o l d s t h e n

i s s a i d t o be concave.

A f u n c t i o n which i s convex as w e l l as concave i s s a i d t o be a f f i n e .

Problem: Given two f u n c t i o n s f,g h with f s h s g ?

: X

-,

fi , can we f i n d an a f f i n e f u n c t i o n

T h i s problem i s e a s i l y solved: The0 rem:

-There _ _i-s

an a f f i n e f u n c t i o n

h : X

n

whenever

n,m E N; n t i=o

such t h a t --

-,

6 with

f I h 5 g

i f and o n l y i f

m

ho,. =

. .,in, xo,. .. , X k

m t k=o

xk

= 1

and

-> 0

n I: xixi i=o

=

xo,

. . . ,xnYXoy.. . ,$, E

n t xkXn k=o

.

Proof: The n e c e s s i t y o f t h e c o n d i t i o n ( * ) i s obvious. Observe t h a t

x

inff

-,

m

t i k g(ik) k=o

1m

E N, i k > o w i t h t with

d e f i n e s t h e g r e a t e s t convex f u n c t i o n Furthermore x

+

sup{

n t i=o

x.1

f ( x . ) I n E N,Xi 1

: X

t Xkxk= -,

: X

1, ik E

-,

x

XI V

2 0 with

7

=

r? w i t h g

with d e f i n e s t h e s m a l l e s t concave f u n c t i o n

Xk

I g.

t xi = 1, xi E X 1 xixi

=

XI

fi w i t h f 5 'i

provided

X

36

Linear Functionals

that

f(x) < +

, since

(*)

(*)

x E X. B u t t h i s i s guaranteed i n o u r case by

for all

w

‘i I

means

4

on

X

. Hence t h e problem i s

reduced t o

f i n d i n g an a f f i n e f u n c t i o n between a concave and a convex f u n c t i o n . Consider t h e cone

Ix

X 8 R+ = {(x,h)

E X,

h > 01

u I01 where t h e cone

operations a r e defined by

X

a(x,x) = ( ~ ~ h . 6f o) r

6 > 0,

o+o

*

= 0, O ( X , A) = 0,o

0 = 0

Y

0

+

(X,X)

+0

= (X,X)

=

(X,X).

X B R+ i f x i s i d e n t i f i e d w i t h p : X Q R, -, 6 and a s u p e r l i n e a r

may be considered as a subset o f

( x , l ) . NOW, d e f i n e a s u b l i n e a r q : X8R++6

= A 4(XL

P(X,X)

Then

-f

I

by

4

yields

f u n c t i o n a l i n between

p(0) = 0 and q(x,h)

= h ?(x

Y

q(0) = 0

.

q I p and t h e Sandwich Theorem g ves us a l i n e a r

.

Restriction o f that linear functional

to

X

gives the desired

h

.

o

Corollary: Let f and g be f u n c t i o n s X + 6 -such t h a t f i s concave ~ and g convex. Then t h e r e i s an a f f i n e f u n c t i o n h w i t h f I h I g i f and - -----

if -

only

-

f I g.

Remark :

If t h e f u n c t i o n f i s n o t d e f i n e d on a l l o f X t h e n we extend i t t o X b y p u t t i n g f ( x ) = - w whenever X i s o u t s i d e t h e domain o f f The theorem t h e n goes o v e r unchanged. As a consequence we o b t a i n : Every convex f u n c t i o n i s t h e p o i n t w i s e supremum o f t h e a f f i n e f u n c t i o n s i t dominates

.

O f course, one recovers t h e Sandwich Theorem from t h e preceding i n t e r -

p o s i t i o n theorem by r e q u i r i n g ready a cone. I f X

f ( 0 ) = g(0) = 0

i n case t h a t

X

i s al-

i s i n a d d i t i o n a v e c t o r space then, i n t h i s s i t u a t i o n ,

t h e i n t e r p o s i t i o n theorem i s due t o Mazur-Orlicz Q210 (see a l s o 208 1). Even i n s p e c i a l cases t h i s r e s u l t has proved t o be v e r y useful .

37

The Sandwich Theorem

One m i g h t ask under w h a t k i n d o f c o n d i t i o n a bounded a f f i n e f u n c t i o n between f

and

g

can be found. We l e a v e as an e x e r c i s e t h a t t h i s can be done i f

and o n l y i f t h e r e a r e constants

whenever

n,m E N

and

such t h a t t h e r e a r e

I n t h i s case we have

1.6

THE TIESZ

- K'ONIG

-

m

< C1 5 C2 < +

m

such t h a t

h o y . . . y ~ n y?lo'..., A, 2 0, xlY...,xny

xoy

ko E

X

C1 I h 5 C2

-

X ~ ~ . . . m~ E X

X

with

.

THEOREM

The aim of t h i s c h a p t e r i s t h e g e n e r a l i z a t i o n o f t h e F i n i t e Decomposition Theorem t o a measure t h e o r e t i c s i t u a t i o n . The r e s u l t which we o b t a i n h e r e i s a r a t h e r powerful g e n e r a l i z a t i o n o f t h e c l a s s i c a l Riesz R e p r e s e n t a t i o n Theorem. We would l i k e t o c a l l t h i s g e n e r a l i z a t i o n t h e Riesz-Konig Theorem s i n c e t h e u n d e r l y i n g i d e a appeared i n a paper o f Heinz Konig [139]. For those readers who a r e n o t f a m i l i a r w i t h t h e n o t i o n s and techniques o f elementary measure t h e o r y we have gathered t h e necessary m a t e r i a l i n t h e

Linear Functionals

38

appendix. F i r s t , some n o t a t i o n . L e t

X

be a s e t and l e t

lydenotes t h e c h a r a c t e r i s t i c f u n c t i o n o f As b e f o r e we denote by

supy

Y

Y

, i.e.

be a subset o f X. Then 1 if xEY ly(X) 0 otherwise

={

t h e f u n c t i o n a l which assigns t h e v a l u e

.

I f Y .t: 0 t h e n supy i s a subs u p y ( f ) = sup f ( y ) t o e v e r y f : X -, li YEY l i n e a r f u n c t i o n a l on t h e cone o f upperbounded f u n c t i o n s on X. By USC(n)

we denote t h e cone o f upper semicontinuous f u n c t i o n s on a t o p o l o g i c a l

.

space

n

1.6.1

Riesz-Konig Theorem:

Consider a cone f f

F and and assume t h a t by - a compact H a u s d o r f f space R ----,T a f u n c t i o n F + USC(n) --___i s g i v e n such t h a t , -for all w E R , -, ? ( w ) i s sublinear on F , Moreover l e t LI : F -,6 _--be l i n e a r such t h a t -

Then t h e r e i s a r e g u l a r B o r e l p r o b a b i l i t y measure

u(f) I

Proof:

R

d.r

for a l l f E F --

T

on n with -

.

The p r o o f i s a s u i t a b l e a d a p t a t i o n o f t h e p r o o f f o r t h e F i n i t e

Decomposition Theorem t o t h e p r e s e n t s i t u a t i o n . So, d e f i n e a s u b l i n e a r f u n c t i o n a l

on

USC(n)

for all for all

p

and a s u p e r l i n e a r f u n c t i o n a l

6

as f o l l o w s :

h E USC( a ) . Here w

E R

. Of

-f

Ih

i s meant p o i n t w i s e , i . e .

course, t h e cone

t h i s pointwise order r e l a t i o n

s

.

a l i n e a r monotone ( w i t h r e s p e c t t o 6 I v I p. Then

USC(n)

f ( w ) 5 h(w)

i s an o r d e r e d cone under

From t h e Sandwich Theorem we o b t a i n I) f u n c t i o n a l v on USC(n) such t h a t

t h e Riesz Representation Theorem y i e l d s a r e g u l a r B o r e l

39

The Sandwich Theorem

measure

T

on

R

with

v(h) = v

Since

and

'I

E C(n) c USC(n).

I inf{v(h)

Ih

continuous w i t h

g I hl =

n

g dr

g E USC(n). A p p l i c a t i o n t o 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 s

-1,

Hence

n

i s monotone we have V(g)

for a l l

j h dT f o r a l l h

1,

yields:

T

(a)

v

= 1. Furthermore t h e monotony o f

i s p o s i t i v e . Therefore

T

obviously implies that

i s a p r o b a b i l i t y measure. The f o l l o w i n g

inequal it y completes o u r p r o o f : u(f) I E ( f ) I

"(7)

I

R

7

dr

,

f E F. 0

I t s h o u l d be observed t h a t t h e assumptions o f t h i s theorem a r e f u l f i l l e d

i n t h e s p e c i a l case when

n i s a s e t o f s u b l i n e a r f u n c t i o n a l s on

F

which i s compact under some t o p o l o g y such t h a t a l l t h e T , f E F , a r e upper semicontinuous. Here, o f course, by ^f we mean t h e e v a l u a t i o n : f(p) = p(f) set

R

for all

o f sub1 i n e a r

p€n

. Specialization

o f this situation t o a f i n i t e

f u n c t i o n a l s (equipped w i t h t h e d i s c r e t e t o p o l o g y )

recovers t h e F i n i t e Decomposition Theorem 1.4.3.: L e t pl, -

...,Pn

Proof:

Take

xi

be s u b l i n e a r -and l e t -

R ={pl,...,pnl

= ~ ( { p ~ } )i=l,...,n. ,

p

be l i n e a r on

and a p p l y 1.6.1. 0

F -such t h a t

Then p u t

Linear Functionals

40

1.6.2

Corollary:

Let -

X be topological Hausdorff space _.and l e t F be a subcone of USC(X). Assume n ___--t o be a compact subset of X -such t h a t :

supn(f)

=

supX(f) f o r a l l

f E F.

Then f o r every l i n e a r u on F with ---p(f)

supX(f) for a l l

I

t h e r e i s a Bore1 p r o b a b i l i t y measure ---p(f) 5

Proof:

Take

=

f

J f n

la

dT

T

f E F ,

on n with -

for all f --

E F.

and apply 1.6.1.

0

Later on, we shall use Corollary 1.6.2 i n connection w i t h boundaries ( t h e n i n t h e c o r o l l a r y will then be c a l l e d a sup-boundary o r even maxboundary s i n c e s u p n ( f ) = max f ( x ) f o r a l l f E USC(n)). XER

A very popular example f o r such a boundary i s given i n t h e following situation: Consider t h e Laplace equation

where f i s a continuous real-valued function on t h e closure G of some open, connected and bounded subset G of I R ~ , such t h a t t h e second p a r t i a l d e r i v a t i v e s o f f e x i s t and a r e continuous i n G. Such functions a r e c a l l e d harmonic. Let a G = E\G denote the topological boundary of G and l e t H ( E ) be the s e t of a l l harmonic functions on E Then

.

H(G) is not only a cone b u t a vector space. An important information about t h e harmonic functions i s contained in t h e following elementary result (of which we include a proof f o r the reason of completeness):

1.6.3

Maximum Princiole:

sup- ( f ) = max f ( t ) G

tEa G

for all

f E H(G).

The Sandwich Theorem

Proof:

If

f

i s c o n s t a n t , t h e n t h e a s s e r t i o n i s t r i v i a l . So, i t remains

t o deal w i t h t h e case when

to = (xl

0

0

, x2,.

.., x l

f

i s n o t c o n s t a n t . Assume t h a t t h e r e i s some

) E G such t h a t

f ( t o )> max f ( t )

t€aG

Then t a k e some (xl-xl)

0 2

I

p

41

> 0

p

.

such t h a t we have

for all

t = (x~,x~,...,x~) E

E .

And choose

E

> 0

so,

that

g(t) 5 f ( t )

we have

for all

t E

E.

But

proves t h a t t h e r e must be some

where

tl E G

g

a t t a i n s i t s maximum.

Elementary d i f f e r e n t i a l c a l c u l u s shows t h a t a necessary c o n d i t i o n f o r a maximum i s t h e f o l l o w i n g :

$2 Since

f

1

t=tl

5

0

for

i s harmonic we have a p p a r e n t l y

contradiction.

i = l,...,n

.

A g = 2~

> 0

NOW, i n t h i s c o n t e x t t h e meaning o f C o r o l l a r y 1.6.2 functions

f €

More p r e c i s e l y :

in

G

, hence

0

H(G)

i s t h a t t h e harmonic

a r e c o m p l e t e l y determined b y t h e i r values on

aG.

a

Linear Functionals

42

1.6.4

Proposition:

For every --

t

there - - i s a probability measure

G

E

f(t)

=

I

aG

f d~~

for all

0” a G -such t h a t

T~

f E H(G)

.

Proof: Use Coro for a l l f E H(G In case t h a t

G i s the open unit disc in

G = { ( r coscp, r sincp) 1 0

the measure

T~

(t d

=

sr

R2

< 1, 0 scp< 2 8 ) (polar coordinates)

( r , o ) ) i s the well-known ~Poisson measure, =

T t (cp)

P(r,o -cp)dm(cp)

where dm i s the Lebesgue measure on the unit c i r c l e and Poisson kernel ~-

P ( r , o ) i s the

n

P(r,o)

1 . 7 A STRASSEN

-

=

1 - rL 1 - 2 r c o s o t rz

TYPE DISINTEGRATION THEOREM

As in the case of the Finite Decomposition Theorem we can generalize the Sum Theorem to a measure theoretic situation. The resulting generalization i s very similar t o the Strassen disintegration theorem[304] which has proved t o be very useful in a couple o f applications. Although we can generalize some of the subsequent arguments t o the case o f 1 inear functionals attaining values in an order-complete vector l a t t i c e we shall r e s t r i c t our considerations t o li- valued 1 inear functionals. We do this in order t o make the results more transparent.

The Sandwich Theorem

We c o n s i d e r a measure space t o be

measurable f u n c t i o n s on

we denote

t h e convex cone o f

t

X

f dm <

may be equal t o in

, where -a

.

ments i n

$(m)

everywhere. I n

ft = sup(f,O). Hence, f o r

k and t h e f u n c t i o n

i s supposed

E- v a l u e d

such t h a t t h e i r p o s i t i v e p a r t ( b u t n o t necces-

X

s a r i l y the negative p a r t ) i s i n t e g r a b l e w i t h respect t o 1.e.

m

where t h e measure

(X,.I,m),

1 (m)

f i n i t e . By

U-

43

-m

m.

On t h e o t h e r hand,

1

f E LE(m)

i s an element o f

the integral

$(m).

J(f-ft)dm X

J f dm e x i s t s X

O f course, two e l e -

a r e considered t o be equal i f t h e y c o i n c i d e

$(m)

m- a l m o s t

we c o n s i d e r t h e p o i n t w i s e o r d e r .

We a r e i n t e r e s t e d i n o p e r a t o r s

P :

F -,k 1 ( m ) , where F i s some convex

cone. As u s u a l l y , such an o p e r a t o r i s s a i d t o be s u b l i n e a r i f i t i s p o s i t i v e l y homogeneous ( i . e .

P(Af) = x P ( f )

for

x

t 0

and

f E F) and

subadditive. I f i t i s superadditive instead o f subadditive then i t i s c a l l e d s u p e r l i n e a r . A l i n e a r o p e r a t o r i s one which i s b o t h s u b l i n e a r and superlinear.

NOW, i n s t e a d o f i n t r o d u c i n g i n

F a s i n g l e o r d e r r e l a t i o n we c o n s i d e r a

l a r g e c o l l e c t i o n o f order structures elements o f

X.

1:

~ I E xX I

b e i n g indexed by t h e

O f course, as b e f o r e , we assume t h a t a l l t h e < x are 1 P : F -,LE(m) i s c a l l e d

c o m p a t i b l e w i t h t h e cone o p e r a t i o n s . An o p e r a t o r

monotone w i t h r e s p e c t t o t h i s l o c a l i z e d o r d e r s t r u c t u r e ( o r fl,f2 E F ,we have

f o r short) if, f o r a l l

P(fl)

X- monotone

5

P(f2)

m- a l m o s t everywhere on { x E X l f l

< x f21.

44

Linear Functionals

Abstract Disintearation Theorem:

1.7.1

F -,6 ---be linear and l e t 1inear operator with

Let -

11 :

(*)

Then there --__ (i.e.

T(f)

u(f)

I

X

P : F

1 -, k(m) -be an

--

P ( f ) dm for a l l

f E F.

i s an X- monotone linear operator T : F 5

P ( f ) for a l l

X- monotone sub-

+

1 LE(m) with T -

IP

f E F) -such that

We should remark that the special case of t h i s theorem where F i s a vector space and where no order relation i s considered i s due to M. Valadier [3141 (see also M. Neumann [2431,M. Wolff[336land H . Konig [ 1991). The proofs of these authors do n o t carry over unchanged t o our situation since by considering cones instead of vector spaces one looses in general the required measure theoretic convergence property i f a 1 inear functional i s decomposed with respect t o an integral over sublinear functionals. To make even for that we have t o combine the usual proof with a maximality argument. The advantage of the somewhat a r t i f i c i a l order structure we introduced will become clear l a t e r on. Before going through a l l the details of the proof we would l i k e t o point out the connection between t h i s result and the Finite Sum Theorem.

1.7.2 Let -

Corollary: F -be_ _a _cone - - and l e t


N

be-a sequence -of order relations on -

n , (F, < n) i s-a preordered cone. Furthermore we sequence pn o f
F -such t h a t , -for each

consider

a

on F ---such t h a t , for a l l f -

E F,

we have --

45

The Sandwich Theorem

W

Z

n=l

Then, __

max(pn(f),O) <

for every linear 1.1 : F ---

tone linear functionals --

+

pn

pn I

p(f) I

Z

n€N

0

r? with

. x pn there are

I

1.1

nEN

-$,-mono-

such t h a t

p n ( f ) for all

f E F

Proof: equal t o N , l e t m be the counting measure on N and l e t the < n define the localized order structure. Let the operator P : F + be given by P ( f ) ( n ) = p n ( f ) . Apply the Disintegration Theorem and obtain Put

X

the

1.1~

G(m)

via

un(f) = T(f)(n).

0

NOW,l e t us return to the proof of our theorem: Proof of 1 . 7 . 1 : Let o be the convex cone consisting of a l l simple x-measurable functions cp : X + F. Here, a function cp i s called simple i f c p ( X ) i s a finite s e t and i f , for every f E F, the s e t {x E Xlcp(x) = f l belongs t o 1 I n o we consider the preorder given by:

.

Then

defines a monotone sublinear functional on

o

,

Linear Functionals

46

And

I

6(v) =

v(f)

if u, i s c o n s t a n t ( w i t h v a l u e f ) on

-

otherwise

00

gives us a s u p e r l i n e a r f u n c t i o n a l on

a,

, with

Sandwich Theorem t h e r e i s a monotone l i n e a r

6 I p.

with

u

u s i n g Z o r n ' s lemma we can f u r t h e r assume t h a t linear functionals v I

,

Ip

if G : o

i.e.

s p then necessarily

V

According t o t h e

. And

6 Iu I p

by

i s maximal among t h e

v

-6

X

i s monotone l i n e a r w i t h

( t h i s i s e x a c t l y t h e argument we used

= v

i n t h e f i r s t p a r t o f t h e p r o o f o f Theorem 1.1.2). Now, f o r

where

A E 1 and f E F

, we

define

lA i s the characteristic function o f

1Af(x) = I f

A

,

i.e.

i f x E A, 0 o t h e r w i s e } . We c l a i m t h a t f o r

f,g E F and

A E t the following are true: (2)

d(A,.)

i s l i n e a r on

(3)

d(.

(4)

v ( f ) -< d(X,f)

(5)

d(A,f) I

(6)

when

(7)

i f An

,f )

i s an a d d i t i v e s e t f u n c t i o n on

A

f <x

x

P(f)dm g

f o r m-almost a l l

x E A

then

d(A,f)

i s a sequence o f p a i r w i s e d i s j o i n t s e t s i n z

u

nEN The a s s e r t i o n s ( 2 )

F

-

A,,f)

= lim inf

M+w

M

1 d(An,f)

n=l

5 d(A,g)

then

.

( 7 ) prove t h e theorem i n t h e f o l l o w i n g way:

( 3 ) and ( 7 ) show c e a r l y t h a t

-

d( ,f)

i s a signed measure on

X . A s s e r t on

47

The Sandwich Theorem

m.

( 6 ) i m p l i e s t h a t t h i s measure i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o f <x

T h i s i s so, because from m(A) = 0 we o b v i o u s l y g e t 0

i x f for

x E A

almost a l l

, and

hence

d(A,f)

= d(A,O)

0

and

= 0.

NOW, we a p p l y t h e Radon-Nikodym theorem t o f i n d a measurable f u n c t i o n

T(x)

such t h a t (8)

d(A,f) =

T(f)dm.

A

Then, because o f ( 5 ) , t h e p o s i t i v e p a r t o f

m. ( J u s t c o n s i d e r i n ( 5 ) A = I x € X I T ( f ) ( x ) 2 0 1 . ) 1 LE(m). A s s e r t i o n ( 2 ) g i v e s t h a t f + T ( f ) i s

w i t h respect t o

So

T(f)

T(f) (i.e. T(f)+) i s integrable

must belong t o

l i n e a r , and f r o m ( 4 ) and ( 5 ) we o b t a i n ( * * )

and

T ( f ) I P ( f ) . To prove

A = { x E X I T ( f ) ( x ) > P ( f ) ( x ) ) . Then ( 5 ) shows t h a t m(A) = 0. F i n a l l y , we show t h a t f + T ( f ) i s i n f a c t X- monotone. Consider f , g € F , p u t B = I x € X I f <x g 1 and assume

the l a t t e r i n e q u a l i t y consider

T(f)(x) > T(g)(x)

x € A c B with

for

g e n e r a l i t y , we may f u r t h e r assume t h a t replace

A

m(A) > 0.

1 T(f)dm

Then, w i t h o u t l o s s o f >

A

( o t h e r w i s e we

--m

by a s u i t a b l e s u b s e t ) . And we have i n c o n t r a d i c t i o n t o ( 6 ) = d(A,g)

.

( 2 ) and ( 3 ) a r e easy consequences o f t h e l i n e a r i t y o f

v

d(A,f)

=

A

T(f)dm >

1 T(g)dm

A

So we a r e l e f t w i t h :

Proof o f ( 2 )

-

(7):

f o l l o w immediately from Then

.

-

6 I v I p.

Let

lA f < lA g and b y monotony o f

So we have a l s o proved ( 6 ) .

f <x

v

g

, and ( 4 )

and ( 5 )

f o r m- a l m o s t a l l

we g e t :

x€A.

4a

Linear Functionals

The proof o f ( 7 ) i s a l i t t l e b i t more complicated and depends e s s e n t i a l l y on the maximality o f a r b i t r a r y cp E

(9)

where

@

v

. So,

P(Q) = v ( l y

Y = X

U An

c~

l e t the

:

n€N

M

cp) t l i m i n f

M

. Then

be as i n ( 7 ) and d e f i n e f o r

An

Z

n=l

+ m

v(1

An

. cp),

i s s u p e r l i n e a r (because o f the i n f i n the

p

p . The u- a d d i t i v i t y o f m i m p l i e s use the f o l l o w i n g obvious i n e q u a l i t y :

l i m i n f ) and

where

I

ZM = X \ ( Y

u

M m U An) = U An n=l n=Mt1

.

Since t h e

p 2

u. To see t h i s we

ZM are decreasing t o

0 we get: l i m sup w(l

M +-

zw

-

cp) I

I

l i m sup p(1 M +zw

l i m sup M+m

This i n s e r t e d i n (10) leads i n f a c t t o

- cp)

I

P(cp(x))(x)dm(x) s ZM

p 2 v

. Now,

o

.

we apply t h e Sandwich

Theorem t o o b t a i n a monotone l i n e a r w i t h p 1 7 I p. Then, because o f p 1 u and t h e f a c t t h a t v was already maximal, t h i s y i e l d s v = Hence p = w I n s e r t i n g t h i s i n ( 9 ) and p u t t i n g cp = 1 u An * we get n a the desired r e s u l t . D

.

.

49

The Sandwich Theorem

1.8 SOME APPLICATIONS 1.8.1 An abstract Flow Theorem In t h i s section we give an abstract F.JW Theorem which l a t e r on wil turn o u t t o be very useful in applications to network theory and to supplydemand models .

All measures in t h i s section are assumed to be k? or R- valued. We consider a measurable space (n,I) (n a s e t and Z a U- algebra in n) and some positive measure T on X = n x n (of course with respect t o the product U- algebra 1 8 Z ) . Since we do n o t want to overload t h i s section with technical d i f f i c u l t i e s we assume, for simplicity, t h a t T is f i n i t e . Furthermore we consider a signed measure !J on (n,x). We recall + t h a t u can be s p l i t u p in the following way !.I = !.I - !.I with positive t imeasures !.I ,!.I, where u i s f i n i t e (Appendix Theorem 1.5.2). We need t o introduce the notion of bimeasure (see[168] or r1941). A bimeasure on X = n x n i s a function v : Ixx -,R being a signed measure separately in each variable, i . e . for fixed A E t the functions B -, v(A,B) and B -, v(B,A) are signed measures on ( ~ , x ) . Sometimes such a bimeasure can be extended t o a n honest measure on the product u- algebra, b u t t h i s i s n o t always possible. For t i g h t measures such an extension does n o t cause any problem ( [ 168, thebrPme 41 or [ 194 , lemma 31). Before we Proceed t o the main r e s u l t of t h i s section we have t o work o u t some technical d e t a i l s . By F we denote the convex cone of measurable, positive and simple functions on n , i . e . every f E F i s of N the form f = I a 1 , where a 2 0 , An E 1 f o r a l l n . An n n= 1

To every f E F we assign a nonnegative function f on the product space X = n x n by defining L

f(wl , w 2 )

=

max(O,f(wl)

-

f(w2))

for all

w1,w2

E R

50

Linear Functional$

I t i s not d i f f i c u l t t o verify:

Observation: (i) The map

(ii)

f --.

Let f , g E -

7 i s sublinear

F

and assume that ---

whenever g(w) >

0

f ( w ) = sup f(E) EER

. Then

n

(ftg)='ft;j.

I n f a c t , only the second assertion i s n o t completely t r i v i a l . (For i t s proof one has t o consider the four different cases which can occur for wIyw2

namely (1) g ( w l ) = 0, g ( w 2 )

(3) g ( q )

*

0

9

4(9) = 0

Y

*

0, ( 2 )

g(w1)

=

g(w2) = 0

* 0 * gbp)).

(4) ! A w l )

We shall need the following technical result: Lemma: -The

following are equivalent:

(i)

u ( A ) I - ~ ( A XCA)

for all

A E 1

(ii)

J f

for a l l

f E F.

(Here

CA

n

dp I

J 7 X

d-r

denotes the complement of

A

in n)

.

Proof: ( i i ) * ( i ) : For an arbitrary A E and from ( i i ) we get: p(A) =

J f

n

dp I

x we take f

-

J f d-r X

= lA

. Then

-

f=

lAx CA

= - r ( A x CA)

( i ) * (ii): An easy computation sho1,'s t h a t every f E F can be written in the following form:

The Sandwich Theorem

51

N f = . x A l n=l An with

Xl

,... ' A N z

and A1

0

,... ,An

E Z such t h a t

NOW, u s i n g s u c c e s s i v e l y p a r t (ii) o f o u r o b s e r v a t i o n we o b t a i n :

-f =

N Z X l n = l n AnXCAn

-

A p p l i c a t i o n o f i n e q u a l i t y ( i ) then y i e l d s :

1f R

du

N =

Z

n=l

u (A,)

An

I

N X An T(A,,x n=l

CA,)

=

I ^f d t .

0

X

Flow Theorem [128]: The f o l l o w i n g are equivalent: (i)

p(A) I .(AX CA)

(ii)There i s

2

for all

A E Z

-_.

bimeasure w

on -

X = nx R

having t h e f o l l o w i n g

properties: (a)

p(A) I w(Aya)

(b)

w(A,B) I T ( A x ( B

(c)

w(A,B) t 0 whenever

P r o o f : (ii) =b ( i ) :

for all A E --

n

CA)) _ f o_r -a l l

*

(ii):

defining, f o r

From ( i i a ) and ( i i b ) we o b t a i n :

We i n t r o d u c e i n

x = f sxg

AyB E 1

AYE E 1 are disjoint.

p(A) 5 w(Ayn) 5 T ( A X (Q (i)

Z

(w1,w2)

f(wl)

E

F

a

R X

R

s g(wl)

n

CA)) = T ( A X CA).

l o c a l i z e d o r d e r s t r u c t u r e on

and

f(w2) 2 g(w2)

.

X =

R X

R

52

Linear Functionals

Now, assume f sX g holds. Then for f(w2) 5 f(wl) we get q b 2 ) 5 g(w1) a n d

If

f(w2) 2 f(wl) then f ( w , , w 2 )

t o !j(w,,u2).

0 and this must be less than or equal

=

So we have i n b o t h cases

-

f(w1,w2) I

?j(w,,w,).

Hence the map f + P ( f ) = f i s X- monotone. Now, consider the linear function : F -, R given by i(f)

=

n

f dp

.

Then, according t o the Lemma, ( i ) i s equivalent t o ;(f) I

IX P ( f ) d T

for a l l f

E

F.

From the abstract Disintegration Theorem 1 . 7 . 1 we then obtain a 1 tone linear map T : F -, LR (T) such t h a t

for all

f

X- mono-

E F.

We define v(A,B)

=

I

T ( l A ) dr

n xB

for A,B

E

z ,

and we claim t h a t this v has the required properties. As an immediate consequence of this definition v i s additive in the f i r s t variable and u- additive in the second.

Consider A,B E 1 with A x E n x B c nx CA.

n

B = 0 . Then 0 sX lA for all

So, from the

X- monotony of

T we obtain

The Sandwich Theomm

53

and ( c ) i s proved. The a s s e r t i o n ( b ) comes d i r e c t l y f r o m ( 2 ) :

=

J l A d.r x=C T ( A xA (B nCA)).

QXB

And ( a ) i s a consequence o f (1): p(A) =

1 du 5

R A

T ( l A ) d T = v(A,n).

RxR

A l l what remains t o prove i s t h e

additivity o f

a-

B E Z and

v a r i a b l e . F o r t h i s we f i x a r b i t r a r y (i.e.

n An

nEN

(3)

=

v

i n the f i r s t

An E X w i t h

An

+

0

0 and A1 2 A2 2 ...? An 2 . . . ) . A l l we have t o show i s :

l i m v(An,B) n-m

=

0.

F o r t h i s we s t a r t by s t a t i n g two h e l p f u l convergence p r o p e r t i e s . F i r s t we consider

B,

E II

such t h a t

0 5 v(An,Bn)

Ann B,

5 T ( A x~ (Bn

Hence

(4) l i m v(An,Bn)

=

= 0

n-co

0

. Then

n CA,))

u s i n g ( b ) and ( c ) we g e t : 5

T ( A ~x Q ) .

.

Furthermore, from u(An) 5 ~(A,,R) we g e t

(5)

l i m v(An,Q) = 0 . n-m

I T(A,x

CA,)

Ir ( A n x

R)

54

Linear Forictionals

Now, we use t h e a d d i t i v i t y of v ( ~ n , ~= ) ,,(A,,B =

v(.)

i n both i t s variables:

n CA,) t ~ ( A , , , B n A,,)

v(An,B n CA,) + u ( A n nCB,B n A n )

= v(An,B

n CA,) + v ( A n K B , B n An)

t

\](An

t

v ( A n n B,U)

Il B,B

n An)

- u(Ann B, C(B n A , ) ) .

The f i r s t , second and fourth term converges t o zero by v i r t u e of ( 4 ) , a n d 0 the t h i r d does t h e same because o f ( 5 ) .

1.8.2 Flows i n networks One of the basic r e s u l t s i n t h e theorey of networks i s t h e Ford-FulkersonTheorem (max - flow - min - cut Theorem) ( s e e 11101 o r 11351 ) . There a r e very many v a r i a n t s and equivalent forms of t h i s theorem which a r e of g r e a t value in several f i e l d s of a p p l i c a t i o n s . One of t h e s e theorems i s the celebrated Gale-Ryser theorem. I t deals w i t h a somewhat more sophisticated s i t u a t i o n than t h e c l a s s i c a l Ford - Fulkerson r e s u l t , h u t i t can be - a t l e a s t f o r f i n i t e networks - obtained from the l a t t e r one. I t was found independently in the context o f networks by D. Gale ( [ 1 3 6 ] a n d in connection with zero-one matrices by H. Ryser ( [ 2 7 9 1 see a l s o t h e survey [ 2801).

I t should he mentioned t h a t usually most of the theorems f o r networks a r e proved by combinatorial arguments, nevertheless most of the underlying problems make sense f o r n o n f i n i t e networks, even f o r nondiscrete networks, where combinatorial arguments c e r t a i n l y f a i l .

I n t h i s section we show t h a t in f a c t the Gale-Ryser Theorem ( f o r a r b i t r a r y networks) i s a special case of our Disintegration Theorem o r r a t h e r t h e a b s t r a c t Flow Theorem. F i r s t we explain the s i t u a t i o n considered in Gale's Flow theorem. There i s given a f i n i t e s e t $2 o f - l e t us say - o i l consumers a n d the consumption of i E $2 i s measured by p i . Negative consumption p i < 0 means production of Illi I .Furthermore, t h e r e a r e pipelines between the consumers.

The Sandwich Theorem

55

measures t h e c a p a c i t y o f t h e p i p e l i n e r u n n i n g f r o m

T~~

Note, t h a t i n general

‘ik t

k

i.

to

(e.g. t h e p i p e l i n e i s g o i n g down a h i l l ) .

T~~

So i t i s b e s t , t o imagine t h a t t h e p i p e l i n e s between

i and

k

p a r a l l e l one-way s t r e e t s . O f course, i f t h e r e i s no p i p e l i n e from then we assume

a r e two k

i

to

t o be zero.

T~~

Now, t h e problem i s , t o g i v e c o n d i t i o n s f o r a p o s i t i v e and p o s s i b l e f l o w which s a t i s f i e s t h e consumption. To be more p r e c i s e , we assume t h a t a f l o w i s r e p r e s e n t e d by numbers

vik

positive i f

i

vik

5

every

T~~

vik

t 0

for a l l

iE n

f l o w i n g from

,

for

i , k E n.

i).The f l o w i s c a l l e d k , and i t i s s a i d t o be p o s s i b l e i f ( f l o w from

k

to

The f l o w s a t i s f i e s

t h e t o t a l amount f l o w i n g t o

t h e consumption i f , f o r

i minus t h e t o t a l amount

i dominates t h e consumption (minus p r o d u c t i o n r e s p . ) a t

Note, t h a t i n t h i s (and a l l t h e f o l l o w i n g ) c o n d i t i o n s t h e numbers do

vii

n o t p l a y any r o l e . U s u a l l y , these numbers a r e d e f i n e d t o be zero.

Now, we i n t e r p r e t a l l q u a n t i t i v e s as measures on

n o r n x n i n order

t o be a b l e t o w r i t e down o u r c o n d i t i o n s i n a more e l e g a n t form. We a b b r e v i a t e f o r

A,B c R :

and kEB

kEB

i:

56

Linear Functionals

Then the condition on t h e flow t o s a t i s f y consumption i s t r i v i a l l y equivalent to: p ( A ) 5 v(A,n)

-

v(n,A)

for all

A c R

.

For A c R we define the import capacity t o be T ( A X C A ) , t h a t i s t h e capacity of a l l pipelines coming from outside A i n t o A . Obviously, a necessary condition for t h e existence of such a flow i s t h a t t h e import capacity i s s u f f i c i e n t in t h e following sense: ,,(A)

I T(A x C A )

for all

A

=R

Note, t h a t t h i s condition a l s o requires t h a t overall production dominates overall consumption, this because of

We a r e now going t o generalize t h i s problem f o r i n f i n i t e consumer s e t s . On the f i r s t view t h i s does not seem t o be a relevant problem. B u t t h i s generalization i s necessary i f one i s - f o r example - i n t e r e s t e d in t h e dynamical behaviour of such a system. Let us f o r example consider t h e above problem f o r i n f i n i t e l y many d i f f e r e n t points T on t h e time s c a l e , w i t h the provision t h a t every consumer i has t h e p o s s i b i l i t y t o s t o r e o i l from tl

t o t2 u p

consumer s e t i s c e r t a i n l y capacity from

(i,tl) to

t o the amount a it n x

T and a it

1, 2

1’ 2

. Then

t h e mathematical

represents the p i p e l i n e

( i , t 2 ) . Certainly a n i n f i n i t e network system!

For t h e i n f i n i t e system i t seems appropriate t o replace t h e q u a n t i t i e s V,T and v by s u i t a b l e measures. So, l e t (nJ) be a measurable space, where R i s c a l l e d the consumer set and where t is a a- algebra on R . We consider a signed consumption measure p on (n,x) measuring t h e consumption and t h e production, respectively. Furthermore, we consider a p o s i t i v e and f i n i t e measure T on X = n x n (with respect t o z 8 x) , and we assume t h a t T ( A X B ) measures the capacity of the pipelines going from B t o A . Therefore T is c a l l e d

57

The Sandwich Theorem

t h e c a p a c i t y measure. A bimeasure

v

X = n

on

x

n

i s called a positive

flow i f -

(1) v(A,B)

for all disjoint

2 0

A,B E 1

.

It i s said t o satisfy

t h e consumption i f (2) The f l o w

v

(3)

-

p ( A ) Iv ( A , C A )

CA,A)

V(

for all

A E

x .

i s c a l l e d possible ( w i t h respect t o the capacity

v(A,B)

IT ( A x

B)

A,B E

for all

O f course, one s h o u l d imagine v(A,B)

T)

if

x .

as t h e f l o w g o i n g from

B

to

A.

F i n a l l y , we say t h a t we have s u f f i c i e n t i m p o r t c a p a c i t y if

(4) As an

p ( A ) IT ( A

x

CA)

for all

A E E

.

immediate a p p l i c a t i o n o f o u r a b s t r a c t Flow Theorem we t h e n g e t t h e

following

g e n e r a l i z a t i o n o f t h e Gale-Ryser Theorem:

The0 rem: t h e consumption There i s a p o s i t i v e and p o s s i b l e f l o w which s a t i s f i e s i f and only if we have s u f f i c i e n t i m p o r t c a p a c i t y . -Proof: -

The n e c e s s i t y of t h e s u f f i c i e n t i m p o r t c o n d i t i o n i s q u i t e t r i v i a l , because from ( 1 ) - ( 3 ) we g e t :

p ( A ) I v(A,

CA)

-

V(

CA,A)

5 v(A,

C A ) IT ( A X C A ) .

For t h e o t h e r i m p l i c a t i o n , we observe t h a t t h e s u f f i c i e n t i m p o r t c o n d i t i o n i s n o t h i n g e l s e t h a n c o n d i t i o n ( i ) i n t h e Flow Theorem. So, l e t us t a k e t h e bimeasure g i v e n b y t h i s theorem. Then from ( c ) we know t h a t i t i s a p o s i t i v e f l o w , and ( b ) t e l l s us t h a t t h i s f l o w i s p o s s i b l e . A f u r t h e r consequence o f (b) i s t h a t

u(n,B)

I 0

for a l l

B E

x . Hence

we g e t f r o m ( a ) :

and because o f t h e a d d i t i v i t y o f v t h e l a s t t e r m i s equal t o v(A, C A ) - V( CA,A) s i n c e t h e v(A,A) cancel. 0

58

Linear Functionals

1.8.3

Supply

-

Demand Problems

I n t h i s s e c t i o n we would l i k e t o t r e a t a supply-demand model ( e v e n t u a l l y w i t h i n f i n i t e commodity s e t s ) where t h e producer as w e l l as t h e consumer has a l t e r n a t i v e s concerning t h e p r o d u c t i o n and consumption. Furthermore we deal w i t h t h e case where these a l t e r n a t i v e s can be used w i t h i n c e r t a i n bounds g i v e n by raw-material and s a t u r a t i o n c o n s t r a i n t s . B e f o r e we a r e going t o make p r e c i s e what we mean we have a l o o k a t a s i m p l e example.

A t a small u n i v e r s i t y f o u r t u t o r s a r e a v a i l a b l e f o r t u t o r i a l s i n A n a l y s i s (A), P r o b a b i l i t y (P), and Numerical A n a l y s i s (N). Each o f them has t o g i v e a1 t o g e t h e r s i x t u t o r i a l

hours i n t h e s e s u b j e c t s . Furthermore t h e r e a r e

f o u r students, and each o f them has t o t a k e s i x t u t o r i a l

hours. I t i s a

fancy u n i v e r s i t y , which means t h a t i n each hour t h e teacher has t o deal w i t h o n l y one s t u d e n t . The dean asks t h e teachers students

( S 1,...,S4)

(T1,...,T4)

and t h e

f o r t h e i r preferences. Here a r e t h e r e p l i e s o f t h e

teachers : TI wants t o teach 4 hours A n a l y s i s and f o r t h e remaining two hours he does n o t c a r e i f he teaches e i t h e r P o r N. T2 wants t o teach 2 hours i n e i t h e r

A

e q u a l l y b o r i n g and t r i v i a l ) * and he

or

P

(which he considers

i n s i s t s i n t e a c h i n g 4 hours

N.

T3 wants t o d i s t r i b u t e h i s a t t e n t i o n e q u a l l y t o t h e e x c i t i n g s u b j e c t s A, P and N Hence he o f f e r s t o teach 2 hours i n each s u b j e c t .

.

T4

i s a charming person. He l o v e s

A

and

P

, hence he o f f e r s t o teach

4 hours i n t h e s e s u b j e c t s ( b u t he does n o t care i f i t i s

A

or

P).

And o u t o f a f e e l i n g o f r e s p o n s i b i l i t y towards t h e mathematical community he i s w i l l i n g t o teach i n t h e remaining two hours whatever i s needed, even Numerical A n a l y s i s . As i t happens q u i t e o f t e n , t h e dean f o r g e t s a l l t h e good reasons which were t o l d t o him and he makes a l i s t :

*

I n t h i s r e s p e c t he r e a l l y shows bad t a s t e .

59

The Sandwich Theorem

(supply o f t u t o r i a l hours) F o r reasons which may be e q u a l l y w e l l founded t h e s t u d e n t s

(S1,

...,S4)

have t h e f o l 1owing preferences :

S i n c e t h e department i s a b i t s h o r t i n computing t i m e t h e dean decides t h a t a t most one o f t h e hours o f t h e hours A v P v N

P v N

( o f f e r e d by

TI)

and a t most one

s h o u l d be g i v e n i n Numerical A n a l y s i s .

And s i n c e t h e department w i l l need l a t e r on s t u d e n t s w i t h some b a s i c knowledge i n p r o b a b i l i t y he decrees t h a t whenever a s t u d e n t o f f e r s t o take

P

a l t e r n a t i v e l y t o something e l s e , he s h o u l d t a k e a t l e a s t h a l f of

these h G u r s i n

P

.

60

Linear Functionals

Then he turns over the two l i s t s t o t h e vice dean and t e l l s h i m " i t can be done". Now, the vice dean could proceed i n two ways. He cculd e i t h e r go t o h i s colleague working i n networks or he could read through t h i s chapter *. B u t s i n c e t h e vice dean i s very clever ( i n h i s p a r t i c u l a r way) he does n e i t h e r of t h i s . He decides t h a t a student has t o take Numerical Analysis whenever he allows t h i s a l t e r n a t i v e and whenever t h i s i s not i n c o n t r a s t t o t h e decrees of the dean. Then immediately he writes down t h e hours t o be taken by the students and t o be given by t h e teachers:

NOW, l e t us s t a r t t o formalize problems of this type. Certain q u a n t i t i e s play a basic r o l e i n our model. These q u a n t i t i e s a r e e s s e n t i a l l y of measure t h e o r e t i c nature, b u t we believe t h a t t h e i r meaning can be b e s t understood in t h e s e t t i n g o f f i n i t e commodity s e t s . Therefore we s t a r t with a f i n i t e commodity s e t X = { 1 , 2 , . . . , n I (the s e t X = IP,N,AI in our example). And we consider one producer and one consumer. The production capacity ( o r r a t h e r the aggregated production capacity) and the consumption d e s i r e a r e measured by functions a and respectively. We look f o r conditions which guarantee t h a t production can s a t i s f y consumption. I t seems natural t o require t h a t has t o exceed , b u t in a r e functions r a t h e r on Po IY c X Y P instead real l i f e a and of functions on X , s i n c e the consumer allows a l t e r n a t i v e s and t h e producer has a t his disposal c a p a c i t i e s (manpower and machines) which he uses according t o the market s i t u a t i o n . In our example the function 6 i s

<

<

+

<

*) Actually, then he would only know t h a t the dean was r i q h t . In order t o

know how t o do i t he should read through Chapter 1.1.9 and use t h e few remarks he can f i n d t h e r e t o make a reasonable program f o r t h e problem.

The Sandwich Theorem

G{A3

g i v e n by t h e sum i n o u r f i r s t l i s t , i . e .

a{A,P,N}

61

=

6, a I A , P I = 6,

= 2 e t c . The demand f u n c t i o n corresponds t o t h e sum i n t h e second

list. Things become even more c o m p l i c a t e d i f one takes i n t o account t h a t u s u a l l y t h e producer i s h i n d e r e d i n u s i n g h i s p r o d u c t i o n c a p a c i t y by some bound which i s determined by t h e amount o f raw-material needed f o r t h e p r o d u c t i o n .

A s i m i l a r bound i s g i v e n on t h e consumer

s i d e . Even i f he a l l o w s a l t e r -

n a t i v e s a c c o r d i n g t o h i s t a s t e , t h e n he may n e v e r t h e l e s s become f e d up b y g e t t i n g t o o many goods o f t h e same k i n d i n s t e a d o f a w e l l mixed v a r i e t y .

0 , given

We w i l l c a l l t h e bound

b y t h i s e f f e c t , t h e s a t u r a t i o n bound.

I n o u r example t h e raw m a t e r i a l bound i s g i v e n b y t h e d e c i s i o n o f t h e dean, which was based on t h e a v a i l a b l e computing t i m e . And t h e s a t u r a t i o n bound corresponds t o h i s decree concerning t h e n e c e s s i t y o f knowlegde i n Probabi 1 it y . The concept can be made more p r e c i s e i f we assume t h a t t h e p r o d u c e r c o n s i s t s U(Y), Y E Po

o f subunits

, where

U(Y)

i s the collection o f a l l factories

iE Y

where, a c c o r d i n g t o manpower and machines, t h e commodities

can

be produced e q u i v a l e n t l y b u t where p r o d u c t i o n cannot s w i t c h t o commodities outside

Y

. Then

by numbers

a

the production capacity function

:

Po

+ R,

a ( Y ) , measuring t h e maximal o u t p u t o f t h e s u b u n i t

i s given

U(Y)

if

no raw-material bound i s given. Then we assume t h a t f o r t h e p r o d u c t i o n o f each

iE Y

a s p e c i f i c r a w - m a t e r i a l i s needed which t o

U(Y)

a v a i l a b l e up t o a c e r t a i n amount, thus p r e v e n t i n g t h e s u b u n i t produce more t h a n

, say,

p(Y,i)

o f t h a t commodity

i

.

i s only U(Y)

to

A s i m i l a r s i t u a t i o n i s g i v e n on t h e consumersside. H i s demand i s g i v e n b y a function demand which

-

: Po

+

, where

R,

;(Y)

measures t h a t f r a c t i o n o f h i s t o t a l

i f no s a t u r a t i o n bound i s g i v e n

-

can be s a t i s f i e d b y t h e

a l l o c a t i o n o f any a r b i t r a r y combination ( o f t o t a l amount commodities o u t o f

Y

s a t i s f y t h e demand

;(Y)

every

.

I f a s a t u r a t i o n bound z ( Y , i )

;(Y))

o n l y those a l l o c a t i o n s a r e p e r m i t t e d which, f o r

i E Y , do n o t g i v e more t h a n

G ( Y ,i) o f t h e commodity

I n o u r example t h e raw-material bound i s g i v e n by ;( IP,Nl,N)=;( and

;(Y,i)

i s g i v e n by

=

OD

f o r a l l o t h e r cases when

o(IP,Nl,N)

of

i s given, t h e n t o

=

3

(and a g a i n

+m

i E Y.

i.

{A,P,Nl,N)=1

The s a t u r a t i o n bound

f o r t h e o t h e r cases when

iEY).

Our problems i s , r o u g h l y speaking, t h e e s t a b l i s h m e n t o f a p r o d u c t i o n p l a n

Linear Functionals

62

b

and a d i s t r i b u t i o n p l a n

such t h a t :

i; observes t h e l i m i t s g i v e n by t h e p r o d u c t i o n c a p a c i t y 6. and t h e r a w - m a t e r i a l bound ,

(1)

i s possible, i.e.

i

(2)

i s satisfactory, i.e.

i

s a t i s f i e s t h e demand

observes t h e l i m i t s g i v e n by t h e s a t u r a t i o n bound

b

(3)

i

and

a r e compatible

, i.e.

and

0 ,

we do n o t d i s t r i b u t e more

t h a n we produce. Again, we have t o make t h i s more p r e c i s e . To do s o , we assume t h a t t h e production plan

p

as w e l l as t h e d i s t r i b u t i o n p l a n

i

a r e maps from

Po x X t o R, . The q u a n t i t y p(Y,i) measures t h e amount o f i b e i n g produced by t h e s u b u n i t U(Y) and i ( Y , i ) measures how much o f i i s a l l o c a t e d t o t h e demand ;(Y).Then (1) - ( 3 ) l e a d t o t h e f o l l o w i n g : Definition: a l l Y E Po

( i ) The p r o d u c t i o n p l a n and i E Y , we have:

z p(Y,k)

(4)*

kEY

(6)*

(iii)

iE Y

and

;(Y)

I

The p l a n s

commodity (8)*

i s s a i d t o be p o s s i b l e i f , f o r

Ii ( Y )

(ii)The d i s t r i b u t i o n p l a n Y E Po

6

, we

i s d e f i n e d t o be s a t i s f a c t o r y i f , f o r a l l

have:

1 i(Y,k)

kEY

and

a r e s a i d t o be c o m p a t i b l e i f , f o r every

i E X , p r o d u c t i o n exceeds d i s t r i b u t i o n , i . e . 1 V(Y,i) Y w i t h iEY

I

E P(U,i) Y w i t h iEY

.

NOW, o u r problem i s : Are t h e r e --

plans which suitable production and d i s t r i b u t i o n plans, i.e. -

The Sandwich Theorem

are possible,

satisfactory

63

compatible?

B e f o r e we proceed t o discuss t h i s problem we l i k e t o s i m p l i f y t h e n o t a t i o n . F o r m a l l y we p u t

And whenever we have a map d e f i n e a measure

A

on

T

: A

-,R,

( w i t h respect t o the

T(B) f

t ?(b), B c A bEB i s the density o f T

Now, we denote by

- - - - - -

a,v,p,o,p,v Po

X

x

U-

we c a n o n i c a l l y

algebra

P(A))

by:

.

.

7

i.e.

A

on a f i n i t e s e t

on

and

a,u

Po

then f o r

and

Po

R c Po

p,~,p,v

t h e c a n o n i c a l measures g i v e n by

X , r e s p e c t i v e l y . And i f

x

B

and

c X

we denote by

i s a measure on

T

~ ( xn

0

and

)

T(.

xB)

t h e measures:

on

X

and

D

+

~ ( f ix

D),

D c X

D

+

T(D

B),

D c Po

Po

x

respectively.

Using t h i s n o t a t i o n e q u a l i t y ( 9 ) * and i n e q u a l i t i e s ( 4 ) * t o ( 8 ) " have t h e f o l l o w i n g s i m p l e form:

(9)

p

( x~

D)

= o(n

x

D)

= p(n x

when

A n D

D) =

= v(R x

0

D)

for all

= 0

A E fi

.

Now we i n t r o d u c e two b a s i c q u a n t i t i e s which immediately l e a d t o a necessary

64

Linear Functionals

condition f o r the existence of s u i t a b l e plans. We consider an a r b i t r a r y nonempty subset Y c X and we c a l c u l a t e the maximal t o t a l amount a*,,,,,(Y) of commodities i E Y which can be produced by observing a l l given bounds. That i s t h e amount produced f o r t h e submarket Y i f the producer concent r a t e s a l l his e f f o r t s on t h e productionof goods o u t of Y , i . e . whenever he has the choice producing something f o r Y instead o f something o u t s i d e Y , he does so. For our example t h i s function i s

a*

12

max 14

8

18

Let us see how this function i s constructed i n general. The production capacity allows t h e production of a ( ? ) i f Y n + 0 , and t h e raw-material bound l i m i t s this- production i n any case t o p(IY1 x Y ) . So, remembering t h a t p({YI x Y ) = 0 f o r Y n Y = 0 , we obtain t h a t the maximal production of U ( v ) f o r Y i s equal t o

v

Summation over the subunits leads t o a -~ t o t a l maximal production f o r Y equal to:

I f we define

(ml

A

m2)

t o be the g r e a t e s t lower bound ( i n the s e t of

p o s i t i v e measures) of the p o s i t i v e measures rewrite:

ml

and m2

then we can

.

Now, l e t us determine the --minimal t o t a l demand f o r goods in Y If Y c Y then, c e r t a i n l y , t h e demand given by can only be s a t i s f i e d by a l l o cation of commodities in Y . B u t due t o the s a t u r a t i o n bound t h e r e may a l s o e x i s t demands ; ( i ) f o r ? & Y which have t o be p a r t i a l l y s a t i s f i e d by a l l o c a t i o n of goods out of Y . The amount of this p a r t i s obviously

;(v)

The Sandwich Theorem

65

equal t o

CY

where

-

X ). For

case i n

"(7)

denotes t h e complement i n t h e s e t under c o n s i d e r a t i o n ( i n t h i s

Y c Y

t h i s q u a n t i t y becomes a u t o m a t i c a l l y equal t o

(consequence o f ( 9 ) ) . So we o b t a i n t h a t t h e minimal t o t a l demand f o r

goods i n Y

has t o be:

And i f we d e f i n e

t o be p o s i t i v e p a r t o f a s i g n e d measure

(m)'

m this

can be r e w r i t t e n as ";Cnin(Y)

(11)

Y

Note t h a t n e i t h e r

a*max(.)

I n general

= (v

+

-

u(.

x

CY

nor

a*,,,(Y)

Y

+

V*~~,,(Y)

i s a s u b a d d i t i v e and

v*min(

a r e measures o n

.)

X.

a superadditive set

function. I n o u r example t h e f u n c t i o n

v*

min

v

i ~s g i v~e n ~by:

*

{A1

{PI

IN1

IA,PI

IA,NI

{P,NI

IA,P,N)

10

7

4

17

14

14

24

Common sense t e l l s us t h a t a necessary c o n d i t i o n f o r t h e e x i s t e n c e o f s u i t a b l e p l a n s i s t h a t f o r e v e r y submarket a*

max

(Y)

Y

, the

maximal t o t a l p r o d u c t i o n

i s more than o r equal t o t h e minimal t o t a l demand

v * , , , ~ ~ ( Y ).

S i n c e we have f o r m a l i z e d o u r problem so f a r i t i s q u i t e obvious how t o f o r m u l a t e i t f o r a general measure t h e o r e t i c s e t u p i n t h e case o f a r b i t r a r y commodity s e t s . We deal w i t h a nonempty s e t and X

on

X

and

p o s i t i v e measures:

X

( c a l l e d commodity s e t ) , CJ- a l g e b r a s zX

Po f { Y c X

IY

i?~} r e s p e c t i v e l y , and w i t h f i n i t e

Linear Functionals

66

-+

a : =Po

(production capacity) R+

(demand measure)

(I

-

Q

Zx-+R+

(raw

: tpQ

tx-,R+

( s a t u r a t i o n bound)

p : . I

m a t e r i a l bound)

0

.

F o r t h e bounds we assume t h a t

(12)

P(R x

D)

= ( I ( Q x D) = 0

when

A

n D

=

b

for a l l

A E n

.

From these measures we d e r i v e t h e q u a n t i t i e s : a*max(Y) = ( a

A P(*

xy))(po)

v* m i n (Y) = ( v

-

XCY))+(P0)

(I(.

,

which a r e c a l l e d t h e ~ maximal t o t_ a l E od uc t_ i o n f o r t h e submarket the ~ minimal t o t a l demand f o r Y , r e s p e c t i v - ely.

Y

and

We a r e l o o k i n g f o r s u i t a b l e p r o d u c t i o n and d i s t r i b u t i o n p l a n s . A plan i s s i m p l y a p o s i t i v e measure B

and

Po

x

X

(with respect t o the

(I-

algebra

z x ) . A p r o d u c t i o n p l a n p i s s a i d t o be p o s s i b l e i f

(14)

p(.

X X )

v

A distribution plan

and

on

Ia

i s s a i d t o be s a t i s f a c t o r y i f

(15)

V

(16)

v I V('XX).

l

.

O

The two plans a r e s a i d t o be compatible i f

(17)

v(Po

x

* ) I p(Po

x

*)

.

Note t h a t p o s s i b l e and s a t i s f a c t o r y plans f u l f i l l ( 9 ) anyway (because o f

( 1 2 ) , (13) and ( 1 5 ) ) .

67

The Sandwich Theorem

Theo rem : There a r e --

p r o d u c t i o n and d i s t r i b u t i o n p l a n s

p,v,

which are possible, -t h e minimal s a t i s f a c t o r y and compatible, -if for _ e v e_ r y Y E Ix '- and _ _o-n l y i f , _ t_ o t_ a l_demand W * ~ ~ ~ ( Y ) i s l e s s t h a n o r equal t o t h e maximal t o t a __ ---~ - - _ _ _ - l ~

p r o d u c t i o n a*(,Y).

F o r t h e p r o o f we need a lemma. We r e c a l l t h a t g i v e n two measurable spaces

(r j J j )

j = 1,2;

a fu ction

t o be a bimeasure on q(A1 x

.

and

cp(.

x

r2

rlx

A2)

cp :

{A1

x

A2

if , f o r every

I A J.

E X.,j J

A1 E

x1

=

1,Zl

and

-,R

A2 E

i s said

x2 ,

both

a r e measures.

Lemma: Let -

cp

rlx

r2

-be_ a

for a l l --

rl

x

r2

p o s i t i v e bimeasure on rlx

(w i_ th _ r e s_ pect A1 E

zl,A 2

w i t h +(A1

E z2 x

z1 Q

to

. Then

A2) = q(A1

r 2 - and -

x 2 ) -such t h a t

let cp(A1

m x

be a measure on A2) I m(A1

t h e r e i s a p o s i t i v e measure x

A2)

for all

A1 E

xl,

x

A2)

G on

A2 E I2 .

Proof: One c o n s i d e r s t h e a l g e b r a o f a l l f i n i t e d i s j o i n t unions o f measurable r e c t a n g l e s [ 151p.1491.

Then one d e f i n e s f o r these f i n i t e d i s j o i n t unions

@( U ( A j x B.)) = x cp(Aj x B . ) . S i n c e cp i s s e p a r a t e l y a d d i t i v e , J J j j t h i s l e a d s t o an unambiguously d e f i n e d f i n i t e l y a d d i t i v e p o s i t i v e measure.

5, by

S i n c e t h i s i s dominated by an honest ( i . e .

a- a d d i t i v e ) measure i t must

i t s e l f be

a d d i t i v e l y extended t o t h e

a- a d d i t i v e . Hence i t can be

U-

a- a l g e b r a generated b y t h e measurable r e c t a n g l e s (compare [151 p.30 and

541).

0

ion has t o-exceed al Proof o f the necessity t h a t t o t a l maximal - p r o d u c t~ - t o t~

minimal demand art): ~ _ (easy p _ Assume t h a t t h e r e a r e p l a n s which a r e p o s s i b l e , s a t i s f a c t o r y and compatible. From ( 1 3 ) and (14) we know: P('XY) I P('XY)

Linear Functionals

68

and p(.

Y E Ex

for

I p(* xX) I a

X Y )

. This c l e a r l y p(.

implies

xY) I a A

P(- x Y ) .

Hence,

(18) Since

v

P(P0

Y) 5 a*,,(Y).

i s s a t i s f a c t o r y ( 1 5 ) and ( 1 6 ) l e a d t o :

From t h i s we o b t a i n :

Since

v

i s p o s i t i v e we can r e p l a c e t h e l e f t s i d e by i t s p o s i t i v e p a r t

without v i o l a t i n g the inequality: (v

-

u(.

x

CY))+IV ( '

Evaluation o f t h i s inequality f o r

(19)

v*min(Y)

Po

X Y ) .

gives:

5 v(Po x Y ) .

I n e q u a l i t i e s (18) and (19), t o g e t h e r w i t h t h e c o m p a t i b i l i t y o f yield:

*

v min(~)

f o r every

Y E Ex

Y ) Ip(p0

I v(p0 x

x

p

and

v

,

Y ) 5 a(*~,)

.

Proof t h a t t h e condition i s s u f f i c i e n t f o r the existence o f s u i t a b l e plans: We t a k e a copy subset o f

P*

P* by

of R*

Po

, and

for

. Then we d e f i n e

r =pOuxuP*. In

r

we c o n s i d e r a

R c

u- a l g e b r a

r

Po we denote t h e corresponding t o be t h e d i s j o i n t union:

69

The Sandwich Theorem

I n1,a2

I = {n1 U B u R; And w i t h r e s p e c t t o t h i s

v(nl u

B

U-

except

on X

x

Po

and

(20 a )

"I(A

x

(20 b )

T(R*

x

u n);

= v(n2)

B E Ixl .

7

,rk

P*

X

-

!.I

i s g i v e n by:

a(nl).

on t h e p r o d u c t

"I

rk(ri x

PO

a l g e b r a a s i g n e d measure

Furthermore a p o s i t i v e measure p r o v i s i o n t h a t on ri

E I

= Po,X7P*)

T

r

x

r

i s given by the

s h a l l vanish everywhere

and t h a t on these s e t s i t s h a l l be equal

to:

when

A E Zx

n) x

,nE I

We now c l a i m t h a t

T

= p(n x A)

A ) = u(n

x

A)

. and

p

do f u l f i l l c o n d i t i o n ( i ) o f t h e Flow Theorem

1.8.1 i f t o t a l maximal p r o d u c t i o n exceeds t o t a l minimal demand. To p r o v e t h a t c l a i m we c o n s i d e r some a r b i t r a r y

u(D) 2 v*

min

= v(n2)

- a(nl)

D E t and we a b b r e v i a t e :

.

we t h e n g e t :

Using ( 2 1 ) one sees t h a t t h i s i m p l i e s

T(D

x

CD)

2 u(D)

.

70

Linear Functionals

Now, l e t us t a k e t h e bimeasure g i v e n by ( i i ) o f t h e Flow Theorem. We denote cp

. Then

p(n

x

A) = cp(A

v(n

x

A) = cp(n*

R E

zp , A

t h i s bimeasure by

we d e f i n e bimeasures

p

and

v

on

Po

x

X

by :

for arbitrary

n)

x

.

E Ex

0

but that

p

v

and

A)

x

Note t h a t

cp

i s n o t a p o s i t i v e bimeasure,

a r e p o s i t i v e . From ( i i b ) o f t h e Flow Theorem we ob-

t a i n w i t h t h e h e l p o f (20):

for

A ) I T(A

( 2 2 a)

p(n

(22 b )

~ ( xn A ) I T ( R *

Q E

z ,A

x

E Ex

. So,

(n n C A ) )

x x

(A

n Cn*)) =

w i t h t h e lemma

n)

= T(A x T(R*

x

= p ( ~ A)

A) = a(n

x

A)

,

, we know t h a t p and v a r e

honest measures and we prove t h a t they a r e t h e r e q u i r e d p l a n s . (22) i m p l i e s t h a t they a r e f u l f i l l i n g (13) and ( 1 5 ) . For t h e p r o o f t h a t f a c t o r y i t remains t o show ( 1 6 ) . To do t h i s we t a k e w i t h (ii a ) and ( i

v

is

satis-

R E 1

b ) (always o f t h e Flow Theorem)

v(Q

( t h e l a s t e q u a l i t y f o l l o w s from (2Q)and t h e d e f i n i t i o n o f

T

)

.

Before we proceed t o prove ( 1 4 ) we observe t h a t ( i i b ) i m p l i e s cp(r

x

D)

5 0

for all

.

D E z

So ( i i a ) immediately leads t o :

A p p l i c a t i o n t o t h e s p e c i a l case

Here, we have used t h a t

cp(n

t h e same reason we g e t t h a t t h i s q u a n t i t y must a l s o be

D = n E z

Cn) I cp(C(nu x

2 0

.

gives:

0

(because o f ( i i b ) and ( 2 0 ) ) . For

X)

x

n)

5 0

. But,

because o f ( i i c )

71

The Sandwich Theorem

Hence cp(

C(n u

X ) x n) = 0

.

Using t h i s we can r e w r i t e (24)

a(n) 2

cp(

Cn

n)

x

X ) + cp(C(n u X )

= p(n x

So ( 1 4 ) i s proved and a l l what remains t o show i s t h a t

x

n) = p ( n p

and

x

v

x). are

compatible. From (23) we o b t a i n f o r

Y E zx

Now, we use ( i i c ) and (iib ) and o b t a i n f o r

Y E zx

the following

inequal it i e s : “Po

x

Y ) = Q(P* = cp(Y

x

Y ) 5 cp(

x

Po)

I p(Po x

+

Y) +

= p(Po x Y )

+

CY

Y ) 5 cp(Y X C Y )

C(Y u

cp(Y

T(Y

x

x

Po))

C(Y lJPo))

0 (because of ( 2 0 ) ) .

T h i s shows t h a t t h e p l a n s a r e compatible. Hence t h e p r o o f i s f i n a l l y 0 finished.

72

1.9

Linear Functionals

ADDITIONAL REMARKS AND COMMENTS

1.9.1

The Hahn-Banach Theorem

I n Chapter 1.1 we discussed t h e Hahn-Banach Theorem, o r r a t h e r a v a r i a n t t h e r e o f , and some o f i t s consequences. The Hahn-Banach Theorem i s c e r t a i n l y one o f t h e most fundamental r e s u l t s i n modern a n a l y s i s , i t i s one o f t h e b e s t i n v e s t i g a t e d i n d i v i d u a l theorems and t h e l i t e r a t u r e about i t covers thousands and thousends o f pages ( f o r example,already t h e r e f e r e n c e l i s t of

[2691, which i s f a r from b e i n g complete, c o n t a i n s 125 t i t l e s ) . It has been proved

and r e f o r m u l a t e d c o u n t l e s s times, and y e t t h e r e i s s t i l l demand

f o r new v e r s i o n s which a l l o w more e f f e c t i v e a p p l i c a t i o n s t h a n b e f o r e . And s u r p r i s i n g l y , new and b e t t e r v e r s i o n s a r e s t i l l found. Some o f t h e s e v e r s i o n s and some v a r i a n t s w i l l be mentioned i n t h e sequel ( o f course, we a r e aware o f t h e f a c t t h a t i t i s i m p o s s i b l e t o do j u s t i c e t o a l l t h e major contributions

-

even i f we knew a l l o f them).

F i r s t some remarks on t h e h i s t o r y o f t h e s u b j e c t (see a l s o [ 551 and E1321). B e f o r e t h e t h e o r y o f Banach spaces ( o r Banach-Wiener spaces as t h e y were o r i g i n a l l y c a l l e d ) was w e l l e s t a b l i s h e d b y Banach's c e l e b r a t e d book [15

3

mathematicians were m a i n l y i n t e r e s t e d i n s p e c i a l normed spaces. The work o f D. H i l b e r t and E. Schmidt l e d t o t h e general moment problem which was s o l v e d i n 1907 by F. Riesz ([2611,[2631) and E. F i s c h e r [lo51 and t h i s s o l u t i o n was about t h e f i r s t c h a r a c t e r i z a t i o n o f t h e dual o f a n o n t r i v i a l Banach space. I n 1912 E. H e l l e y [I591 s i m p l i f i e d t h e s o l u t i o n and t r e a t e d t h e problem i n such a general and t r a n s p a r e n t way t h a t one can almost speak o f t h e f i r s t appearance o f t h e Hahn-Banach Theorem. ( S i n c e H e l l e y was o n l y i n t e r e s t e d i n normed sequence spaces i n s t e a d of general Banach spaces h i s i m p o r t a n t c o n t r i b u t i o n s were almost f o r g o t t e n a f t e r t h e e s t a b l i s h m e n t o f t h e general theory; i t s h o u l d be n o t e d t h a t he was n o t o n l y v e r y c l o s e t o t h e Hahn-Banach Theorem b u t a l s o t o t h e Banach-Steinhaus Theorem, see [I641 and [ 611).

Phrased i n modern t e r m i n o l o g y , H e l l e y d i d i n 1912 a l r e a d y use

t h e b i n a r y i n t e r s e c t i o n p r o p e r t y o f t h e r e a l s t o extend a l i n e a r f u n c t i o n a l ( w i t h o u t i n c r e a s i n g i t s norm) from a space w i t h codimension one. L e t us r e c a l l t h a t a Banach space i s s a i d t o have t h e b i n a r y i n t e r s e c t i o n p r o p e r t y

if every c o l l e c t i o n o f b a l l s has a nonempty i n t e r s e c t i o n whenever any two of them have nonempty i n t e r s e c t i o n .

73

The Sandwich Theorem

The Hahn-Banach Theorem f o r real valued linear functionals, in i t s present day form,was firmly established in 1927 by H . Hahn [14gI and independently in 1929 by S. Banach [141. The proof was the same as the standard proof today: Extension of the dimension,of the space on which the linear functional under consideration i s defined,in each step by one, and then obtaining the full result by t r a n s f i n i t e induction. Of course, the same proof works for (locally) convex topological vector spaces, one only has to replace the norm by the gauge functional given by a suitable absolutely convex neighbourhood of the origin. Since non-convex vector spaces in general do n o t have sufficiently many continuous sublinear functionals one cannot expect t h a t they have separating d u a l spaces (even i f they are Hausdorff-spaces). Of course, many of them do have separating duals: I n fact,whenever the weak and the Mackey topologies are different then there i s a non-convex topology between them 11461 (then,trivial ly,having the same dual space).

B u t l e t us go back t o the history of the Hahn-Banach Theorem.Extensions of positive linear maps defined on a subspace of a vector l a t t i c e and attaining values in an order complete vector l a t t i c e were already studied in 1937 by L.V. Kantorovitch 11821 (see also 12221 and 12231). Then in 1950 L . Nachbin [2381 discovered t h a t the binary intersection property i s a crucical tool for generalizations of the Extension Theorem (see also Goodner 11431). His celebrated result was that one can replace the reals ( i . e . the image space of the linear functionals) by a normed space E i f and only i f E has the binary intersection property. One direction of t h a t result i s again proved by extending the dimension of the space where the linear functional i s defined step by step by one. The other direction i s a rather ingenious argument. Let us give a brief sketch (see also [1321). Take a collection of closed balls B ( x i , r i ) ( i n the normed space E ) with radii ri around xi E E , where i E I . Any two of them have nonempty intersection i f and only i f (*)

II xi

Then one defines in

-

x.II

Iri t

J

E x

R

r

j

for a l l

a seminorm by

i,j

.

Linear Functionals

74

Because o f t h e i n e q u a l i t y ( * ) we have always

NOW, t a k e t h e o p e r a t o r T : (x,O)

F

7

= {(x,O)

Ix

E El

.

-,

x

II (x,O)ll = I I x II d e f i n e d on t h e subspace

.

I f t h i s o p e r a t o r can be extended t o an o p e r a t o r

: E x R -,E w i t h o u t i n c r e a s i n g i t s norm t h e n ? ( O , - l ) must be i n t h e i n t e r s e c t i o n o f a l l t h e B(xi,ri). Hence, if t h e Hahn-Banach Theorem i s v a l i d f o r l i n e a r maps w i t h values i n

E

then

E

must have t h e b i n a r y

i n t e r s e c t i o n p r o p e r t y . O f course t h e r e a r e spaces w i t h o u t b i n a r y i n t e r s e c t i o n p r o p e r t y having a Hahn-Banach p r o p e r t y w i t h c a r d i n a l i t y r e s t r i c t i o n s . I n p r e c i s e terms: F o r a g i v e n c a r d i n a l i t y Banach space

E

B l i n e a r operator

normed space t o a l l of property o f

B

, without binary , h a v i n g a dense T :A

-,E

rl

there i s a

i n t e r s e c t i o n p r o p e r t y , such t h a t f o r e v e r y subset o f c a r d i n a l i t y 5

on a subspace

q

a continuous

A c B can always be extended

w i t h o u t i n c r e a s i n g i t s norm (see [2481). O f course, t h i s E

corresponds t o a r e s t r i c t e d i n t e r s e c t i o n p r o p e r t y .

Another c h a r a c t e r i z a t i o n o f spaces w i t h t h e Hahn-Banach p r o p e r t y (which, by Nachbin's theorem, i s e q u i v a l e n t t o t h e b i n a r y i n t e r s e c t i o n p r o p e r t y ) i s t h a t they a r e t h e i n j e c t i v e elements i n t h e c a t e g o r y o f Banach spaces ( w i t h o p e r a t o r s o f norm I 1 as morphisms) (compare [76 I).Here, a g a i n one can o b t a i n v a r i a n t s and m o d i f i e d r e s u l t s by changing t h e c a t e g o r y under c o n s i d e r a t i o n (e.g. v e c t o r l a t t i c e s w i t h l i n e a r l a t t i c e homomorphisms [220] [156],[216] ,[63];

PA

-

spaces [157],

spaces o f continuous f u n c t i o n s [3] ,[4] [271],

[272],

, [174] ,

[334];

separable Banach spaces [ 3 4 3 ] ) .

Actual l y , t h e r o l e of t h e b i n a r y i n t e r s e c t i o n p r o p e r t y becomes even more t r a n s p a r e n t i f one considers s e t - v a l u e d mappings. So, l e t

E

and

F be

K and c o n s i d e r a nonempty t r a n s l a t i o n (i.e. A E A, u E F A t u E A ) o f F.

v e c t o r spaces over t h e same f i e l d i n v a r i a n t c l a s s A o f subsets

Furthermore, l e t a s u b l i n e a r s e t - v a l u e d map

=$

Y : E -,

A

be g i v e n ( s u b l i n e a r

Y ( k x ) = k ~ ( x ) ,' Y ( x t y ) c ~ ( x t) ~ ( y ) f o r a l l x,y E E, k E K ) . has t h e h i n a r y i n t e r s e c t i o n p r o p e r t y then t h e f o l l o w i n q r e s u l t o f B. Rodriguez-Salinas and L . Bou h o l d s :

means If A

,

75

The Sandwich Theorem

Theorem 12701: that -

i ( x ) E v(x)

7to a ll of

of -

Let

?

_be_a

c E

for all

x E

-_.

E

subspace of E

and l e t 7 --

. -Then- t h-e r e_ i s ~a

:

E

+

F

such

l i n e a r extension

f

f ( x ) E ~ ( x ) -f o r a l l x E E.

such t h a t

__I_

T h i s seems t o be an a p p r o p r i a t e s t a r t i n g p o i n t f o r an i n v e s t i g a t i o n o f Hahn-Banach-property versus b i n a r y i n t e r s e c t i o n p r o p e r t y . The s t r u c t u r e o f spaces w i t h b i n a r y i n t e r s e c t i o n p r o p e r t y was comp l e t e l y c l a r i f i e d when K e l l e y [1921

showed t h a t a Banach space w i t h b i n a r y

i n t e r s e c t i o n p r o p e r t y i s i s o m e t r i c a l l y isomorphic t o

C(S)

,

S

extremally

disconnected and compact ( f o r a s i m p l e p r o o f o f t h a t f a c t see [1321, f o r t h e e x t e n s i o n o f f u n c t i o n a l s a t t a i n i n g values i n a complex l i n e a r space one s h o u l d c o n s u l t t h e work o f Hasumi [1551 and Hustad [1721). Taking i n t o account t h a t such a space

i s an o r d e r complete v e c t o r l a t t i c e we see

C(S)

t h a t i n a l l g e n e r a l i t y t h e Hahn-Banach theorem o n l y h o l d s i f t h e l i n e a r maps a t t a i n values i n an o r d e r complete v e c t o r l a t t i c e . The f i r s t Hahn-Banach theorem f o r a d d i t i v e maps on a commutative semigroup is

-

as f a r as we know

-

due t o I. H a l p e r i n [152], who s t u d i e d t h e problem

i n 1952. A f t e r t h a t Hahn-Bach theorems f o r semigroups were s t u d i e d by:

, R.

,

M. C o t l a r [80]

G. Aumann [12]

H. Dinges [87]

P. Kranz [203],[204],

t h i s book [116

and many o t h e r s . The d i s c o v e r y t h a t dominated e x t e n s i o n s a r e

Kaufmann [186],[187],

G. Seevers [288]

F. Topsoe [313], t h e f i r s t a u t h o r o f

more s u i t a b l e t h a n p r o p e r e x t e n s i o n s f o r t h e case o f f u n c t i o n a l s d e f i n e d on semigroups appears i n [ 1 8 7 l f o r t h e f i r s t t i m e . Monotone a d d i t i v e f u n c t i o n a l s on o r d e r e d semigroups a r e considered s i n c e t h e paper o f Aumann [121. R e c a l l t h a t an a d d i t i v e f u n c t i o n a l subaddi t i v e f u n c t i o n a l

p

on a group

p

G

, which i s dominated by a

, has a u t o m a t i c a l l y t h e same m o n o t o n i c i t y proper-

t i e s as p. T h i s i s no l o n g e r t r u e i n t h e case o f semigroups. So i t i s n a t u r a l t h a t t h e q u e s t i o n whether as

p

has t h e same m o n o t o n i c i t y p r o p e r t i e s

t u r n s up i n t h e c o n t e x t o f semigroups. T h i s q u e s t i o n i s s a t i s f a c t o r i -

p l y answered by t h e Sandwich Theorem we have discussed i n t h i s c h a p t e r . The p r o o f o f t h i s theorem i s taken from [ 1341. I n 1968 Heinz Kdnig became i n t e r e s t e d i n a p p l i c a t i o n s o f t h e Hahn-Banach

Theorem. I n a few papers [197]

-

[199] he developed a c o n v i n c i n g and e l e g a n t

t h e o r y o f s u b l i n e a r f u n c t i o n a l s on v e c t o r spaces. The a p p l i c a t i o n s he gave showed t h a t t h e b a s i c elements o f a t h e o r y o f s u b l i n e a r f u n c t i o n a l s can be

76

Linear Functionals

considered a s a e f f e c t i v e tool i n modern a n a l y s i s . For vector spaces, a l l t h e main theorems of our chapter 1.1, l i k e the Sum-Theorem, t h e F i n i t e Decomposition Theorem and t h e Riesz-Konig Theorem, a r e already contained in Konig's paper [199].From t h a t point on, the extension of these basic f a c t s t o the case of preordered semigroups was j u s t a matter of routine, this routine work was done in [116]by t h e f i r s t author of this book. B u t t h e s t o r y of t h e Hahn-Banach Theorem would be incomplete without mentioning the many important contributions t o the theory of sub1 i n e a r functionals which can be found i n some p a r t s of a b s t r a c t potential theory, e s p e c i a l l y i n t h e work of H . Bauer and G. Choquet. Although t h e i r r e s u l t s will play a major r o l e i n t h e second p a r t of t h i s book, we l i k e t o give a few examples: For vector spaces t h e Sum Theorem, and i t s r e l a t i o n t o the Strassen Disintegration Theorem, can already be found as a remark [ 72, p.2731 i n Choquet's 1968 Princeton l e c t u r e s . Here one a l s o finds the Sandwich Theorem f o r vector spaces as a consequence of t h e Hahn-Banach Theorem. And, generally speaking, t h e observation t h a t many r e s u l t s r e l a t e d t o the Hahn-Banach Theorem carry over t o cones instead of vector spaces o r i g i n a t e s i n potential theory. Of course, f o r cones one has t o replace e q u a l i t i e s i n a s u i t a b l e way by i n e q u a l i t i e s . In potential theory i t is well known t h a t many r e s u l t s concerning harmonic functions can be t r a n s f e r r e d (under s u i t a b l e modifications consisting mainly i n replacing e q u a l i t i e s by i n e q u a l i t i e s ) t o the convex cone of subharmonic functions. In f a c t , on a more elementary l e v e l , the same holds t r u e i f t h e continuous functions a r e replaced by t h e semicontinuous functions. This observation successfully penetrated t h e i n v e s t i g a t i o n s which were made i n the context of compact convex s e t s ( s e e f o r example the work of H. Bauer and D.A. Edwards). In section 11.5 the same idea leads us t o a new level of a b s t r a c t i o n . There we show t h a t e s s e n t i a l portions of the topological aspects of t h e theory o f continuous functions can be verbatim generalized f o r convex cones of a r b i t r a r y functions.

77

The Sandwich Theorem

1.9.2

Other E x t e n s i o n Theorems

I n t h i s s u b s e c t i o n we p r e s e n t a v a r i e t y o f e x t e n s i o n theorems r e l a t e d t o t h e Hahn-Banach Theorem. Since we have seen [ 2 4 8 ] t h a t under r e s t r i c t i o n s on t h e c a r d i n a l i t y o f t h e f a m i l i e s w i t h t h e i n t e r s e c t i o n p r o p e r t y one o b t a i n s theorems somewhat weaker t h a n t h e Hahn-Banach Theorem, i t seems n a t u r a l t o s t u d y spaces where t h e b i n a r y i n t e r s e c t i o n p r o p e r t y i s r e s t r i c t e d t o f i n i t e f a m i l i e s . T h i s s i t u a t i o n was c o m p l e t e l y i n v e s t i g a t e d and c h a r a c t e r i z e d b y J. Lindens t r a u s s 12123 : Theorem ( L i n d e n s t r a u s s ) : Let -

X

be a Banach space, -then t h e following are equivalent:

Every of f o u r mutually i) _ _ c o l l e c t i o n -non-empty i n t e r s e c t i o n --

i n t e r s e c t i n g -balls i n

.

ii)

Let Z T : Y

X

-has _

a

be any Banach space, t h e n f o r e v e r y ___ compact o p e r a t o r X

from a subspace -

Y c Z

into -

t h e r e i s a compact l i n e a r e x t e n s i o n

II Tll 5 (1 + c ) I I T I I iii)--The dual space L- space. X"

iv)

The bidual --

v)

The _ b i du a_ l has _ -

X'

7

X

: Z

and f o r e v e r y --+

X

E

> 0

with

. i s isometrically -

isomorphic to an a b s t r a c t

has the binary i n t e r s e c t i o n property.

--

t h e p r o p e r t y s t a t e d i n i).

L e t us g i v e a b r i e f s k e t c h o f t h e main ideas i n v o l v e d i n t h e p r o o f o f t h i s theorem. F i r s t o f a l l one observes by i n d u c t i o n t h a t i)i m p l i e s t h a t e v e r y f i n i t e family o f mutually intersecting b a l l s i n

X

has a nonempty i n t e r -

s e c t i o n . Then u s i n g i n an a p p r o x i m a t i v e way t h e arguments which appear when t h e Hahn-Banach Theorem i s proved v i a t h e b i n a r y i n t e r s e c t i o n p r o p e r t y one o b t a i n s ii).The a spac.e

E

comes i n t o t h e game s i n c e i n e x t e n d i n g t h e o p e r a t o r t o

Y , where t h e dimension i s i n c r e a s e d by one, one has t o deal w i t h

n o n - f i n i t e f a m i l i e s of b a l l s such t h a t e v e r y f i n i t e s u b f a m i l y has nonempty i n t e r s e c t i o n , and f o r f i n d i n g a conimon p o i n t

i n the intersection o f a l l

t h e s e b a l l s one has t o i n c r e a s e t h e i r r a d i i s l i g h t l y . Now, f r o m t h e b i n a r y

70

Linear Functionals

intersection property i n X f o r f i n i t e families one e a s i l y obtains ( v i a weak compactness) the binary i n t e r s e c t i o n property f o r X " . We already know t h a t i f X" has t h e binary i n t e r s e c t i o n property then i t must be isometrically isomorphic t o an order complete C ( K ) - space. B u t then X"' must be an a b s t r a c t L- space and using t h e canonical projection P : XI1'-, X ' ( r e s t r i c t i o n of the functionals x"' E XI" t o X ) one obtains t h a t X ' i s an a b s t r a c t L- space. Using t h e f a c t t h a t the dual of an L- space i s an M- space one obtains v ) . And a f i n a l application of t h e p r i n c i p l e o f local r e f l e x i v i t y 1207 p. 2301 leads t o i ) .

As another consequence of intersection properties in Banach spaces we can consider K Donner's [891 , [901 theorem about extensions of p o s i t i v e

.

operators in LP- spaces: Theorem: -

-

15 q 5 p < Lp(p) . Then --and consider spaces L q ( p ) -- for a p o s i t i v e continuous l i n e a r operator T : H -, L q ( p ) on-a subspace H of L p ( p ) t h e r e i s a p o s i t i v e l i n e a r extension : Lp(p) + Lq(,,) with -

Let -

II 711

=

and

II TI1 i f and only i f

for every family ( h i ) i E I in H , where V -denotes t h e supremum operation and h t t h e positive p a r t .

In f a c t t h i s r e s u l t can be proved by a ( s k i l l f u l l ) application of the theorem of Rodriguez-Salinas and L . Bou (which we mentioned e a r l i e r ) , a1 though Donner's o r i g i n a l proof proceeds d i f f e r e n t l y . He uses the HahnBanach theorem f o r hypolinear functionals due t o 6. Anger and J . Lembcke. They c a l l a sublinear functional p (defined on a convex subcone P of a l o c a l l y convex topological vector space E ) hypolinear i f i t a t t a i n s values i n R U I + -1. In t h i s s i t u a t i o n a theorem l i k e our Sandwich Theorem does n o t hold s i n c e Simons showed t h a t t h e r e i s a hypolinear functional which dominates no l i n e a r f u n c t i o n a l . Anger and Lembcke proved t h a t a hypolinear functional p : P + R U I + -1 dominates a continuous l i n e a r functional f : E -,R on P i f and only i f p i s lower semi-

The Sandwich Theorem

79

( c f . 17 1 , [ 8 1)

continuous a t t h e o r i g i n

A r a t h e r powerful v e r s i o n o f t h e c l a s s i c a l Hahn-Banach Theorem f o r v e c t o r spaces i s due t o Heinz Konig (L200Iand [133]):

Let

Theorem:

a function numbers

p : E

a,B > 0

Inf W€T

be s u b l i n e a r on t h e v e c t o r space

R

-,

: T +R

T

on- a_nonvoid _ _ - subset

with -

-

p(w-au- Bv)

T(W)

linear p : E Then t h e r e i s 2 ---

-,

+

a T(u)+

R

with

Ip

w € T.

for all

0

~ T ( v )I

v

and c o n s i d e r

E

such t h a t t h e r e a r e ----

T c E

U,V

E T

.

and ~ ( w )2 ~ ( w ) f o r a l l -

Here t h e emphasis l i e s on t h e f a c t t h a t i t s u f f i c e s t o have a r b i t r a r y a,B > 0

.

F o r t h e case a + B = 1 t h e theorem e a s i l y can be o b t a i n e d from

t h e Sandwich Theorem, b u t t h i s does n o t work f o r t h e general case. T h i s r e s u l t has been l i f t e d t o a new l e v e l o f a b s t r a c t i o n by G. Rode [ 2671. He proved a theorem - which seems t o be something l i k e t h e u l t i m a t e v e r s i o n of t h e Sandwich Theorem

-

n

mapping f i n i t e powers o f

: Xn

n(xl,.

-,X,

.. ,xn)

: Xm

p

-+

X

f o r commuting r e l a t i o n s . Consider maps

we l i k e t o w r i t e

if

whenever

r

xik

E X,

i=l,...,n

n

i

and

X

into

n

and we say t h a t

(xi)

k = I , ...,m.

X

.

Instead o f

and

p

commute

Now, c o n s i d e r a f a m i l y

o f m u t u a l l y commuting maps o f t h i s k i n d . O f course, t h e degree o f t h e

power o f

X

on which

y E

r

a c t s may b e d i f f e r e n t f o r each y

.

Theorem (Rode): Let -

cp : X

G,F : X -+

be two maps w i t h G ---6 w i t h G I cp I F -such t h a t -+

I F.

Then t h e r e ---

is

function

n

whenever

sly.. . ,an 2 0 ,n E T

and (xl,.

. . ,xn)

E Xn

-such t h a t

80

Linear Functionals

Extensions o f l i n e a r f u n c t i o n a l s which a r e maximal on g i v e n subcones have t u r n e d o u t t o be r a t h e r u s e f u l (see Andenaes [ 5 1 ) . One may even c o n s i d e r extensions w i t h s t r o n g e r p r o p e r t i e s , f o r example b e i n g extreme p o i n t s i n t h e s e t o f a l l extensions. Since these extensions a r e u s u a l l y connceted w i t h ( t r a n s f i n i t e ) a l g o r i t h m s f o r t h e c o n s t r u c t i o n o f e x t e n s i o n s we p r e s e n t one r e s u l t o f t h i s k i n d . Let

Gyp : F -,fi

F be a f i x e d cone. F o r

we w r i t e

4

--I

fi

i f the

following three conditions are s a t i s f i e d :

i)

irij

ii)

2;

i s s u p e r l i n e a r and

iii) ij(tl) We w r i t e

6

-.I p

t

4(t2) I

i f either

4

-1

i s sublinear

F(tl

tt2) f o r a l l

or

6 = i.

tlyt2E F.

And b y Lin(q,p) we denote t h e s e t o f l i n e a r f u n c t i o n a l s p : F + 6 with q I p I p Nowy p E Lin(q,p) i s s a i d t o be an exposed e l e m e n t , i f whenever

4

q 5

.

6

Ip

, then

t h e r e i s some

to E F w i t h :

Theorem [ 1221: i s superlinear and p i s s u b l i n e a r . -Then t h e r e is an exposed p E Lin(q,p). 9a d d i t i o n : if p i s monotone w i t h r e s p e c t which i s compatible w i t h t h e cone then p t-o some p r e o r d e r 4 ---- s t r u c t u r e i s a l s o monotone. -Let q -

I

p

such t h a t

q

C l e a r l y t h i s theorem general izes t h e Sandwich Theorem. B u t i t general izes 11. 3.1) as w e l l s i n c e t h e exposed [1221 t o be extreme p o i n t s o f L(q,p). I n

t h e Krein-Milman Theorem (see s e c t i o n elements o f

L(p,q)

turn out

f a c t t h e y even have a s t r o n g e r p r o p e r t y t h a n b e i n g extreme. The s e t o f

81

The Sandwich Theorem

exposed p o i n t s i s what we w i l l c a l l l a t e r on (see s e c t i o n 11. 3.1) t h e minimal f i x p o i n t boundary o f by t h e evaluationsof F.

Lin(q,p)

w i t h respect t o the functions given

L e t us e x p l a i n how these exposed elements

u

" a l g o r i t h m " f o r t h e Sandwich Theorem. L e t

4

functional

we denote b y

p

[p,q](t)

infIfi(tts)

NOW, f i x some

=

def

= [p,61,

2

with

p

5 p

.

p.

This

i t i s given by

-

to E F. Then t h e f u n c t i o n a l

q(s)

Is

Tt

(6)

0

i s a g a i n s u p e r l i n e a r and f u l f i l l s

;i 5

be s u p e r l i n e a r w i t h

p

Then t h e r e i s a unique maximal s u b l i n e a r

a r e r e l a t e d t o an

E Lin(q,p)

5 Tt

0

(4)

5 p

E FI

.

given by

.

I n f a c t , i t s v a l u e on

i s equal t o t h e maximum o f { p ( t o ) I y E L i n ( 4 , p ) I . Having d e f i n e d t h e s e to q u a n t i t i e s t h e a l g o r i t h m f o r o b t a i n i n g t h e exposed elements i n Lin(q,p) @ c F

i s e a s i l y e x p l a i n e d . Take a subset

which generates

i s equal t o a l l f i n i t e sums o f elements i n

@

c o e f f i c i e n t s ) . Then c o n s i d e r any w e l l o r d e r i n g q

where

i,(t)

=

def

supIqy(t)

if

IY

4

of

@

E @

with

Y

< cp}

(i.e.

F

and d e f i n e

i s t h e minimal element o f

cp

F

w i t h nonnegative

for all

@

t E F.

Then i t t u r n s o u t t h a t U(t) = supIqw(t)

I cp E

@}

defines an exposed element o f

Lin(q,p).

exposed elements o f

come o u t o f such a c o n s t r u c t i o n . C l e a r l y ,

if

@

Lin(q,p)

Furthermore one can p r o v e t h a t a l l

i s countable, t h e n t h i s c o n s t r u c t i o n p r o v i d e s an a l g o r i t h m f o r t h e

Hahn-Banach Theorem.

82

Linear Functionels

A t t h e end o f t h i s survey about o t h e r v e r s i o n s o f t h e Hahn-Banach Theorem we l i k e t o mention t h a t t h e r e a r e a l s o Hahn-Banach Theorems f o r s t r u c t u r e s which a r e n o t l i n e a r (see

, for

example, C i g n o l i [ 7 5 1 ) . Furthermore

t h e r e has been c o n s i d e r a b l e progress concerning o p e r a t o r - V a l ued HahnBanach Theorems (Wi t t s t o c k l 3 3 2 1 , [3331).

1.9.3

A l g o r i t h m s f o r Networks

There a r e theorems whose f o r m u l a t i o n s g i v e an immediate i d e a how t h e p r o o f s have t o proceed. The Sandwich Theorem i s an example f o r such a theorem and, f o r f i n i t e s e t s , t h e Flow Theorem i s a n o t h e r example. L e t us e x p l a i n t h a t i n d e t a i l . Take t h e statement o f t h e Sandwich Theorem, s a y i n g t h a t whenever

q 5 p

then t h e r e i s a l i n e a r map i n between. I f t h a t statement i s

t r u e i n general t h e n i t must a l s o be t r u e f o r a minimal s u b l i n e a r b t q instead o f p B u t then t h e theorem g i v e s a l i n e a r t 1-1 t q which i s a

.

s p e c i a l case o f a s u b l i n e a r f u n c t i o n a l . Since

6

= p

.

was minimal we t h e n have

T h i s argument shows t h a t f o r t h e p r o o f o f t h e s t a t e d theorem one

has t o show t h a t every minimal sub1 n e a r

2 q

i s i n f a c t l i n e a r . This

p r o o f then i s a m a t t e r o f r o u t i n e .

A s i m i l a r argument i n f a c t provides a p r o o f f o r t h e theorem on f l o w s i n networks which we gave i n s e c t i o n 1 8.2. Under t h e h y p o t h e s i s t h a t t h e existence o f a s u i t a b l e flow i s equivalent t o s u f f i c i e n t import capacity l e t us c o n s i d e r a minimal system among t h e p i p e l i n e systems such t h a t s t i l l t h e s u f f i c i e n t i m p o r t c o n d i t i o n h o l d s . Then t h e r e i s a p o s s i b l e and p o s i t i v e f l o w i n t h a t minimal system. B u t t h e n a l l t h e p i p e l i n e s have t o work on f u l l c a p a c i t y , o t h e r w i s e we would have a c o n t r a d i c t i o n t o t h e m i n i m a l i t y o f t h e p i p e l i n e system. Hence, t h e e s s e n t i a l argument i n t h e p r o o f o f t h e theorem c o n s i s t s i n showing t h a t t h e c a p a c i t i e s o f a minimal p i p e l i n e system define a p o s s i b l e p o s i t i v e f l o w . A c t u a l l y t h i s procedure has some advantages over t h e p r o o f we have given, s i n c e i n a d d i t i o n i t p r o v i d e s an a l g o r i t h m f o r f i n d i n g t h e d e s i r e d f l o w . T h i s a l g o r i t h m c e r t a i n l y c o n s i s t s of making c o n s e c u t i v e l y each p i p e l i n e as smal.1 as p o s s i b l e w i t h o u t v i o l a t i n g t h e i m p o r t c a p a c i t y c o n d i t i o n . T h i s a l g o r i t h m has t h e f u r t h e r advantage t h a t t h e flow one o b t a i n s by t h i s procedure i s an extreme p o i n t i n the s e t o f a l l possible p o s i t i v e flows.

The Sandwich Theorem

In t h e case of n o n f i n i t e as i n the f i n i t e case, s t h s case a ( t r a n s f i n i t e o f the algorithm f o r the ceding s e c t i o n .

1.9.4

83

network systems t h e s i t u a t i o n i s not as simple nce measurability conditions a r e involved. B u t i n algorithm can e a s i l y be obtained by a p p l i c a t i o n Sandwich Theorem which we described i n t h e pre-

Remarks about Networks and t h e Disintegration Theorem

Around 1960 ( s e e 11881 t o 11911) a systematic i n t e r e s t s t a r t e d i n t h e existence of product measures with prescribed marginal measures .when additional conditions a r e imposed upon t h e product measure. These additional conditions require t h a t e i t h e r t h e product measure has t o admit a d i s i n t e g r a t i o n w i t h r e s p e c t t o given t r a n s i t i o n p r o b a b i l i t i e s o r i t has t o be dominated by a given measure on the product space. In f a c t these two conditions a r e intimately connected. In t h e p u r s u i t of t h i s problem V . Strassen wrote h i s celebrated paper 13043 where he gave the following the0 rem: The Strassen Disintearation Theorem:

(X,x,m) -be a measure ~space with p o s i t i v e -~ f i n i t e measure m . ---Assume t h a t f o r every x E X a sublinear Let -

E be-a separable ---Banach space and l e t

functional

0" E ---__ i s given such t h a t

px

s u p I l p , ( s ) l / s E E, IIsIII l l < m -f o_r_a l l

i)

II pxll

ii)

functions X 3 x -, II pxII and x -,p x ( s ) f_ or _ a l l s E E , functions in 1 L (dm).

Then f o r every u ---

E E'

=

with -

~ ( sI ) J p,(s) dm(x) X

there are --

pX

for all -

s

E E

such t h a t --

E E'

i)

lJx 5 P,

ii)

x

-,

for all

u,(s)

x

E X

is,for all

s

E E

,in

1 L (dm)

are,

x

E X

Linear Functionals

84

iii)

~ ( s =)

I X

u x ( s ) dm(x) -for a l l s E E.

This theorem has a wide range o f applications ([173],[229]). The d i f f e r e n c e between t h i s theorem and appropriate s p e c i a l i z a t i o n s of t h e Disintegration Theorem ( a pre- Strassen Theorem i n t h e sense of [2431 i s t h a t here the sublinear L 1- valued operator r e a l l y dominates t h e l i n e a r one whereas i n our case we only considered domination almost everywhere. Of course, combining our r e s u l t with a monotone l i f t i n g (when such a l i f t i n g e x i s t s ) one obtains t h e preceding theorem. This approach t o the StrassenTheorem i s due t o Heinz Ktinig [1991 and has been considered subsequently by other authors ([2431,[2451 ,[3361). Actually the pre - Strassen Theorem i s completely s u f f i c i e n t f o r t h e treatment of s u i t a b l e product measures with prescribed marginals, s i n c e i n t h i s case one i s only i n t e r e s t e d i n almosteverywhere e q u a l i t y . We have l e f t out applications concerning marginal measures, s i n c e these applications a r e standard by now. In t h e case of f i n i t e measure spaces t h e r e is a very intimate connection between combinatorics and network theory on one h a n d and product measures with prescribed marginals on the o t h e r hand. We have shown t h a t Gale's fundamental existence theorem f o r flows i n networks can be obtained from a s u i t a b l e Disintegration Theorem. The o t h e r d i r e c t i o n , namely t h a t r e s u l t s about measures w i t h prescribed marginals can be obtained from network theory and combinatorics, has been observed before. For example i n [911 the Marriage Theorem i s used as an e s s e n t i a l tool and i n 11531 t h e FordFul kerson Theorem plays t h e same r o l e . That, f o r example, the Marriage Theorem and the Ford-Ful kerson Theorem a r e closely r e l a t e d i s we1 1 known [ 1751. In f a c t , the Ford-Ful kerson Max-FlowMin-Cut-Theorem can be used as the s t a r t i n g point f o r network theory and p a r t s of combinatorics. This theorem deals w i t h a f i n i t e network X where only one point s E X i s allowed t o have nonzero production and only one t E X i s allowed t o have nonzero consumption. All t h e other points a r e only s t a r t i n g and endpoints f o r pi pel i nes Then, under prescribed pi pel i ne capacity, t h e question i s t o determine t h e maximal flow from s t o t along the pipe1 ine system. The Ford-Ful kerson Theorem [1101 s t a t e s t h a t t h i s max-flow i s equal t o the min-cut of t h e system, where t h e min-cut i s the minimal import capacity of s e t s A c X with t E A and s d A.

.

85

The Sandwich Theorem

I n f a c t , even G a l e ' s theorem (which c e r t a i n l y l o o k s much more s o p h i s t i c a t e d and has t h e Ford-Fulkerson Theorem as immediate consequence) can be obt a i n e d w i t h t h e h e l p o f max-flows and min-cuts. We d i d n o t show t h a t f o r f i n i t e networks ( w h i c h were n o t i n t h e focus o f o u r i n t e r e s t anyway) one can always have i n t e g e r - v a l u e d f l o w s , p r o v i d e d t h a t t h e consumption measure and t h e c a p a c i t y measure a r e i n t e g e r - v a l u e d .

I n f a c t , t h i s r e s u l t does n o t f o l l o w w i t h o u t a f u r t h e r argument. To o b t a i n t h e i n t e g e r - v a l u e d f l o w one has t o c o n s i d e r an extreme p o i n t o f a l l possible flows. I n f a c t , the r e s u l t i n i t s e l f i s n o t surprising a t a l l i f one l o o k s a t i t f r o m t h e v i e w p o i n t of a l g o r i t h m s , s i n c e t h e a l g o r i t h m we have b r i e f l y d e s c r i b e d i n t h e p r e c e d i n g s u b s e c t i o n c e r t a i n l y l e a d s t o integer-valued flows.

I n t e g e r - v a l u e d f l o w s a r e i m p o r t a n t f o r a r e f o r m u l a t i o n o f a s p e c i a l case o f t h e Flow-Theorem i n terns of(0,l)-

m a t r i c e s ( m a t r i c e s w i t h o n l y zeros

and ones as e n t r i e s ) . T h i s v e r s i o n o f t h e theorem was found i n 1957 b y

A.J. Ryser [2791: Consider m a t r i c e s o f s i z e m S = ( s l , ... ,sn)

and

be t h e s e t o f a l l vector

S

. An

by

n. For given vectors

R = ( r l , . . . , rm)

w i t h nonnegative i n t e g e r s as components l e t

( 0 , l ) - m a t r i c e s w i t h row sum v e c t o r

element o f

A(R,S)

R

A(R,S)

and column sum

can be c o n s i d e r e d as a network, where

t h e e n t r i e s correspond t o t h e p i p e l i n e c a p a c i t i e s and where

R

and

S

p l a y t h e r o l e o f t h e p r e s c r i b e d m a r g i n a l measures. Then t h e e x i s t e n c e o f a s u i t a b l e f l o w corresponds t o

A(R,S)

0

.

I n order t o translate the

i m p o r t c a p a c i t y c o n d i t i o n t o t h e s i t u a t i o n under c o n s i d e r a t i o n we i n t r o d u c e , f o r a g i v e n row sum v e c t o r

R = (rl,.. .,rm) t h e maximal m a t r i x

MR

, that

i s t h e m a t r i x where i n each row w i t h i n d e x j t h e f i r s t r e n t r i e s a r e one j be t h e column sum v e c t o r and t h e o t h e r s a r e zero. L e t SR = (S1,...,~,,) of

MR

. We

say t h a t

of

S

and

SR

-

s1 2

-s 2

2

...2

-

S < SR

(SR

dominates

a r e rearranged such t h a t

sn

, we

have

S)

i f , p r o v i d e d t h e elements

s1 2 s 2 2...1 sn

and

86

Linear Functionals

i)

and

k

z si

i=1

n ii) z si i=1

k I 1 si i=1

for a l l

k = l,...,n-1

n

=

x Si

i=1

Theorem: ( [2791 o r [2801) : A(R,S)

i s non --

empty i f , and o n l y i f ,

S < SR

SECTION 1.2 ORDER UNITS AND LATTICE CONES

The theory of vector l a t t i c e s , and s u i t a b l e subcones of vector l a t t i c e s , permeates many f i e l d s of a n a l y s i s . Therefore t h e main emphasis of t h i s chapter l i e s on t h e study of l a t t i c e cones. These a r e ordered cones such t h a t t h e suprema of f i n i t e l y many elements e x i s t and such t h a t the d i s t r i butive law holds f o r these suprema. This notion turns out t o be t h e natural generalization of vector l a t t i c e . Natural examples f o r l a t t i c e cones a r e : The upper ( o r lower) semicontinuous functions, t h e convex f u n c t i o n s , o r the cones given by t h e sub- or the s u p e r l i n e a r f u n c t i o n a l s . Special a t t e n t i o n will be given t o the l a s t two examples.

B u t before we begin a systematic t r e a t i s e of l a t t i c e cones we s t a r t w i t h a fundamental notion, namely with order u n i t cones. These a r e cones where a l l elements can be measured from above by means of a c e r t a i n i n v e r t i b l e element. This element plays t h e same r o l e as t h e constant function lX i n t h e case o f a concrete cone c o n s i s t i n g of bounded functions on some s e t X . Order u n i t s give r i s e t o t h e introduction of s t a t e s and s t a t e spaces. S t a t e spaces will play an e s s e n t i a l r o l e throughout t h i s book, they a r e more o r l e s s the s t r u c t u r e spaces of order u n i t cones. (Those readers i n t e r e s t e d i n how our s t a t e s a r e r e l a t e d t o s t a t e s o f , say, C*- algebras a r e r e f e r r e d t o chapter 11. 6 . 4 ) .

-

Then we turn our a t t e n t i o n t o special examples of order u n i t cones, where the cones a r e a c t u a l l y vector spaces w i t h an additional l a t t i c e s t r u c t u r e . F i r s t we prove the basic r e s u l t (Kakutani - Krein - Stone - Yosida Theorem) t h a t a l l order u n i t vector l a t t i c e s a r e isomorphic t o dense subspaces o f C ( X ) , where X is t h e compact s e t o f l a t t i c e s t a t e s . Then, i f ' i n addition the order u n i t vector l a t t i c e is order complete, then the well known r e s u l t i s shown t h a t t h e s t r u c t u r e space X i s extranallydisconnected o r , i n o t h e r words, a Stone space. This, of course, c l a r i f i e s the nature of 87

88

Linear Functionals

the order complete vector l a t t i c e s which we introduced a t the beginning of t h i s book as image spaces for the functionals under consideration. A t the end of section 2.3 we deal w i t h the basic construction (due to Loomis and Sikorski) of a U- montone l a t t i c e homomorphism from the space of a l l bounded Baire functions on X ( X compact and extremally disconnected onto C ( X ) . This construction will be important when vector valued representing measures are treated (Chapter 11.4). We continue by discussing under which conditions the cone of a l l monotone linear functionals on a given cone F i s a l a t t i c e cone (with respect to the pointwise order on F ) . This leads t o study the Riesz property and the f i n i t e sum property (FSP) f o r cones. As a byproduct of these considerations we obtain a powerful version of the Cartier-Fell-Meyer The orem. In Chapter 2.6 we consider duality theory for convex cones. Generally speaking, convex cones do not admit a duality theory which i s a s nice as in the case of vector spaces. The reason f o r t h a t deplorable f a c t i s t h a t we can introduce too many different order relations f o r convex cones whereas in the case of vector spaces the structure of order relations i s unique insofar as every order relation i s given by a suitable subcone (which then automatically bears the order which we call natural order). In 2.8 free l a t t i c e cones are treated. The free l a t t i c e cones are the universal elements which come u p when arbitrary order unit cones are embedded into l a t t i c e cones. These free l a t t i c e cones give r i s e t o the introduction of simplicial cones. A cone (F, < ) i s said t o be simplicial i f for i t s positive dual there i s a linear embedding I, : FT + (LF); (positive dual of the free l a t t i c e cone generated by F) such t h a t composition with the restriction map t o F i s the identity. A standard example for a simplicia1 cone i s a cone having a Choquet simplex as base. Another important notion coming o u t o f the consideration of free l a t t i c e cones i s the notion of character. These are s t a t e s on a given cone characterized by a unique extension property t o the assigned free l a t t i c e cone. An important observation i s t h a t they coincide with the extreme point s e t of the s t a t e space. Therefore i t i s n o t a t a l l surprising t h a t in concrete situations they coincide in one way or another with characters

89

Order Units and Lattice Cones

d e f i n e d i n t h e usual way.

F i n a l l y , i n 1.2.11 we r e v i e w t h e t o p i c s discussed i n t h e p r e c e d i n g s e c t i o n s and we t r y t o e x p l a i n a t some examples t h e i n t e r r e l a t i o n o f o u r n o t i o n s and r e s u l t s w i t h t h e c l a s s i c a l t h e o r y . F o r example we prove K a k u t a n i ' s c o n c r e t e r e p r e s e n t a t i o n o f AM- spaces and we p r e s e n t t h e K o r o v k i n theorem for

C(K).

2.1

ORDER UNIT CONES

I n a cone

F we do n o t always have n e g a t i v e elements. But, o f course, i t

f f F

may happen t h a t a s p e c i a l with

f

+ g

= 0. Since

t h a t we denote i t by

g -f

i s invertible, i.e. there i s a

r) €

F

i s t h e n u n i q u e l y determined i t i s q u i t e n a t u r a l

, and

one e a s i l y sees t h a t t h e usual r u l e s o f

c a l c u l a t i o n f o r n e g a t i v e elements h o l d . I n p a r t i c u l a r we t h e n have t h a t t h e v e c t o r space

I R f = t a f l a € IRI

i s a subset o f

F.

I n the following

an i m p o r t a n t r o l e i s p l a y e d by t h e i n v e r t i b l e elements which a r e s t r i c t l y p o s i t i v e . Here, an element s t r i c t l y positive i f

0< f

f

o f a p r e o r d e r e d cone holds, and i f

Now, o u r b a s i c d e f i n i t i o n reads as f o l l o w s :

f < 0

(F,<

)

i s s a i d t o be

does n o t hold.

Linear Functionals

90

2.1.1

Definition:

(F, < ) be a preordered cone. An i n v e r t i b l e , s t r i c t l y p o s i t v e I E F i s c a l l e d -o r d e r u n i t i f f o r every g E F t h e r e i s some A t 0 such t h a t g < A I. ( F , i , I) i s t h e n c a l l e d --o r d e r u n i t cone ( i i ) The s u b l i n e a r f u n c t i o n a l given, f o r g E F, b y (i)

Let

element

SI(g)

= infIa E R

on t h e o r d e r u n i t cone

(F, < I )

i s c a l l e d t h e -order u n i t functional.

And t h e s e t

R =

tp

: F

+k I

p

I g < a I1

l i n e a r , monotone,

1.1 5

SIl

i s c a l l e d t h e -s t a t e space o f o u r o r d e r u n i t cone, i t s elements a r e c a l l e d

states.

Since t h e o r d e r u n i t f u n c t i o n a l i s l i n e a r on t h e subspace I R I i t must be equal ,on t h i s subspace,to e v e r y l i n e a r f u n c t i o n a l i t dominates (see c h a p t e r 1.2). Hence, f o r every s t a t e v E n and e v e r y a E R we have p ( a I ) = a.

F can always be represented b y a f u n c t i o n cone on a s u i t a b l e s e t . The answer i s yes i n t h e sense t h a t t h e r e i s a n a t u r a l o r d e r u n i t cone o f f u n c t i o n s on t h e s t a t e space n o f F such t h a t t h e s t a t e s i n R i d e n t i f y t h e elements of F w i t h c o n c r e t e f u n c t i o n s on 62 T h i s i s e a s i l y e x p l a i n e d : For f E F we d e f i n e 7 : n -,I6 t o be t h e e v a l u a t i o n ? ( p ) = v ( f ) f o r a l l u E n. The f u n c t i o n 7 i s s a i d t o be t h e Gelfandtransform o f f . I n p a r t i c u l a r we have 7 = 1 , and e v e r y f, f E F i s upperbounded on R . Note, t h a t t h e G e l f a n d t r a n s f o r m i s l i n e a r , i.e.

One may ask i f an o r d e r u n i t cone

.

A t A2f2) =

(Alfl

Because the set

h p1

R

t

f(Aplt(l-A)p2)

n

A2T2

( 1 - x ) ~E ~n whenever

must be convex and t h e

f u n c t i o n s on ul,p2

t

Alil

T,

. (f

: n

= ~

f p , () + ( l - A ) f ( p 2 )

+nT

f

i s affine i f f whenever

0 5 A I 1 and

E n). Now we use t h e G e l f a n d t r a n s f o r m t o endow n w i t h a t o p o l o g y , ^F = { ? I f E F} c o n s i s t s o f

namely t h e c o a r s e s t t o p o l o g y such t h a t

91

Order Units and Lattice Cones

k- valued continuous f u n c t i o n s (here, t h e t o p o l o g y o f lk i s , o f course, It--,a)la

understood t o be t h e t o p o l o g y which has

F

t h e open s e t s ) . Since by d e f i n i t i o n

E

R) as a subbase f o r n this

separates t h e p o i n t s o f

t o p o l o g y must be H a u s d o r f f . And i t i s easy t o deduct from T y c h o n o f f ' s theorem ( o r by d i r e c t l y showing t h a t e v e r y u l t r a f i l t e r on .R

converges)

n i s compact under t h i s t o p o l o g y .

that

^F

Note, t h a t i f we endow then

w i t h the pointwise order r e l a t i o n I

(F, I, 1, ) i s a g a i n an o r d e r u n i t cone and

sup,

on

R

i s the associated

order u n i t functional. 2.1.2

Lemma:

The G e l f a n d t r a n s f o r m ---i s an o r d e r u n i t homomorphism f r o m (F, (F, I, I , ) , i . e . (i)

it i s linear

(ii)

^f

(iii)

I= 1 ,

(iv)

SI(f)

whenever

5

f
A

= max,f

= sup,f

-

for -

4

onto ,I ) -

f,g E F

-for all

f E F.

Proof: We a l r e a d y know ( i ) and ( i i i ) . A s s e r t i o n ( i i ) i s an immediate consequence o f t h e f a c t t h a t s t a t e s are assumed t o be monotone, and ( i v ) i s a d i r e c t consequence o f t h e Norm Theorem (1.3.3. ) Let

(F, <

o

, IF) and

(G, < , I G ) be two o r d e r u n i t cones and c o n s i d e r

G.

i s c a l l e d an --o r d e r u n i t cone homomorphism i f i t

+

Q

t h e cone o p e r a t i o n s and o r d e r r e l a t i o n s , t h a t i s , i f f2) =

x1

Q (fl)

+

h2 Q ( f 2 )

for all

and whenever Q

and

fl < f 2

,

fl,f2 E F

i s c a l l e d an --o r d e r u n i t cone isomorphism 0-l

if

fl,f2 E F, X1,x2

E IR+ ;

. Q

i s b i j e c t i v e and

Q

a r e o r d e r u n i t cone homomorphisms.

So, i n case t h a t t h e G e l f a n d t r a n s f o r m i s i n j e c t i v e t h e o r d e r u n i t cones

92

Linear Functionals

(F,<,

I)

and

(r,

5

, In) are

isomorphic. I n case t h a t

^F

space, we can check vhether F and

F

i s a vector

a r e isomorphic by l o o k i n g a t t h e

f o l 1owing semi norm

A t r i v i a l consequence o f 2.1.2 ( i v ) i s t h a t ISIl(f)

2.1.3

= supnlfl

f E F.

for all

Lemma:

L e t F --be a v e c t o r space. I f lSIl -

i s a norm t h e n

(F,<,

I) and

(F, I, In) a r e isomorphic, -and i n p a r t i c u l a r t h e p r e o r d e r < --must be a n t i symme t r ic

.

Proof:

lSIl

Let

SI(f-g)

b e a n o r m and assume = SI(g-f)

= 0. Hence

T

=

G.

Then f r o m 2.1.2

ISIl(f-g)

= 0

( i v ) we conclude

and we must have

f = 9.

The antisymmetry o f t h e p r e o r d e r < f o l l o w s immediately from t h e f a c t t h a t 5

i s antisymmetric on

For v e c t o r spaces

F.

F one

can g i v e another u s e f u l e q u i v a l e n t d e f i n i t i o n

f o r t h e s t a t e space. 2.1.4

Proposition:

L e t F be a v e c t o r space. Then t h e s t a t e space o f (F,< , I ) i s equal t o ------_.---

111 : F - r R

I !.I

linear,

!J

s lSIl, ~

( 1 )= 11

.

Proof: We know a l r e a d y t h a t f o r s t a t e s we have

p ( 1 ) = 1 and

So, what remains t o prove i s t h a t a l i n e a r f u n c t i o n a l p e r t i e s s t a t e d above i s i n f a c t monotone and f u l f i l l s

!J

s

SI I ISI!

.

u h a v i n g t h e pro!.I

s

SI

. Indeed,

Order Units and Lattice Cones

let

g< f

Then

0<

x

p(f)

Now, s i n c e and

a E W

xI

I -(f-g)

x

Thus

x

and c o n s i d e r

and we g e t :

i

=

S I ( h I ) t SI(XI

2

u(x I + g - f )

2 p(g)

and

xI .

(f-g) <

such t h a t

t 0

93

p

g - f ) = ISIl(XI + g - f )

+ p(g) - u(f).

= A

must be monotone.

i s monotone we have w i t h f < a 1 . Hence p

u(a I

-f

) 2 0

for

arbitrary

U ( f ) I i n f I a E IR I f < a 1 1 = S I ( f ) .

2.2

THE

f E

F

0

KAKUTANI-KREIN-STONE-YOSIDA THEOREM

We assume i n t h i s s e c t i o n t h a t

(F,<)

i s a v e c t o r l a t t i c e . L e t us r e c a l l

F whose p r e o r d e r < i s a n t i symmetric and w i t h every f,g € F t h e v e c t o r space F c o n t a i n s t h e supremum and infimum o f f and 9 . L a t e r on, i n s e c t i o n 2.4., we s h a l l t h a t a v e c t o r l a t t i c e i s a v e c t o r space

g e n e r a l i z e t h e n o t i o n o f v e c t o r l a t t i c e s t o l a t t i c e cones. L e t us mention again some b a s i c i d e n t i t i e s whose p r o o f s a r e s t r a i g h t f o r w a r d and a r e l e f t t o t h e reader:

(11

suolf,gl

=

-

(2)

sup(xf, xg)

=

x sup(f,g),

(3)

sup(f+h ,g+h) = sup(f,g)

-

(4)

f = f+

where

f+ = sup(f,O)

2.2.1.

Lemma:

f-

for all

,

Assume t h a t ( F , < , I ) ~-

f E F.

infl-f,-gl

f-

=

+ h,

.

f, g E F

X t 0, f,g E F f,g,h

E

F

f E F

(-f)

t

i s a vector l a t t i c e w i t h order u n i t ---____---

u with u(f-)

= 0

and -

I -and l e t

94

Linear Functionals

If SI(-f) -

= 1 ----then t h e r e i s a s t a t e

with

v

and

v(f+) = 0

v ( f - ) = v ( f ) = +1. Proof: By Lemma 2.1.2

xI2

then

1=

We o b t a i n

( i v ) there i s a state

f . Hence p(f)

xI= + s u(f )

S i m i l a r l y , t h e assumption v

with

with

P

5 SI(f

SI(-f)

v ( f + ) = 0, v ( f - ) = t 1

+) =

5 1 and t h u s

.

for a l l

f, g E F. Note t h a t sup(f,g)

i s c a l l e d l a t t i c e homomorphism i f = (f-g)'

Hence i n o r d e r t o prove t h a t a g i v e n s t a t e = sup(p(h),O)

n be t h e s t a t e space o f

F

we have d e f i n e d a t o p o l o g y on

R

Let

.

1 > 1

If i.e.

+ p(f )

=

P

p(sup(f,g))

= 1

SI(ft)

.

1, u ( f - ) = 0.

= sup(P(f),u(g))

+ g f o r a l l f, g E F. i s a l a t t i c e homomorphism

for all

h E F.

. Recall

t h a t i n t h e preceding s e c t i o n under which t h e s t a t e space i s compact.

The s e t o f a l l l a t t i c e homomorphisms on

2.2.2

+

0

p

U(ht)

= f

1 ensures t h e e x i s t e n c e o f a s t a t e

A state

i t s u f f i c e s t o show

P(f) = 1

sup(x I ,O) t sup(f,O)

F

i s a subset o f

n

.

Lemma:

{U : F -, k I 1.1

Proof: -

l a t t i c e homomorphism}

-i s

a compact _ subset~ o f

I

~

.

F i x f, g E F Then c l e a r l y 111 E R I P (sup(f,g)) = s u p ( u ( f ) , I - ( g ) ) l i s closed, hence compact. So t h e a s s e r t i o n f o l l o w s from 111 :

F

-, fi

I

l a t t i c e homomorphism} =

)J

Now, l e t S denote a compact H a u s d o r f f space. Then we denote C ( S ) = i f : S -+ IR f c o n t i n u o u s } . I f C ( S ) i s equipped w i t h t h e p o i n t w i s e order

I then

(C(S),

I

w e l l known sup-norm on We show t h a t (F,*)

S

, Is)

i s an o r d e r u n i t cone and

lSIl

(which we denote i n t h e f o l l o w i n g by

i s the

I I . II ) .

( C ( S ) , 5 ) i n c l u d e s a l r e a d y a l l examples o f v e c t o r l a t t i c e s

with order u n i t

w i t h respect t o

lSIl

I such t h a t

. Indeed,

lSIl

we have:

i s a norm and

F

i s complete

-

95

Order Units and Lattice Cones

2.2.3

[2 1 ) :

Theorem f'Kakutani-Krein-Stone-Yosida

Let (F,< , I ) -be a _vector- l a t t~i c e w i t h an order u n i t . Then t h e r e i s a compact Hausdorff space S ---and an order u n i t homomorphism @ from (F, i ,I ) i n t o ( c ( S ) , I , Is) -such t h a t @ ( F ) i s sup-norm dense in

_ .

we have C(S). Moreover, --

I] @ ( f ) l l= I S , l ( f ) 5 --a norm,then 0

-for all f

In p a r t i c u l a r , i f ISI\ from F onto a dense subspace o f C(S).

E

F.

i s an order u n i t isomorphism

Proof: Let n be t h e s t a t e space of ( F , < , I ) . F i x some f E F with 1 S I l ( f ) = 1 and assume without l o s s of g e n e r a l i t y t h a t S I ( f ) = 1. We show t h a t t h e r e i s a l a t t i c e homomorphism K(f-) = 0 :

g E F Then

Consider a s t a t e u w i t h

the compact s e t

u E no

cp =

= Cu E

g We claim t h a t

.

To prove t h i s we take

where

62

+ inf{g.1 I

A c

i E A}.

n

with

li

U(f)

1 v(ft)

=

K(f+) = 1 and

u ( f + ) = 1 and define f o r

+

1, u(g )

c

u(g-) = 01.

n n...n n * 0 f o r a l l g1 ,... ,gn E F . 91 n { l ,...,n } t o be a maximal subset with p(cp) > 0 , We can assume A + 0 s i n c e otherwise

n n...n n . For 0 < h we define n 91 t l ( h ) = sup{p(k) 0 < k < h and t h e r e i s x > 0 with k < x c p ) Then v i s l i n e a r on the cone of a l l p o s i t i v e elements of F and can t h u s be extended t o a l i n e a r function on F which will be c a l l e d v again. The proof of t h e l i n e a r i t y of v follows from an elementary computation: Indeed, v i s s u p e r l i n e a r by d e f i n i t i o n . So i t remains t o show t h e suba d d i t i v i t y of u Therefore, l e t h l , h 2 z 0 and assume O < k< h l + h 2 . p E

.

.

Then d e f i n e k l and

k = kl

+

k2

=

inf(k,hl)

.

and

k2 = k

-

kl

. We have

From t h i s observation we derive

v(hl

0
+ h 2 ) I v(hl) +v(h2),

which i s t h e s u b a d d i t i v i t y of u . We have u(w) = v(cp) > 0 and, f o r a l l O < h , 0 Iv(h) Iu(h) by d e f i n i t i o n of v and monotony o f u

From f + < ( l + c ) I , f o r every

E

> 0 , we get

Hence u ( f + ) = ~ ( 1 2) ( S I ( c p ) ) - l v ( w ) > 0 F i n a l l y , we claim t h a t

p =

(u(I))-'u

0

I v(1-f

+)

+

I u(1-f ) = 0

in view of t h e monotony of i s in

n , i = 1,..., n . gi

. .

v

.

96

Linear Functionals

We g e t

p ( f t ) = 1 from

quence o f

u(1) = v(f+)

v ( g 71) = 0, i = 1, ..., n.

and p ( g i )

-

p(g;)

i s a conse-

= 0

The l a t t e r f a c t f o l l o w s f r o m t h e ob-

h = g- , t h e u ( k ) must be equal t o zero f o r those j considered i n t h e d e f i n i t i o n o f v (because k = 0 i f j E A and

servation that, i f k

t

because p(inf{cp,gjl)

= 0

otherwise).

n i s compact. We have K(g+) K ( q ) =

since

~ ( g ' ) = s ~ p ( ~ ( g ) , O ) .T h i s shows t h a t Furthermore, we have

f o r a11

E

n

SEF

R

g

g E F which means

o

. Let

s

c

n be t h e s e t

i s a compact H a u s d o r f f space.

: F -,C(S) by @ ( f ) ( w ) = v ( f ) f o r a l l v E S and a l l f € F . i s c l e a r l y an o r d e r u n i t homomorphism. The p r e c e d i n g argument shows

Define

@

f E F with

that for K(f-) =

o

IS,l(f)

=

or

i s dense i n

IS,l(f)

K ( f - ) = 1 and

11 @ ( f ) I I

for all

u n i t isomorphism from

2.3

K(f-) = S

o

K

i s a l a t t i c e homomorphism.

K

K ( f t ) = 1 and

o f a l l l a t t i c e homomorphisms, i . e .

0

Hence t h e r e i s some

C(S)

= 1 there i s

K(ft) =

f E F.

F onto

o ,

K

i.e.

E S

IK(f)l

T h i s proves t h a t

@(F) i f

with

o

=

K ( f t ) = 1 and

1

. Hence

i s even an o r d e r

ISI/ i s a norm. F i n a l l y ,

o(F)

which i s a consequence o f t h e Stone Weierstrass theorem.

ORDER COMPLETE VECTOR LATTICES WITH ORDER U N I T

Recall t h a t a vector l a t t i c e

(F, <)

i s c a l l e d o r d e r complete i f f o r any

A c F which i s o r d e r bounded from above ( i . e . t h e r e i s f E F w i t h a < f f o r a l l a E A ) t h e supremum o f A , sup(A), e x i s t s . Then, c e r t a i n l y , f o r e v e r y subset B c F which i s o r d e r bounded from below t h e infimum o f B, i n f ( B ) , e x i s t s s i n c e i n f ( B ) = - sup(-B). L e t S be a compact H a u s d o r f f space. Then C ( S ) equipped w i t h t h e p o i n t subset

97

Order Units and Lattice Cones

wise order if

I i s a ( n o t n e c e s s a r i l y o r d e r complete) v e c t o r l a t t i c e . Here,

f,g E C(S),

t h e supremum o f

supremum, i . e .

A c C(S)

.

in

(C(S),s)

i s j u s t the pointwise

for all

s E S. However, i f

i s an i n f i n i t e bounded subset whose supremum sup(A)

(C(S),<),

in A

f,g

= max(f(s),g(s))

sup(f,g)(s)

then

sup(A)

For instance, l e t

and c o n s i d e r

S = [0,11

A = { g E C(S) 1 0 Ig 5 1, g ( 0 ) = 0). Here, sup(A) = lS hand t h e p o i n t w i s e supremum f i s d e f i n e d b y

i

f(s) =

1

s 9 0

0

s = o

exists i n

need n o t be t h e p o i n t w i s e supremum o f t h e f u n c t i o n s b u t on t h e o t h e r

I n o r d e r t o be a b l e t o d i s t i n g u i s h between these suprema we denote t h a t

A

p o i n t w i s e supremum of

by

Sup(a)

and, s i m i l a r l y , t h e p o i n t w i s e infimum

by I n f ( A ) .

A compact Hausdorff space

U

i s open whenever

c S

i s c a l l e d e x t r e m a l l y disconnected i f

S

i s open

( 0

denotes t h e c l o s u r e o f

0

U i n S ).

Simple examples o f extremally disconnected spaces a r e a l l f i n i t e s e t s endowed w i t h t h e d i s c r e t e t o p o l o g y . The n e x t p r o p o s i t i o n p r o v i d e s us w i t h many o t h e r i m p o r t a n t examples :

2.3.1

p o s i t io n -.(Stotie .-P r o-

[ 302] :

is order complete ifand only if --

C(S)

S

i s e x t r e m a l l y disconnected.

Proof: A t first, l e t

C(S)

that

We show,

6* S . , an

x B

6

16 o f

function

h

Certainly, 0 Ik I1

u c U

for a l l

fi .

x

= 0 on

. We

Therefore,

Now, assume t h a t

i

, gx(x)

= 0

and

. C(S)

i n view o f o u r assumption.

S \ i . F o r any u E i t h e r e i s k E C(S) w i t h S\U and k ( u ) = 1 . Hence k I h s i n c e k Ig X

conclude t h a t

h

must be open. Thus, S

0 Igx I1

such t h a t

which e x i s t s i n

h = 0 on

,k

i s open b y p r o v i n g t h a t t h e c h a r a c t e r i s t i c

gx E C(S)

inf g xaii

=

U c S, which i s open, such

i s continuous. By Urysohn'slemma t h e r e i s , f o r e v e r y

element

gx(u) = 1 f o r a l l Put

be o r d e r complete. Consider

i s the characteristic function o f S

i s extremallydisconnected.

i s extremallydisconnected. L e t

A c C(S) be bounded

98

Linear Functionals

f

from below and l e t

be t h e p o i n t w i s e infimum o f

A

which i s upper

semicontinuous, n o t necessarily continuous. We have , f ( S ) c [ - N, N ] some

. Consider

N > 0

define

U. J ,m

x

-1 = f (]-m,x

j ,m

. Furthermore,

m

=

j ,m

- N

,j

2N j

t

= 0,1,

U

[ ) . Then we o b t a i n

o,m

...,m y c

U

m E

for

N, and

c... c U

l,m

m ,m

U a r e open. L e t fjYm be t h e j ,m c h a r a c t e r i s t i c f u n c t i o n s o f t h e open and c l o s e d s e t s Uj,m'Uj-l,m

for all

j

=

-

. (We p u t

O,l,...,m

d i s j o i n t . Define

f,

=

all

U-l,m

=

k3 ) . These s e t s are,of course,

m

.

E x

j , m fj,m a r e continuous. We c l a i m t h a t (f,) t h i s i t s u f f i c e s t o prove t h a t , f o r

Hence we c o n s i d e r

j =O

x E S

fm i s continuous s i n c e a l l

By d e f i n i t i o n t h e s e t

whereas

Xj+l,m

and we assume x E

0 5

for a l l

If

x E

-

f(X)

for a l l

-

t h e n (**) i m p l i e s

.

f o r snme

1 I 2N;Fiy 0 I

fn(X)

[ X ~ , ~ , X ~ + ~ , ~ ~ .

.

fm Hence

-

1 f(X) 5 2 N 5 S

,

, hence

hEC(S) i s such t h a t

US

It i s w e l l known, t h a t

BX

2 a

for a l l

w i t h more examples o f e x t r e m a l l y d i s c o n n e c t e d B(X)

B(X)

i s t h e Stone

be t h e Banach space o f a l l

* .

bounded, r e a l v a l u e d f u n c t i o n s on a s e t

, where

S.

U

compact H a u s d o r f f spaces. Indeed, l e t

C(B X )

h2 a

g

on

h I g on a dense subset o f S and t h e r e f o r e i s t h e g r e a t e s t l o w e r bound o f A , and hence

P r o p o s i t i o n 2.3.1 p r o v i d e s

sup-norm.

'u j , k ) )

('j,k

x E Uktl ,n\Uk,n

So, ( * ) i s proved on a dense subset o f

on S T h i s shows, g C(S) i s o r d e r complete.

to

j=l

Now, t h e value o f f m ( x ) i s equal t o

fn i n s t e a d o f

a€A, s i n c e f = I n f ( A ) . I f

j ,m

n

denotes t h e l i m i t o f t h e fm then, by c o n t i n u i t y and ( * * )

g

a E A

3.

f,(X)

U k=m,n

= S'

has t o be some v a l u e i n t h e i n t e r v a l

The same argument holds f o r

(**)

s

and

s i s dense i n s .

f(x)

f

i s a norm Cauchy sequence. To show n 2 m and a l l x E S

i n o r d e r t o be sure t h a t x E Ujtl,m~~j,m j and k

.

mutually

-

X 0 I n B(X) we c o n s i d e r t h e i s i s o m e t r i c and o r d e r isomorphic Czech c o m p a c t i f i c a t i o n o f

X

99

Order Units and Lattice Cones

endowed with the discrete topology. On the other hand , B ( X ) i s clearly order complete. So, 13 X must be extremally disconnected by Proposition 2.3.1. I n particular, we mention the case X = N . Next, we prove another useful property of extremally disconnected sets which i s implicitly contained already in the proof of 2.3.1. Recall t h a t a subset V of a topological Hausdorff space S i s called meagre i f there are subsets W n c S, n E N , with V = U Wn and interior(p ) = 0 for a l l n . Meagre n

n a

sets are stable under countable unions, t h a t i s , the union of countably many meagre sets i s meagre. 2.3.2

Proposition ([338]) :

Let S be an extremally disconnected compact Hausdorff space and assume

that

i s bounded from below. Then

A c C(S)

{s E S

I (Inf

A)(s) > ( i n f ( A ) ) ( s ) } i s meagre.

Proof: Let

wn

Then a l l

=

n {s E S I a(s)

1

2F +

aEA Wn are closed and

(inf(A))(s)l,n E N

Is E S I ( 1 n f A)(s) > ( i n f ( A ) ) ( s ) }= U Wn with

-

UcW,

interior(in) * b

. Since

S

.

nm

. Assume there

Then there exists an open s e t

i s extremallydisconnected,

the characteristic function 10 of inf(A)

1 1- t inf(A) n U

5 -

.

I

U

i s some n E N

@ * U c S with

i s open and closed. Thus,

i s continuous. We obtain a for a l l

a E A ,

a contradiction. Let K be any compact Hausdorff space and denote by Baire ( K ) the vectorspace of a l l realvalued bounded Baire measurable functions on K (see Appendix A 1 ) . Recall t h a t Baire ( K ) i s the smallest vector space F of realvalued bounded functions on K which contains C ( K ) such t h a t InfIf, I n E N ) E F whenever f n E F and f n 2 g for a l l n E N and some g E F.

100

Linear functionals

Recall t h e d e f i n i t i o n o f a 2.3.3

-

Theorem (Loomis

C(S)-valued measure i n A 5 .

S i k o r s k i l‘3391):

be an e x t r e m a l l y disconnected compact H a u s d o r f f space. -Then f o r Let S -

e v e r y f E B a i r e ( S ) --t h e r e i s a unique L ( f ) E C ( S ) -such t h a t is E S 1 f ( s ) L(f)(s)l i s meagre. -The map L : B a i r e ( S ) -, C(S) -___ i s linear and d e f i n e s a C ( S ) - valued measure --u- a l g e b r a o f t h e B a i r e s e t s on S Proof: L e t M = I f E Baire(S)

I there

.

m

0”

w i t h respect t o the

S

i s L ( f ) E C(S)

.

S3

d i f f e r o n l y on a meagre subset o f

such t h a t

L(f)

and

f

M i s , by d e f i n i t i o n , a v e c t o r space s i n c e t h e u n i o n o f two meagre s e t s i s meagre. For f E M , l e t L(f), i ( f ) E C(S) be f u n c t i o n s which d i f f e r

. Then

Cs E S l L ( f ) ( s ) + % ( f ) ( s ) l i s meagre. From B a i r e ’ s c a t e g o r y theorem i t f o l l o w s t h a t I s E S I l ( f ) ( s ) = i ( f ) ( s ) l i s dense i n S. Hence L ( f ) = i ( f ) s i n c e

from

f

o n l y on a meagre subset o f

S

b o t h f u n c t i o n s a r e continuous. T h e r e f o r e L ( f ) Hence l :

M

-,

I t remains t o show t h a t

O f course, all

C(S) c

M

.

M = Baire(S): L e t f n E M and

n E N. Then L ( f n ) 1 L ( g )

i n f ( L ( f n ) ) E C(S), n inf(L(fn)) n

i s unique.

i s linear.

C(S)

Inf(L(fn)) n

and I n f ( l ( f n ) ) n

for a 1 and

g E

M such t h a t f n 2 g f o r

n E N

Inf(fn). n

. We c o n s i d e r By P r o p o s i t i o n 2.3.2,

d i f f e r on y on a meagre s e t . Furthermore,

I n f ( l ( f n ) ) and I n f ( f n ) d i f f e r o n l y on a meagre s e t s i n c e t h e c o u n t a b l e n n u n i o n o f meagre s e t s i s meagre. We o b t a i n f = I n f ( f n ) E M and n L ( f ) = i n f ( l ( f n ) ) . The corresponding a s s e r t i o n i s t r u e w i t h n Inf

M

i f the

= Baire(S)

Bc S

fn a r e bounded by some

g E M

linear. Let

Bn

For any B a i r e s e t

be t h e c h a r a c t e r i s t i c f u n c t i o n . C l e a r l y

D e f i n e m(B) = L ( l B ) . I n p a r t i c u l a r , we have

,n E

N

, be

instead o f

from above. We conclude,

from t h e remark p r e c e d i n g Theorem 2.3.3.

, l e t lB

Sup

m(0)

= 0

lB E Baire(S).

, since

L was

a sequence o f m u t u a l l y d i s j o i n t B a i r e s e t s

101

Order Units and Lattice Cones

in

. Then

S

1

"

n aBn

clearly

sES.

for a l l

( s ) = sup i n f { l (s) I n 2 m l m a Bk Bk k=l k=l

(s) = l i m inf n +-

n=l

1

I n view o f t h e p r o p e r t i e s o f

m

m( U Bn)

1n

=

"

L( 1

n a

) = Bn

m a

n

Hence

2.4

is a

m

n t L(lB ) k=l k

sup i n f I

= l i m in? + W

t

n

E m(Bk)

k=l

C ( S ) - valued measure.

we o b t a i n

I

n tm 1

.

0

LATTICE CONES

We cannot a v o i d t o b e g i n w i t h some elementary d e f i n i t i o n s . 2 . 4 . 1 D e f i n i t i on:

(F, < ) be some o r d e r e d cone ( w i t h an a n t i s y m m e t r i c o r d e r r e l a t i o n ) . have l a t t i c e cone i f any two elements f,g E F ( i ) ( F , i ) i s c a l l e d a ~-

Let

a supremum

sup(f,g) h

+

in

F such t h a t t h e d i s t r i b u t i v e l a w h o l d s , i . e .

sup(f,g)

= sup(h+f,

for a l l

h+g)

f,g,h

E

F.

( i i ) A l a t t i c e cone i s s a i d t o be complete i f e v e r y nonempty upper-bounded set

A c

F has a supremum sup(A)

such t h a t t h e u n r e s t r i c t e d d i s t r i b u -

t i v el a w holds, i . e . f

+ sup(A)

= supff + a

Ia

E A}

f o r every

O f course, upper-bounded means t h a t t h e r e i s some for a l l

f E

g E F with

F

. a -c g

a E A.

The r e a d e r s h o u l d r e a l i z e t h a t , i f

F

i s a v e c t o r space, t h e requirements

o f t h e d i s t r i b u t i v e laws a r e redundant. Furthermore, i t i s q u i t e obvious, t h a t i n t h i s s p e c i a l case we g e t immediately t h e a p p r o p r i a t e i n f i m a by going over t o

-f, - g

or

-A

r e s p e c t i v e l y . That i s n o t t r u e i n t h e

general case. Even i n f i m a of f i n i t e s e t s must n o t e x i s t i n general l a t t i c e

102

Linear Functionals

cones. O f course, i n a complete l a t t i c e cone e v e r y lower-bounded subset has an infimum (equal t o t h e supremum of i t s l o w e r bounds), b u t t h e p o i n t s i s , t h a t even f i n i t e subsets do n o t always have l o w e r bounds. F o r instance, t a k e t h e cone

Conv(F) o f 1.2.2. (v) c o n s i s t i n g o f a l l nonF endowed w i t h t h e p r e o r d e r c g i v e n b y i n c l u s i o n . because f o r e v e r y s e t ( C o n v ( F ) , c ) i s a complete l a t t i c e cone, {Ai I i E 11 o f nonempty convex subsets Ai o f F , I any i n d e x s e t , t h e

empty convex subsets o f

supremum i s g i v e n by

I E Xiai iEJ

I

J

c

ai E Ai,

I finite,

x. 20, z x i =

'

i€J

11

.

Here, t h e u n r e s t r i c t e d d i s t r i b u t i v e l a w h o l d s t r i v i a l l y . B u t f o r any two d i s j o i n t elements

A

and

B of

Conv(F)

t h e r e i s no l o w e r bound.

So, t h e n o t i o n " s e m i l a t t i c e - c o n e " o r even "meet-semi1 a t t i c e - c o n e " i n s t e a d

o f " l a t t i c e cone" would have been more precise. B u t f o r t h e sake o f b r e v i t y (and by abuse o f language) we have chosen t h e n o t i o n s as above.

-I n analogy

t o 1.2.2 we denote b y

(F, < )# , ( F , < ) #

and

R- valued monotone s u b l i n e a r , s u p e r l i n e a r and l i n e a r maps on

t i v e l y . I n case t h a t < i s equal t o = we j u s t w r i t e 2.4.2

#

F

)*

(F,<

, F#

F and

the

, respecF*.

Remarks

i)(F, <)# i s a complete l a t t i c e cone under t h e p o i n t w i s e o r d e r on F : I f A c (F,< )# i s upper-bounded ( i . e . sup{a(f)la E A1 < + for all f E F) then t h e supremum i s g i v e n by t h e p o i n t w i s e supremum: s u p ( A ) ( f ) = s u p I a ( f ) l a E A),

f E F.

ii) (F,< )# i s i n general n o t even d l a t t i c e cone w i t h r e s p e c t t o t h e p o i n t w i s e o r d e r on F. B u t i t i s a complete l a t t i c e cone w i t h r e s p e c t t o t h e i n v e r s e o f t h i s o r d e r i n g . With r e s p e c t t o t h i s i n v e r s e o r d e r i n g e v e r y nonempty subset A i s upper-bounded and t h e supremum i s g i v e n by t h e p o i n t w i s e infimum F 3 f -, i n f l a ( f ) I a E A)

.

We r e c a l l t h a t i f i n a s e t preorder

<'

F a p r e o r d e r < i s given, t h e n t h e i n v e r s e

i s d e f i n e d by:

Order Units and Lattice Cones

iff

g

f

I t i s obvious t h a t i f

(F,<)

g< f

103

. (F,< ' )

i s a p r e o r d e r e d cone t h e n

i s also

a p r e o r d e r e d cone. I n t h e above examples t h e suprema were g i v e n by p o i n t w i s e o p e r a t i o n s . T h i s i s n o t t h e case

f o r t h e i n f i m a . F o r example t h e p o i n t w i s e infimum

o f two s u b l i n e a r maps i s i n general n o t a t a l l s u b l i n e a r . B u t t h e s e i n f i m a ( i f t h e y do e x i s t s ) a r e e a s i l y c o n s t r u c t e d w i t h t h e h e l p o f envelopes. Def in i t $ o n :

2.4.3

(F, < )

Let on

F.

be a p r e o r d e r e d cone and l e t cp

Then c o n s i d e r t h e f o l l o w i n g maps on

-R-

valued

map

F:

n cpv(f) = i n f { t cp ( f k ) l n E N , fl,...,fn k=l d ( f) =

be some

E F

such t h a t

n f < X fk} k=l

n n t c p ( f k ) l n E N , f l,...,fn E F such t h a t t f k < f3 k=l k=l

SUP{

R i s f o r m a l l y d e f i n e d t o be a r e c a l l e d t h e l o w e r and upper (F,* ) - en-

where t h e supremum o f an unbounded subset o f +

m

. The maps

and cp"

(p"

velopes.

2.4.4

Proposition:

are monotone

(i)

cp"

(ii)

V A cp
cp"

(iii) _Lf ~ " ( 0 )

*-

m

then

cp"

&

sublinear

(iv)

rf.

(v)

I f p I cp

(vi )

_Lf q 2

(vii)

cp

E (F, < )# i f 2nd o n l y

(viii)

cp

E (F, < )# i f and o n l y i f cp

cp"

does no t a t t a~ i n t h e v al u e +m then -

cp

,

2 monotone and 3 monotone

i t i s superlinear

then p 5 cp" super1 i n e a r then q 2 cp" i f cp = cp" and cp * sublinear

= cpA

Linear Functionals

104

Proof: ( i ) and ( i i ) a r e immediate consequences o f D e f i n i t i o n 2.4.3.

(iii) The d e f i n i t i o n o f cpv immediately i m p l i e s ( p V ( f t g ) 5 cpv(f) tcpv(g) f,g E F , and moreover we have cpv(0) I 0. I f ~'(0) 2 0 t h e n

for V

Q

(0) =

V

cp

gives

( O t o ) I ( ~ " ( 0 ) t cpV(o)

, and

we must have ~ ' ( 0 ) = 0

cpV(o) =

--m

, SO i f

, ~ ~ ( $0 0)

T h i s proves t h a t cpA

-m

must be s u b l i n e a r .

cpv

( i v ) From t h e d e f i n i t i o n we c l e a r l y g e t d ( 0 ) 2 0 and CPA ( f t g ) 2 (p"(f) t cp"(g) i f c p " ( f t g ) < +-m . SO, i f cp" values i n

cpV(o)+

only attains

i s i m p o s s i b l e because cp"(0) = cpA(n.0)2n ~ ~ ( 0 ) .

i s then superlinear.

( v ) and ( v i ) a r e immediate consequences o f D e f i n i t i o n 2.4.3.

and t h e i n e -

q u a l i t i e s which h o l d f o r s u b l i n e a r and s u p e r l i n e a r f u n c t i o n a l s . ( v i i ) I f cp E (F,< )# t h e n Using ( i i i ) we see t h a t cpv

b y ( v ) and (ii)g i v e s cp = cp". and cp(0) + - a i m p l y t h a t cpv i s sub-

Q I cpv

= cp

l i n e a r (and monotone by (i)). T h i s shows cp E (F,<

)

#

.

( v i i i ) i s e s s e n t i a l l y proved i n t h e same way as ( v i i ) .

0

Rephrasi ng y i e l ds : 2.4.5

Theorem:

Consider i n (F,<)#

(i)If ~ ' ( 0 )

+

cpA

(ii)

fimum o f --

2.4.6

I

P

I

order

.

of )#-supremum -

(F,<

cpl.

I q E (F, < )#

I

cp I

ql

k then

.

cpA

i s the --

(F, < )#

-in-

Corollary:

( i ) pl,...,pn for -

the p o i n t w i s e

a t t a i n s values o n l y i n

Consider -again i n

pl(fl)

)#

t h e n cpv -i s the

--m

#

IP E (F, <

and (F, <

t...t

(F,<

)# and (F, < ) #

E (F, < )# has an infimum i n pn(fn) 2 0

f E F, g i v en by

whenever

0 < fl

the p o i n t w i s e

order.

(F, < ) # ---i f and o n l y i f

t...t

fn. ---T h i s infimum i s then,

105

Order Units and Lattice Cones

n

o(f) = inf{ (ii)

k=l

E (F,<

ql,...,qn

f o r every --

pk(fk)

1

n

I fl,...,fn

E F, f < 1

k=l

has a supremum i n (F,<)#

)#

fk3. i f and o n l y i f ,

,

f

.

T h i s supremum i s t h e n g i v e n b y t h e f u n c t i o n a l q Proof: (i)

Take

t o be t h e p o i n t w i s e infimum o f

cp

o and

finition of

@"

V

we have cp I o I 9.

pl,

...,pn

.

Then, by de-

o i s c l e a r l y subadditive o(0) 2 0

i s equivalent t o and monotone. The c o n d i t i o n , g i v e n i n (i), Hence

0

o

fore

i s s u b l i n e a r . P r o p o s i t i o n 2.4.4

. Now,

= @"

Ep E (F, < )

supremum o f of

p1,

(i), @"

by Theorem 2.4.5. #

". 'Pn .

Ip

o

(v) yields

( i i ) i s proved w i t h P r o p o s i t i o n 2.4.4 e s s e n t i a l l y t h e same way as ( i ) .

and t h e r e -

I (p"

i s equal t o t h e p o i n t w i s e

which must be t h e

I cp)

.

(F,< ) #

( v i ) and Theorem 2.4.5

- infimum

(ii)i n

0

We conclude t h i s c h a p t e r w i t h a t e c h n i c a l r e s u l t which w i l l have some i m portance l a t e r on. The arguments f o r t h i s theorem a r e c l o s e l y r e l a t e d t o t h e p r o o f o f t h e Sum Theorem. 2.4.7

Theorem:

L e t (F, < ) be a l a t t i c e cone and c o n s i d e r f i x e d Then there a r e pl,p2 E (F, < )* y itJ 91'92 E F. u(suP(glYg2))

=

!q(q)+

p

E (F , < )

p1 t p2

= p

*

and such t h a t

"(92).

Proof: Consider

F

x

F

F

x

F

= { ( f ,f )

1

2

I f 1 ,f 2

E F}, t h e f u n c t i o n s f r o m

into

i s an o r d e r e d cone under p o i n t w i s e o p e r a t i o n s and p o i n t w i s e o r d e r

(see p r o o f o f t h e Sum Theorem 1.4.1). n a l on

I1,2)

F

x

F

by

D e f i n e a monotone s u b l i n e a r f u n c t i o -

F.

Linear Functionals

106

G

Now c o n s i d e r t h e subcone

= I(ftxgl,

f txg2)

If

E F,

x

2

01

of

F

x

F.

Then because o f t h e d i s t r i b u t i v e l a w ( D e f i n i t i o n 2.4.1 ( i ) ) p i s l i n e a r on G. So, t h e r e i s a monotone l i n e a r v 5 p w i t h (Dominating p I G 5 vIG

.

Extension Theorem 1.3.1).Put

pl(f)

= v(f,O),

u 2 ( f ) = v(O,f),

f u n c t i o n a l s have t h e r e q u i r e d p r o p e r t i e s . 2.4.8

t h e n these

0

Remark:

By u s i n g t h e Norm Theorem i n s t e a d o f t h e Dominating Extension Theorem one

can a v o i d t o use t h e d i s t r i b u t i v e law f o r t h e f o l 1owing weaker r e s u l t: Let

be an ordered cone such t h a t e v e r y two elements have a

(F,< )

supremum, Then f o r f i x e d u1,p2

2.5

)*

E (F,<

)=1+ !

+

(F, < ) . I n t h i s case one g e t s

with

p E

v1

t

(F, < ) *

u2

5

and

g1,g2 E F t h e r e a r e

u such t h a t u(sup(g1,g2))

=

492)

RIESZ PROPERTY AND FINITE SUM PROPERTY

I n a cone F we c o n s i d e r some s p e c i a l o r d e r r e l a t i o n s which a r e canonic a l l y a s s o c i a t e d w i t h subcones G o f F. 2.5.1 Let

Definition: G

be a subcone o f

F.

( i ) The o r d e r r e l a t i o n given, f o r a r b i t r a r y fl

i s called

fl,f2 E F

< f2 i f f there i s a g E G w i t h

t h e G- o r d e r (sometimes denoted by <

, by f 2 t g = fl

,

).

A f u n c t i o n a l which i s monotone w i t h r e s p e c t t o t h i s o r d e r i s s a i d t o be G- monotone. A f u n c t i o n a l which i s monotone w i t h r e s p e c t t o t h e i n v e r s e o r d e r i s c a l l e d G- a n t i t o n e . (ii) The o r d e r i n F g i v e n by t h e i n v e r s e o f t h e F- o r d e r i s c a l l e d t h e n a t u r a l p r e o r d e r , i . e . h e r e we have f 1 4 f2 i f f t h e r e i s g E F w i t h fl + g = f2

.

I a7

Order Units and Lattice Cones

( i i i ) We say t h a t G generates F i f for every f E F there i s some g E G such t h a t f + g E G ( i v ) A preordered cone ( F , < ) i s said t o be negatively generated i f i t i s generated by F- = If E F I f < 01 .

.

We hope that the following remarks will c l a r i f y the role of the notions we have defined so f a r . All the remarks are immediate consequences of Definition 2.5.1 2.5.2

Remarks :

( i ) Let ( E , < ) be some preordered vector space. Then E i s endowed with the E- - order, where E- i s the negative cone E- = If E Elf < 01. The positive cone E+ = {f E Elf > 01 of E i s always equipped with i t s natural order. E- (or equivalently E + ) generates E i f and only i f E = E+ + E- , or, in other words ,if E i s negatively qenerated. ( i i ) In the general case, a linear functional i f and only i f p ( g ) I 0 for a l l g E G.

u

on

F

is

G- monotone

( i i i ) Every order unit cone i s negatively generated. ( i v ) Of course, f o r any subcone cone.

G of

F

,

(F,


i s a preordered

The following notions are important for the investigation of l a t t i c e structures in dual cones. 2.5.3

Definition:

( i ) A cone F i s said t o have the f i n i t e sum property (FSP in short) i f for a l l f l , f 2 , g 1 , g 2 E F with f l + f 2 = g1 + g2 there are ~~

h i k E F , i,k

=

1,2;

such t h a t

fi = h i l

+ h i 2 , g.1

=

h l i + h 2 i y i = 1,2.

( i i ) A preordered cone ( F , < ) i s said t o have the Riesz property i f f o r f , g , h E F with f < g + h there are always g,lh E F with < g , i< h such that f = + h

.

The next theorem shows t h a t these properties are closely related. Recall, a cone F s a t i s f i e s the cancellation law i f f f l + g = f 2 + g implies f l = f 2 for a l l fl,f2,g E F .

Linear Functionals

108

2.5.4 __ Let

Theorem:

<_ be _ t h_ e n_ a t_ u r a_ l preorder

&I

F

(i)

Lf

(ii)

L e t t h e c a n c e l l a t i o n --law hold i n --

t h e n ( F y < ) has t h e Riesz p r o p e r t y . F. Then (F, < ) --has t h e Riesz

F -has t h e FSP

_.--

p r o p e r t y i f and o n l y i f -_.--

F _ has _ -t h e

FSP

.

Proof: (i) f
?

g2 means t h a t t h e r e i s an

t

( D e f i n i t i o n 2.5.1). h21 t h22 = g2

The FSP g i v e s us

and

i n particular that

hll hll<

t

h21 = f

?.

f t

E F such t h a t

. But t h e f i r s t

h21 < g2

and

g1

hik

E F with

= g1 t g2

hll

t

h12 = g l Y

two e q u a l i t i e s i m p l y

( i n the natural preorder !).

( i i ) We have t o prove t h a t under t h e c a n c e l l a t i o n l a w t h e Riesz p r o p e r t y f o r the natural order implies the This gives with

hll

FSP

fl t f2 = g1

+

f l < g1 t g2 and from t h e Riesz p r o p e r t y we o b t a i n t h12 = fl

such t h a t

hll

< g1

t i o n of t h e n a t u r a l p r e o r d e r t h e r e a r e and

. Consider

h12 t g22 = g2

. I t remains

and

hyli

t o show t h a t

we i n t r o d u c e t h e e q u a l i t i e s from above i n t o fl t f2 = hll

t

h12

4

h22 w i t h

. By t h e

g2

hll

92'

yhll

h12 defini-

t h21 = g1

f 2 = h21 t h22

. To

fl t f 2 = g1 t g2

do t h i s

and g e t

h21 t h12 t h22 = fl t h21 t h22

A p p l i c a t i o n of t h e c a n c e l l a t i o n l a w y i e l d s t h e a s s e r t i o n .

0

Although t h e

FSP was o n l y d e f i n e d f o r sums w i t h two terms i t e a s i l y c a r r i e s over t o a r b i t r a r y f i n i t e sums: 2.5.5

Lemma:

Suppose t h a t --

F -has t h e FSP

k = ly...yn,

with

m

z fi

- i=l

and c o n s i d e r

_ .

n

=

E

k=l

gk

fi,gk

. Then t h e r e

E F, i=l,...,my are

hik

E F with

109

Order Units and Lattice Cones

and

Proof: We use i n d u c t i o n over Hence, s i n c e

ntm

F has t h e

.

FSP

The a s s e r t i o n i s t r i v i a l i f

,

the assertion holds i f

Then assume, t h e lemma i s proved f o r a l l n a t u r a l number Now, l e t

no 2 4

ntm = notl

o u t l o s s of g e n e r a l i t y f l y

-..fm,2y( f m - l + f m ) ,

k = l,..., n ; wl,...,wn (1)

f. =

(3)

gk =

'

. . Hence

m

or

n

n,m

. This

91, ...,gn

yields

,

'

hi,k "k

,k

Since we assumed

; i = Iy...,m-2;

hi,k

m > 2

we have

2

t

i = l,...,m-2

.

= l,...,n

n < mtn = n

0

t

1

. Therefore

a p p l y t h e h y p o t h e s i s a g a i n t o ( 2 ) . T h i s p r o v i d e s us w i t h such t h a t hiYk; i = m - 1 , m ; k = l,...,n;

Inserting (5) i n (3) yields

izl

m

(6)

gk =

f o r some

with hi,k

i=1

0

m > 2 . Apply t h e i n d u c t i o n h y p o t h e s i s t o

n

m-2

m=l

n t m 5 4.

nt m s n

with

or

i s g r e a t e r t h a n 2. Assume w i t h -

r

k=l

n = l

hi,k

, k

= l,...,n

.

we may

.

Linear Functionals

110

Now, ( l ) , ( 4 ) and ( 6 ) show t h a t t h e a s s e r t i o n i s t r u e i f T h i s proves Lemma 2.5.5.

ntm = notl. 0

The r e a l importance o f t h e o f t h e upper

-

FSP

l i e s i n i t s consequences f o r t h e s t r u c t u r e

and l o w e r envelopes.

2.5.6 Theorem: L e t F -have t h e FSP. -

rf

(i)

(F, < )

has t h e Riesz ---

property -then f o r

q E (F, < )

t h e lower

# --

i s l i n e a r i f and o n l y i f q v ( 0 ) + - m . -----( i i ) I f f o r the inverse preorder < ' o f < -t h e cone (F, < ' ) -has t h e Riesz p r o p e r t y -t h e n f o r p E (F, < )# t h e upper envelope ph -~ i s linear i---f and o n l y i f ph ----does n o t a t t a i n t h e v a l u e + m . envelope

qv

_.-

Proof: (i)

P r o p o s i t i o n 2.4.4

qv(0)

*

-m

t e l l s us t h a t

. We c l a i m t h a t qV(fltf2)

2

i s s u b l i n e a r i f and o n l y i f FSP and Riesz p r o p e r t y t o g e t h e r i m p l y t h a t

qv(fl)

t

T h i s t h e n c l e a r l y proves t h a t qv To see t h i s we c o n s i d e r a r b i t r a r y fl

t

f 2 < hl

hl < h l,...,hn

t...t

qv

qv(f2)

for a l l

E F

flyf2

.

i s not only sublinear but linear. hl,...,hn

E F

such t h a t

hn. Then from t h e Riesz p r o p e r t y ( i n d u c t i o n ) we 9 e t


such t h a t

fl

t

f 2 = hl

t...t

.

And t h e

k=l,

...,n )

FSP

(in

t h e s t r o n g e r form o f Lemma 2.5.5)

yields

Now, t a k i n g i n t o account t h a t t h e d e f i n i t i o n o f qv :

i s monotone and s u p e r l i n e a r we g e t w i t h

q

h . (i=1,2; 1, k n

hn

with

111

Order Units and Lattice Cones

Taking t h e infimum o v e r a l l p o s s i b l e qV(f1+f2) 2 qV(fl) (ii)

+

we o b t a i n i n f a c t

hly...,hn

qV(f2).

Again P r o p o s i t i o n 2.4.4 t e l l s us t h a t

i s s u p e r l i n e a r i f and

ph

o n l y i f ph does n o t a t t a i n t h e v a l u e t m . The p r o o f t h a t i n t h i s case t h e f u n c t i o n a l i s a l s o s u b l i n e a r i s i n complete analogy t o ( i ) . L e t flyf2 be a r b i t r a r y and c o n s i d e r fl

t

f2 r

g i v e s us

hl +...+

E F

such t h a t

Then t h e Riesz p r o p e r t y f o r t h e i n v e r s e o r d e r i n g

hny

hl > hly...,h

hly...,hn

n

> hn

such t h a t

fl

f2 = hl +...+

t

hn

.

Then

w i t h t h e same p r o p e r t i e s as above. And b y ,k monotony and s u b l i n e a r i t y we f i n d :

with the

FSP we g e t

n

h

n

n

n

NOW, t a k i n g t h e supremum o v e r a l l p o s s i b l e hl,...,h inequality

p"(fl

+f2)

I ph(fl)

i.

n

we g e t t h e d e s i r e d

ph(f2). 0

T h i s r e s u l t immediately y i e l d s an i n t e r p o s i t i o n theorem which appears i n v a r i o u s forms i n t h e l i t e r a t u r e (Ando [ 6 I, A s i m o v - E l l i s [111, Edwards [971, see a l s o 11161 and [431)

2.5.7

.

I n t e r p o s i t i o n Theorem:

L e t F -have t h e FSP and assume ~ t h a t F --has t h e Riesz p r o e r t y e i t h e r w i t h -r e s p e c t t o i --o r i t s i n v e r s e i ' . -Then f o r p E ( * a 7 _ .

q E (F, < ) # $ t J p I q ---t h e r e i s always a

u

E (F,<

)*

-such t h a t

Linear Functionals

112

Proof: L e t us assume t h a t

(F, < )

{b

E (F, < )# I I q } i s qv (Theorem 2.4.5 ( i ) ) . Hence LI = qv does t h e j o b . p e r t y one can work w i t h ph same arguments go through. set

p

has t h e Riesz p r o p e r t y . Because i s n o t empty and i t s

(i)

(F, < ' )

I n case t h a t

and Theorems 2.4.5

q

the

)#- supremum

(F,<

B u t a c c o r d i n g t o 2.5.6

5

i s linear.

qv

has t h e Riesz pro-

( i i ) , 2.5.6

( i i ) . Then t h e

(F, < ) * has almost

The r e a l c o n t e n t o f t h e I n t e r p o s i t i o n Theorem i s t h a t

t h e s t r u c t u r e o f a complete s e m i l a t t i c e . More p r e c i s e l y : 2.5.8 Theorem: Let (i)

F -have t h e FSP. (F,< ) --has t h e Riesz p r o p e r t y -----t h e n every l o w e r bounded subset o f

If

#or---

T h i s infimum remains t h e same i f we (F, < )* has an infimum~ i n (F,<-)*. -t a k e i t i n e i t h e r o f t h e f o l l o w i n g cones: F*, (F, < ) -

-_____--

(ii) I f F has t h e Riesz p r o p e r t y ----w i t h respect t o the inverse preorder < ' (F, 4 ) * has 5 supremum & then every upper bounded subset o f --(F, < )*. T h i s supremum remains the same i f we take it in either of the ~

f o l l o w i n g cones:

F*,

(F, < ) # o r F#

.

I

Proof:

)*. We d e f i n e a monotone s u p e r l i n e a r cp by cp(f) = i n f I p ( f ) I u E A 1 f o r a l l f E F. L e t cpv be t h e l o w e r envelope o f cp w i t h r e s p e c t t o (F, ). Since A was (i)

Let

A

be a l o w e r bounded s e t i n

(F,<

*

l o w e r bounded cpv property f o r

4

equal :

i s s u b l i n e a r ( P r o p o s i t i o n 2.4.4). implies that f o r arbitrary

infI

n I: 'P(fk) k=l

jnf{

n E cp(fk) I fl,...,fn k=l

1 fl,...,fn

f

NOW, t h e Riesz

the following infima are

<

E F with

f

E F

f =

with

n

Z

k=l n k=l

fkl =

fkl

-

So, qV i s t h e g r e a t e s t element I cp i n (F, < ) # as w e l l as i n ( P r o p o s i t i o n 2.4.4). O r , i n o t h e r words, cpV i s t h e infimum o f A

F# in

F ~ .

Order Units and Lattice Cones

Since

(p"

i s monotone and l i n e a r (2.5.6

of

in

(F, < )*. T h i s i s an obvious consequence o f

A

(i))

113

i t must a l s o be t h e infimum

(F, < ) * and

t h e same h o l d s f o r a l l cones between

(F, < ) * c F#. And

F#.

( i i ) T h i s i s a g a i n t h e same p r o o f . The o n l y d i f f e r e n c e i s t h a t one has t o

c o n s i d e r u ( f ) = s u p t p ( f ) I p E A 1 and uA i n s t e a d . Then as above, a p p l i c a t i o n of 2.5.6 (ii) yields the assertion. 0

2.6

THE POSITIVE DUAL CONE

F i r s t , we d i s c u s s t h e r e s u l t s o f t h e l a s t s e c t i o n i n a more c o n c r e t e

F the natural preorder <

s i t u a t i o n . We c o n s i d e r i n a g i v e n cone

(F,

denote

i) *

F;

by

=

.

F;

{pip : F

-,R+

T h i s cone we endow w i t h t h e

A c F;

every s u b s e t subset

tpl,...,pnl

FE

Then c l e a r l y linear}

and

i s equal t o

. F

p o i n t w i s e o r d e r on

i s l o w e r bounded, namely by i s upper bounded b y

p1

+...+

. And we 0

, and

pn

remark t h a t every f i n i t e

.

2.6.1 Lemma: L e t F have t h e FSP , then F i e v e r y nonempty

A c FZ

5 complete l a t t i c e cone. And f o r A

t h e infima o f

i n F:

and F# -a r e equal

Proof: F

has t h e Riesz p r o p e r t y w i t h r e s p e c t t o t h e n a t u r a l p r e o r d e r (Theorem

2.5.4

f o l l o w f r o m Theorem 2.5.8 ( i ) F: and t h e u n r e s t r i c t e d d i s t r i b u t i v e l a w i s a m a t t e r o f d i r e c t v e r f i c a t i o n . ( i ) ) . So t h e l a t t i c e p r o p e r t i e s of

NOW, l e t E,

E

= {x E E

*

be a v e c t o r l a t t i c e and c o n s i d e r i t s p o s i t i v e cone i s equipped w i t h t h e n a t u r a l o r d e r , t 01. Then E,

Ix

(E+)n = (E,,s)*,

and t h i s cone can be t r i v i a l l y i d e n t i f i e d w i t h

We denote t h i s cone by O f course,

Eo

E:

and c a l l

ET

- Er

=

Eo

SO

(E,I)*.

t h e o r d e r dual o f

i s endowed w i t h t h e p o i n t w i s e o r d e r on

E+

.

E.

Linear Functionals

114

2.6.2

Lemma:

A c E*t -the infima

E*t i s-a complete ~l a t t i c e cone, -and f o r nonempty

(i)

* i n E, (ii)

(E+)#

are equal.

i s an o r d e r ---

Eo

complete -___ vector l a t t i c e .

Proof: ( i ) Consider a r b i t r a r y 0 I hl I h and

Then E,

f,g,h 0 I f

E E,

- hl

with 9 g.

f I gth.

And because

Take

hl = sup(f-g,0).

f = (f

-

hl) t h l

f u r t h e r t e l l s us t h a t

must have t h e Riesz p r o p e r t y . Theorem 2.5.4

E,

has t h e

FSP. NOW, e v e r y t h i n g f o l l o w s from 2.6.1.

( i i ) Let

A c Eo be a l o w e r bounded s e t w i t h l o w e r bound b. Hence Take t h e i n f i b2 where bl,b2 E ET Therefore, b2 t A c E:

b = bl mum o f of

A

2.6.3

.

.

-

b2 in

+A EO.

in

, inf(b2

E:

t

-

A). i n f ( b 2 t A)

b2

i s t h e n t h e infimum

0

Remark:

T h i s lemma shows t h a t a c t u a l l y many o f t h e p o p u l a r spaces a r e o r d e r comp l e t e v e c t o r l a t t i c e s . F o r example the appropriate

Lq(p) (p-' t q-'

signed Bore1 measures on a compact bidual o f

qR(K)

Lp(v) = 1)

, or

(1 < p I m ) as o r d e r d u a l s o f Y R ( K ) , t h e space o f f i n i t e

K (as o r d e r dual of

(as t h e o r d e r dual o f

YR(K)).

k(K)),o r

the

One consequence o f these

c o n s i d e r a t i o n s i s , t h a t f o r s p e c i a l l a t t i c e cones (namely t h e p o s i t i v e cones o f v e c t o r l a t t i c e s ) t h e dual cone always has t h e

FSP and t h e Riesz

p r o p e r t y . One may ask if t h i s remains t r u e f o r general l a t t i c e cones. To f i n d t h e answer t o t h i s problem we have t o make a d e t o u r i n t o elementary a1 gebra. L e t us f i r s t r e c a l l how t o make a v e c t o r space T h i s i s done by an elementary procedure. Take (al,bl), f o r some

(a2,b2) E F

x

F

x

F.

F and d e f i n e two p o i n t s

al t b 2 + c = a t bl t c 2 t h e corresponding e q u i v a l e n c e c l a s s .

t o be e q u i v a l e n t i f

c E F. Denote by [al,bll

( F - F) o u t o f a cone F

Order Units and Lattice Cones

115

Consider i n

(F

- F)

=

def

I

I[a,bl

(a,b) E F

x

FI

the f o l l o w i n g class operations: x[a,bl

-

(F

Then

F)

seen t o be

. If

[b,al

(F

-

F)+

x,y E ( F

for

x

E R+

becomes a vectorspace. The n e g a t i v e o f

=

de f

-

<)

(F,

canonically a preorder i n

i.e.

xbl ,

= [xa, def

i s easily

i s a p r e o r d e r e d cone t h e n we d e f i n e

( F - F) I[a,bl

[a,bl

v i a t h e f o l l o w i n g p o s i t i v e cone

I a,b

E

F, b < a 1 ,

F ) we have x < y

i f and o n l y i f

y

- x

E (F

-

F)+.

F t h e c a n c e l l a t i o n l a w h o l d s F can be c o n s i d e r e d as a subcone o f ( F - F ) . More p r e c i s e l y : a -, [a,Ol i s then an i n j e c t i v e l i n e a r map f r o m F i n t o ( F - F ) . I n t h e general case t h e map I n case t h a t f o r

p

: F

+

(F

-

F)

g i v e n by

P -,

a

[a,O]

i s s t i l l l i n e a r (and monotone),

b u t i t f a i l s t o be i n j e c t i v e s i n c e p ( a ) = p ( a ) whenever t h e r e i s some . B u t t h e i n j e c t i v i t y i s n o t so i m p o r t a n t s i n c e b E F with b + a = b +

a

we have t h e f o l 1owi ng e l ementary

2.6.4

.

Universal Prooertv:

n:F

E be a-l i n e~ a r map - from - F i n t o a v e c t o r space A : ( F - F ) 3 E -such t h a t A p = t h e r e i s a unique l i n e a r (i)

-

L

-

~

_

-$

_

.

_

i s a preordered ~ v e c t o r- space - and_i f

L -

i s monotone t h e n -

L_

A

E ,Then

.If

i s monotone.

Proof: Let

(a1,bl),

f o r some

(a2,b2) E F c E F X(al)

x

F

be a r b i t r a r y w i t h

. Then by l i n e a r i t y o f ;i we - n(bl) = n ( a 2 ) - n ( b 2 ) .

get

al

+

b 2 + c = a 2 + bl+C

E

116

Linear Functionals

Hence,

-

h(a)

( a , b ) E [ a l , b l l . Let us denote

i s constant f o r a l l

i(b)

t h i s constant by A[a,b]. By d e f i n i t i o n A i s l i n e a r s i n c e A i s l i n e a r . The map A i s monotone whenever i s monotone. And we have A * p = A . I t remains t o show t h a t A i s unique. So, l e t A2 : ( F - F ) -+ E be l i n e a r with

Hence,

2.6.5

-

-

A2

p = A

.

Then by l i n e a r i t y

.

A2 = A

n

Lemma:

Let ( F , < ) be 5 preordered

cone.

given by ( F - F) t -i n (i)rf < & antisymmetric then the preorder -(F - F ) i s again antisymmetric. (ii) If F i s a -l a t t~ i c e cone ( F - F) i s a vector e. - then -l a t t i c~ (iii) If F i s negatively generated then ( F - F ) i s negatively generated. Proof: ( i ) One e a s i l y sees t h a t the preorder i n ( F only i f t h e p o s i t i v e cone ( F - F ) + i s sharp, zero whenever x and -x a r e elements of ( F x, -x E ( F - F)+. Then by d e f i n i t i o n t h e r e a r e with

and a 2 < b2

bl < a l

Then because of

x = [al,bll

f o r some c E F. Let

such t h a t x =

=

F ) i s antisymmetric i f and i . e . x must be equal t o - F ) + . So assume al,bl,a2,b2 E Ft

[ a l , b l l , -x

=

[b2,a21

.

we have a t b t c = a2 t b l + c 1 2 We claim a 1 t 6 = b 1 t 6 . The i n e q u a l i t y

[a2’b21

6=altb2tc

.

al t E >bl + E follows immediately from al > b l , and t h e missing p a r t is seen via Now, x =

a 1 t 6 = ( a l t b 2 t c ) t a l = ( a 2 t b l t c ) t a l < b 2 t b l t c t a - b t 6. 1- 1 [a 1’b 1I = [a 1 + 6 , b l t 6 1 = 0 since a 1 + 6 and b l + d a r e equal.

( i i ) Consider a r b i t r a r y

z

x

=

[al,bll, y

= [ s u p ( a l + b 2 , a 2 t b l ) , bl t b 2 1

Abbreviate

c

=

=

[a2,b21. We claim t h a t

i s t h e l e a s t upper bound of

sup(al t b 2 , a2 t b l )

and consider

x,y

i n (F-F).

Order Units and Lattice Cones

z

-

x = [c,bl+b

1 - [a 1, b 11

2

=

This i s clearly an element o f

117

[c + b l y b l + b 2 + a l l = ~[c,b2 + a,]. (F

.

- F)+ since b2 + a l < c

I n the same

way one shows t h a t z - y i s in the positive cone. So, z i s an upper bound of x and y. Let i = [a,D] be another upper bound. Then because z - x E(F - F)+ , we get a + bl > p + a l , and similarly a + b2 z 13 + a2.

.

This gives a + bl + b e > 13 + a l + b 2 and a + bl + b 2 > + a 2 + bl Hence, a + bl + b 2 > sup(B + a l + b 2 , 13 + a 2 + b l ) = 13 + c. And t h i s yields

i-

[c,bl + b21

Hence 2

-

=

[a + bl + b2, 13 + b, + b21 + [ p + bl + b2,c + 131

=

[a + bl + b2,c + 81 E ( F

z E (F

-

F)+

and

z

-

F)+

.

must be the l e a s t upper bound.

( i i i ) Let x = [ a , b l E ( F - F ) . Since F i s negatively generated there Hence, are a-,b- E F- such t h a t a t a- and b + b- are in F-

.

x = [ a + b - + a-

,b +

b- + a-I

=

[a + b - + a - , O l - [ b + b - + a - , O ]

shows that x i s the difference of two elements in ( F valently in ( F - F)+) i . e . ( F - F)- generates F - F.

- F)-

(or equi0

The reader should note t h a t for the proof of part ( i i ) of the l a s t lemma i t was absolutely essential t h a t we required the validity o f the d i s t r i butive law for l a t t i c e cones. One might wonder i f ( F - F) i s always a complete vector l a t t i c e i f (F, < ) i s a complete l a t t i c e cone. Here, the answer i s negative. 2.6.6

Definition

I p(f)

for a l l f E F I denotes the cone consisting o f the R- valued elements in (F, < ) * . FIT i s called the positive dual cone of (F, < ) and, f n o t mentioned otherwise,it i s endowed with i t s natural order. FT

= { p E (F,<

)*

> -

OD

-

Let us compare F: with the positive dual cone ( F - F): of ( F F). I f p : F -,( F - F) i s the canonical ( n o t necessarily injective) embedding

Linear Functionals

118

of

F i n t o (F

element of

-

F:

F) then f o r

.

(F

p E

-

*

F)+

the map vp

= p o p

And t h e Un versa1 Property 2.6.4 implies t h a t

i s an p

-B

and Ff . Furthermore, we remark i s a b i j e c t i v e map between F - F): t h a t by d e f i n i t i o n every l i n e a r b i j e c t i o n between cones i s monotone with respect t o t h e natural preorders. So, we have 2.6.7

Lemma:

are

FT and ( F - F): i d e n t i c a l i n the sense t h a t t h e r e i s a monotone bij-____.-e c t i v e l i n e a r map between them. 2.6.8

Theorem:

Let ( F , < ) be a l a t t i c e cone. Then i t s p o s i t i v e dual cone FT has t h e l a t t i c e cone. FSP --and t h e Riesz property and i t i s a complete ~Proof: In view of t h e above lemma i f s u f f i c e s t o prove t h e theorem f o r ( F - F): instead of Ff In 2.6.5 we s t a t e d t h a t ( F - F) i s a vector l a t t i c e . 0 So, everything follows from 2.6.2 ( i ) .

.

2.7

2.7.1

DUAL ORDERS AND THE CARTIER

-

FELL - MEYER THEOREM

Definition:

Let F be a cone and l e t @ be a subset of F*. ( i ) For f , g E F we w r i t e f s @ g i f p ( f ) I u ( g ) This preorder i s c a l l e d t h e ( i i ) For f , g E F we w r i t e

f =

n z fi (fi

i =1 for all p E i s called the (F,

)

E F,

n

E N)

@-

p E @

.

pointwise order. g i f f o r every decomposition

f <

@

t h e r e a r e gi

, we have p ( f i ) 0-

for all

I

p(gi),

with i

=

g

=

l,...,n.

n

E gi i =1

such t h a t ,

The preorder

<

decomposition preorder.

i s always a preordered cone. In c o n t r a s t t o t h i s the r e l a t i o n

in general i s not compatible w i t h the cone operations.

Order Units and Lattice Cones

2.7.2

119

Lemma:

L e t F -have t h e FSP , t h e n (F, < @ ) -i _ s a preordered cone, i . e . the Q- decomposition -o r d e r i s c o m p a t i b l e --w i t h t h e cone o p e r a t i o n s . Proof: Only t h e c o m p a t i b i l i t y of assume

< @g1

fl

n

Because

Y

k,

.

i=l1

hi,kE

p(hi

-

f.

Z

fl+ f2

and

f2 <

The

@

FSP

such t h a t

fk

we get

hi,k

with

p E @

. NOW,

gk

Y

k,

gi(i=l,...,n)

for all

fi =

n hYil

91

Recall t h a t i f

92'

+

F

+ hiY2

n 1 hiYk=

i=1 put

g1

and

gk

ii= hiyl

i s t h e d e s i r e d decomposition f o r

<@

i s n o n t r i v i a l . So,

(Lemma 2.5.5) y i e l d s

k=1,2)

<@

F

g2 and c o n s i d e r a decomposition

i=l,...,n;

Ip(hi

fl+ f2

with addition i n

+

, i.e.

o

vQ

+

hi?.

g2.

Let

Then

Hence,

has t h e

FSP

then i n

F*= (F,=)*

t h e supremum o f

(F,=)*).

t h e Riesz p r o p e r t y

S o y i n t h i s case, f o r

@ c

F*

,let

t h e s e t o f a l l suprema o f upper bounded f i n i t e subsets

= {sup(ul

,... ),,I.?,

1nE

upper bounded E @ I .

N, p l y . ..,un

Propos it i on :

2.7.3 _ .

V Q

F -have t h e FSP and c o n s i d e r

f s V Qg

,

i.e.

p ( f ) Iu ( g )

f,g E F with f <

for all --

g

1-1 E V Q .

.Then

Proof: From Theorem 2.5.8 sup(p1,...,pn)(h)

hi,k.

CI

is trivially fulfilled for us denote b y

i=1

such t h a t

upper bounded subsets does always e x i s t (Theorem 2.5.8,

of

fk =

( i i ) and C o r o l l a r y 2.4.6 = supt

f o r any upper bounded s e t

n E pi(hi) i=1

{ul,..

(ii) we o b t a i n

I hly...,hn

. ,pn}

c @

E F with

.

n h = z hi] i=1

120

Linear Functionah

Looking i n t e n s i v e l y a t t h i s e x p r e s s i o n one immediately d i s c o v e r s sup(p l,...,pn)(f) 2.7.4

5 sup(p1,...,un)(g)

whenever

9.

f

0

Proposition:

with L e t F have t h e FSP and assume t h a t i t has t h e Riesz p r o p e r t y r e s p e c t t o t h e i n v e r s e o f s @ . Consider f , g E F w i t h f s a g. Then -

%

f

g.

Proof: I f f s @ g and

f =

n

t h e n g (<@)If; t h e Riesz p r o p e r t y t o g e t h e r

1 fi

i=1

w i t h i n d u c t i o n t e l l us t h a t t h e r e a r e gi Hence,

g.

f

('@)Ifi

such t h a t

g =

n 1 qii'

i=1

0

2.7.5 Theorem: Let -

FSP ---___-and assume t h a t i t has t h e Riesz p r o p e r t y with

F -have t h e

r e s p e c t ---t o the inverse o f

svo . Then t h e p r e o r d e r s <

@

.nd

sVQ

are

t h e same. -Proof:

P r o p o s i t i o n 2.7.3 y i e l d s t h a t (applied t o V @ instead o f But

f

g

f Q)

clearly implies

s

v@g when

gives f

f <

~

g. And P r o p o s i t i o n 2.7.4

f < v@ g when


f

s v@ g.

0

On t h e f i r s t view t h e assumptions i n Theorem 2.7.5 seem t o be r a t h e r o u t l a n d i s h , b u t we w i l l see t h a t t h e y a r e n a t u r a l l y f u l f i l l e d f o r t h e p o s i t i v e dual cone o f a l a t t i c e cone. We want t o extend t h e n o t i o n o f l a t t i c e homomorphism we i n t r o d u c e d i n 2.2. Consider a l i n e a r

@

: F

+

G

, where

(F, < )

i s a l a t t i c e cone and

) i s some o r d e r e d cone. @ i s c a l l e d a l a t t i c e cone homomorphism ( o r s i m p l y l a t t i c e homomorphism) i f @ ( F ) = { @ ( f ) I f E F I

(G,

4

Order Units and Lattice Cones

121

i s a l a t t i c e cone such t h a t o(sup(fl,f2))

= sup(@(fl),

cp(f2))

for a l l

E F.

fl,f2

@-'

o i s c a l l e d a l a t t i c e cone isomorphism o r l a t t i c e isomorphism i f b i j e c t i v e and cp and a r e l a t t i c e cone homomorphisms.

o

is

Note, t h a t e v e r y l a t t i c e homomorphism i s n e c e s s a r i l y monotone. Let

(F, < )

be a l a t t i c e cone and c o n s i d e r

t h e p o s i t i v e dual cone (Theorem 2.6.8). subsets o f

(Ff)*

(FT)*, t h e l i n e a r f u n c t i o n a l s o n

(F, < ) . R e c a l l t h a t

FT o f

Hence ( b y Theorem 2.5.8.)

^f

FSP

t h e supremum of upper bounded -+

3

from

F into

FT

.

(Ff)*,

i s d e f i n e d by P(p)

2.7.6

has t h e

does e x i s t w i t h r e s p e c t t o p o i n t w i s e o r d e r on

Now, we c o n s i d e r t h e c a n o n i c a l l y g i v e n map f where

Ff

for all

= p(f)

E F:

.

Theorem:

The map f --

+

3 --i s a lattice

homomorphism.

Proof:

f

It i s q u i t e c l e a r t h a t

t o be an upper bound o f yields for arbitrary

7

i s monotone and l i n e a r . So, s u m has {?,GI. Hence, s u m 2 sup(i?,'j). Theorem 2.4.7 +

II E Ff :

The l a s t i n e q u a l i t y f o l l o w s f r o m C o r o l l a r y 2.4.6 and Theorem 2.5.8 Hence, s m = sup(?,G) f o r a r b i t r a r y f , g E F. 0 Combining t h i s w i t h Theorem 2.7.5

2.7.7 Let -

General C a r t i e r

_-----

-

Fell

-

(ii).

we g e t : Meyer Theorem:

F be a l a t t i c e cone and l e t

@ c

F be such t h a t

V@ = F. Then, for

122

Linear Functianals

, the following a r e equivalent:

v , v E F:

(i)

v

(ii)

p ( f ) s v ( f ) -f o r a l l f E F.

v

Proof:

W e know t h a t Ff has t h e FSP with respect t o t h e inverse of sider

,, (sF)I

v1

+

v2,

. We

claim t h a t FT has t h e Riesz property To prove t h i s claim we conI v~ = s

.

i.e.

( - p ) ( f ) s ( - vl) ( f )

-t

( - v2) ( f ) f o r a l l

Now consider t h e inverse order

i n F , then

f E F.

- p , -v1,-u2

E (F,


Since (F,< ) i s a l a t t i c e cone t h e cone (F,< I ) must have t h e SIP ( c f . chapter 1 . 4 ) . Hence (1.4.2 and 1 . 4 . 1 ) we f i n d .( '-monotone l i n e a r functionals 3l s - v l , S2 s -v2 such t h a t - p = S1 -t v2 -

.

Put p1

pl =

-Gl,

p2 =

-G2

then

( s F ) # ul, ,,2 ( s F ) l v2. Since

that

FT

p1,v2

E (F,<

u,vl,v2

)* w i t h

p

= pl

+

p2

and

were a r b i t r a r y chosen t h i s shows

has the Riesz property with respect t o t h e inverse of

s

F '

Now, application of Theorem 2.7.5 t o t h e ( n o t necessarily i n j e c t i v e ) embedding o f Q and F i n t o F: gives t h e a s s e r t i o n of t h e theorem.

2.8

FREE LATTICE CONES

Throughout t h e l a s t chapters we have seen t h a t l a t t i c e cones do have a very r i c h mathematical s t r u c t u r e . So, the question a r i s e s whether every cone can be canonically embedded i n t o a s u i t a b l e l a t t i c e cone. 2.8.1

Definition

Let (F,< ) be a preordered cone. A l a t t i c e cone (G, < ) i s c a l l e d a f r e e l a t t i c e cone over (F, < ) i f t h e r e i s a monotone l i n e a r map j : F -, G such t h a t f o r every monotone l i n e a r map 6 : F -, L i n t o a l a t t i c e cone L t h e r e e x i s t s uniquely a l a t t i c e cone homomorphism 6*:G+

Order Units and Lattice Cones

with

123

6 = 6*0 j . In other words the diagram

can a1ways be completed t o j F-----I>G

L where

! means t h a t

i s uniquely determined.

6*

Here we have used the following notation: monotone 1 i near

_____L)

D

l a t t i c e cone homomorphism.

Of course, these diagrams are always assumed t o be commutative.

For convenience we sometimes write in the following f v g instead of sup(f,g), and we recall t h a t a l a t t i c e cone homomorphism 6* : G -+ L s a linear map with 6*(gl v g2) = 6*(g1) v 6 * ( g 2 ) for a l l g1,g2 E G

.

Before going any further into the details we 1 i ke t o adopt the follow notation :

-*

_____+

,

-

w

b

monotone sub1 inear monotone surjective sublinear or 1 i near, resp. i njecti ve l a t t i c e cone subhomomorphism,

124

Linear Functionals

where a l a t t i c e cone subhomomorphism rp between two l a t t i c e cones and ( L , < ) i s a map which s a t i s f i e s : cp(lg)=

m ( d Y (P(s1

f

92) < (P(91)

(P(SUP(91Y92)) = SUP((P(91)YV(92))

We use the symbol

>

a

f

and

(P(@

for a l l

(G$)

9Y91’92 E G ; A E R,

*

9 for a l a t t i c e cone isomorphism.

Before going into the construction of the free l a t t i c e cone, l e t us play the usual game. 2.8.2

Proposition:

( i ) ---Free l a t t i c e cones over (F, < ) are uniquely determined up t o l-a t t i c e cone isomorphisms. ( i i )& j ( G , j ) ( j the embedding F + G ) --__--be a free l a t t i c e cone over (F, < ) . Then a l l elements of G ---are of the form j ( f l ) v v j(fn),

...

with

n

E N,

fl,

...,f n E

F.

Proof: ( i ) Let (G,j) be a free l a t t i c e cone over I : G + G then completes the diagram

(F, < ) . The identity map

Because o f the uniqueness of this completion we conclude that I i s the only l a t t i c e cone homomorphism G + G with j = 1 0 j . Now, consider two free l a t t i c e cones ( G l , j l ) and ( G 2 , j 2 ) over ( F , < ) . Then look a t the diagrams jl j2

125

Order Units and Lattice Cones

and observe that

J1

3, = I

morphi sms

.

I G2

( i i ) Let

=

j,

...v

{j(fl)v

, sup(gl,g,)

E

i).

E Pi,

are l a t t i c e cone

j;

and

jf

j(fn)I n

(i.e.

G

= j;

j, = J,

and

, which implies t h a t

sub-lattice cone of g1,g2 E 7;

0

J,

and

j,

j , = J1

homomorphisms with and

*

= jy

0

j;

j,

.

Hence,

J

- I

1-

I+

are l a t t i c e cone iso-

fl,...,fn

E

FI , then

-

G

is a

G i s a subcone of G and for every Look a t the following diagram

Then according t o Definition 2.8.1 j * and 6* are uniquely determined. Replacing 6* by I makes the same diagram commutative, hence 6* = I . This proves t h a t I o j * = I , or G = G. 0 I

NOW, l e t us construct the free l a t t i c e cone f o r a given cone (F, < ). Recall that a s e t A c F i s called convex i f x f + (1-X)g E A whenever f,g E A and 0 I A I 1. For a given s e t B , the smallest convex s e t containing B i s denoted by < B > and i s called the convex hull of B . For a f i n i t e s e t I f l , ...,f n I c F the convex hull i s given by

--

n {

i =1

hi f i

I x1

2 0

,...,A n

2 0,

I n example 1.2.2(v) we have seen t h a t C o n v ( F ) = I A l A convex,

@

* A c FI

i s a cone i f one defines A+B=Ia+blaEA,bEBI A A = {xal a E A I

n

r xi i =1

=

11

.

126

Linear Functionals

I t i s r e l a t i v e l y t r i v i a l t o see t h a t

Convo(F) = { A i s a subcone o f We w r i t e a

ib.

A < B

IA

Conv(F). NOW, we endow t h i s cone w i t h a p r e o r d e r r e l a t i o n .

B

i

t h e r e i s some

b E B

such t h a t

i s t h e n a preordered cone under t h i s p r e o r d e r r e l a t i o n .

We c o n s i d e r two elements

<

a E A

i f f o r every

Convo(F)

A < B and

i s t h e convex h u l l o f a f i n i t e s e t }

A

E Conv(F),

A

.

A, B E Convo(F)

t o be e q u i v a l e n t

(A

- B)

[ A 1 we denote t h e equivalence c l a s s o f A .

By

-

if

Since

was c o m p a t i b l e w i t h t h e cone o p e r a t i o n s i t t u r n s o u t t h a t a d d i t i o n

and R,-

m u l t i p l i c a t i o n a r e class operations, i . e . i f

A2

then

N

B2

for all

A1,

A1

x2

2 0

B1

and

I

< i s a c l a s s o r d e r i n g , i . e . A < B i f and o n l y i f r\ 4 B f o r a l l r\ E [ A ] and B E [ B l . P u t t i n g a l l t h i s i n f o r m a t i o n t o g e t h e r we o b t a i n t h a t t h e q u o t i e n t o f (Convo(F), < ) w i t h r e s p e c t t o i s an ordered cone. To be more p r e c i s e , Moreover

-

L F = {[A]

1A

E Convo(F)l

i s a cone i f we d e f i n e

and

5

g i v e n by

[A1 5 [ B l

if

A< B

i s an o r d e r r e l a t i o n which i s c o m p a t i b l e w i t h t h e cone s t r u c t u r e . If n o t o t h e r w i s e mentioned

L F w i l l always be endowed w i t h t h i s p a r t i c a l u a r

order r e l a t i o n . It i s c l e a r t h a t

[A1 v [ B l

= [<

A

u

LF

B >] ,

i s a l a t t i c e cone, s i n c e

Order Units and Lattice Cones

The map

F 3 f

[Cfll

-,

We denote t h i s map by 2.8.3 Let

F

i s a monotone l i n e a r map from

L and i n s t e a d o f

[ I f } ] we w r i t e

into

LF

.

*f.

Lemma: be a l a t t i c e cone and l e t

G

_ .

127

TI

: F

-,

G

_be_a

monotone s u b l i n e a r map.

Then t h e r e i s a u n i q u e l y determined -l a t t i c e cone subhomomorphism ---L F -+ G w i t h TI* o L = T I . -For every [A1 E L F _ we -have rr*[Al

=

Ia

sup{n(a)

from -

TI*

E A}.

Proof:

A E [ A 1 and d e f i n e

Pick

n*[Al = s u p ( n ( A ) ) . Then

sup(n(A))

T I * [ A I i s unambiguously d e f i n e d . To see t h i s we t a k e a b E B

that for

t h e r e i s an

a E A

with

b
, and

i s t h e convex h u l l o f a f i n i t e number o f elements by monotony o f TI t h a t TI(b ) Hence s u p ( n ( A ) ) = s u p ( n ( B ) ) .

I sup(n(al),.

IT* i s a l a t t i c e cone subhomomorphism w i t h

L

. Then

T o

t

> E [A]. This implies

= TI

[A]

T[AI

Hence, SO,

T

=

TI*,

=

for

=

[A] E L F a E a1

sup{*a

suplT(*a)

and

TI*

Ia

>

we g e t

= sup n(A).

= TI

NOW, c o n s i d e r a second l a t t i c e cone subhomomorphism < a

A =
since

a = l a l,...,anl

. . ,n(an)) n*o

e x i s t s and

B E [ A ] and r e c a l l

. : LF

T

we f i n d a f i n i t e s e t

-,

a c F

G with

with

.

E a} = s u p l n ( a )

1a

i s unique.

E a1

=

n*[Al

.

0

S p e c i a l i z i n g t h i s t o monotone l i n e a r maps we o b t a i n :

2.8.4 (LF,L)

Consequence: i s a f r e e l a t t i c e cone o v e r

(F, < ).

To a v o i d an abundance o f n o t a t i o n we w r i t e t a c i t l y assume t h a t by t h e map L

.

L F instead o f

( L F , L ) and we

F i s ( n o t n e c e s s a r i l y i n j e c t i v e l y ) embedded i n L F

Another consequence of 2.8.3 F

i s t h a t f o r a l a t t i c e cone

L

---I)LF

b

G

t h e diagram

Linear Functionals

128

can be uniquely completed t o

F

D -

e LF

G

Let us add: 2.8.5 Remark:

( i ) I f I i s an order u n i t of ( F , < ) then *I i s an order u n i t of L F . ( i i ) I f ( F , < ) i s negatively generated then L F i s negatively generated. Proof: ( i ) I t i s obvious t h a t *I i s s t r i c t l y p o s i t i v e . Let [ A ] E L F and pick Then t h e r e a r e a f i n i t e set I f l , ...,f n ) = a c F such t h a t < a > E [ A ]

.

x1 ,... , A n

(by d e f i n i t i o n )

Put 6

=

n

then

t hi,

i =1

2 0

such t h a t

fi < hi I , i=l,...,n.

f i < 6 1 , i = I , ...,n . Hence,

( i i ) Again f o r an a r b i t r a r y

[A] E L F

take a f i n i t e s e t a

=

{fl,...,fn)

i n F such t h a t [ < a > ] = [ A ] . Since F i s negatively generated t h e r e a r e g l , . . .,gn < 0 such t h a t gi t f i < 0, i = l , . . . , n .

Put g

n

gi i =1 This shows t h a t = E,

Y

then g LF

t fi < 0

f o r i = l ,...,n.

i s negatively generated.

Hence *g t [ A 1 5

0

0

A t the end of t h i s s e c t i o n l e t us return t o Lemma 2.8.3. I t s content i s t h a t f o r a l a t t i c e cone G t h e r e i s a uniquely determined map *

.

129

Order Units and Lattice Cones

(mapping

n on n*) f r o m t h e cone o f s u b l i n e a r monotone maps

LF

i n t o t h e cone o f l a t t i c e cone subhomomorphisms

n

n*

=

2.8.6

o

+

F

+

G

such t h a t always

G

1 holds.

Proposition:

The map * i s sublinear. -Proof: For an a r b i t r a r y

[A1

that

[A] E L F

= [].

s u b l i n e a r monotone

a = {fl,

pick a f i n i t e set

...,f n l c F ...,n l

Then we have n*[A1 = s u p I n ( f i )

I

n : F

i s homogeneous.

+

G

. This

*

shows t h a t

i=l,

such

f o r any

Moreover t h i s f o r m u l a y i e l d s :

(n, + n2)* [ A 1

= sup{n,(fi)

t n2(fi)

< sup{n1( f i )

I

= n;

[ A ] + n;

I

...,

i=l, n l

..., .

i=l, n}

[A]

I i=l, ...,nl

+ sup{n2(fi)

Rather u s e f u l i s t h e f o l l o w i n g c r i t e r i o n 2.8.7

Theorem:

be monotone -l i n e a r maps F L e t u1 ,u2 value

(iii)

not -i s _

- m

there i s a ---

+

R

,

i.e.

here

a t t a i n e d . Then t h e f o l l o w i n g

A t 0

such t h a t

u1

=

R

= R

and t h e

I_-

equivalent:

x ( u l + p2)

.

Proof: (iii) + (i)

i s a consequence of t h e f a c t t h a t

(1-A) u l = x u z (i)

=+

(ii):R e c a l l , t h a t f o r

p*(*f

*

i s homogeneous, because

i n view o f ( i i i ) .

v *g) = max(u*(*f),

f,g E F we have i n general u*(*g)) = max(u(f),u(g)).

130

Linear Functionals

Application

t o our special case y i e l d s :

max(ul(f),vl(g))

+

max(v2(f), v 2 ( g ) ) = m a x ( ( q

which has ( i i ) as an immediate consequence. ( i i ) * ( i i i ) : P u t v = u1 t p 2 and assume t h a t v ( f ) and v ( g ) a r e both either

I 0

o r 2 0. Then v ( ) v ( f ) l g ) = v ( \ v ( q ) l f ) , and ( i i ) implies

! q ( l v ( f ) l g ) = q ( I v ( 9 ) l f ) . Hence,

whenever v ( f ) and v ( g ) have t h e same s i g n . I f v ( g ) and v ( f ) have opposite s i g n s , then say v ( f ) . And ( * ) applied t o this case gives

v ( f t g ) has the sign o f ,

Using t h e f a c t t h a t v i s a d d i t i v e , we see t h a t ( * ) is valid f o r a l l f , g E F. I f v = 0 then ( i i ) shows y1 = 0 and ( i i i ) i s t r i v i a l . So assume v(g) 0 f o r some g E F . I f v ( g ) < 0 then by ( i i ) we get pl(g) I 0 , and v ( g ) > 0 implies pl(g) 2 0. In any case

q(9) ~

v(9)

-

x

2 0

. Inserting

t h i s i n ( * ) gives the desired

result. 2.9

SIMPLICIAL CONES

Let (F,< ) and ( G , < ) be s t t i c e cones and denote by A ( G ) and t h e cones of 1 a t t i c e cone homomorphi sms , o r subhomo'morphisms S A (F,G) respectively, from F t o G ; i . e . A ( F , G ) = {u

SA(F,G) =

1 u ~p I

: F

p : F-GI

GI

.

Recall t h a t L denoted t h e canonical embedding o f F i n t o L F . Then we have t h e following remarkable property

Order Units and Lattice Cones

2.9.1

131

Theorem:

There i s a l i n e a r map u : S A ( F , G )

---__.-

S A ( L F ,G)

--f

-with the

following

properties: (i)

~ ( p o) L = p

(ii)

u

for all --

A(F,G)

into

p E SA(F,G) A(LF,G)

Proof: Consider

I* g i v e n by L F

-b

LF

( I

and d e f i n e

~ ( p =) p

o

I*. Then u

I i s t h e i d e n t i t y map from F onto F )

has t h e r e q u i r e d p r o p e r t i e s .

n

This property gives r i s e t o the f o l l o w i n g d e f i n i t i o n : 2.9.2

Def in i t i on:

A preordered cone (F,< ) i s c a l l e d s i m p l i c i a l i f t h e r e i s a l i n e a r map z : FT -+ ( L F ) : from t h e p o s i t i v e dual cone o f F i n t o t h e p o s i t i v e dual cone o f L F such t h a t z ( p ) o L = 11 f o r a l l LI E F: . R e c a l l , t h a t t h e p o s i t i v e dual cone monotone l i n e a r maps from

F

into

F: R

of (not

(F, < )

R),

was t h e cone o f a l l

and t h a t we assumed t h i s

cone t o be endowed w i t h i t s n a t u r a l o r d e r . So, a c t u a l l y , t h e n o t i o n " s i m p l i c i a l " depends on what k i n d o f image space Therefore, i f we have s e v e r a l

R

, we

R we have considered.

speak o f R- s i m p l i c i a l i n s t e a d o f

s i m p l i c i a l . F o r s i m p l i c i a l cones t h e map

z

i s u n i q u e l y determined

( t h e r e f o r e we may c a l l i t t h e s i m p l i c i a l map). T h i s f o l l o w s from t h e

f o l l o w i n g more general r e s u l t : 2.9.3 Theorem: L e t L -be t h e danonical embedding o f (F, < ) i n t o L F and l e t IT : FT (LF): z ( ~ o ) L = i.1 f o r a l l be a l i n e a r map w i t h p E F: -----+

.

132

Linear Functionals

-Then f o r a r b i t r a r y , with

v o

1

=

~3

*

u E F,

for all

cp

we have

E LF

t(p)(cp) =

s u p I ~ ( c p ) I v E (LF)*,

.

Proof: Consider an a r b i t r a r y element and l e t

<

CV E (L F

be t h e supremum o f

Theorem 2.4.7

there are

v~,.,.,v

that ;(*fl

(*)

Put

pi

Put

3,

(*I:

= v

0

i = t(pi).

L

, then

v

... v

)T I

n E (LF);

v

R

0

with

ui(*fi) Vi(*fi)

F

= p l . Accordinq t o v1

+...+ un = V

such

.

= ui(fi)

v1 +...+ un

and

= p . ( f . ) = ;.(*f.) 1

..., fn E

fl,

with

n *f ) = II Vi(*fi) n i=l

we have

Then because

*fn E L F

*flv...v

1

1

1

=

v.

we may r e w r i t e

n

;(*f,

v...v

*f ) = t Gi(*fi) i=l

Here, t h e i n e q u a l i t y was a consequence of t h e monotony

.

t h e l a s t e q u a l i t y came from t h e l i n e a r i t y of II Hence, B u t 3 2 r ( p ) f o l l o w s immediately from t h e d e f i n i t i o n o f Note t h a t , by d e f i n i t i o n , every l i n e a r map from

F:

to

o f the

V

V

iii , and

I t(1.1).

.

(LF):

0

i s mono-

tone w i t h r e s p e c t t o t h e n a t u r a l orders. We want t o show n e x t t h a t f o r a l a r g e c l a s s o f cones ( t h e n e g a t i v e l y generated ones) s i m p l i c i a l i t y i s c o m p l e t e l y c h a r a c t e r i z e d by t h e t h e p o s i t i v e dual cone. A t f i r s t , Theorem 2.9.1 immediately y i e l d s :

2.9.4

Remark:

Every l a t t i c e cone i s s i m p l i c i a l .

FSP o f

133

Order Units and Lattice Cones

2.9.5 Let

_ .

Theorem:

F be n e g a t i v e l y generated --and assume t h a t

has t h e FSP. Then --

Ff

i s simplicial.

F

B e f o r e we proceed t o t h e p r o o f o f t h i s theorem we l i k e t o r e c a l l some f a c t s which a r e h i d d e n i n c h a p t e r 2.6. Consider

@+

= {v

Iv

: Ff

R+ l i n e a r }

+

Ff

those l i n e a r f u n c t i o n a l s on

and

@ = @+-

@+,

t h e cone o f

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

functional s. 2.9.6

Remark:

Ff

( i ) If

FSP

has t h e

then

i s an o r d e r complete v e c t o r l a t t i c e

0

( w i t h r e s p e c t t o t h e p o i n t w i s e o r d e r on F f ) . ( i i ) I f F i s n e g a t i v e l y generated t h e n t h e r e i s a monotone l i n e a r map E

:(Fa<)

+ (0,

Ff)

p o i n t w i s e o r d e r on

E(f)(u) = p(f)

for a l l

with

II E Ff,

f E F

.

Proof: ( i ) Lemma 2.6.1 y i e l d s t h a t

i s a complete l a t t i c e cone. NOW, i t i s a

@+

standard procedure (see p r o o f 2.6.2 vector l a t t i c e out o f l o w e r bounded subset

@

. This

A c

where

b

-

b) + b

A.

i s a l o w e r bound o f

( i i ) We know t h a t

Ff = ( F

i s done b y d e f i n i n g t h e infimum f o r a

t o be

@

inf(A) = inf(A

( i i ) ) t o make an o r d e r complete

- F):

(Lemma 2.6.7).

Denote by

cl

the

- F) and d e f i n e c ( f ) t o be FT 3 u + p ( t l ( f ) ) = v ( f ) . Since ( F - F) i s n e g a t i v e l y generated (Lemma 2.6.5 ( i i i ) ) .cl(f) can be w r i t t e n as 'p, - 'p2 where 'p,,'p2 E ( F - F)+.

canonical monotone embedding

F

+

(F

Hence, c ( f ) i s t h e d i f f e r e n c e o f two p o s i t i v e l i n e a r f u n c t i o n a l s , i . e . E maps F i n t o o . 0

Linear Functionals

134

P r o o f o f 2.9.5: Consider

o as i n remark 2.9.6.

t h i s remark and d e f i n e

13 : FT

Let

: F + Q

E

by

+

be t h e map o f p a r t (ii) of

p(p)(cp) = ~ ( p ) ,p E FZ, cp E

a.

Then

(*) NOW, l e t L , +

La

, and

B ( ~ o)

E = p

for a l l

p E

be t h e c a n o n i c a l embedding o f

FT

.

( a , FZ- p o i n t w i s e o r d e r )

c o n s i d e r t h e l a t t i c e cone homomorphism

p :

LF

+

L

given

by t h e fol1 owing diagram

F

D

e LF

a i s a l a t t i c e cone (2.9.6 ( i ) ) t h e r e i s a l i n e a r map : a: + ( L Q): w i t h i ( v ) o La = v f o r a l l v E a: (Theorem 2.9.1).

Since

2

D e f i n e a l i n e a r map 1 : F:

+

(LF);

by

Then

Hence, 2.9.7

F i s simplicial.

0

Theorem:

L e t (F, < )

simplicial.

Then

FT

has t h e

FSP

and

FT

2a

lattice

cone (with respect t o- _ i t s_n a_ t u-r a l o r d e r , -o f course). Proof: We know t h a t

(LF)?

has t h e

FSP and i s a l a t t i c e cone (Theorem 2.6.8).

135

Order Units and Lattice Cones

Let

x

: Ff

( L F):

-,

d e f i n e s a l i n e a r map into

v

be t h e s i m p l i c i a l map. We remark t h a t (LF)f

-,

FT

e

, where

-, v o

i s t h e embedding o f

L

F

LF.

E Ff

NOW, c o n s i d e r a r b i t r a r y

x(il)

Z(vl) + x(p2) =

Then (i,k

= 1,2)

x(u)

o

Since

with

I =

LI

U.

0

ik

l(pi)

( i , k = 1,2) w i t h

+ Z(G2)

and t h e

FSP

gives

= v il + v i 2 3

x(;k)

= vlk

+ V2k

+

pl

vik

p2 =

G1 + i2.

E ( L F ) ~

Because

*

we g e t :

L E Ff , t h i s cone must have t h e FSP.

I t remains t o prove t h a t Ff i s a l a t t i c e cone. Consider a r b i t r a r y pl,p2 E Ff . We c l a i m t h a t a = (x(pl) v x ( p 2 ) ) o L i s t h e l e a s t upper

bound o f

u1

and

t h i s shows t h a t since L

in

p2

Ff

FT

. Since

t h e d i s t r i b u t i v e l a w i s easy t o check

i s a l a t t i c e cone.

a

i s an upper bound o f

i s monotone. Now l e t 1-1 be a second upper bound.

{pl,p2}

Recall t h a t

i s monotone w i t h r e s p e c t t o t h e n a t u r a l o r d e r s s i n c e i t i s l i n e a r . must be an upper bound o f Hence,

~ ( p 2 ) Z(p,)

x ( p l ) , Z(u2).

v x(p2).

Application o f

Hence f o r a n e g a t i v e l y generated cone valent t o the

FSP o f

F:

.

F

L yields

simpliciality

p 2

of

a

F

.

E X(v)

So

0

i s equi-

A t t h e end we would l i k e t o mention t h a t s i m p l i c i a l i t y f o r a g i v e n cone

can be proved by l e t t i n g an a r b i t r a r y l a t t i c e cone p l a y t h e r o l e o f

F

LF.

This, o f course, i s no s u r p r i s e s i n c e i t i s an immediate consequence o f t h e u n i v e r s a l n a t u r e o f t h e f r e e l a t t i c e cone. 2.9.8

Theorem:

L e t T : (F, i) -, ( G , < ) -be l i n e a r map from i n-t o a _ a monotone ~ -F l a t t i c e cone G . ~ Assume t h a_ t t h_ e r e_i s _ a l_ i n e_ a r map ~__ _ -u : F: + GI: between t h e p o s i t i v e ---dual cones such t h a t ~ ( p )0 T = p -f o r a l l p E FT . Then F i s simplicial.

Linear Functionals

136

Proof: Consider

g i v e n by

T*

R

-D

F

LF

n

b

and d e f i n e x(p) = u(p)

Then z

0

T*

.

i s t h e s i m p l i c i a 1 map.

0

Under a d d i t i o n a l assumptions one can a l s o prove t h e e x i s t e n c e o f such a l i n e a r map 2.9.9 Let -

-, G:

: FT

(J

. L e t us

consider the f o l l o w i n g

Situation: T : F

(F,<,I)

+

monotone l i n e a r map from a s i m p l i c i a 1 o r d e r u n i t cone

G

i n t o a l a t t i c e cone

G.

Assume f u r t h e r m o r e

that R

= R

and t h a t

(i) G -~ i s l a t t i c e generated by T(F), i . e . G ---i t s e l f i s t h e Only l~~a t t i c e subcone of G which c o n t a i n s T(F)

.

(ii)

For

p E

-we have u ( f l )

F:

w i t h T(fl)

1. T ( f 2 )

Iv ( f 2 )

whenever

.

2.9.10 Lemma:

-I n the

s i t u a t i o n 2.9.9 we have f o r a r b i t r a r y

u(u)(g)

=

def

fl,f2 E F

sup{w(g)

Iv

E GT w i t h

* and

u

E F,

v

0

T = PI

g E

G

137

Order Units and Lattice Cones

Proof: Some obvious remarks f i r s t . Because o f 2.9.9 o r d e r u n i t , namely

IG = T(1). L e t

a unique monotone l i n e a r

: T(F)

+

such t h a t

R

p = ~(1)s

consider the sublinear

G

on

IG

( i ) t h e cone

, then

E FI

p

. Then

t h e E x t e n s i o n Theorem t h e r e i s a monotone l i n e a r v o T = p

.

But then t h i s functional

i t does n o t a t t a i n

g = T(fl)

--

v . . . v T(fn).

...,fn E

F

The element

v = v1

E FT and

g EG

VOT= v

with

+...+

pi

.

P IT(F)

v : G +

6

g

of

G

u(p)

Now, and by

such t h a t GT

because

a r e o f the form ( i ) ) . Hence

i s nonempty.

g E G.

vn

= vi

g = T(fl)

i s o f the form

. Theorem

2.4.7

v . . . v T ( f n ) . NOW, t a k e any

g i v e s us

v1

,...,v n

E GT w i t h

such t h a t n

where

5

p

ZG(v) 2 u ( v ) :

Proof t h a t

u E GT

p

T =

(consequence o f 2.9.9

t h e s e t appearing i n t h e d e f i n i t i o n o f Consider a r b i t r a r y

( i i ) there i s

must be an element o f

v

s i n c e a l l elements fly

o

has an

G

by 2.9.9

o

T

n

and

v

= pl

+...t

pn.

Put

L F 3 cp = &(fl)v

and we g e t f r o m t h e l i n e a r i t y of t h e s i m p l i c i a 1 map I: t h a t :

...v & ( f n )

Linear Functionals

138

Proof t h a t Take any

Gl,...,G

where v

i

~pE

d(f,)v

cp =

o

u( p ) 2 ZG(11):

n

pi

T =

...v

LF

with

d(fn)

E LFf

= pi

Gi o and

T*(V) = g. Then

(Proposition

with

x(p) =

G1

+...+ V-,

must be o f t h e form Again t a k e

such t h a t

. I n p a r t i c u l a r p = x p i . Now, n p u t v = x vi . Then v o T = p d

i=1

5

(

take

vi E

with

GT

and we o b t a i n :

n

n

2.9.11

~p

2.8.2).

n

x

i=1

vi)

(T(fl)

v...v

T(fn)) = v(g)

.

Theorem:

I n t h e s i t u a t i o n 2.9.9 ----t h e r e i s a unique l i n e a r u : FT -u ( p ) o T = p f o r a l l p E F: . Furthermore -we have ( 9 ) = sup{v(g) f o r a l l p E FT --

and g -

E

1v

E

Gr w i t h

v o T =

+

GT wit.h

PI

G.

Proof: Let

u : FT

Consider

-,GT

v E

be l i n e a r w i t h

~ ( p 0 )

T =

GI: and g E G , say, g = T(fl)v

p

for all

...v

1-1 E

T ( f n ) , fl

FT

.

,...,fn E F .

Order Units and Lattice Cones

2.4.7 we g e t

Again, from Theorem such t h a t

Put of

T

u.1

and

implies: v(g) =

Now, c o n s i d e r t h e

CI

= V. o T

i=1

u

u

then

1

n 1 ui(fi) 5

Hence ( * ) must h o l d and

with

tl =

vl+

...t

vn

. And

= u1 t...t un

~ ( u ) ( T ( f 1 ) v ...v T ( f n ) )

the l i n e a r i t y

= u(v)(q).

must be unique i f i t e x i s t s .

g i v e n i n Lemma 2.9.10.

u ( u ) i s by d e f i n i t i o n

Then

I I G ( u ) (same lemma) i s by d e f i n i t i o n super-

s u b l i n e a r . But t h e f u n c t i o n a l l i n e a r and f u r t h e r m o r e t h e map t y o f t h e s i m p l i c i a 1 map. Since

2.10

*

E G+

vl,...,vn

n

p = v o

u

139

1-1

xG(u)

+

i s l i n e a r because o f t h e l i n e a r i -

EG(u) = u ( u )

e v e r y t h i n g i s proved.

o

CHARACTERS

L o f a p r e o r d e r e d cone

Again we s t u d y t h e canonical embedding i n t o t h e f r e e l a t t i c e cone

LF

LF

g i v e n by

we c a l l t h e map on

F

generated by

F i s equal t o

2.10.1

II : F

i s always s p e c i f i e d t o be

a monotone l i n e a r f u n c t i o n a l p

.

p*

-,6

: LF

.

F o r a g i v e n map rp on

L the restriction o f

rp o

NOW, c o n s i d e r a monotone l i n e a r f u n c t i o n a l o u r g e n e r a l cone

F

6

(F, < )

-,

li

F

to

rp

( i n t h i s section

! ) . We know t h a t t h e r e i s

such t h a t i t s r e s t r i c t i o n t o

Definition:

A monotone l i n e a r f u n c t i o n a l

: F

-,

Ii

i s s a i d t o be a general c h a r a c t e r

: L F + l i such t h a t if t h e r e i s o n l y o n e monotone l i n e a r f u n c t i o n a l i; o L 2 p . I n case o f an o r d e r u n i t cone c h a r a c t e r denotes a qeneral

c h a r a c t e r which i s a s t a t e . Before we c o n t i n u e we l i k e t o r e c a l l t h a t f o r a monotone l i n e a r (L

l a t t i c e cone) t h e c o r r e s p o n d i n g

2.10.2

If -

p*

p

: F

i s m i n i m a l . More p r e c i s e l y :

-

Proposition:

T : LF

+

L

i s monotone -

linear with ___-

T

o

L

2 p

then -

T(rp) > p*(rp)

-,

L

.

Linear Functionals

140

-for all

cp

E L F.

Proof:

.

Consider a r b i t r a r y f l y . . , f n E F. Then T ( * f l v .

- *f n ) .V

> T ( * f i ) > P ( f i ),

i = 1, ...,n . Hence T(*fl v . . . v * f n ) > max(p(fl) ,...,p ( f n ) )

=

p*(*fl v . . . v * f n ) .

This proves the desired i n e q u a l i t y s i n c e a l l elements of k i nd.

L F are of this

0

Characters a r e closely r e l a t e d t o maximal monotone 1 i n e a r f u n c t i o n a l s . Where a monotone l i n e a r functional p : F + fi i s s a i d t o be maximal i f we have v = LI whenever v : F -,fi i s monotone l i n e a r with 1-1 5 v

.

Observe t h a t a general character p i s maximal s i n c e II < p* 2 v* We study the following s i t u a t i o n :

*v

.

implies

Let G be a l a t t i c e cone such t h a t t h e r e i s a s u r j e c t i v e l a t t i c e cone G . Furthermore we assume t h a t f o r every homomorphism y : L F monotone l i n e a r v : L F -, 6 t h e r e i s a monotone l i n e a r G : G + 6 w i t h w I o y. By 6 we denote the map y o 1 : F -,G .

<

2.10.3 Lemma:

a r e equi Val e n t : The fol 1 owing -

5a

general c h a r a c t e r of F

(i)

p

(ii)

P = i; o 6

G-6.

.

, where G -----i s maximal and a l a t t i c e cone homomorphism

Proof:

Let ii be any monotone l i n e a r map G + f i w i t h p* 5 G o y (by assumption there i s a t l e a s t one such G ) . p i s a general c h a r a c t e r , hence we have f o r a l l these G t h a t p* = i; y Since y i s s u r j e c t i v e , so t h e r e i s only one such i An t h i s equation uniquely determines immediate consequence of t h e uniqueness i s t h a t 11 i s maximal. Now, consider a r b i t r a r y g l , . . . , g n E G. Then t h e r e a r e cp,,...,cp,, E LF ( i ) 3, ( i i ) :

.

with y(cpi) obtain

=

g i . Since

y

and

p*

.

a r e l a t t i c e cone homomorphisms we

Order Units and Lattice Cones

i s a l a t t i c e cone homomorphism.

So, (ii)

141

v

( i ) : Let

be any monotone l i n e a r map

LF

.

-,6 w i t h

w o

L

t v,

ii be as i n ( i i ) . Since 11 o y i s a l a t t i c e cone homomorphism L F -, E w i t h G o y o 1 = p we have V o Y = LI? and from P r o p o s i t i o n 2.10.2 we g e t ii o y = LI*I w . Now, by We have t o prove t h a t

assumption on Hence,

11

0

w = LI* L e t

t h e r e i s a monotone l i n e a r

G

y = LI*I u I 5

j e c t i v e ) , and t h e r e f o r e

2.10.4

o

y

i; =

, which

3 : G

-,

iI

U

implies

v I3

with

(because y

V since 11 was maximal. Thus

.

U*=Y

o

y.

i s sur-

Corol l a r v :

2

(F, < ) ---be a l a t t i c e cone. Then LI : F -, r? a general character if and o n l y i f p i s maximal and a l a t t i c e cone homomorphism. --Let -

_ ~ - _ - -

Proof: We have t o show, t h a t here t h e r e q u i r e d assumptions o f Lemma 2.10.3 on hold f o r

G = F. L e t

I

be t h e i d e n t i t y map

F

-,F

G

I*

and consider

given by:

Put Y = I*. I n o r d e r t o apply Lemma 2.10.3 we must show t h a t f o r every monotone 1 i n e a r

v : LF

+

6

t h e r e i s a monotone l i n e a r

: F

-,

6

with

U O I * 2 " .

A t f i r s t we have

L

o

I*(q) 2

(P

for a l l

cp E L F

i n e q u a l i t y f o l l o w s ifwe apply P r o p o s i t i o n 2.10.2 since, c l e a r l y ,

L*

must be t h e i d e n t i t y map o f

by 2.10.2. to

LF.

L =

Indeed, t h i s

LF, T

=

L

o

I*

142

Linear Functionals

So i f

<

v : LF

<

+ E i s monotone, l i n e a r , then

S o I* 2 v . Since we have here 0 the a s s e r t i o n follows from 2.10.3.

= v o L ,has t h e property t h a t

y o L = I*

o

L = I

Application of 2.10.3 and 2.10.4 t o the case G 2.10.5

: F + f i ,given by

=

LF

y i e l d s immediately:

ProDos i t i on:

The general

characters of L F t o F. -

2

F -are the restrictions o f t h e general -characters~

B u t s t i l l another i n t e r e s t i n g c h a r a c t e r i z a t i o n of c h a r a c t e r s i s a v a i l a b l e . F i r s t a d e f i n i t i o n . Let f be an element of a preordered cone ( F , < ). We say t h a t f l i e s i n an extreme ray i f t h e r e i s a 1 2 X > 0 with e i t h e r g1 = x f or g2 = x f whenever f < g1 + g2, g1,g2 E F. Of course, i f -_.

6 > 0 {

6f

I

then 6 > 01

d f l i e s i n an extreme ray whenever f does, and i s then c a l l e d the extreme ray generated by f.

Now, l e t A be a convex subset of F. Then a E A i s c a l l e d an extreme point of A i f a = f l = f 2 whenever f l , f 2 a r e elements of A and 0 <

x

< 1 with

a < x f l + (l-x)f2

.

2.10.6 Theorem: Let be a preordered --cone and endow - (F, <) order on F . Let !.I E (F, < ) * -i)

( F , < ) * -with t h e pointwise

Consider t h e following conditions: (1) p & a general character, l i e s- i n-an~ extreme ( 2 ) !.I _ -ray-and- we-have e i t h e r o r R c !.I(F) = I u ( f ) I f E FI.

1-1 =

0

Then i n general In case t h a t 1-1 ---does not a t t a i n the - (1) implies ~ ( 2 ) . --we have i n addition t h a t ( 2 ) implies (1). value - = --ii)

I f ( F , < ) ------i s an order u n i t cone then the following are equivalent: (1) (2)

(3)

!.I !.I

1-I

i s an extreme point ---of the s t a t e space character i s a s t a t e and belongs ---t o a n extreme ray.

jsa

~

143

Order Units and Lattice Cones

The p r o o f i s , more o r l e s s , a s t r a i g h t f o r w a r d a p p l i c a t i o n o f 2.8.7. Nevertheless, f o r m a l l y , it becomes a b i t c o m p l i c a t e d because i n general l i n e a r

-m

- m . And

f u n c t i o n a l s can a t t a i n t h e v a l u e

i?

plays a very u g l y r o l e i n

s i n c e t h i s element d e s t r o y s t h e c a n c e l l a t i o n law. Proof:

i)(1) =+ (2): F i r s t we show t h a t R c character. I f p =

0

since

does n o t c o n t a i n a

p(F)

m

< 6< 0

t h e m a x i m a l i t y . So t h e r e i s some

s

u2,

pl t

pl,p2

canonical e x t e n s i o n s t o p

*

+

p* = p i

p;

I p;

,in

I p(f)

F

FP = I f E p

u2)*

I( p l t

3,

hence

=

p =

x2 0

Thus

p

E (F, < ) *

. Since

particular

> -

m}

p = pl t p2.

< p(f2) < 0

and

p*,pT,

the

p;

Now c o n s i d e r t h e cone

11 , p.l .,p2

the restrictions o f

G1= x1

o f 2.8.7 y i e l d s

x 1 , A2

x1

> 0. I f

-

F = F,

=

x2

. So,

=

0 then

F

11

,

p

,

p I 0

xl>o.

we can assume

which y i e l d s , by d e f i n i t i o n o f

I

-1

x1 u1

.

must b e l o n g t o an extreme r a y . L e t us p o i n t o u t what d i f f e r e n c e t h e r e s t r i c t i o n o f p

i s i n an extreme r a y and

p I

n E

i=1

ui

i n d u c t i o n and t h e d e f i n i t i o n o f extreme r a y , t h e r e a r e pi

=

xi

p

i=l, ...,n.

Take an a r b i t r a r y monotone l i n e a r Then

u ( f l ) > 0.

contradiction t o

i s a general c h a r a c t e r t h i s i m p l i e s

p

i s maximal) and

p

= h i 1 i;,

r e a l values makes: I f

such t h a t

m

and denote by

and denote b y

for suitable

i)(2) ;+ (1):

then, by

-

,a

LF. Then, b y P r o p o s i t i o n 2.8.6, we have 1-1;

t

(since

b

This gives

4

which i m p l i e s

p 5 0

fl E F w i t h

< 2p

f2 E F w i t h

,pl,p2 t o t h a t cone.Application

and

p

i s a general

X 2 f 2 ) IXl,h2 2 0).

R = { p ( X l fl+ p

then

*0

p

then

6 > 0

i s maximal. Hence t h e r e i s some

p

NOW, i f p ( F ) c o n t a i n s no -

Assume

whenever

p(F)

p* 5 v

LF

-,i?

with

v o

L 2

, pi

v . . . v * f n ) I p* ( * fl

fly...,fn E F. Theorem 2.4.7

v...v

to

E(F,<~

xi

p.

( P r o p o s i t i o n 2.10.2) and i t remains t o prove v(*fl

for arbitrary

v :

p

*fn)

g i v e s us l i n e a r monotone

t 0

Linear Functionals

144

...,v n on

v1, Put

pi

1

1

=

LF w i t h v = v l t . . . t v n

v

i p?(*f.),i=l,...,n.

o

such t h a t

pi

=

t h i s implies

x

xi

the

L . Then i n p a r t i c u l a r z Since

p

x 1. ~ , i=l,...,n.

n I: xi i=1

5

and

pi 1 p

Put

= 1, and i n case

Xi then

A = I:

p i ( f . )1 =vi(*fi)

x

we can have

LI = 0

E vi(*fi).

ray there are

belongs t o an extreme

=,

p

* ( *f l V ...V * f n )

I

x

U*(*

=

Xi

2 0

t p . I f R c p(F)

p

x

= 1 by changing

...V*f,)

flV

=

lJ*(*

( 3 ) : T h i s i s an immediate consequence o f i) ( 1 )

( 1 ) : Take s t a t e s pl,p2 w i t h i i ) (3) p belongs t o an extreme r a y t h e r e i s a Evalution a t 1 2 p2 =

i

p2 t

I yields 1

=

. Another

p2

i>

extreme r a y g i v e s a =

u(*fl v . . . v * f n ) =

s u i t a b l y . Now, we have

,

ii)( 2 )

such t h a t

x

and

x

(2).

i,

ult(l-x)p2,

0 .c

p2 =

x

X > 0 such t h a t , say,

p.

< 1. Since X

=X

pl

p.

T h i s gives

p1 = p .

a p p l i c a t i o n o f the f a c t t h a t

with

0

p I

...V*f,).

flV

i s an

p

Again, e v a l u a t i o n a t

i

I gives

1 and p 2 = p .

i i ) ( 1 ) * ( 2 ) . Take a monotone l i n e a r v : LF + 6 w i t h v o L 1 p . Then u must be a s t a t e s i n c e I i s an o r d e r u n i t f o r LF. Since p* 5 v ( P r o p o s i t i o n 2.10.2) we have t o show t h a t where

fl,...,fn

... ,Gn

U(W)

I p*(lp)

a r e a r b i t r a r y . Theorem 2.4.7

LF such t h a t

f o r cp=*flv

...v * f n

g i v e s monotone l i n e a r

Gi

and v ( q ) = I: Gi(*fi). When -1I f Ci(I) I 0 then, and v i = xi vi. Gi(I) > 0 t h e n we d e f i n e xi = ;i(I) 3i S O . I n t h i s case we p u t s i n c e we a r e d e a l i n g w i t h an o r d e r u n i t cone,

v1,

on

E F

v = I:

So, we have found s t a t e s vi and X 1. t 0 w i t h i E v a l u a t i o n a t i *I g i v e s again v 5 I: x . v and v ( q ) s 1 xi vi(*fi). i i I: a . = 1. NOW, proceed as b e f o r e . D e f i n e s t a t e s on F by pi = uio t A. = 0 1

and

u = p*.

1

Then I:

xipi

t p

.

we o b t a i n

Since

p

i s an extreme p o i n t we have

n

n 1 x.p.(f.) i=1 1 1 1

v(w) I I: AiUi(*fi)

=

n I: i=1

v...v

i=l

I

xip

* ( *fl

*f n )

=

pi

n I: hiU(fi) i=l

I !.I*(@)

=lip

.

and

Order Units and Lattice Cones

This proves Hence

p

v 5 11

* .

must be a general character.

Sometimes a given cone (F,< ) subcone of a given l a t t i c e cone given functional i s a character, L F by L . I n f a c t , under mild First, some prerequisites. 2.10.7

145

can be considered in a natural way as a

( L , < ) . Then, for deciding whether a i t seems desirable t o replace the role of additional assumptions t h i s can be done.

Definition:

.

Then L i s said Let (F, < ) be a subcone of the l a t t i c e cone ( L , < ) t o be l a t t i c e - generated by F i f the map I*: L F + L given by

F

I\,

e

A LF

$I* L

i s surjective. Of course, t h i s definition i s a fancy way of saying t h a t a l l elements are of the form cp = f l v . . . v f n , with n E N and f l , . . . , f n E F

cp E L

If F i s negatively generated, and i f L i s l a t t i c e - generated by F then i t i s quite obvious t h a t L i s generated by LMoreover, any order unit of F i s then also an order unit of L

.

.

2.10.8

Lemma:

Let L -be l a t t i c e - generated by i t s subcone F . (i) If F i s negatively generated, --then for every some u E L z such that p o I* 2 v . -

v E (L

(ii) If F is an order unit cone, then for every v i s some p E ( L , < ) * -such t h a t p o I* t v --

.

E

F):

-there

( L F , < )*

is

there

.

Linear Functionals

146

Proof: ( i ) We claim t h a t

-,

L 3 g

q(g)

=

sup{~(cp)\v E LF

with

I*(cp) < g l

i s a monotone IR- valued superlinear functional. The only d i f f i c u l t point i s t o show t h a t q ( g ) < m for a l l g E L . For t h i s i t suffices t o show q ( g ) 5 0 for

(*)

This i s so, because q(gl + g2) 2 q(gl) cp 5 0

L +

, and therefore

fl,...,fn E F 0 > *fi

. Hence

for a l l

q, q ( 9 ) 5 0

.

g E L-

i s negatively generated and we always have q ( g 2 ) , g1,g2 E L B u t for I*(cp) < 0 we have

.

~ ( c p ) 5 0.

Indeed

0 > I*(@) > f i

i , hence

cp 5 0

cp =

*fl v...v

for a l l

. Thus,

if

* f n for some

i = l,...,n

g E L-

. Therefore

, by definition of

'

Now, consider the sublinear p ( g ) = - q(g). This i s monotone w i t h respect t o the inverse preorder < ' . We know, t h a t there must be a monotone linear I p . Obviously C ( g ) 5 0 i f g < 0 . Hence, 1; does not a t t a i n the value - a0 ( L i s negatively generated!). So N = - i i s a monotone linear !R- valued functional with LI 2 q . Thus IJ o I* 2 v by definition < I -

of

q

.

( i i ) This case i s much easier. Because now q ( 9 ) i s dominated by the monotone sublinear functional, which i s given by ?, times the order unit functional , where A i s the image of the order unit ( i n L F ) under v Hence, by the Sandwich Theorem we find a monotone linear IJ 2 q . 0

.

For completeness i t should be remarked t h a t i f one i s interested in the transfer o f functionals from F t o G (instead of L F t o G ) then one gets a very similar result (even i f G i s n o t a l a t t i c e cone).

Order Units and Lattice Cones

147

Lemma:

2.10.9

Let (F, < ) be a subcone of

(G, < ).

If

(i)

v

G i s generated by F- , --then for every v E F: E Glf -such t h a t p ( f ) 2 v ( f ) -for a l l f E F.

there -_-

i s some

(ii) If J E F i s an order unit of G , --then for every v E (F, < )* there i s some u E ( G , < ) * such t h a t p ( f ) 2 v ( f ) -for a l l f E F. -__.

Proof:

on G by

( i ) Define p p (9)

=

infI- v ( f ) l g t f s 0 , f E F-)

for a l l

g E G

p i s monotone and sublinear. Moreover we have p ( f ) 2 v ( f ) for a l l f E F, since v i s less t h a n or equal t o zero on FHence (Corollary t o 1.3.1) there i s a monotone linear 1-1 I p with p L v

Then

.

on F. ( i i ) Take the same functional p . Then even i f - v ( f ) i s not always < + the inf s t i l l i s , because there i s always a negative multiple - nJ of the order unit such that g - n J < 0 . NOW, use the same argument as above. 0 B u t l e t us return t o characterize characters by means o f a l a t t i c e cone G:

2.10.10

Let -

Theorem:

G -be lattice-generated by- _ i t s_subcone _

(i)If F is an order unit cone, (character) of F i f a n d only i f general character (character) of

F.

i s- a- general character i t i s the restriction t o F o f some

then -

p

-__-___-

--

G.

(ii) Let F be negatively generated, --and assume t h a t p does n o t a t t ain the value Then v --i s a general character of F -----i f and only i f i t i s -the restriction t o F --of some general character of G -

.

.

~

148

Linear Functionals

Proof: ( i ) Lemma 2.10.8 shows t h a t the assumptions of Lemma 2.10.3 are f u l f i l l e d . Hence, everything i s a consequence of 2.10.3 and 2.10.4. ( i i ) The proof of Lemma 2.10.3 can be carried over completely t o the situation where the assumption on G i s only f u l f i l l e d for those v E L F which do not a t t a i n the value , i f one r e s t r i c t s the assertion of Lemma 2.10.3 t o those LI which do n o t a t t a i n the value - m Hence, one can 0 use the same arguments as above.

-

.

149

Order Units and Lattics Cones

2.11

SOME EXAMPLES

I n t h e preceding s e c t i o n s we considered many d i f f e r e n t n o t i o n s and we gave theorems showing how these n o t i o n s a r e i n t e r r e l a t e d . But we d i d n o t g i v e rtiany examples t o m o t i v a t e t h e t o p i c s we considered. I n t h e case o f v e c t o r spaces t h e r e s u l t s we discussed t h e r e a r e e i t h e r w e l l known

or t h e y a r e c l o s e l y

r e l a t e d t o w e l l known theorems ( s e e t h e standard t e x t [283] ) . O c c a s i o n a l l y t h e c o n n e c t i o n w i t h t h e c o r r e s p o n d i n g " c l a s s i c a l I' r e s u l t s may be hidden s i n c e we use e x t e n s i v e l y t h e n o t i o n o f l a t t i c e cones i n s t e a d o f v e c t o r 1a t t i ces. We l i k e t o emphasize t h a t t h e main r e s u l t s we have proved, i n p a r t i c u l a r Theorem 2.10.6,

t u r n o u t t o be v e r y u s e f u l l a t e r on, e.g.

i n s e c t i o n s 11. 3

and 11. 5. Therefore, t h e main a p p l i c a t i o n s of c h a p t e r I . 2 w i l l be found i n these s e c t i o n s .

A t present, c o n c l u d i n g c h a p t e r I.2, we want t o d i s c u s s some s i m p l e and w e l l known consequences i n o r d e r t o i l l u s t r a t e t h e i n t e r r e l a t i o n between these t o p i c s and t h e t h e o r y o f v e c t o r l a t t i c e s . Although we have i n mind m a i n l y a p p l i c a t i o n s t o Banach l a t t i c e s we cannot h e l p s t a r t i n g a g a i n w i t h cones. Then we i l l u s t r a t e o u r technioues a t Banach l a t t i c e s (among o t h e r examples ) .

2.11.1

Absorbin9 F u n c t i o n a l s and Banach L a t t i c e s .

, c o n s i d e r a preordered cone (F,< ) and l e t S : F +IF+ be an

As usual

arbitrary function. S (fn.Pn) with

E F x IP,

N

1

n=l

pn

,

such t h a t

fn< g

-

i s s a i d t o be a b s o r b i n g i f f o r ever-v seouence forall

1

nQN

Q,

S (f,)

converges, t h e r e i s some

a E F

N E P4.

Examples: i)

L e t (F,< ) be a preordered cone,

i s some

fo E F

with

p : F +IF+ s u b l i n e a r such t h a t t h e r e

f < fo whenever

p ( f ) 5 1.

150

Linear Functionals

Then p i s absorbin!. m

x

1

:=

Indeed, l e t

p(fn) <

hn

m.

such t h a t

( ~ ~ , Ef IP~, )x F

Then we have

n=l

N

1

Hence

ii)

n=l

X n fn I ( X t l ) f 0

for all

N

L e t (F, < , I ) be an o r d e r u n i t cone.Recal1 t h a t t h e o r d e r u n i t

f u n c t i o n a l i s d e f i n e d by N

c o n s i d e r a subcone

Fc F

= inf

SI(f)

such t h a t

I X E IR I f < X I I , f E F. NOW, SI i s IR,-valued on F. Then N

S r l i~s absorbing. T h i s i s a d i r e c t consequence o f t h e f i r s t example. iii) L e t (F, < , I ) be a g a i n an o r d e r u n i t cone such t h a t F i s a v e c t o r w i t h I S I l ( f ) = max (Sr(f),SI(-f)) for all space. Then t h e f u n c t i o n a l lSIl f E F is absorbing. T h i s f o l l o w s a g a i n from t h e f i r s t example. iv)

Let

E

be a Banach l a t t i c e . Then t h e norm

11.11

on E i s absorbing.

--

R e c a l l , a Banach space E i s c a l l e d a Banach l a t t i c e i f l a t t i c e and if I l x l l I~l y l l whenever 1x1 = sup(x,-x).

Then, c l e a r l y , t h e s e r i e s

1

n4N v)

N pn

lxnl 1

1

n=l

p

n

x,y IE w i t h I x l s l y l , where

(xn,pn) E E x IR,

Let

x

n

1

pn

nQN for a l l

(E, 5 ) i s a v e c t o r

1

such t h a t

lxnl

pn

nQN converges i n E

I l x n l l converges. and we have

N E IN.

The same argument as b e f o r e shows t h a t 1 1 . ll

i s absorbing on t h e cone

o f a l l p o s i t i v e elements o f a g i v e n Banach l a t t i c e . The n o t i o n o f absorbing f u n c t i o n a l s i s c l o s e l y connected w i t h d o m i n a t i o n o f monotone l i n e a r f u n c t i o n a l s . Indeed, we have Lemma 1. F -* li? _be_a monotone l i n e a r f u n c t i o n a l . absorbing, t h e n t h e r e i s some 0 < c <- such t h a t Let

p :

__.-

p ( f ) Ic S ( f )

-f o_r _a l l

f E F.

S : F

-*

IP,

&

Order Units and Lattice Cones

151

Proof: Assume, CS does n o t dominate fn E F

f o r any c o n s t a n t c. Then t h e r e a r e

p

such t h a t U ( f n ) > 22n S ( f n ) 2 0.

I t f o l l o w s t h a t U ( f n ) > 0. We c l a i m

7, =

Indeed, o t h e r w i s e w i t h

t h e r e would be an element We would o b t a i n U ( g ) 2 N a contradiction. Put

pn =

S ( f n ) > 0.

fn f o r a l l

g E F with

u (f,)

1 s

k,

k€iN N 1 f k=l

g

N E

for a l l

-

(fk) = 0 = N

9

fn

IN, which means

.

Hence for a l l

N E IN.

U ( f n ) I 0,

Thus S ( f n ) > 0.

( Zn S ( f n ) ) - ’ .

Then

1

pn

naN

S(fn)

converges and t h e r e must

be some g w i t h g > Hence

p(g) 2

N

1

pn

n=l N

1

n=l

fn f o r a l l N

pn

u(fn) >

1

n=l

N E IN.

Zn

for a l l

N E IN, a c o n t r a d i c t i o n .

This implies

f o r some c o n s t a n t

c > 0.

D

An immediate a p p l i c a t i o n o f Lemma 1 t o Banach l a t t i c e s i s C o r o l l a r y 2.

Every monotone l i n e a r f u n c t i o n a l

p on a Banach l a t t i ce E i s continuous ~ C o r o l l a r y 2 can be g e n e r a l i z e d . Indeed, we have:

C o r o l l a r y 3 ( [ 72, p.2941):

a

L e t ( E , < ) be p r e o r d e r e d t o p o l o g i c a l v e c t o r space such t h a t E is i) __ E = E+ - Et, where Et = I x E E I O < XI. If n e g a t i v e l y generated, i . e . E, has an i n t e r i o r --p o i n t t h e n every monotone l i n e a r f u n c t i o n a l p on E

i s continuous .

~

-

L

_

-

152

ii)

Linear Functionals

Let

that -

.

(E, II II )

-

E = E,

Et

i s absorbing. -

< E such norm-closed. Then 11.11

be a Banach space. Consider a preorder

and t h e p o s i t i v e cone --

Hence

&

E,

-e _ v e_ r y_montone _ _ _ _l i n e a r f u n c t i o n a l

p : E

-,IR

con t inuouil Proof: i) L e t

x o be an i n t e r i o r p o i n t o f 1 (xo-E,), i.e. functional o f p(x) = i n f C h t 0 I x E C l e a r l y if p ( x ) 5 1 t h e n

x E xo

and d e f i n e

E,

-x2 ( x o -E + ) I

-

E,

p

t o be t h e gauge

and t h e r e f o r e

xo

t h a t p has t h e p r o p e r t i e s r e q u i r e d i n Example i ) .Thus by Lemma 1, we have

u I Since

xo

-

E,

6 p

f o r some

x

I x <x 1. 2 0

= i n f IxtO

*

x. We o b t a i n i s absorbing and,

p

6 > 0.

i s a neighbourhood o f zero, p i s continuous. Hence

p

must

be continuous.

ii) For Ilxlll

x E E

consider

:= i n f Imax ( IIxlII

t inf I

71

5

71

llxll

Since IIxII I 2 llxlll w i t h some

x

( IIxlII + IIx211 ) I x = x1-x2’

for all

x E E

x

IIxII

Et 1

xlYx2€

t h e open mapping theorem p r o v i d e s us

1

n=l

for a l l

absorbing. Indeed,

m

In IIxnII1

<

m

.

1

n=l

An IIynII

<

m.

x E E. consider

(xn,xn) E

Then t h e r e a r e yn E E,

m

such t h a t

1

E E+

> 0 such t h a t

i s , by d e f i n i t i o n ,

such t h a t

x1,x2

.

IIxII I 2 Ilxlll d

II Ill

IIx211 ) I x = x1-x2’

Hence

m

1

n=l

An y n E E+

wfth

since

E,

IR, x

E

yn > xn

i s closed

Order Units and Lattice Canes

153

We have

c xnxn n=l N

II Ill

Thus

m

1

n=l

xn

for all

yn

1 I II

and, t h e r e f o r e ,

Lemma 1 shows t h a t

N E IN.

a r e absorbing. Another a p p l i c a t i o n o

i s continuous.

1-1

For a g i v e n Banach l a t t i c e E l e t

E,

0

C x E E I x 2 0 1

=

and denote by E l

t h e Banach space o f a l l continuous ( i . e . norm bounded) l i n e a r f u n c t i o n a

S

E + IR. R e c a l l , t h a t t h e o r d e r d u a l

,E:

Eo was d e f i n e d t o be equal t o

ET

-

where ET = { p : E

-, IR

I

I

l i n e a r and monotone

p

(compare t h e remarks f o l l o w i n g Lemma 2 . 6 . 1 ) . C o r o l l a r y 4:

I f E -i s _ a Banach ~ - l a- t t i c e

E' = E

then

o

= E,

*

-

* E+

*

Proof:

If P E E '

i s g i v e n we have

I

LI

sub1 i n e a r monotone f u n c t i o n a l on t h e r e i s a monotone l i n e a r 1-1 I v

v

I

x

11.11

x

II It

-, IP,

v : E,

1 0 . I [ . I1

with

.

E,

on

x

is a I E, T h e r e f o r e , by t h e Sandwich Theorem,

.E,

f o r some

E = E+

has a n a t u r a l monotone l i n e a r e x t e n s i o n t o a l l o f

called Hence

u P

.

Put

pl=v,

.

E Eo

Conversely, i f

p

p2 =

E Eo, then

v -

p =

p

.

Then

u1 - u 2

ply

, ul,

continuous by t h e preceding c o r o l l a r y . Hence We always c o n s i d e r t h e o r d e r

u L Lemma 2 . 6 . 2

iff

u

t e l l s us t h a t

I in

E' i s

C l e a r l y , E ' i s a Banach space

E'

p(x) 2

p2

E EY

E E: E E'.

generated by U(X)

and

p2

p

for all

-

,E,

again

u

and

v1,

= u1

p2

-

p2.

are 0

E,

*

, that i s x E E,

.

an o r d e r complete v e c t o r l a t t i c e .

and i t i s easy t o show, t h a t E ' i s a Banach

154

Linear Functionals

l a t t i c e under t h e g i v e n norm. Hence we o b t a i n w i t h Theorem 2.7.6.: Theorem 5: If E i s a-Banach attim i s an o r d e r complete Banach l a t t i c e - l~ -t h e n a s u b l a t t i c e . I .e. , with e,f E t; and E , regarded as 2 subspace __ o f E", sup(e,f) E E , where sup(.,.) is t h e supremum -t a k e n i n E". E l

O f course,

E i s i d e n t i f i e d w i t h a subspace o f E" by t h e e v a l u a t i o n map, i . e .

x E E

every

A

i s i n t e r p r e t e d as a l i n e a r f u n c t i o n a l

x(x') = x'(x) for a l l 2.11.2 Let

- - _ _ .

A

x

on

E'

by

x' E El.

AM- and AL-Cones (F, < ) be a preordered cone such t h a t

0

f

.(

for a l l

f E F.

Furthermore, c o n s i d e r an absorbing

monotone s u b l i n e a r f u n c t i o n a l S : F

(F, < ,S) i s s a i d t o be an AM-cone,

i f (F,<

s (F,<

(sup(f,g))

, S)

= max ( S ( f ) ,

S(g))

) i s a l a t t i c e cone and

for a l l

i s d e f i n e d t o be an AL-cone

i'f

fyg E F

F bas

t

S(g)

for a l l

Define the sublinear functional

.

t h e FSP and t h e Riesz

< and i f S i s a d d i t i v e , i . e .

property w i t h respect t o the inverse order o f S(ftg) = S(f)

-,R+

f,g E F.

11. lls

on

FT

The l a s t e q u a l i t y f o l l o w s from t h e f a c t t h a t

by

S

i s homogeneous. S i n c e

S

i s assumed bo be absorbing we have, i n view o f Lemma 1 o f t h e l a s t s e c t i o n ,

I I ~ I
*

Here

Ff. =

Ft {

Ff.

E F:

.

i s t h e p o s i t i v e dual cone o f E

(F,<

* )

i s equal t o (F,< ) of

p

=

*

I u(f) >

-

for a l l

s i n c e a l l elements

(F,< ) * a r e Rt- valued.

F, i . e . f E F 1

( D e f i n i t i o n 2.6.6).

This

f E F a r e > 0, i . e . a l l elements

O f course h e r e we t a k e

R

= R

.

Order Units and Lattice Cones

>O

f

E F:

T

with

f E F,

for a l l

pointwise order, i . e .

p

+

T

= u

.

F+, i . e . p I v, p , u E F+ , I n o u r s i t u a t i o n , s i n c e we assumed

the natural order i n for

Iu

p

p,

*

*

We consider, as before, t h e n a t u r a l o r d e r i n iff there i s

155

u E FT

*

F+

coincides w i t h the

iff

for all

p(f)lv(f)

f E F. Remark: Note t h a t t h e c o n d i t i o n t h a t F has t o have t h e Riesz p r o p e r t y w i t h r e s p e c t t o t h e i n v e r s e o r d e r , i n t h e d e f i n i t i o n o f AL-cone,

F

whenever

i s t r i v i a l l y satisfied

i s t h e cone o f a l l pos t i v e elements o f an ordered v e c t o r

space.

II I1

We observe f u r t h e r m o r e t h a t f u n c t i o n a l on

F:

.

i s an a b s o r b i n g monotone s u b l i n e a r

Indeed, monoton c i t y and s u b l i n e a r i t y f o l l o w f r o m

s(g) I 1 I .

I I ~ I =I ~sup I p ( g ) I g E F,

1

Let

pn

, on

E F:

1

p n IIpnlls converges. T h i s i m p l i e s t h a t v ( g ) := nQN ntlN converges f o r a l l g E F. C l e a r l y u E FT and u 2 1 p n nIN

E RI,

such t h a t

pn

pn(g)

p

f o r a l l N E IN.

n

One e a s i l y v e r i f i e s t h a t t h e cone o f a l l p o s i t i v e elements o f any s u b l a t t i c e o f a C(K)-space i s an AM-cone w i t h r e s p e c t t o t h e sup norm.

S i m i l a r l y , if F

1

i s t h e cone o f a l l p o s i t i v e elements o f an L (m)-space

w i t h r e s p e c t t o a measure space

(F, 5,s) i s an AL-cone. and

S

i s g i v e n by

Here

S(f)

=

(X,

1,m)

( c f . Appendix A 4 ) , t h e n

I i s t h e almost-everywhere p o i n t w i s e o r d e r

1f X

dm.

The n e x t theorem shows t h a t AM- and AL-cones a r e dual n o t i o n s : Theorem 1: (F,< ,S)

i)

I f (F,< ,S) ii) Furthermore, ment o f --

t h e n (FT, -i s_an AM-cone -is an AL-cone -then (F:, 4 .

F*+

ll.lls)

is an AL-cone -,

rl.llS)

i s an AM-%.

-___

i s an o r d e r complete ____-l a t t i c e cone, and S i s an e l e -

and i t i s an o r d e r u n i t -------

for

FT - FT

Proof:

*

i) F+ hence

1 s t h e p o s i t i v e cone o f t h e v e c t o r l a t t i c e F,

*

F+

-

*

F,

(Theorem 2.6.8.),

has t h e FSP. By t h e p r e c e d i n g remark, t h e i n v e r s e o f t h e n a t u r a l

156

Linear Functionals

*

has t h e Rl‘esz p r o p e r t y . We claim, t h a t II.lls i s a d d i t i v e on F. F, The s u b l i n e a r i t y i s a d i r e c t consequence o f t h e d e f i n i t i o n o f II.lls . On t h e order i n

I g E F, S ( g ) I 1 I . The c o n d i t i o n on S,in t h e d e f i n i t i o n o f AM-cone,implies t h a t {p(g) I g E F, S(g I l l Ilplls = sup I

o t h e r hand, we have

p(g)

i s upwards d i r e c t e d ( r e c a l l p i s monotone). From t h i s we o b t a i n t h a t II.lls i s i n a d d i t i o n s u p e r l i n e a r . T h i s proves t h a t II.lls i s a d d i t ve on F. Hence,

.

(F, II lls) i s an AL-cone.

*

ii) By Theorem 2.5.8. ii), F, i s an o r d e r complete l a t t i c e cone, s i n c e here t h e n a t u r a l o r d e r i n FT c o i n c i d e s w i t h t h e p o i n t w i s e o r d e r . * then by C o r o l l a r y 2.4.6. and Theorem 2.5.8. Furthermore, i f p , w E F, we have

Hence IISUP(M,V)ll

Imax(

=

sup I s u p ( l l , v ) ( g )

I S(g) I 1 I

=

IIIJII~,I I W I I ~ ) .

We obtai’n Ilsup(p,v)ll I

max(

Ilsup(p,v)ll 2 max(

* Hence , F, , II.l$) c 1ear 1y , F, F:

*

-

SE

F.: 0

IIplls, Ilvlls) and t h e o t h e r t n e q u a l f t y ,

Ilvlls, llvlIs), i s t r i v i a l s i n c e II

i s an AM-cone. S i n c e

S

(Is i s monotone.

i s monotone and a d d i t i v e ,

I n view o f 2.11.1 Lemma 1 S must be an o r d e r u n i t o f

Order Units and Lattice Cones

*

L e t us Stay w i t h t h e s i t u a t i o n where

*

o r d e r u n i t of

*

i s on

F,

-

F,

F,

IIp II

11~111 = i n f { Note, t h a t

*

F,

-

, then

II.lls

equal t o

t o S, i . e .

*

.

x

Let

> 0 I IuI 5

*

F,

x S

1

I1 1 ~ lls1 = I1 1 ~ 1II

=

II

Theorem 2.2.3

* - F+* , and

F,

Using t h e

II.

i s an AL-cone, then

IIpII = i n f

{ h

II 11) i s

C(K)-space, K a compact H a u s d o r f f space. hence

C o r o l l a r y 2: <,S)

.

we see t h a t (F,-F,

NOW, an a p p l i c a t i o n o f P r o p o s i t i o n 2.3.1 shows t h a t

If (F,

*

F+

-

C(K)

*

F,

,is

K

i s extremely

order-complete. We o b t a i n

endowed w i t h t h e norm g i v e n by

> 0 I 1p1 I A S 1

i s i s o m e t r i c a l l y -and o r d e r isomorphic -t o some C(K)-space, where extrema11 y d i sconnected compact Hausdorf f - s pace. The element -~

i s an

II II be t h e o r d e r u n i t norm w i t h r e s p e c t S I LI I x S1. Then we have

i s o m e t r i c a l l y isomorphic t o some disconnected s i n c e

S

- x

i s complete under

Kakutani-Krein-Stone-Yosida

i s a l a t t i ' c e cone. I f

F,

the order u n i t functional

infIh 2 0 I

=

157

S E F:

corresponds --t o the one-function on K

K

i s an --

.

As a consequence we o b t a i n t h e c e l e b r a t e d c h a r a c t e r i z a t r o n of AM-spaces, due t o Kakutani ( [ 1813):

A Banach l a t t i c e E i s c a l l e d AM-space, i f ( E + , I E, c o n s i s t s o f a l l p o s i t i v e elements o f E.

, 11.11)

i s an AM-cone, where

i s an AL-cone and, f u r t h e r m o r e , w i t h o * E" = ( E )+ - (EO): i s a C(K)-space. I n v i e w o f 2.11.1 Theorem 5, E i s a s u b l a t t i c e o f E". Hence we o b t a i n According t o Theorem 1 (E,:

<,I1 II)

Theorem 1 and C o r o l l a r y 2,

Theorem 3 ([181I ) :

6Banach lattice E is an AM-space i f and o n l y ----compact C(K)-space, - K -a _ _ _ H a u s d o r f f space.

if

__

E i s a sublattice _ o f_a

Theorem 3 may be regarded as another v e r s i o n o f t h e Kakutani-Krein-StoneYosida Theorem 2.2.3.

Here we do n o t r e q u i r e t h e e x i s t e n c e o f an o r d e r

Linear Functionals

158

u n i t b u t we need i n s t e a d t h e e x i s t e n c e o f a norm such t h a t

( E , II II ) i s

a Banach l a t t i c e . 2.11.3

Characters o f AM-Cones

Throughout t h i s s u b s e c t i o n l e t as i n 2.11.2.At 2.10.1):

, S)

(F,<

be an AM-cone.Let

II lls be d e f i n e d

f i r s t , we s l i g h t l y extend t h e n o t i o n o f c h a r a c t e r ( D e f i n i t i o n

An element

p

*

w i t h Ilplls = 1 i s c a l l e d a c h a r a c t e r o f F i f

E F,

-1-1

t h e r e i s o n l y one monotone l i n e a r f u n c t i o n a l

*

5 S

on t h e f r e e l a t t i c e

cone LF o f F such t h a t

Here L : F

+

Lemma 2.8.3)

LF denotes t h e canonical embedding ( c f . remarks p r e c e d i n g

and

S* : LF

+

R I,

i s t h e unique l a t t i c e cone subhomomorphism

d e f i n e d by

R

F -bLF

I n t h i s s i t u a t i o n the notion o f character i s only a s l i g h t extension o f D e f i n i t i o n 2.10.1.

Here we do n o t r e q u i r e t h e e x i s t e n c e o f an o r d e r u n i t

and, t h e r e f o r e , cannot speak o f s t a t e s . I u n i t and

F

F

i s endowed w i t h an o r d e r

i s t h e o r d e r u n i t f u n c t i o n a l then c l e a r l y every c h a r a c t e r o f i s a c h a r a c t e r i n t h e sense o f D e f i n i t on 2.10.1. S

Now, l e t us s t u d y t h e b i d u a l

By 2.11.2 C o r o l l a r y 2 we know t h a t

E

can be i d e n t i f i e d w i t h a C(K)-space

w i t h extremal ly disconnected compact K. L e t j : F + E be t h e c a n o n i c a l embedding a l a t t i c e homomorphism. Recall, t h a t unit for

( F ~ , s l l Ils)

E. The norm 11.11

i s an AL-cone, on

(F,),

**

. Then,

by Theorem 2.7.6,

**

hence II lls E (F+)+

j is

i s an o r d e r

i s equal t o t h e o r d e r u n i t f u n c t i o n a l .

159

Order Units and Lattice Cones

I1 lls t h i s means i n p a r t i c u l a r

I n view of t h e d e f i n i t i o n o f I, Ilj(f)ll = i n f I A E R = inf { A >

I j ( f ) I II.lls 1 =

o

x

I p(f)

= S(f

I

for all

I I ~ I I f ~o r a l l

f E

p E

I

F:

F.

Remark: The compact Hausdorff space K was i d e n t i f i e d w i t h t h e s e t o f a l l r e a l v a l u e d l a t t i c e homomorphism on E (Theorem 2.2.3).

S i n c e e w y linear functional

on a v e c t o r space i s maximal, C o r o l l a r y 2.10.4 and Theorem 2.10.6 t e l l us t h a t , f o r t h e v e c t o r l a t t i c e E w i t h o r d e r u n i t II lls, t h e c h a r a c t e r s c o i n c i d e w i t h t h e extreme p o i n t s e t o f t h e s t a t e space and w i t h t h e s e t o f a l l l a t t i c e homomorphisms. C(K)-space E , t h e n t h e s e t o f a l l l a t t i c e

Conversely, i f we c o n s i d e r a

u on

homomorphisms

E

with

D i r a c f u n c t i o n a l a t k ).Indeed,

(tik

homomorphisms on E. Then A

and A

let

Q

A

K =

I

6k

I k E K 1

be t h e s e t o f a l l l a t t i c e

i s compact (when endowed w i t h t h e w*-topology)

i s a compact subset o f

K

we would f i n d an element k E K

A

1 coincides w i t h

p(1) =

A

.

f E E, f

If

^K

9 A

0, b u t

which i s c l e a r l y i m p o s s i b l e . Hence

k

t h e n by Urysohn's lemma 0 = gk(f) = f(k) for a l l = A

.

P r o D o s i t i o n 1: L e t ( F , < ,S) be an AM-cone, then t h e f o l l o w i n g are equivalent: i) u & 2 c h a r a c t e r f F

ii)

&a

p

maximal l i n e a r f u n c t i o n a l

iii)There i s a c h a r a c t e r

A

1-1

and a_ l a t_ t i c e homomorphism -_ A o f t h e b i d u a l E such t h a t = 1.1 0 j

-

IS

-

-~

.

Proof: i) Let

ii): The m a x i m a l i t y f o l l o w s d i r e c t l y from t h e d e f i n i t i o n o f c h a r a c t e r . : LF

+

IR be t h e unique monotone l i n e a r dominated e x t e n s i o n o f

which appears i n t h e d e f i n i t i o n o f c h a r a c t e r . I n view o f t h e f o l l o w i n g diagrams

u

160

Linear Functionals

e

e

F -----E>LF

F --bLF

*

we obtain 7 = u . Indeed, every element in LF is of the form ~p= sup(lfl ,... Pfn) with f . E F, i=l,...,n. Hence 1

v*(efi)) = max im * * ( S (Lfi)) = S (cp).

u*(w) = max ( i5n =

max iIn

*

(u(fi))

5

max (S(fi)) iIn

*

(Recall p and S are lattice homomorphisms and reppectively). Hence, by uniqueness o f T , LI = *

Moreover, we have

subhomomorphisms, N

p.

e

where I* is a surjective lattice homomorphism. (Recall, F was assumed to be a lattice). By the uniqueness of T , we obtain

*

Hence, if fl,f2 E F, then there are cpl, cp2 E LF with I (cpi) = fi , i = 1,2 . Therefore

This proves that 1-1 i s a lattice homomorphism.(Note, the preceding proof i s essentially the same as the proof o f Lemma 2.10.3 i)4 ii).)

161

Order Units and Lattice Cones

ii)

=+

i ) :The m a x i m a l i t y o f

implies

p

a monotone l i n e a r f u n c t i o n a l on Again, m a x i m a l i t y i m p l i e s

. We

rp = s u p ( L f l,...,Lfn)

p =

I\

=

LF such t h a t i;

o

.

L

Let

1

.

Let

p I S*

p ( f ) 5 G(Lf)

flY...,fn E F

be

f o r a l l fEF. and c o n s i d e r

obtain

...,f n ) )

= u(sup(fl,

111-1

= max(p(f1),...,p(fn))

i

for all

.

This implies max(G(Lfi)) i
. ,efn)) ,

= ;(sup(efly..

i.e.

;

Hence

ii i s unique. T h i s proves t h a t

LF and t h e r e f o r e

i s a l a t t i c e homomorphism on

11 =

U*

.

i s a character.

p

i) iii) i s an easy e x e r c i s e w i t h t h e Dominated E x t e n s i o n Theorem. iii)* ii): By t h e remark p r e c e d i n g P r o p o s i t i o n 1 C can be i n t e r p r e t e d as a D i r a c f u n c t i o n a l . S i n c e j i s a l a t t i c e cone homomorphism, t h i s proves t h a t

i s a l a t t i c e cone homomorphism on

p

such t h a t

p I v IS

Furthermore,

v

-

p

. Then

*

(F+,II

This implies

lls)

v

1's = 1 , s i n c e

.

Let

II p l l s

v

E FT be

= 1.

E FZ and

1 = II V l l s = II since

II

F

i s an

II v - p I I s = 0

p

l l s + II v - p l l s = 1 -k II V -!JIIs

Y.

AL- cone.

, hence

u

v =

and

p

must be maximal. 0

Corollary 2: F o r every --

f E F

t he r-e i s a c h a r a c t e r -

p

such t h a t

p(f)

= S(f)

.

Linear Functionais

162

Proof: Indeed, K

j(f)

F

of

can be regarded as a continuous f u n c t i o n on t h e c h a r a c t e r s

, i . e . as an element of

functional

C(K) = E. Hence t h e r e i s a D i r a c

dk w i t h 6 k ( j ( f ) ) = 11 j ( f ) l l

d k i s a c h a r a c t e r on

2.11.4

E

, hence

= S(f)

dk

y =

o

j

Kakutani’s Characterization o f Sublattices o f

C(K)-

There i s another famous c h a r a c t e r i z a t i o n o f s u b l a t t i c e s of Kakutani. L e t us i l l u s t r a t e t h e s i t u a t i o n :

Consider

C( [0,11)

F

i s a c h a r a c t e r on

0

Spaces C(K)

due t o

and d e f i n e

E = { f E C([0,11) It i s easily v e r i f i e d , t h a t

I f(0) E

=

1

1

7 f ( l ) , f(T)

i s a s u b l a t t i c e of

=

1

3

3 f($}

C(t0,ll)

. hence an

AM- space. The f o l l o w i n g theorem s t a t e s t h a t , conversely, an

AM- space

i s always a

space o f f u n c t i o n s on some compact H a u s d o r f f space which s a t s f y re1a t i o n s

.

similar

Theorem 1( [ l S l l ) :

-A Banach space E & i s o m e t r i c a l l y isomorphic g s u b l a t t i c e of g C(K)a compact Ha u s d o r f f space, -i f and only ----i f t h e r e i s a subset space, K A c K

x

K

[0,11

x

E

such t h a t --

I f E C(K)

I

f(kl)

(Note, i n t h e preceding example

=

,

x

f(k2)

for a l l

(klYk2,x)

E A1

.

1 A = {(O,l,~),

B e f o r e we prove Theorem 1 we have t o c o l l e c t some b a s i c f a c t s about subl a t t i c e s o f C(K) f o r compact K .

163

Order Units and Lattice Cones

Lemma 2: Let E be a s u b l a t t i c e _o f_a C(K)- -space which separates t h e p o i n t s o f K e r e_ i_ s _ exa c t l y one p r o b a b i l i t y Then f o r every p E E: w i t h II u l l = 1 -t h_ --I

measure

with

T

u(f) =

f

dT

II Ek

I

for all

f E E.

K

Moreover, _ we -have

if

Conversely,

I

E K

T({k

116

T

T

It

klE

({k E K

-i s_ =

1

E

II

=

11)

= 1

a p r o b a b i l i t y ~measure on

1))

=

then

1

l l l ~I I =

.

K

1, where p ( f ) =

K

f d.r

E E

for all f Proof: Let

be a p r o b a b i l i t y measure on

T

jf

p(f) =

D

I

= Ik E K

I I 4(

II

IE

s 1-tl.

T(D)

< 1

, and

f o r the functionals T

Hence

~

c > 0

T(D)

Assume,

B

c K

II

II= 1

,

T

. We ~

have

= 0

T~

=

p1

+

0.

and

Then d e f i n e

T~

.

T(D)

T~

y ~ 2 on

T(D)

2

=

1- T ( D )

t

t

(T-

r(D)-rl)

= 1. I n any case we have

(1 - T ( D ) ) T ~

E

d e f i n e d by t h e i n t e g r a l s w i t h r e s p e c t

II vz11 5 1 and, by d e f i n i t i o n o f

II p l l s T ( D ) ( 1 - t )

IIlJII = 1

if

where

and p u t

T(D)

T

to

. Fix

f E E

for all

f o r a l l Bore1 s e t s

q(B) =

if

dT

K

such t h a t

K

(l-r(D))

T

~

II,

plll 5

1-

t.

1 which c o n t r a d i c t s t h e assumption

164

Linear Functionals

Hence

T({k

E

K

I

II 6

T({k

E

K

I

I1 6

Assume now, t h a t J f K

T~

dTl

IE IE

, T ~

=

II > 1 -

&I)

= 1 for a l l

II = 11)

= 1

.

> 0

, i.e.

are two probability measures with

f dT2 = u ( f )

K

E

for all

f E E.

arbitrarily. Then by regularity there i s a compact D c { k E K I II d k J II = 11 such t h a t E

Fix

; . 1( B )

E

> 0

= ri(B

n 0 ) for a l l Bore1 sets B

as linear functionals on

C(K)

.

c K

(Here we regard

T~

,;i

, therefore we can consider I1 T~ - T i I1 ) .

Note, the restrictions of the functions in E

to

D , E

.

ID

are a sub-

l a t t i c e of C ( D ) Moreover, for any k E D there i s , by definition of D , for any 6 > 0 an element f E E with 1 = Ilf I I Z f ( k ) t 1

-

6 .

Using the f a c t t h a t E i s a sublattice of C ( K ) we see t h a t , for any f i n i t e subset { k l , . . . , k n l c D and any 6 > 0 , there i s f E E with 1 = IIfll 2 f ( k i ) t 1 - 6

for all

i = l,...,n

This means, by Dini's Lemma, the sup-norm closure E-

ID

of

. EID

is a

vector l a t t i c e with an order unit norm. We obtain, by the Kakutani-KreinStone-Yosida Theorem 2.2.3 and the f a c t t h a t E separates the points of K , E I D = '(Dl = C ( K ) I D ( c . f . the remark preceding Proposition 1 of 2.11.3) This implies

II ;l(f)

-

T2(f)ll I 2c 11 fll

for a l l

. f E E , and we get

165

Order Units and Lattice Cones

II Y1 -y211 5

;,,;,

(Recall

a r e supported by

II T~ -r211 5 4

Thus

To prove t h e converse, assume

, for

Then, by r e g u l a r i t y

I

D c Ik E K

II

4

II

I E

D ).

which i m p l i e s

E

=

any

T

E

.

2 E

r1

=

T

~

since

E

was a r b i t r a r y .

i s a p r o b a b i l i t y measure on

> 0

K

with

t h e r e i s a compact

11 w i t h

r(D) t 1 - E

.

As b e f o r e , u s i n gD i n i ' s Lemma and t h e f a c t t h a t E i s a s u b l a t t i c e o f C(K)

, we

II u D l l t 1 - E

Therefore

II

We have since

E

EID = C(D).

have

p

- p D II

5

E

was a r b i t r a r y

, where

, where

pD E

ET i s d e f i n e d by

v(f) =

J

K

f dT

for all

f E E.

Hence,

,

P r o o f o f Theorem 1: The s u f f i c i e n c y o f t h e c o n d i t i o n on

E t o be a s u b l a t t i c e o f C(K) i s

easily verified. We prove t h e n e c e s s i t y . Let

E

be a s u b l a t t i c e o f some

the points o f Let

A c K

x

K

K

.

x

[0,1]

C(K)- space and assume t h a t

(Otherwise t a k e a s u i t a b l e q u o t i e n t o f

f(kl)

be t h e s e t o f a l l t r i p l e s (kl,k2,x) =

x

f(k2)

for all

f E E

.

K

E separates instead). such t h a t

166

Put

Linear Functionals

i.

= Cf E C(K)

We have

E

c

I f(kl)

i c C(K).

=

x

for all

f(k2)

I t remains t o show t h a t

whenever

p

llE

=

be r e g u l a r Bore1 measures on f E

Let

1

that

ji

4(

i; =

- F

If

+

E, = E

K with

E K

R lK c C(K). Every c h a r a c t e r

of E,1

by p u t t i n g

El

since

El

i;(lK)

p

1

=

II

4(

IE

=

and l e t

"1E

I II 4( of

. By

_I1 IE

x-'

morphism, hence a c h a r a c t e r o f k E K w i th

\

IE

. Therefore,

for a l l

-1 6 = A

k gI E

"I E E

.

E

have t h e same norm as

K

. This

pi

dTi

, i=1,2,

By t h e p r e c e d i n g Lemma we o b t a i n

F of

,S)

-i s_an

C(K)

with

k ' E K.

i s maximal and a l a t t i c e homo-

means

f(k')

2 , we

for all

II

=

T~ = T~

--

AM- cone and j : F

canonical embedding, ----then t h e r e a r e a s e t

II f ( k )

IE

II

have

E wi

for all

II = II $ , - I 1 E

E ET w i t h

f E E, i=1,2,

and, o f course,

C o r o l l a r y 3: If (F,< -

~

Theorem 2.10.10 we o b t a i n

By t h e preceding Lemma t h i s means t h a t t h e f u n c t i o n a l s

1f

T

From t h e preceding o b s e r v a t i o n we g e t

by construction o f

=

,

can be extended t o a

k E K.

vi(f)

T~

11, i=1,2.

=

generates a l a t t i c e subcone

II t: 0 t h e n

i.1 , we have

E

= ~ i ( f ) for a l l

f

K

= ri(Ik

'IE

By t h e S e p a r a t i o n

p1 , p 2

C(K) (Stone-WeierstraB theorem). Now t a k e any

h =

f E E

v

So assume

E, i=1,2, and T ~ ( K )= II pill

character

F

.

"1E

i.

E =

Theorem 1.5.8 i t i s enough t o show t h a t , f o r a l l p l = p2

E A).

(kl,k2,x)

, i.e.

** -,(F+)+ A c K

v1 = pllE=

u1 = u2

- IF:): x

K

pZlE=

.

0

= C(K) x

[0,11

u2

i s the

such th_ at _ __

.

167

Order Units and Lattice Cones

(j(F)

-

j(F)

2.11.5

denotes the closure of

j(F)

-

j(F)

with respect to II I( )

Korovkin's Theorem

So far we have seen t h a t l a t t i c e cone homomorphisms give much insight in

the structure of special cones. I n this section we shall give a further result which emphasizes the usefulness of l a t t i c e homomorphisms. We prove a classical theorem due t o Korovkin ([202 I ) , which s t a t e s t h a t i f Tn i s a sequence of positive linear operators from C([O,ll)into C ( [ O , l I ) such lim T n f i. n-,

that

x

for all

E

=

f i , i=1,2,3, where f l ( x ) = 1, f 2 ( x )

[0,11 , then

lim Tn f = f n+

for all

= x,

f3(x)

f E C([O,ll)

=

xL

.

We s t a t e and prove t h i s result in the slightly more general form which can be found in the l i t e r a t u r e . Let K be a compact Hausdorff space. Then a subset M c C ( K ) i s called a Korovkin family i f for every sequence of uniformly bounded positive linear operators Tn : C ( K ) + C ( K ) and every l a t t i c e cone homomorphism S : C(K)

+

lim T n f n-,

C(K)

= S f

we have t h a t lim T n f = S f n+ for a l l f E C ( K ) .

for a l l

f E M

implies

Note, in the preceding formulation of the classical Korovkin Theorem we have M = I f l , f 2 , f 3 ) . If

M c C(K)

, then we denote M2 = I f 2 1 f E MI

M

.

i s said to separate the points of

there i s

f E M with

f(kl)

9

f(k2)

, i f for every

K

k l y k 2 E K , kl*

k2,

and assume that M contains

the

.

Now we have 2.11.8 Let -

Theorem ([283 I ) :

M c C(K)

separate the points of

K

168

Linear Functionals

function

lK . Then M U

fi a K o r o v k i n

M2

family o f

C(K).

Proof: Tn : C(K)

Let

+

l i m Tn f = S f n-, m

C(K)

be a sequence o f p o s i t i v e l i n e a r o p e r a t o r s such t h a t

, where

f E M u M2

for all

S : C(K)

+

C(K)

i s a lattice

cone homomorphism. Note, t h a t t h i s assumption a l r e a d y i m p l i e s t h a t t h e

Tn

a r e u n i f o r m l y bounded s i n c e we have c l e a r l y , i n view o f t h e p o s i t i v i t y o f i s an o r d e r u n i t f o r C(K), t h e Tn and t h e f a c t t h a t lK

II Tn l K l l = II TnII Fix

k E K

on

C(K)

for all

.

. Then we c l a i m t h a t t h e r e i s e x a c t l y one monotone l i n e a r such t h a t

k ). T h i s means, o f course, t h a t

Dirac functional a t

(dk

n

p =

6k

y

.

Proof o f the claim: Consider a monotone l i n e a r hf = ( f ( k ) l K by

M u M2

-

f)2

, which

!J

with (*)

.

Fix

f E M and

i s c o n t a i n e d i n t h e subspace o f

define C(K)

generated

.

From ( * ) we o b t a i n p(h ) = 0 f

Furthermore we have

for all II pi1 supK

!J 5

f E M

.

By t h e Riesz-Konig Theorem 1.6.1 t h e r e i s a r e g u l a r p o s i t i v e Bore1 measure T

on

K

such t h a t p(g) =

J

K

g dr

for all

Hence we o b t a i n supp

T

c

n

f EM

hi1 (0)

g E C(K)

.

.

Order Units and Lattice Cones

The l a t t e r s e t must be equal t o We conclude

tkl

since

M

169

separates t h e p o i n t s o f

K

4(

lJ=

which proves t h e c l a i m . We know (Remark preceding P r o p o s i t i o n 1 o f 2.11.3), homomorphisms C ( K )

%,

functionals

c o i n c i d e w i t h nonnegative m u l t i p l e s o f D i r a c

-, R

K .

k E

Hence, if k E K

t h a t the l a t t i c e

i s fixed, then

,

= A ,Q

\ O S

f o r some

k' E K

, and

A t 0.

NOW, f i x to

f E C(K)

. Then

Sf

(**) where

N

and assume

there are

-

I ( T n f ) (k,)

E

(Tnf)

i s n o t converging ( i n sup-norm)

> 0, and a sequence

I>

( S f ) (k,)

for

E

i s a s u i t a b l e i n f i n i t e subset o f

On t h e o t h e r hand, we have by assumption

f o r every

g E M

u

M

2

4(

n

o

Tn

such t h a t

n E N

.

,

.

T h i s i m p l i e s , by t h e compactness o f of

N

kn E K

extends a

w*-

cluster point o f

p o i n t s c l e a r l y a r e o f t h e form

El(

must be a c l u s t e r p o i n t o f t h e kn p r e v i o u s remark, i s equal t o We d e r i v e from t h e c l a i m

, t h a t any w*- c l u s t e r p o i n t

K

o

S

IM u M~

. The

x $ , for

61(

o

n

I M u ~2

f o r some

functional some

S

k'E K

4(

k E K 0

. But

and t h i s

S , by t h e

and

x

such

t0

.

k

170

if

Linear Functionals

i s s u i t a b l y chosen. T h i s c o n t r a d i c t s (**)

n E N

II Tn f - S f II

-,0 , as

n

-,a

f o r a l l f E C(K)

.

. Therefore 0

2.12 REMARKS AND COMMENTS S e c t i o n 2.1:

The n o t i o n o f s t a t e p l a y s an i m p o r t a n t r o l e i n t h e s t r u c t u r e

theory o f

algebras and i n a p p l i c a t i o n s o f t h i s f i e l d , l i k e a b s t r a c t

C*-

s t a t i s t i c a l mechanics (see [581,[2761,[2771). For a C*- a l g e b r a a s t a t e i s a p o s i t i v e l i n e a r f u n c t i o n a l o f norm 1. I n s e c t i o n I 1 6.4 we w i l l show, t h a t t h i s i s a s p e c i a l case o f t h e n o t i o n we have d e f i n e d here. S t a t e s o f o r d e r u n i t v e c t o r spaces have been s t u d i e d e x t e n s i v e l y (compare [ 2 I), t h e i r importance f o r a s t r u c t u r e t h e o r y o f these v e c t o r spaces was d i s covered i n t h e 50's and 60's (see [ 521,[ 961,[ 97

I [loll).

r e q u i r e s i n t h e d e f i n i t i o n o f an o r d e r u n i t v e c t o r space space i s Archimedean, i . e . f o r for all

x

a E A

we have

I n general one

(E,I) t h a t t h e

a 5 0 whenever

a 5

x

I

> 0. B u t t h i s p r o p e r t y i s o n l y i m p o r t a n t i n t h e c o n t e x t o f

d u a l i t y theory, because, when

E

i s embedded i n t h e second dual t h e non-

archimedean elements f a c t o r o u t . Since o u r concept i s so general t h a t t h e embedding i n t h e second dual i s anyway n o t i n j e c t i v e we d i d n o t r e q u i r e t h e o r d e r u n i t cones t o be Archimedean.

S e c t i o n 2.2:

Theorem 2.2.3 was independently proved i n [181 3,[2o51,[3oo1

[342]. I n s t e a d o f Kakutani-Krein-Stone-Yoshida

Theorem t h e r e s u l t i s

sometimes c a l l e d Kadison-Kakutani-Stone Theorem because o f Kadison's i m p o r t a n t paper [179 1. U s u a l l y t h e theorem i s s t a t e d as a r e p r e s e n t a t i o n b y f u n c t i o n s on t h e extreme p o i n t s o f t h e s t a t e space ( i n s t e a d o f f u n c t i o n s on t h e s e t o f l a t t i c e s t a t e s ) . That t h e l a t t i c e s t a t e s and t h e extreme p o i n t s a r e t h e same w i l l become obvious l a t e r on ( s e c t i o n 2.10).

171

Order Unizs and Lattice Cones

Extremely disconnected compact spaces a r e a l s o c a l l e d

S e c t i o n 2.3:

___S t o n i a n spaces. They have a v e r y r i c h t o p o l o g i c a l s t r u c t u r e s i n c e t h e c l o -

s u r e o f each open subset f i c a t i o n of

G

. This

G

i s homeomorphic t o t h e Stone-Czech- compacti-

f a c t has t h e i m p o r t a n t consequence t h a t e v e r y bounded

continuous f u n c t i o n on an open dense s u b s e t has a continuous e x t e n s i o n (c.f.

[2831). L a t e r on we see t h a t Choquet's Theorem can b e extended t o

measures t a k i n g values i n

C(S), S

S t o n i a n (Chapter 11.4). The Loomis-

S i k o r s k i Theorem 2.3.3 has been proved i n d e p e n d e n t l y i n [2181 and [2941 (see a l s o 12951). The Loomis-Sikorski-Theorem p r o v i d e s a d i r e c t approach t o M a i t l a n d W r i g h t ' s e x t e n s i o n o f t h e Riesz R e p r e s e n t a t i o n Theorem t o C ( S ) valued measures, S S t o n i a n ( c . f . [3381 t o [3411). The L o o m i s - S i k o r s k i Theorem can be e a s i l y o b t a i n e d b y combining P r o p o s i t i o n 2.3.2 w i t h t h e B i r k h o f f - U l a m theorem. T h i s r e s u l t s t a t e s [1501 t h a t f o r e v e r y B o r e l - s u b s e t B o f a compact space symmetric d i f f e r e n c e

X

t h e r e i s a r e g u l a r open s e t

B A f(B) = (BLf(B)) U ( f ( B ) 4 )

f(B)

such t h a t t h e

i s meagre. ( R e c a l l

t h a t a r e g u l a r open s e t i s a s e t which i s t h e i n t e r i o r o f i t s c l o s u r e ) . The s e t

f(B)

i s t h e n n e c e s s a r i l y unique and

f

defines a

u- homo-

morphism h a v i n g t h e meagre s e t s as k e r n e l . For t h e sake o f completeness we i n c l u d e h e r e t h e B a i r e c a t e g o r y theorem which p l a y e d a c r u c i a l r o l e i n t h e

B a i r e space A t o p o l o g i c a l space X i s s a i d t o be a -if t h e complement o f e v e r y meagre s u b s e t i s dense ( i . e . 0 i s t h e o n l y

p r o o f o f Theorem 2.3.3.

meagre open s e t ) . Then t h e B a i r e c a t e g o r y theorem says t h a t e v e r y l o c a l l y compact space and every c o m p l e t e l y m e t r i z a b l e space i s a B a i r e space [2841. S e c t i o n 2.4:

The o b s e r v a t i o n t h a t t h e d i s t r i b u t i v e l a w i s i m p o r t a n t f o r

l a t t i c e semigroups can be found i n most s t a n d a r d t e x t s i n l a t t i c e t h e o r y ( f o r example [ 3 9 1 ) . The c o n s t r u c t i o n o f t h e upper and l o w e r envelopes and cpv [

2

'YU

i s adapted from t h e c o n s t r u c t i o n o f convex and concave envelopes

, p.41.

These q u a n t i t i e s i m p l i c i t e l y always t u r n up when Edward's

s e p a r a t i o n theorem [ 971, o r a g e n e r a l i z a t i o n [1161, i s proved. Theorem 2.4.7 w i l l be i m p o r t a n t i n c o n t e x t w i t h f r e e l a t t i c e cones and c h a r a c t e r s , i t i s a g e n e r a l i z a t i o n o f a f o l k l o r e r e s u l t about v e c t o r l a t t i c e s [283

p.721.

S e c t i o n 2.5:

A cone F i s n e g a t i v e l y generated i f

i t i s downwards

directed w i t h respect t o the following modified order r e l a t i o n :

,

Linear Functionals

172

f 5 g

i f f there i s

h

4

0 with

g t h = f.

O f course, f o r v e c t o r spaces t h a t i s t h e same as b e i n g upwards d i r e c t e d

w i t h r e s p e c t t o t h e g i v e n o r d e r . L a t e r on

-

- i n connection w i t h representing

downwards d i r e c t e d s e t s w i l l be i m p o r t a n t . F o r t h i s reason we i n t r o d u c e d t h e n o t i o n " n e g a t i v e l y generated" i n s t e a d o f " p o s i t i v e l y

measures

generated". The Riesz p r o p e r t y i s an a d a p t i o n o f t h e w e l l known Riesz Decomposition p r o p e r t y [ 2 1 f o r v e c t o r spaces. A v e c t o r space has t h e Riesz Decompos i t i o n p r o p e r t y i f i t s p o s i t i v e cone has t h e Riesz p r o p e r t y i n o u r sense. The importance o f t h i s p r o p e r t y was d i s c o v e r e d by F. Riesz i n 1940 (see [2661). F o r v e c t o r spaces t h i s p r o p e r y i s e q u i v a l e n t t o t h e Riesz i n t e r p o l a t i o n p r o p e r t y , where E i s s a i d t o have t h i s p r o p e r t y i f f o r x1,x2,y1,y2 that

E E

with

x. I z 5 yk 1

xi I yk, i , k = 1,2,

, i,k

t h e r e i s always some

z

such

= 1.2.

I f f o r a cone F t h e c a n c e l l a t i o n law h o l d s and i f F i s endowed w i t h t h e F- - o r d e r t h e n t h i s p r o p e r t y i s e q u i v a l e n t t o t h e FSP. Hence, a1 1 these p r o p e r t i e s a r e e q u i v a l e n t f o r v e c t o r spaces.

A l o c a l l y convex ordered v e c t o r space w i t h Riesz p r o p e r t y such t h a t i s l o c a l l y compact i s a u t o m a t i c a l l y a v e c t o r l a t t i c e

2

,

p. 851.

E,

has a compact base t h e n i t i s , by a r e s u l t o f K l e e [1951, E, a u t o m a t i c a l l y l o c a l l y compact.

And i f

S e c t i o n 2.6:

T h i s s e c t i o n does n o t c o n t a i n any new i n f o r m a t i o n

s i n c e t h e p o s i t i v e dual cone can be considered as t h e p o s i t i v e dual cone o f a s u i t a b l e ordered v e c t o r space (Lemma 2.6.7).

A l l the r e s u l t s o f t h i s

s e c t i o n a r e c o n t a i n e d i n standard textbooks about o r d e r e d v e c t o r spaces. S e c t i o n 2.7:

The r e s t r i c t i o n o f theorem 2.7.7 t o compact convex s e t s

(geometric s i t u a t i o n ) was p u b l i s h e d b y C a r t i e r , F e l l and Meyer [ 6 2 1 i n 1964. Since t h e F i n i t e Sum Theorem p l a y s an e s s e n t i a l r o l e i n t h e p r o o f o f 2.7.7

one can g e n e r a l i z e t h e d e f i n i t i o n o f t h e

0- decomposition o r d e r

Order Units and Lattice Cones

173

by t a k i n g i n t e g r a l s i n s t e a d o f f i n i t e sums. T h i s approach leads i n a n a t u r a l way t o t h e n o t i o n of " d i l a t i o n " . Working w i t h t h e A b s t r a c t D i s i n t e g r a t i o n Theorem one t h e n o b t a i n s a s t r o n g e r r e s u l t t h a n 2.7.7.

This

approach was c a r r i e d o u t i n S t r a s s e n ' s paper 13041 (compare a l s o [ 2 , p.271 and [ 8 7 1 ) . S p e c i a l v e r s i o n s o f theorems o f t h i s k i n d d a t e back t o Hardy-Li t t l e w o o d - P o l y a ' s i m p o r t a n t book on i n e q u a l i t i e s [154 1. The r e s u l t s were t h e n improved by B l a c k w e l l [ 4 2 1 and Shermann 12911. Theorem 2.7.7 was p u b l i s h e d i n t h i s form i n [1161.

S e c t i o n 2.8:

The n o t i o n o f f r e e l a t t i c e cone seems t o be new.

Free v e c t o r l a t t i c e s have been considered b e f o r e b y E.C. K. Baker [ 131, S. Bernau [ 341, P.F. Conrad

781, R.D.

Weinberg [3231, B l e i e r [ 4 5 1 and

o t h e r s . Many o f t h e r e s u l t s about r e p r e s e n t a t i o n o f f r e e v e c t o r l a t t i c e s can be o b t a i n e d v i a t h e m a t e r i a l presented i n s e c t i o n 2.8. B u t we d i d n o t want t o g e t t o mucli i n v o l v e d i n t h a t m a t t e r s i n c e we need f r e e l a t t i c e cones o n l y as a t o o l t o i n t r o d u c e s i m p l i c i a l cones. The dual cones o f o u r f r e e l a t t i c e cones a r e c l o s e l y r e l a t e d t o f r e e

compact convex s e t s i n t r o -

duced by Z. Semadeni [289 1. S e c t i o n 2.9:

G. Choquet

[67]

i n t r o d u c e d i n 1956 i n f i n i t e dimensional

s i m p l i c e s . O f course, t h e t y p i c a l example o f a s i m p l i c i a l cone i s a cone h a v i n g a Choquet simplex as base. T h i s f o l l o w s e a s i l y w i t h Theorem 2.9.5 and t h e f a c t t h a t i f

K

i s a Choquet simplex t h e n

Decomposition p r o p e r t y [2, p.911.

A ( K ) has t h e Riesz T h i s example o f a s i m p l i c i a l cone w i l l

be t r e a t e d e x t e n s i v e l y i n s e c t i o n 11. 3.5. Since o u t n o t i o n o f s i m p l i c i a l cone covers cones w i t h o u t bases as w e l l , our concept g e n e r a l i z e s c o n s i d e r a b l y t h e concept o f Choquet simplex. Theorems 2.9.3 and 2.9.11 a r e corresponding t o t h e a s s e r t i o n t h a t i n case o f a Choquet simplex maximal measures a r e u n i ue. Theorems 2.9.5 and 2.9.7

are a generalization o f the

c h a r a c t e r i z a t i o n o f a Choquet simplex by means o f t h e Riesz Decomposition P r o p e r t y . That,

n t h e d e f i n i t i o n o f s i m p l i c i a l cone, t h e r o l e of t h e f r e e

l a t t i c e cone can be r e p l a c e d by o t h e r s u i t a b l e l a t t i c e cones i n a consequence o f 2 9.8.

174

Linear Functionals

Section 2.10: I t is well known t h a t f o r a compact convex s e t K t h e extreme points a r e characterized by the property t h a t a l l extensions t o C ( K ) of t h e evaluation maps a r e vector l a t t i c e homomorphisms C ( K ) + R . For example, t h i s follows e a s i l y from t h e f a c t t h a t an extreme point k € K has % as unique representing measure.Section 2.10. generalizes t h i s concept t o a r b i t r a r y cones. Since a r b i t r a r y cones a r e not always subcones of l a t t i c e cones we had t o find l a t t i c e cones in which they can always be embedded. This r o l e i s played by the f r e e l a t t i c e cones. The theorems of t h i s s e c t i o n show t h a t , sometimes under additional assumptions, extreme points (and extreme rays) can be characterized by l a t t i c e properties and by unique-extension properties as well.

One of t h e most important theorems of t h i s s e c t i o n , Section 2.11: Theorem 2.2.3, and i t s modification, Theorem 3 of 2.11.2, goes back t o a famous paper of Kakutani ([181]). The second c r u c i a l c h a r a c t e r i z a t i o n of s u b l a t t i c e s of C ( K ) , Theorem 1 of 2.11.4, can a l s o be found i n Kakutani's paper. The l a t t e r theorem gives r i s e t o he following natural g e n e r a l i z a t i o n : i s c a l l e d a G- space f there i s a compact Hausdorff 2 xa) E K x K x [-1,1], a € A , space K and a s e t of t r i p l e s ( k a1, ka,

A Banach space A

E

some index s e t , such t h a t E = I f E C(K)

I f ( k 1a )

= ha

f(kt) for all

I t i s c l e a r t h a t E i s no longer a s u b l a t t i c e of

.

a E A}

C ( K ) when

x a attains

negative values. On t h e o t h e r hand, of course, every AM- space i s a G- space. G- spaces ( a n d therefore AM- spaces) have many i n t e r e s t i n g f e a t u r e s . The d e f i n i t i o n of G- spaces appeared i n [1471. There i t was shown t h a t the duals of a l l G- spaces a r e isometrically isomorphic to L1- spaces. In thesame paper the question was raised whether t h e Banach spaces whose duals a r e L1 a r e always G- spaces. I t was shown i n 12123 t h a t , i n f a c t , the answer i s negative. AM- spaces and

G- spaces a r e closely connected, e . g . a l l

complemented i n s u b l a t t i c e s of

G. spaces a r e

C ( K ) - spaces (12173). There i s a

Order Units and Lattice Cones

175

c h a r a c t e r i z a t i o n o f G- spaces which corresponds t o Theorem 1 o f 2.11.4:

A Banach space E i s a G- space i f f t h e r e i s a compact H a u s d o r f f space K such t h a t E c C(K) and min(fl,f2,f3) + max(fl,f2,f3) E E whenever f l y f 2 s f j E E.

([2171)

.

G- spaces (and t h e r e f o r e AM- spaces) have drawn i n t e r e s t i n r e c e n t y e a r s

because they a r e sometimes isomorphic t o

A

s e p a r a b l e G- space i s isomorphic t o a

dorff-

C(K)-

spaces:

C(K)- space,

K

a compact Haus-

( 130 1).

space

However, t h e same a u t h o r has shown: There i s a (non-separable) AM- space which i s n o t i s o m o r p h i c t o any space ( [ 31 1 ) .

C(K)-

There e x i s t corresponding, s u r p r i s i n g l y simple, a b s t r a c t c h a r a c t e r i z a t i o n s o f concrete Banach spaces. F o r example, a Banach l a t t i c e E c a l l y isomorphic t o an

L1- space i f f t h e norm i s a d d i t i v e on t h e cone o f

a l l p o s i t i v e elements o f [1801)

.

E ( i . e . i f f t h i s cone i s an AL- cone) ([481, [491,

S i m i l a r l y , a Banach l a t t i c e

1I p <-

iff

i n f ( x , y ) = 0.

i s isometri-

IIx+yll

E

i s i s o m e t r i c a l l y isomorphic t o an L - space,

= Ilxllp

+

IlyIIp

for all

x,y E E

P with

( [ 481).

F o r a d e t a i l e d and comprehensive t r e a t m e n t o f Banach l a t t i c e s t h e r e a d e r

i s advised t o c o n s u l t [215] and [283]. One o f t h e theorems which have i n i t i a t e d a l o t o f r e s e a r c h a c t i v i t i e s i n r e c e n t y e a r s i s K o r o v k i n ' s theorem ( i . e . Theorem 2.11.8).

I n fact, i n

numerous p u b l i c a t i o n s many a u t h o r s g e n e r a l i z e t h i s theorem t o Banach l a t t i c e s o r l o c a l l y convex l a t t i c e s and d i s c u s s n o t i o n s c l o s e l y r e l a t e d t o t h i s theorem. The whole s u b j e c t deserves t o be t r e a t e d i n d e p e n d e n t l y For references t h e r e a d e r s h o u l d c o n s u l t [ 26 ] o r [ 1081, [337]. The most i n t e r e s t i n g case, o f course, i s t h e s i t u a t i o n when t h e u n d e r l y i n g space possesses a f i n i t e K o r o v k i n f a m i l y . T h i s i s e.g. t r u e i n t h e

C(K)-

176

Linear Functionals

2

"classical" case C ( 0 , l ) , here { l , x , x 1 constitutes a Korovkin family. An interesting result in this respect i s due t o Wolff ([3351), which can be stated even for general Banach l a t t i c e s : A Banach l a t t i c e E possesses a f i n i t e Korovkin family i f f E i s finitely generated, i . e . i f there i s a f i n i t e s e t M c E such t h a t the smallest sublattice of E containing M i s E i t s e l f .

C H A P T E R

I1

REPRESENTING MEASURES SECTION 11.1 COUNTABLE DECOMPOSITION

The aim o f t h e second p a r t o f t h i s book i s t o s t u d y those l i n e a r f u n c t i o n a l s on c o n c r e t e cones ( i . e .

cones o f

6-

X ) which can b e r e p r e s e n t e d as i n t e g r a l s . T h i s problem has two p a r t s . We have t o v a l u e d f u n c t i o n s on some s e t

l o c a l i z e the l i n e a r functionals w i t h respect t o

and, i n d o i n g t h i s , we

X

have t o proceed i n such a way t h a t t h e e x t e n s i o n s t o t h e measurable functions have t h e Monotone Convergence P r o p e r t y (see Appendix). I f one i s i n t e r e s t e d o n l y i n f i n i t e l y a d d i t i v e measures t h e n one c o u l d e i t h e r proceed w i t h t h e Sandwich Theorem o r w i t h t h e F i n i t e Decomposition Theorem. B u t u n f o r t u n a t e l y honest measures have t o be u- a d d i t i v e , and t h i s f o r good reasons. NOW, one has t o keep i n mind t h a t f o r a l i n e a r f u n c t i o n a l

1-1

, given by the i n t e g r a l m on

w i t h r e s p e c t t o some

(u- a d d i t i v e ) p r o b a b i l i t y measure

a g i v e n sequence

n E N , o f nonempty measurable s e t s w i t h

Yn,

one can e a s i l y decompose

u

into

u

=

1 un

, where

pn I

T h i s means t h a t , w i t h r e s p e c t t o t h e s u b l i n e a r f u n c t i o n a l s

supy

X, and f o r

u Yn n

,n E

= X N.

supy, Y c X,

i n t e g r a l s have a s t r o n g e r decomposition p r o p e r t y t h a n t h e p r o p e r t y g i v e n b y t h e F i n i t e Decomposition Theorem. Roughly speaking, f o r t h i s case we have t o c a r r y o v e r t h e F i n i t e Decomposition Theorem t o c o u n t a b l e s e t s of subl i n e a r f u n c t i o n a l s . L a t e r on i t t h e n t u r n s o u t t h a t t h i s i s a l s o s u f f i c i e n t f o r a l i n e a r f u n c t i o n a l t o be an i n t e g r a l , i . e .

i f we can f i n d f o r a l i n e a r

f u n c t i o n a l a s u i t a b l e decomposition f o r c o u n t a b l e s e t s o f s u b l i n e a r f u n c t i o n a l s t h e n t h i s f u n c t i o n a l can be r e p r e s e n t e d b y an i n t e g r a l . And, t h e l o c a l i z a t i o n can a l s o be done by decomposition t e c h n i q u e s . F o r t h i s reason

we s t a r t t h i s c h a p t e r by an i n v e s t i g a t i o n o f t h o s e cones

where a Countable Decomposition Theorem h o l d s ; these cones w i 11 be c a l l ed D i n i cones. I n s e c t i o n 1.3 we study permanence p r o p e r t i e s o f D i n i cones and i n s e c t i o n 1.4 we c a r r y o v e r o u r arguments t o a p r o p e r t y which l a t e r on

w i l l be neccessary f o r r e p r e s e n t a t i o n b y s i g n e d measures.

178

Representing Measures

1.1 PRELIMINARIES The main o b j e c t i v e o f c h a p t e r 11.1 i s t h e g e n e r a l i z a t i o n o f t h e F i n i t e Decomposition Theorem t o sequences o f s u b l i n e a r f u n c t i o n a l s . To be more p r e c i s e , we s t a r t w i t h a sequence

o f s u b l i n e a r f u n c t i o n a l s and we (p,) search f o r c o n d i t i o n s which ensure t h a t f o r e v e r y l i n e a r p I sup pn nm there are l i n e a r

pn Ipn

such t h a t

p Ix .

n '

nEN

P a r t i c u l a r a t t e n t i o n w i l l be g i v e n t o t h e case when

, the

F

cone under

c o n s i d e r a t i o n , c o n s i s t s o f bounded r e a l v a l u e d f u n c t i o n s on some s e t

X

and

c o n t a i n s t h e c o n s t a n t f u n c t i o n s , and when t h e sequence of s u b l i n e a r f u n c t i o n a l s i s a s s o c i a t e d t o subsets a r e g i v e n by t h e

Yn-

Even i n t h i s case t h e s t a t e s

pn

p

n

= sup

'n

weak sense as supported b y

Yn

Yn c X

i n t h e sense t h a t t h e

suprema. I supy

.

n

can be imagined o n l y i n a v e r y

1.1 Remark: Let Y c X

and assume t h a t

state with

1-1 5

sense t h a t

p(f) I

I supy

u(g)

i s t h e p r e o r d e r g i v e n by

p Isupy

then

;1

whenever f sYg

= p

, i.e.

i s maximal

. Then

p

f,g E F such t h a t iff

whenever

i s supported b y

f ( y ) Ig ( y )

f

i

is a

Y

i n the

sYg , where sY

for all

y E Y.

The a s s e r t i o n i s an immediate consequence o f t h e Sandwich Theorem s i n c e t h i s r e s u l t gives a

sy- monotone s t a t e ii w i t h

leaves o u t t h e assumption t h a t

p

i s e a s i l y found. F o r example t a k e f o r f : R+ + R +

, take

Y = [0,11 and

p

F t h e cone o f decreasing f u n c t i o n s g i v e n by

p(f) = f(3).

So, i n a c e r t a i n sense, t h e d e s i r e d decomposition up t h e

1-1

ii I sup

I f one Y ' has t o be maximal t h e n a counterexample p 5

i n t o terms which a r e l o c a l i z e d on t h e

p I

Yn.

1

pn

nEN

for

p

splits

The s t u d y o f these

179

Countable Decomposition

decompositions t h e n l a t e r on y i e l d s an immediate access t o r a t h e r general r e s u l t s c o n c e r n i n g t h e e x i s t e n c e o f r e p r e s e n t i n g measures. Throughout c h a p t e r 11.1 we assume t h a t a l l l i n e a r and s u b l i n e a r f u n c t i o n a l s do a t t a i n values i n

r? ( i n s t e a d o f a general o r d e r complete v e c t o r l a t t i c e ) .

T h i s r e s t r i c t i o n i s necessary, s i n c e most o f t h e r e s u l t s do n o t h o l d f o r t h e general case o f v e c t o r l a t t i c e v a l u e d f u n c t i o n a l s . L e t us s t a r t by r e c a l l i n g some conventions and d e f i n i t i o n s . always be a preordered cone. Let

(rn)

Let

pn

co

I: p n ( f ) E n=l G c F

Let

+

i? be some f u n c t i o n a l s . We say,

r?

for all

f2, fl,f2 E F

iff

G- monotone i f f

t o be

m

x

exists, i f

pn

n=l

f E F.

be a subcone and l e t

the d e f i n i t i o n o f the fl>

m I: rn = l i m i n f x r n=l m +co n = l n' co

be a sequence o f r e a l numbers. We d e f i n e

: F

will

(F,<)

G- o r d e r

there i s u ( g ) I0

be a l i n e a r f u n c t i o n a l on

IJ i

( D e f i n i t i o n I . 2.5.1):

g E G for all

with

fl + g = f2

F , Recall

.

u was s a i d

g E G.

We c o n s i d e r now t h e f o l l o w i n g s i t u a t i o n :

E, n

pn : F

Let

be l i n e a r . D e f i n e F-

-

E N

, be s u b l i n e a r f u n c t i o n a l s on F and l e t

F- = { f E F I p n ( f ) I 0

monotone i f f u ( f ) 5 0

F- = { f E F I f

whenever

hence iF 01, -

for all

n E NI.

pn(f) 5 0 f o r a l l

the notion o f

11

Hence

n E N

.

: F + fi

u

is

We have

F- h e r e concides w i t h t h a t o f

Ch. 1.2.5.

1.1.2 Let (i)

Definition: (p,)

be a sequence o f s u b l i n e a r f u n c t i o n a l s o n

A linear

are l i n e a r 115 1 nEN

p

pn I pn

tnIJn,

F.

i s c a l l e d decomposable ( w i t h r e s p e c t t o and r e a l numbers

tn 2 0

with

where t h e sum i s assumed t o e x i s t

I: tn

.

n€N

(p,)) 0

i f there such t h a t

180

Representing Measures m

I f we have in addition

p =

z

n=l

tnpn

then we call

p

s t r i c t l y decomposable.

( i i ) If there are a real t > 0, n E N and some F--monotone linear v with 11 5 t p, + v then we call u partially decomposable. Note that we require t

+

0 in i i ) and z tn + 0

in i ) .

Examol e: Let F = C(0,l) be the vectorspace of a l l continuous realvalued functions 1 $,(f) = f(n) for on [0,11. Let 6, be the Dirac measure a t 1 , i . e . a l l f E F. Assume t h a t u : F -* R i s linear and monotone. By the RieszKonig Theorem 1.1 can be regarded as a positive measure on [0,11, hence v i s ( s t r i c t 1 y ) decomposable (with respect t o ( a n ) ) i f and only i f s u p p 11 c { 1 n E N 3 . I f p = T t 6n , where T i s the Lebesgue measure on [0,11 a then 11 i s partially decomposable b u t n o t decomposable. Now, consider an arbitrary compact Hausdorff space X and l e t F = C ( X ) (the continuous realvalued functions on X ) . Let p be any positive measure on X ( t h a t i s , a monotone linear functional on C ( X ) and take a disjoint measurable covering (Y,) of X. Then clearly, p i s decomposable with respect t o pn = supy , j u s t take pn t o be the restriction of the n measure 1-1 on Y n . I t will turn o u t l a t e r on t h a t here we can get rid of the measurability assumptions on Y n .

I

1.1.3 Proposition: Assume t h a t --

pn(f)

for a l l

5 0

-._

f

E

F and n

E N.

Then the following --

are equivalent: decomposable.

i)

p

ii)

p ( f ) 5 0 for a l l

with

m

z

k=l

p(fk)

f E F ; --and for every sequence ( f , ) , f n E F , rn > we have sup(inf pn( fk))> - m n m k=l

.

181

Countable Decomposition

Proof: i)

*

ii)

ii):

This i s a d i r e c t consequence o f D e f i n i t i o n 1.1.2.

i):

We c l a i m t h a t t h e r e i s some

1

f ) I - max(pl(f) ,...,p

P

no E N

such t h a t

for all

(f))

f E

F.

Then, using the F i n i t e Decomposition Theorem I 1.4.4 we o b t a i n

I+,

and l i n e a r

I

pn

such t h a t

,

x =A

E

n=l

An 2 0

no

n and

This proves i)i f we take

P S

pn =

0

1

n=l

xnpn.

0, An = 0 f o r a l l

n

7

no

.

NOW, we prove t h e claim: Let

n n ( f ) = max(pl(f),

f o r every

n

there i s

Fn

We m u l t i p l y

p2(f),..,,pn(f))

Tn

E

for all

..

f E

F and assume t h a t

1 0 2 P ( f n ) > Fi nn(fn).

F such t h a t

w i t h an a p p r o p r i a t e p o s i t i v e number and o b t a i n

fn E F

with

o

t p(fn) 2

- -21 n

m

Hence we have X, p(fn) t n=l

1 1 > n n n ( f n ) t -n n k ( f n ) 1 - m ,X -2

n=l n

>

m

m

m

m

-

m

for

k 5 n

.

and

s u p ( i n f pn( t f k ) ) I s u p ( i n f nn( ,X f k ) ) n m k=l n m k=l 5 sup(inf

n

I

m

(

sup(inf ( n mtn

E

n n ( f k ) ) ) I sup(inf( t n n ( f k ) ) ) k=l n m>n k=n

-

m 1 " 1 E ) ) = sup(- t K) = k=n n k=n

-

m

. This

contradicts i i ) . 0

182

1.2

Representing Measures

THE MAIN DECOMPOSITION THEOREMS

Here again l e t pn : F

be sublinear functionals for a l l

+IF

n E N,

where (F,s) i s a preordered cone. Let G c F be a subcone. Recall that G generates F i f for every f E F there i s some g E G such t h a t f t g E G. (Definition I . 2.5.1) 1.2.1 Example: Let F be some cone and assume t h a t p : F + 6 i s sublinear such t h a t there i s go E F with p ( g o ) < 0 Let G = { g E F I p ( g ) 5 01. G i s a

.

subcone of F. If f E F then for suitable x > 0 we obtain p(f t x go) I p ( f ) t x p ( g o ) 5 0 . Hence f t x go E G and thus G

.

generates F The following lemma i s a variant of the statement of the Uni versa1 Property I . 2.6.4. 1.2.2. Lemma: Assume that --

G

unique linear --

generates F. : F - + R with

5

!.I

G IG =

(Note t h a t we require v ( g ) > -

: G !.I

.

for a l l

OD

+ IR

be linear. ---Then there i s a -~

g E G)

Proof: Let f E F and g E G with f t g E G. Then define ; ( f ) = v(ftg) - u ( g ) . This definition i s independent of the choice of g . Indeed, i f g1 E G with

f t g1 E G

!.I(f+g)

-

v(g) =

then f P(f t

t

g t g1 E G and we have

9 + 91)

-

!.I(g1)

-

v(9) = u ( f

Since ;1 i s certainly linear, we obtain t h a t unique linear extension of v 0

.

G

t

91)

-

"91).

i s the well defined

1.2.3 Lemma: Assume t h a t G generates F . Let v : F + R be linear, -and l e t -TI : F + 6 be sublinear. -Then the following are equivalent: i)

v(g)

5

n(g)

for a l l g E G --

.

183

Countable Decomposition

i i ) There i s a linear functional : F - * R -such t h a t and i ( g ) I 0 fo r a l l g E G . --

p I

+ n

Proof: Only i ) * i i ) requires a proof. Let p = n - LI. Then p i s sublinear and p ( g ) t 0 for a l l g E G. By the Dominating Extension Theorem I 1.3.1 there i s a linear functional v Ip with v ( g ) 2 0 f o r a l l g E G. Let f E F and g E G be such that f + g E G Then v ( f ) 2 - v ( g ) > Hence we may define i = v

-.

.

As before, l e t

F- = {f E F I p n ( f ) I 0

for a l l

- .

n

E NI

.

Partial Decomposition Theorem:

1.2.4

Let u : F Assume that F- generates F . ~1 owing are equi Val ent : Then the fol --

+ R

be F-- monotone -~ and l i n e a r . -

i s partially decomposable

i)

p

ii)

For every sequence ( f k ) in F- with --

sup(inf p n ( z f k ) ) > n m k=l

m

Proof: We only have t o combine Proposition 1.1.3 and Lemma 1.2.3.

0

Note t h a t in 1.2.3 and 1 . 2 . 4 as well as in the following theorem we For a function p : F + 6 we denote r e s t r i c t ourselves t o realvalued p by p i0 the fact t h a t !.I i s n o t dominated by 0 , i . e . there i s some f E F with p ( f ) > 0.

.

184

Representing Measures

1.2.5

DecomDosition Theorem:

Assume t h a t G c F- 2 a subcone g e n e r a t i n g F and t h a t f o r each n E N -such t h a t pn(g) < 0 . -Then t h e f o l l o w i n g are equivalent: t h e r e i s g E G --Every

i)

G-monotone l i n e a r

1.1

with -

1.1 i0

is -

-,R with

1.1 i0

is -

: F -tlR

p a r t i a l l y decomposable

ii)

Every

G- monotone l i n e a r

1.1 :

F

decomposable For e v e r y --

iii)

G- monotone l i n e a r

1.1

: F +IR OD

f o r every --

sequence

1 v(gk) >

gn E G w i t h

LI i0

with -

- k=l

-

and -

we have

Q)

-

Proof:

ii) i)i s , b y d e f i n i t i o n , e v i d e n t . i)3, iii)i s an immediate consequence o f t h e P a r t i a l Decomposition Theorem 1.2.4.

iii)-+ ii)F i x

1.1

as i n iii).P r o p o s i t i o n 1.1.3 g i v e s t h a t

decomposable w i t h r e s p e c t t o t h e

p

.

C a l l a sequence

"IG o f nonnegative r e a l numbers a r e p r e s e n t a t i o n i f

1.1

(tn)

IG

is

,n E

N

,

m

'

n = l tn P n j G

*

Observe t h a t i f

(tl,...yt~,t~+l,...)i s a r e p r e s e n t a t i o n t h e n

(tl,...,t~,O,O,O

e t c . ) i s a g a i n a r e p r e s e n t a t i o n . The reason f o r t h i s i s

that a l l the with

1 tn

Denote by

*0

p

"G

0. Observe f u r t h e r t h a t t h e r e i s a r e p r e s e n t a t i o n

5

since

1.1

IG

i s decomposable.

n t h e s e t o f a l l r e p r e s e n t a t i o n s . Consider i n n t h e p o i n t w i s e

order, i . e . f o r two r e p r e s e n t a t i o n s (t,)

5 (sn)

iff

( t n ) , (sn)

t, 5 s,

we w r i t e

for a l l

m E

N

.

Countable Decomposition

185

We claim: (1) There i s a representation which i s maximal with respect t o the pointwise order of n .

( 2 ) For every (t,) E n there are linear u, : G+R-= I r E R ~ r I O l with vn I pn,. for a l l n E N and with G m

Furthermore the un are maximal, i . e . i s linear with vn I p 5 p n I G

.

whenever

vn = p

p

F i r s t we show how these assertions imply t h a t IJ i s decomposable. Let (t,) E n be maximal and consider the vn as in ( 2 ) . Then the Dominating . Since the vn Extension Theorem gives us linear Gn 5 p, w i t h u I < "IG

-

are maximal, t h i s implies vn = u n l G and the 3, must be the unique extensions of

vn (given by Lemma 1 . 2 . 2 ) .

Furthermore

<=

aD

Z

n=l

tn Gn

must

be the unique extension of v , this f o r the same reason as before. I f we can show t h a t 1.1 I G then the proof i s finished. Lemma 1.2.3 gives us a : F + R such t h a t p I 3 t i; . If 5 0 then linear G- monotone obviously P I V. Therefore we may assume i; i0. Then by i i i ) and Proposition 1.1.3 ; i s decomposable, i.e. there are s,, 1 0 with IG z s n + 0 such that m

Hence v I G I mality of

m

I

n=l

( t nt s , ) p

"G

.

Clearly a contradiction t o the maxi-

(tn).

Proof of (1): Let tl = sup{s L Ol(s,O,O,O, ...,0 ) E 01. This supremum exists since there i s some g E G with p l ( g ) < 0 . Obviously (tl,O,O,O ...) E a . NOW, we continue in the obvious way. We define inductively for a l l n E N tntl = supis t 0 I ( t l , t 2 , . ..,t,,s,O,O,. ..) E n l . Then the sequence (t,) i s a maximal element of n Proof o f ( 2 ) : This assertion i s a consequence of the Sum Theorem I 1.4.1 combined with a maximality argument. Fix N E N and define a sublinear

.

186

Representing Measures

EN =

m

z t n p n I G . Observe t h a t

n=N

G1,...,V

*

N '

such t h a t

'n

I

-,6 with C1 N- 1

1 t

plG I

n=l

m

u p the SN

-

G

I

with

", I G

z t p N

'IG

I

n Vn +

instead of

z

n=l

The Sum Theorem gives l i n e a r

-

p l I ,...,VN-l I p N - l I G and SG I p N I G G GN . Repetition of t h e procedure, by s p l i t t i n g

m

pIGI

pN I 0.

plG I

tnOn.

m

pIGI

z t n p n I G etc., yields linear

n=l

Actually, we f i r s t obtain t h i s i n e q u a l i t y

m

lim sup z t p on t h e r i g h t s i d e , b u t this term N - + m n=N 'IG can be dropped since i t i s 5 0. NOW, replace the Vn by maximal

w i t h an additional

Vn

I vn 5

p

"G

, then s t i l l we have

m

p

I'

<

z tnvn

- n=l

.

Finally, we observe t h a t a l l the vn a r e R- - valued, t h i s i s so because a t t a i n s only values i n R and because of p I 0 for all n € N "G ( i f tn = 0 p u t vn = 0 ) . Let u s add a somewhat more technical r e s u l t which wil be needed i n Corollary 1 . 2 . 7 below and i n various o t h e r s i t u a t i o n s l a t e r on. We c a l l a subset 0 c F downwards d i r e c t e d i f f o r f g E ways some h E @ with h 5 f and h 5 g . 1.2.6 Lemma: be-a s u b l i n e a r Let - p : F -,fi downwards d i r e c t e d . ----Then t h e r e and inf p ( f ) = f€O Proof: Let a

=

inf p ( f ) . If fE@

a > -

m

monotone functional -and l e t @ c F be i s a l i n e a r monotone p : F + 6 with p inf p ( f ) f€@

then define a s u p e r l i n e a r 6

with x f superlinear since 0 was downwards d i r e c t e d . I f 6(g) = supiA alX 1 0 , t h e r e i s

there i s al-

@

f E

Ig )

.

a

-

=

5

p

by

6 is indeed m

then define

I

p

Countable Decomposition

'(9)

1

=

yields

if

- m

g + O

if g = 0

0

a l i n e a r monotone

.

I n any case, t h e Sandwich Theorem I 1.2.5

,: F - 6

with

a = i n f 6(f) = i n f u(f) = i n f p(f).

f€Q

fE@

I p

6 5

. Then

0

fE@

I n t h e f o l l o w i n g c o r o l l a r y t o t h e Decomposition Theorem we show t h a t i n t h e case o f an o r d e r u n i t cone one can d r o p t h e r e s t r i c t i v e assumption t h a t a l l the

under c o n s i d e r a t i o n have t o a t t a i n values i n R

p

(instead o f

6 ) . 1.2.7 Let -

Corol 1a r v :

( F , <,I) be an o r d e r u n i t cone and assume t h a t -_--____-__

sublinear w i t h pn I S I va 1ent :

for all n E --

Every l i n e a r u : F -,6 with -~ ii) -Every l i n e a r M : F + 6 w i t h iii)-For e v e r y d e c r e a s i n g sequence

i)

u

N

. -Then t h e

I S

p I

S

-.

6

are

f o l l o w i n g -a r e equi-

I -i s p a r t i a l l y decomposable i s decomposable I -

( f n ) , fn E F

sup ( i n f p n ( f m ) ) = i n f SI(fm) n m m

pn : F

, -we have

.

P-roof : i s an immediate consequence o f D e f i n i t i o n 1.1.2.

(ii) * (i)

(i) ( i i ) L e t us f i r s t e s t a b l i s h t h e s i t u a t i o n we considered i n t h e Decomposition Theorem 1.2.5. F i x some

u

F, = { f E F since

S 0

G = I f E F,

I

SI and c o n s i d e r i t s r e s t r i c t i o n

I U(f)

>

- -1.

u

=

Clearly, t h i s r e s t r i c t i o n

; ( I ) = 1. D e f i n e a s u i t a b l e subcone f < 01. The cone

G

generates

F

U

G =

IF, of

t o t h e subcone i s r e a l v a l u e d , and F,

by

G t {a I] a E R I

since

-IE G. And, o b v i o u s l y , f o r each pn t h e r e is some g E G w i t h p , ( g ) < 0 , j u s t t a k e g = - I . Observe t h a t t r i v i a l l y e v e r y l i n e a r u I S

I

is

G- monotone. Now, e v e r y l i n e a r

I SI

on

FP can be

188

Representing Measures

extendedto a l i n e a r V(f) =

-w

linear

v

SI on a l l o f F by p u t t i n g

v I

f E F x Fp

for with

. Therefore

i s decomposable. Hence, 1.2.5 i m p l i e s t h a t

v $ 0

i s decomposable. So, c e r t a i n l y

(iii) * ( i) : As b e f o r e f i x holds f o r Then

. Let

F

i)i m p l i e s t h a t e v e r y G- monotone

,p

p I

LI

1 FLI

i s decomposable.

SI and c o n s i d e r G W

1 p(gk) >

g, E G such t h a t

k=l

. Assume

-w

. Put

that =

f,

iii)

m

1 gk.

k=l

i s decreasing and from iii)we o b t a i n

(f,)

m

m

W

Using t h e P a r t i a l Decomposition Theorem 1.2.4 we conclude t h a t

p

is

p a r t i a l l y decomposable. (ii) inf

m

* (iii):L e t fn E

and

= inf S ( f ) I m

p(fm)

F be decreasing. L e t

m

p 5

p :

S (Lemma 1.2.6).

I

F

+6

be l i n e a r w i t h

Then t h e r e a r e

tn 2 0

W

such t h a t

Since

p I

-p(I)

1 t p n = l n n'

= p(-I) I

Note t h a t

p n ( * I ) 5 SI(*I)

t tn p ( - I ) I - t tn n=l n=l

= i 1.

we o b t a i n

I tn I 1. n=l

Hence m

i n f SI(fm) m

= i n f u(fm) I

m

I

T h i s proves iii).

0

1 tn i n f pn(fm) I s u p ( i n f pn(fm)) I

n=l

i n f SI(fm) m

m

n

.

m

189

Countable Decomposition

DIN1 CONES

1.3

We want t o give some applications of the preceding section t o cones F(X) of upperbounded functions f : X -,li? containing a l l constant functions. ( X * 0 i s an arbitrary s e t ) . Hence t h r o u g h o u t t h i s section we assume t h a t F(X) i s an order unit cone whose order unit i s the function l X ,i . e . for all

f E F(X)

there i s

such that

X > 0

f

5

we consider the pointwise order with respect t o

X l X . Of course in

F(X)

x .

Definition:

1.3.1

i s called -Dini cone i f for every pointwise decreasing sequence

F(X) with

f n E F(X) ( ( f , ) t

(f,)

i n short) we have

supX(inf f n ) n a

=

inf supX(fn)

nEN

(By inf f n we mean the pointwise infimum).

n a

We call this condition the Dini

-

condition.

If F(X) = C ( K ) , the vector space of a l l continuous realvalued functions on a compact Hausdorff space K ,then, clearly, the Dini condition i s equivalent t o : Whenever f n E C ( K ) , (f,) t and (f,) converges pointwise t o a continuous function

f

on K then

(f,)

converges uniformly t o

f.

Dini's Lemma: of a l l upper semicontinuous functions on-a compact Hausdorff space i s- a- -Dini cone.

The cone U S C ( K ) --

1.3.2 Remark:

A max-stable order unit cone F(X) ( i , e . max(f,g) E F(X) whenever f , g E F ( X ) ) i s a Dini cone i f and only i f every decreasing sequence in

F(X)

converging pointwise t o zero converges uniformly to zero.

Proof: i ) Assume t h a t F(X)

i s a Dini cone and take f n E F ( X ) , (f,) t , which

Representing Measurss

190

converges p o i n t w i s e t o zero. The D i n i c o n d i t i o n i m p l i e s which c l e a r l y means t h a t

(f,)

i n f supX(fn) = 0 nEN

converges u n i f o r m l y t o zero.

ii) Assume t h a t every decreasing sequence i n F(X)

which converges p o i n t -

wise t o zero a l s o converges u n i f o r m l y t o zero

fn EF(X), (f,) c

Since

supX(inf fn) nEN

t h a t t h e assumption

i n f supX(fn) nEN

I

s u p X ( i n f fn) < i n f supX(fn) nEN nE N r E R

s u p X ( i n f fn) < r < i n f s u p X ( f n ) nEN nE N

- r,O)

for all

n E

i n f (supXfn) n a

F(X),

N , Then

- r

i n f (supxgn) = 0 nEN

= i n f (supX(fn

nEN

-

leads t o a contrasuch t h a t

.

i s max-stable and c o n t a i n s t h e c o n s t a n t s . We have by assumption on

.

i s always t r u e i t s u f f i c e s t o show

d i c t i o n . Therefore assume t h a t t h e r e i s

D e f i n e gn = max(fn

. Let

gn E F(X)

since

F(X)

gn J. 0 and t h e r e f o r e ,

. Since

r ) ) 5 i n f ( s u p X gn) nEN

weobtain t h e c o n t r a d i c t i o n i n f (supx fn) 5 r < i n f (supx f n ) nEN nEN

1.3.3.

Lemma:

a t t a i n s i t s maximum on L e t F(X) ---be a D i n i cone. -Then every f E F(X) -__-Proof: D e f i n e gn =

n(f

-

supX(f))

a t t a i n i t s maximum, t h e n

. Then

i n f gn(x) = nEN

gn E F(X), gn 2

-

0

for all

CJ,~

x E X.

.

If

.

f

X.

does n o t

Countable Decomposition

Hence

s u p X ( i n f gn) = n

-

03

, but

inf n

191

,a

supx(gn) = 0

contradiction. 0

The c o n d i t i o n o f Lemma 1.3.3 does n o t c h a r a c t e r i z e D i n i cones. The r e a d e r

w i l l f i n d a counterexample a f t e r Lemma 1.3.5. Let

VF(X)

be t h e max-stable cone generated by

I

V F ( X ) = {max(f l,...,fn)

fl

,..., fn E

F(X), i . e .

F(X), n E N I

.

Then we have 1.3.4

Lemma:

The r e. e q u i v a l e n t : -f o l l o w i n g a_ (i)

VF(X)

_i s _a -D i-n i cone

is_ a Dini _ --

(ii) F(X)

cone

(iii)For e v e r y sequence

fn I 0 i n F(X)

we _ -have

n m i n f supx ( IT f k ) = supx( I f k ) n k=l k=l

.

Proof: ( i i i ) a r e t r i v i a l . So i t remains t o show: ( i ) =b ( i i ) ( i i i ) + ( i ) : L e t F(X) be a D i n i cone and l e t (gn) t Define

13 = supx ( i n f 9,)

n

Hence i t remains t o prove we may assume

a I 13

supx(gn) Ia +

subsequence). Every

1 fn,...,f:n

and

E F(X)

gn

.

1

if

for all

Take a n u l t r a f i l t e r

decreasing sequence o f non empty s e t s :

n

such t h a t

a > -

i s o f the form

consequence o f t h e m a x i m a l i t y o f

. We

a = i n f supx(gn) n

n

always have

. Without

m

.

VF(X)

in

a 2 8.

loss o f q e n e r a l i t y

( o t h e r w i s e we go o v e r t o a

gn = max(f

o of

X

Xn = I x E X

1 ,,,..., fr;”

) where

containing the following

I gn(x)

o we f i n d a number

z a -

pn I kn

1

. As

f o r every

a

Representing Measures

192

pn 1 Yn = IX E X Ifn (x) 2 a -

E Q

.

T h i s f o l l o w s from t h e f a c t t h a t

and t h a t

h = ym E

m

x

n {Yi

u

xI

fi(x) 2 a -

=

Q

i s an u l t r a f i l t e r . Put

fn

n=l

kn

X,

j=l

. Then

Ii5

ml

cx E

I

1 fn = n ( f i n

-

a

- -)n1

pn 1 fn I gn Ia + n

because

f, 5 0

1

and

. Choose x,EX

(which i s nonempty). Then ( i i i ) p r o v i d e s us w i t h

such t h a t 1 pn (x,) IT Ti (fn n=l

-

OD

a

- -)1 '

-

1

t -ii ( p + n=l

E

-

a) >

--

n

pn i n f fn (x,) n

Combining t h i s i n e q u a l i t y w i t h m

2 i n f sup

r

for all

E

>0

X

u :C

- 1z

n

.

inf(hm(ym)) m

(XI

= tf :

x

+

tx E X R

1f

I f(x)

1

E )

-

1 2

we o b t a i n

n( xo ) I p 0 3 .

Since

1

t -= n=l n

we have

(i) o ( i i ) a f t e r Theorem 1.3.6.

be a t o p o l o g i c a l space. A non n e g a t i v e f u n c t i o n

t o -vanish a t i n f i n i t y i f Put

(h ) m

Ii n f g

We s h a l l g i v e a second p r o o f o f Lemma 1.3.4 Let

X

f : X

+

i s compact f o r a l l

i s said

R E

non negative, upper semicontinuous, vanishing a t i n f i n i t y }

.

> 0

.

Countable Decomposition

1.3.5

193

Lemma:

be a D i n i cone c o n s i s t i n g o f upper semicontinuous f u n c t i o n s on L e t F(X) ---t h e t o p o l o g i c a l space X . Then F(X) + U : C ( X ) __-i s a Dini cone. Proof: We can assume, t h a t hn = fn t cpn

(*)

B

, fn E =

F(X) F

i s max s t a b l e ( 1 . 3 . 4 ) . L e t

, cpn

€ U: C

(X)

Assume t h a t we have found i n t e g e r s functions be m kl

1

gn = max(fk ,a) + n > k,

such t h a t

h

t-

fkm+l

f

kmtl

(y) I a

hn h

kmtl

n

such t h a t t h e 5

m

f o r a l l elements y

kmt 1

I

...< km

are decreasing f o r a l l

considering the r e s t r i c t i o n s o f q u a l i t y together w i t h

,

1 = k 1 < k2 <

Otherwise D i n i ' s Lemma would proviide

with

and assume

> a > supx(inf(hn)) n

i n f supX(h,) n

(hn)+

. There must of

us w i t h a c o n t r a d i c t i o n t o ( * ) by t o t h e compact s e t hk

I

1 5 max(fk , a ) mtl m

Y

. This

ine-

implies

in

t

1 m

.

1 Hence b y i n d u c t i o n we o b t a i n a decreasing sequence qn = max(fk ,a) t si n i n F . I t remains t o show supx(gn) 2 B f o r a l l n E N T h i s means t h a t

.

(gn)s

contradicts t h e Dini c o n d i t i o n o f

i n f supx(gn) t B > a = s u p X ( i n f gn) n n

Assume t h e r e f o r e

since

F(X)

.

) = 6 < B f o r some no

sup ( g "0

Representing Measures

194

*) implies t h a t a l l

Then t h e l e f t s i d e o f

hm(m 2 kn )

I x E X I wk ( x ) 2 p

maxima on t h e compact s e t D i n i ' s Lemma. U

attain their

0

- 61 and ( * ) c o n t r a d i c t s

i s a Dini cone. I n c o n t r a s t t o 1.3.5 I n p a r t i c u l a r we have t h a t U C:(X) we observe t h a t i n general t h e sum o f two D i n i cones i s n o t a D i n i cone. Exampl e : Let

F = I ( r n ) ( r nE R ,

e v e n t u a l l y c o n s t a n t and

l i m rn = r13 n-m

.

Both cones may be considered as cones of f u n c t i o n s on N I t i s easy t o see, t h a t F and G a r e D i n i cones. B u t F d G c o n t a i n s fn

and F+G

1.3.6

(O,O,

...0,

1,1, ...) +n c

supN ( i n f fn)= 0 n

for all

n.

We have

(fn)r

, inf

sup,

= 1

(f,)

n

. Hence

F+G

i s n o t a Dini-cone. On t h e o t h e r hand,

s a t i s f i e s t h e maximum c o n d i t i o n o f Lemma 1.3.3.

Theorem:

The f o l l o w i n g _ a r e. e q u i v a l e n t : i) F(X)

i s a D i n i cone

ii) For e v e r y sequence functional

u

I supx

m

with

-

I: A = 1 n=l n

for all n --

0

E N.

.

m

P

Yn c X with U Yn = X --and e v e r y l i n e a r n=l

there are l i n e a r

, -such t h a t u

vn : F(X)

-,k

and x n

t0

m

5

I: in pn n=l

and -

pn

I sup

yrl

195

Countable Decomposition

Proof:

i) i i )

. Put

fn E F(X)

Let

p,

,

= supy

(fn)+

. Then

n

. Then we

pn

i s s u b l i n e a r and

m)

m

Then ii)f o l l o w s

f r o m C o r o l l a r y 1.2.7.

ii)

E F(X), (f,)+

E)

i ) . Let

f,

= s u p X ( i n f fm) =

m

. Then t r i v i a l l y

Assume, t h e r e i s y with

m

a
with

.

i n f p ( f m ) = i n f supx (f,) m m

.

.

= p

a = sup ( i n f fm) I i n f supx (f,)

' m

By Lemma 1.2.6.

Define

Yn

=

Ix E X

there i s

I fn(x)

LI

I Y)

5 supx

.

W

Then

U Yn = X. n=l

We i n f e r f r o m o u r assumption t h a t t h e r e a r e

m

z

n=l

A n = 1 such t h a t

a contradiction.

p

c

m

z xn sup

n=l

1.3.7 Let

(i)

H

3

i)p r o v i d e s us w i t h a d i f f e r e n t

(ii) ( i ) cs ( i i ) :

be a D i n i cone. Then we k n o w , i i ) o f Theorem 1.3.6 h o l d s .

( I n t h e p r o o f o f t h i s f a c t we d i d n o t use Lemma 1.3.4) LetVFB

,

0

Another p r o o f o f Lemma 1.3.4 F(X)

An 2 0

. Hence

'n

An e x t e n s i o n o f t h e argument used i n i i ) p r o o f o f Lemma 1.3.4

.

have

Ii n f supx ( f

s u p ( i n f pn (f,)) n m

pn ISI = supx

be t h e bounded f u n c t i o n s i n V F ( X )

and

put

E = V FB

- V FB

196

E

Repmenring Measures

i s a v e c t o r l a t t i c e w i t h o r d e r u n i t . We deduce from Theorem I 2.2.3

t h a t t h e r e i s a compact Hausdorff space

C(S)

S

such t h a t

( w i t h r e s p e c t t o t h e supremum norm). Now, l e t

D e f i n e g,

= max(f,,-m).

g, E E

Then

.

m and (g,)+

for all

Lemma we o b t a i n a l a t t i c e homomorphism v : E

E i s dense i n fmE V F(X), (f,)+.

+

R

w i t h u ( l x ) = 1 and

i n f v(gm) = i n f sups(gm) = i n f supx(gm) = i n f supX(fm) = B

m

Define p

m

p

: V F(X)

m

+ i by

m

m

There a r e

xn 1 0 w i t h

max(p(hl),

p(h2))

OD

p

I

z xn supy

n=l

n

which shows t h a t 1.4

for a l l

1.4.1

I

Yn = I x E X

fn(x) s y l .

t xn = 1 such t h a t n=l

V F(X)

hl,h2

(i.e.

E V F(X))

p(max(hl,h2))

.

=

, hence

the contradiction P = i n f p(fm) I Y

i s a D i n i cone

m

.

0

i s a cone of bounded

r e a l v a l u e d f u n c t i o n s on

X,

must n o t c o n t a i n t h e constants. Definition:

---

F(X) i s c a l l e d a weak D i n i cone if f o r every sequence (fn,an) E F(X) x R , n E N , such t h a t (an + f n ) + and have

a
WEAK D I N 1 CONES

I n t h i s s e c t i o n F(X) F(X)

i s linear,

p

i n t h e preceding p r o o f assume

. Again we deduce V F(X)

Then

.

0

i s a l a t t i c e homomorphism on

p

m

. As

and d e f i n e

( a = supx ( i n f f,))

m

p ( f ) = i n f v(max(f, -m)).

I supx and i n f p(fm) = B

Using D i n i ' s

(an

-

fn)+

, we

197

Countable Decomposition

supx(inf(an t l f n l ) ) = inf supX(an t I f n ' ) n n The l a t t e r condition i s called weak - Dini - condition . Consider F = F(X) x R and introduce the following order relation : ( f , r ) I (g,s) i f supx( If - g l ) I s - r . Clearly, ( F , I ) i s a preordered cone. Then ( f n , r n ) E F i s decreasing (with respect t o I) i f and only i f (rn- f n ( x ) )

( r n t f n ( x ) ) and

are decreasing in IR for a l l

x

E

X.

Furthermore, l e t I = ( 0 , l ) E F(X) x R . Then ( F , I , s ) i s an order unit cone and S I ( f , r ) = s u p X ( l f l ) t r . Define ^x = X x I01 U X x C11 and X with

identify ^

^

F(X)

=

X x IO) c

I f : ^x

-+

1 flXx

R

^x . P u t Iol

E

F(X)

, f(x,o)

=

- f ( x , l ) for a l l xEX1

Then ( F Y I y < )can be identified with the following cone

(F(x) t R , 1-xy <- )

of concrete functions on a s e t

ii

.

Hence we have: 1.4.2 F(X)

1.4.3

Lemma: i s a weak Dini cone __---___--

i f and only i f

F _ i s_ a -Dini -

cone.

Theorem:

are equivalent: The f o l 1 owing i s a weak Dini cone.

i)

F(X)

ii)

For every linear for a l l f -with

-

with

-

m

U

n=l m

1

n=l

p

E F(X)

Yn = X

xn

=

:~ F(X) + R with p ( f ) 5 supx( l f l ) ,

* Yn

c X

and linear

p,,

, --and for every sequence 0 there are --

An 2 0

1 and p n ( f ) I supy ( I f l ) for a l l

n

:F(X) + R

f E F(X)

198

Representing Measures

-such t h a t

proof:

Let

Hence supy on

F

n

with

m

t 1, n=l

P 5

. We

Yn c X

F(x)+

on

Pn

define IR

yn

= Yn

x

{o} u

yn

x

{l}c

^x

.

corresponds t o t h e s u b l i n e a r f u n c t i o n a l

p n ( f , r ) = supy ( I f ! ) t r n

pn

.

The theorem f o l l o w s now f r o m Theorem 1.3.6 and t h e p r e c e d i n g lemma.

1.5

o

REMARKS AND COMMENTS

The decomposition theorems of t h i s s e c t i o n a r e due t o t h e f i r s t a u t h o r (see [1201,[1231 and [ l l g l ) . A c t u a l l y , t h e y a r e s t a t e d here i n a s l i g h t l y generalized version, b u t t h i s generalization i s n o t essential f o r the a p p l i c a t i o n s we have i n mind. The D i n i cones we i n t r o d u c e d here s h o u l d n o t be mixed up w i t h t h e D i n i l a t t i c e s considered by C. P o r t e n i e r 12531. A p a r t f r o m b e i n g v e c t o r l a t t i c e s t h e y have much s t r o n g e r convergence p r o p e r t i e s : A l o c a l l y convex v e c t o r i s s a i d t o be a -~ D i n i l a t t i c e i f every downwards d i r e c t e d n e t

lattice E

( f i ) i E I o f p o s i t i v e elements converges t o z e r o i f f whenever

P

: E

+

i n f u(fi) iE1

= 0

i s a continuous l a t t i c e homomorphism. These cones were

R

a l s o considered by G o u l l e t de Rugy [1451 (see a l s o [1071). L e t us mention an i m p o r t a n t n o n t r i v i a l example where t h e P a r t i a l Decompos i t i o n Theorem a p p l i e s . Consider an RNP- s e t and bounded subset

X

o f a Banach space

c a l l e d Radon-Nikodym-Property. p r o b a b i l i t y space

(fi,E,m)

X

, that

i s a convex, c l o s e d

B such t h a t X has t h e so-

This property requires t h a t f o r every

and e v e r y

B- valued measure

T

: t

+

B

t h a t i s o f bounded v a r i a t i o n and a b s o l u t e l y continuous w i t h r e s p e c t t o and has average range

m,

Countable Decomposition

IE

tm-’(E)?(E) contained i n cp :

function

.I,

E

m(E) > 01

, there i s a density

X

n

+

E

B

199

cp

, i.e.

a Bochner i n t e g r a b l e

such t h a t

cp dm =

T(E)

for all

E E z

I n s e v e r a l r e s p e c t s t h i s p r o p e r t y i s i m p o r t a n t . F i r s t o f a l l y t h e RNP i s e q u i v a l e n t t o a s u i t a b l e m a r t i n g a l e convergence p r o p e r t y ( C h a t t e r j i 1 64 1). And secondly, Banach spaces w i t h RNP have v e r y i n t e r e s t i n g geometric p r o p e r t i e s ( f o r i n f o r m a t i o n see t 861,t 851 and 11701).

So l e t

be such an RNP- s e t . Consider i t s extreme p o i n t s

X

F = Con(X), t h e upper semicontinuous,

X

+

k.

xo E X

F i x some

g i v e n by

po(f) =

servation that

and c o n s i d e r t h e l i n e a r f u n c t i o n a l

f ( x o ) , f E F.

u0

and

po

: F

+:

Then i t seems t o be an i n t e r e s t i n g ob-

i s p a r t i a l l y decomposable w i t h r e s p e c t t o any sequence

o f sublinear functionals,

Yn

ex(X)

upper bounded, convex f u n c t i o n s

pn, n E N, g i v e n by with

a r e non empty subsets o f

U

nEN

pn = supy

n Yn = ex(X).

, where

T h i s a s s e r t i o n i m p l i e s t h a t t h e Krein-Milman Theorem h o l d s f o r

X

the

.

It i s

unknown t o us whether o r n o t t h i s p r o p e r t y ( f o r a l l closed, bounded B ) i s e q u i v a l e n t t o t h e RNP o f

convex subsets o f

B

.Of

course, f o r

dual spaces t h e answer i s a f f i r m a t i v e , s i n c e f o r dual spaces t h e RNP i s e q u i v a l e n t t o t h e Krein-Milman P r o p e r t y [ 169. Using t h e d e n t i n g p r o p e r t y t251

I

o f RNP- s e t s one can g i v e a d i r e c t proof

o f c o n d i t i o n ii)i n t h e P a r t i a l Decomposition Theorem v i a a s l i g h t modif i c a t i o n o f t h e decomposition techniques d e s c r i b e d i n [ 1171.

A s h o r t p r o o f can be g i v e n by u s i n g t h e f o l l o w i n g s t r o n g r e s u l t o f Ch. S t e g a l l :

X

be an

upper bounded f u n c t i o n

f : X

Theorem [2991 : L e t

b* E B*

(dual o f

exposes

X

.

B)

with

RNP- s e t . F o r e v e r y upper semicontinuous, + R

and f o r every

I1 b * l l 5 6 such t h a t

6 > 0

there i s a

g = f t b*

strongly

200

Representing Measures

"Strongly exposesl'means t h a t there i s some xo E X such t h a t g attains i t s supremum on X a t xo , and i f { x n ) i s any sequence in X with g(xn)

-,

g(xo)

then

II x,

-

xoll

+

0.

SECTION 11.2 REPRESENTING MEASURES

I n t h i s s e c t i o n we a p p l y t h e decomposition techniques, we have developed so f a r ,to t h e problem o f i n t e g r a l r e p r e s e n t a t i o n o f l i n e a r f u n c t i o n a l s . I n t h e f i r s t paragraph we show t h a t a s t a t e

on a v e c t o r l a t t i c e

LI

an i n t e g r a l r e p r e s e n t a t i o n i f and o n l y i f

LI

E(X)

admits

has t h e decomposition p r o -

p e r t y . T h i s f a c t immediately y i e l d s a proof of t h e D a n i e l l - S t o n e Theorem by means o f t h e Riesz R e p r e s e n t a t i o n Theorem. I n t h e second paragraph we p r e s e n t t h e c r u c i a l r e s u l t o f t h i s s e c t i o n ( R e p r e s e n t a t i o n Theorem 2.2.1). We show t h a t e v e r y s t a t e o f F(X) has an i n t e g r a l r e p r e s e n t a t i o n i f and o n l y i f F(X) i s a D i n i cone. L a t e r on we see t h a t t h i s r e s u l t g e n e r a l i z e s t h e c e l e b r a t e d Choquet Theorem. I n paragraph 2.3 we a p p l y t h e R e p r e s e n t a t i o n Theorem t o r e p r e s e n t a r b i t r a r y l i n e a r f u n c t i o n a l s b y s i g n e d measures. And i n 2.4 we s t u d y i n t e g r a l r e p r e s e n t a t i o n s o f l i n e a r f u n c t i o n a l s by means o f w e i g h t f u n c t i o n s . F i n a l l y , i n 2 . 5 we p r e s e n t a l o c a l v e r s i o n o f t h e R e p r e s e n t a t i o n Theorem. I n t h e a p p l i c a t i o n s s e c t i o n 2.6 we s t a r t b y g i v i n g a s l i g h t g e n e r a l i z a t i o n o f t h e Riesz-Konig Theorem. Then, b y means o f w e i g h t f u n c t i o n s , we e x t e n d some r e s u l t s t o t h e case o f r e p r e s e n t a t i o n b y unbounded measures. We e x t e n d t h e n o t i o n o f adaptedness i n such a way t h a t H e w i t t ' s Theorem about i n t e g r a l r e p r e s e n t a t i o n o f l i n e a r f u n c t i o n a l s on

C(X)

,

X

t o p o l o g i c a l space, i s

i n c l u d e d . And we conclude t h e paragraph by an i n v e s t i g a t i o n o f r e p r e s e n t a t i o n o f l i n e a r f u n c t i o n a l s (which a r e p o s i t i v e on t h e monotone f u n c t i o n s ) on c o n t i n u o u s l y o r d e r e d compact spaces. 201

202

Representing Measures

2.1

DECOMPOSITION PROPERTIES AND MEASURE THEORY

L e t us r e c a l l some elementary measure t h e o r e t i c a l f a c t s .

For a more

d e t a i l e d account o f measure t h e o r y t h e reader s h o u l d c o n s u l t t h e Appendix Y

cp :

-,Z

. Consider

Y be a s e t and 1 (pl(.X) = Icp- ( S ) 1 S E 11 i s t h e s m a l l e s t

a measure space

be a map. C l e a r l y

u- a l g e b r a i n

Y

such t h a t

This

T

on

cp

J

Z

by

f d-r =

Y

f

o

cpdr

cp-'(SZ).

=

n cp(Y) =

= 0 t h e n we may T(S) f o r a l l S E z

r(S1)

=

r ( S 2 ) whenever

S

f :

z +iT

.

Assume i n a d d i t i o n t h a t

Z

i s a compact H a u s d o r f f space, t h a t

o f t h e B a i r e subsets o f

Z

and t h a t

Then i n view o f t h e r e g u l a r i t y o f

2.1.1

.

Moreover we t h e n have

f o r a l l measurable

cp

i s z- mea-

~,(cp-l(S))

S E 1 and

cp-'(Z)

d e f i n i t i o n makes sense s i n c e we have

S 1 ,S 2 E z and cp-l(S,)

f

f o cp i s measurable whenever

s u r a b l e . I f T ( S ) = 0 whenever d e f i n e a measure

(Z,zy~), l e t

z consists

i s a p r o b a b i l i t y measure on

T

Z

.

(see A 3.5 Appendix), we have

T

Remark:

The f o l l o w i n g a r e e q u i v a l e n t : i)

T(S) = 0

for all

ii)

T(G) = 0

f o r a l l compact

iii) T(F) = 1 (Note t h a t

f o r a l l open

G c Z

is

G6

iff

cp(Y) n S = 4

with

S E Z

G

G6- s e t s

Fa- s e t s F F

c

cZ

= Z\G

Z with

Fa)

is

cp(Y) r l G = 4

with

F

3

q(Y)

.

.

o f upper bounded valued c o n s i d e r again a cone F(X) c o n s i s t i n g - 6-functions and suppose f u r t h e r m o r e t h a t F(X) c o n t a i n s t h e c o n s t a n t s

-L e_t us

.

L

2.1.2

Definition:

i)

A countable family

IYn

In

E

N)

c a l l e d countable c o v e r i n g o f

X

o f nonempty subsets o f if

X

=

U Yn

nEN

.

X is

203

RepresentingMeasures

Let EF be the smallest u- algebra on

ii)

X

in F ( X ) are measurable (see Appendix). Let linear with p 5 supX '

such t h a t a l l functions p

: F(X)

-+

Ir? be

has the decomposition property i f 1-1 i s decomposable with respect to (supy ) for a l l countable coverings C Y n I n E N1 of n

p

iii)

A positive IF- measure

on X

for u

T

on X

X.

i s called a representing measure

if

p ( f ) 5 J f d.r f o r a l l X

f E F(X)

.

I n particular i f we have equality for all bounded functions then T i s called s t r i c t l y representing.

f EF('X)

Note that for a representing measure T we always have T ( X ) = 1 since lX,-lXE F ( X ) . Indeed, then p ( f l X )= supX(* l X )= i 1 and therefore 1

J l Xd r = r ( X ) .

= p(lX) =

2.1.3

X

Examples:

Under the assumptions of the Riesz-Konig Theorem ( I 1.6.1) we obtain t h a t forsuch a cone F(X) every 1-1 has a representing measure (see also I 1 . 6 . 2 . ) . I n the Riesz-Konig Theorem the compactness of X was essential 1 as one sees from t h e following example: Let X = ,C I n E N 1 and consider the cone F(X) lim

n+

03

consisting of a l l bounded functions f : X

f(i)exists. Define

linear and

p

+

R

such t h a t

lim f(A) for a l l f E F(X) . Then p i s n-+ 03 5 supx b u t there i s no representing measure T tbr p on X . p(f)

=

I n the following section we want t o characterize those cones F(X) on which every linear functional p , with p I supx , has a representing measure. I t turns o u t t h a t these cones are j u s t the Dini cones. A t f i r s t we prove this result under additional restrictions thus obtaining a representation theorem which contains the Daniel1 - Stone Theorem as a special case. All proofs are based on the decomposition theorems of the previous chapter

204

Representing Measures

and the Riesz Representation Theorem (see I . 1.6 and Appendix).

Recall, a s t a t e p : F(X) + k i s a linear functional, monotone with respect t o the pointwise order in F ( X ) , such t h a t p 1. supx ( I 2 . 1 ) .

For the next proposition l e t us consider the following Situation: Let v : F X ) + g be a s t a t e . Consider a countable covering { Y n a)

max f , r )

b)

If

In

E N1

of

.

X

Furthermore assume:

F(X) whenever f E F ( X ) , r E R .

E

tn t 0 , n

E N

OD

, with It1

=

X tn I 1 n=l

and

00

JJ

1 tn supy t (1 - Itl) supx then there are states

I

n

n=l

,n

vn 1. supy

n

u

E N

, and

v

m

=

z tn v n

n=l

t

with (1 - Itl) v

.

Then we can formulate Proposition:

2.1.4

The following

are equivalent:

fi decomposable with

i) u

respect to

(supy ) n

i i ) For every positive decreasing sequence sup (inf sup ( f ) ) = 0 -we have

n

m

'n

.

(f,)

inf p(f,)

with

F(X) =

0

.

Proof: i)

4

i i ) : By assumption there are linear m

v

I

1 tn v n

n= 1

. Since

F(X)

pn

I sup

and

'n contains the constants, all

tn 2 0 such t h a t pn

must be states

205

Represenring Measures m

Now, t a k e

m

z

1 = ~ ( 1 )=

and

tn ~ ~ ( 1= ) z tn n=l n=l

(f,)

as i n i i ) . L e t

m

such t h a t for a l l

x

n=N+l n=l,

tn u,,(fl)

...,N.

E

take

m so l a r g e t h a t supy (f,) n

N

m

T h i s proves

Ie

Hence

0 Ii n f u ( f k ) I z tn un(fm) I t, supy (f,) k n= 1 n= 1 n I2 E

N E N

> 0 be a r b i t r a r y and t a k e

. Then

It

.

(Recall, i n f V(f,) m

N z

fmIfl, = 0

n=1

since

t,, I1)

+

OJ

tn vn(fl)

n=Nt1

I

.

> 0 was a r b i t r a r y .

E

* i):

ii)

Note t h e f o l l o w i n g two f a c t s :

(1) L e t for

p

u,,

and

v

be t h e s t a t e s d e s c r i b e d i n b ) . Assume t h a t ii)h o l d s

and assume t h a t I t l = z tn < 1. Then, s i n c e ii)deals o n l y w i t h

p o s i t i v e f u n c t i o n s , i i ) must a l s o h o l d f o r

(2)

Consider t h e subcone

i n s t e a d of

v

v

.

F-

F- = I f E F(X)

I supy

n

( f ) I0

for all

n 1

which p l a y s an e s s e n t i a l r o l e i n t h e P a r t i a l Decomposition Theorem. Then F- = I f E F(X) and

F-

generates

To prove i)l e t

1f

5 01

F(X).

(t,)

be as i n b ) . Assume t h a t t h e sequence

(t,)

is

maximal w i t h r e s p e c t t o componentwise o r d e r (compare t h e p r o o f o f 1.2.5). Such a maximal sequence i n e a s i l y found by i n d u c t i o n : t

1=

SUPIS

2 0

I

!J

I s supy

tn+l=Sup{s20jLlI

n

1

+

(1 - s)supxl

z t p p

k=l

'k

+ssupy

n

n+l

+ ( l - s - xtk)supXl. k=l

206

Representing Measures

According t o b ) we can f i n d s t a t e s

and

vn

v

m

If I t = 1 then, by d e f i n i t i o n , n n=1 m

I tn < 1 n=l i n f v(f,) m

. Hence,

i s a sequence as i n ii), we have

(f,)

if

LI i s decomposable. Therefore, assume

= 0 (consequence of ( 1 ) ) .

We claim,

v

i s p a r t i a l l y decomposable.

P r o o f o f t h i s c l a i m : We can a p p l y t h e P a r t i a l Decomposition Theorem 1.2.4 m

to Fix

F-

with

supy

n

instead o f

N > 0 a r b i t r a r i l y and p u t

Then

f,

2 0

(fm) +

and

m s u p ( i n f pn( 1 g k ) ) = k=l n m

s u p ( i n f pn(fm)) = 0 n m

m I gk) 5 k=l

-

m i n f v ( 1 gk) = m k=l

-

inf m

pn:

V(

f,

Let

= max(0,

.

-m

Hence t h e r e i s

with

m

1 gk

k=l

+

1 v(gk) >

k=l

and thus

i n f v(f,) m

.

= 0

n=l n*no

.

T h i s means

N . I n o t h e r words we have o b t a i n e d t h e c o n t r a d i c t i o n

since

N was a r b i t r a r y . So we can conclude,with v

the

i s p a r t i a l l y decomposable

(supy ) . n s > 0

m

1

01

,that

f,

and

no E N

such t h a t

v 5 s

supy

+ (l-s)supx

We then have lJ I

-

N ).

implies,by d e f i n i t i o n o f t h e

P a r t i a l Decomposition Theorem 1.2.4,that w i t h respect t o

gk E F-

tn supy

n

+ (s +

tn )supy 0

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

(tn

+ (1 -

m

z tn - s)supx

n= 1

0

.

207

Representing Measures

2.1.5

Corollary:

Assume t h a t F(X) i s max-stable --

-has t h e

SIP. Let

covering

i)

of

(Y,)

X

. Then

V F ( X ) = F(X))

(i.e.

be a s t a t e o f

LI

ii) I f (f,)

c

l i m LI (f,) wm

i n F(X) = 0

and t h a t --

F(X)

a countable

the following are equivalent: (supy ) n

decomposable w i t h r e s p e c t t o

LI

and c o n s i d e r

F(X)

and fmt 0

whenever

-f o_r_a l l

m

i n f supy (f,) m n

,then = 0

for all n E --

N

Proof: T h i s i s a d i r e c t consequence o f P r o p o s i t i o n 2.1.4 s i n c e 2.1.4.a) h o l d s and a repeated use o f I . 1.4.1. for

F(X) i n view o f t h e

2.1.6

SIP.

trivially

y i e l d s b ) s i n c e I . 1.4.1 ( * ) i s t r u e 0

Proposition:

be a ve c t o rl a t t i~ c e and c o n s i d e r a s t a t e p : E -* R . Then t h e r e i s a r e p r e s e n t i n g xF- measure T for v 0” X i f and o n l y

L e t E = F(X) i f LI -

has t h e decomposition p r o p e r t y

Remark : Note t h a t

i n t h i s case

T

E

i s s t r i c t l y representing since

i s in

p a r t i c u l a r a v e c t o r space (see t h e remarks a f t e r I. 1.2.5).

Proof: A t f i r s t we prove t h e n e c c e s s i t y :

If all

T

i s such a measure t h e n l e t

m E N

. Assume

(f,)

s u p ( i n f supy (f,)) n m n

+ =

in 0

F(X)

and

f,

for

2 0

f o r some s u i t a b l e

Yn

m

U Y = X . T h i s i m p l i e s f, n n= 1 f, dT = 0 l i m p(fm) = l i m W m wm X

1

+

0

p o i n t w i s e . Hence

b y t h e Monotone Convergence P r o p w t y

with

Representing Measures

208

( s e e Appendix

A 2).

Next, we prove the s u f f i c i e n c y : Let S = { p : F(X) -t R I p l a t t i c e homomorphism3 In I 2 . 2 . 2 we showed t h a t S i s a compact Hausdorff space i f we take the c o a r s e s t topology on S such t h a t the Gelfand transform 7 : S R ( i .e. Lf(p) = p ( f ) f o r a l l f E F(X). p E S ) i s continuous f o r a l l Since = {?If E E = F(X)3 i s dense i n C(S) with respect t o thesupnorm ( I 2.2.3) the Riesz Representation Theorem (Appendix) y i e l d s a unique probability measure ? on S which represents p i . e . p ( f ) = d.r S for all f E F(X).

.

-f

From this we want t o obtain a Let

x

E

cp :

X

xF- measure

T

on X which represents

p

S be the natural i n j e c t i o n , i . e . cp(x)(f) = f ( x ) f o r a l l f E F ( X ) . Nowy consider B o ( S ) , t h e u- algebra of a l l Baire

-t

X and

.

.

s e t s in S W e r e c a l l t h a t t h i s i s t h e u- algebra generated by C(S) Hence B o ( S ) is a l s o generated by E because of t h e density of E in

C(S). So we must have

xF s i n c e

c~’(B,(s)) =

Let Kn

(^focp)(x) =

f(x) for all

x E X, f E E

be a sequence o f compact subsets such t h a t

c S

We claim t h a t

m

;(

U

n=l

Kn)

=

m

U Kn n=l

3

. cp(X)

.

1.

Then we can define, according t o the introductory remarks of t h i s s e c t i o n and Remark 2.1.1, a measure T on XF by T(w-’(A)) = ; ( A ) f o r a l l A E Bo(S).

Proof of the claim: be such a sequence of compact subsets and define (K,) -1 Y n = cp (K,) c X . The decomposition property of p y i e l d s s t a t e s Let

vn

I

IJ

=

supy n

and

m

Z

n=l

A n pn

An L 0

. Another

for all

n E N with

W

1

n=l

An = 1

and

application of the Riesz Representation Theorem

:

209

Representing Measures

yields probability measures for a l l

f E F(X)

f E F(X)

.

This means :( p(f) =

and n

on Kn

T,,

E N ,

with

Hence p ( f ) =

pn(f)

S

^f

Thus , in view of the uniqueness of U

n= 1

-.

J f drn Kn

=

OD

z

d(

n=l

,;

A,

for a l l

)

T,,

OD

=

Z

n=l

Xn

T,,

.

K n ) = 1 and proves the claim. Therefore we have

J f d r for a l l X

f E F(X).

0

Theorem (Daniel1 - Stone)

2.1.7

F(X) a vector l a t t i c e and assume, Then the following equivalent:

Let E -

=

are

1-1

: F(X)

i)

There i s a unique representing IF-measure for

ii)

Whenever f, E F(X) , (f,)

J.

and

_.

A!

+

a state.

R

0” X

.

inf fm = 0 then lim v(f,) m n +a

=

0

.

Proof: 2.1.7 i s a combination of 2.1.5 and 2.1.6.

The uniqueness of the representing IF measure - for u can be seen as fol 1ows : Assume T~ and 72 are such measures. Then consider If E B d

zF(X) I

If X

d.rl =

I f dr2 1 X

. .

Obviously, this s e t i s a U- complete l a t t i c e and contains E Hence i t must be equal to B d zF X ) , since B d zF(X) i s the smallest u- complete l a t t i c e containing E

see Appendix A 1 ).

0

210

Representing Measures

Let

F(X)

66

be a g a i n a general cone o f upper bounded

and assume t h a t

F(X)

i s max-stable. L e t

l i n e a r f u n c t i o n a l . We c a l l

1.1

1.1

: F(X)

valued f u n c t i o n s be a monotone

+

! l i r J continuous i f i t s a t i s f i e s ii)o f

Theorem 2.1. 7, i . e . :

l i m v(fm)

0 whenever

Ill+=

f,

E F(X),

(f,)

4

and

i n f f, m

= 0

.

D I N 1 CONES AND REPRESENTING MEASURES

2.2

X

As b e f o r e we assume t h a t

6- valued 2.2.1

Representation Theorem:

The f o l l o w i n g i )

F(X) i s a cone of upper bounded

i s a s e t and

functions containing the constants.

F(X)

are e q u i v a l e n t :

i s a D i n i cone

ii) -Every s t a t e u : F(X)

. -+

iii)-Every l i n e a r f u n c t i o n a l

fi -has _ a representing 1.1 :

r e p r e s e n t i n g IF measure - on

F(X)

X

+

6

.

with

-

-

IF measure on 1.1 5

sup

X

.

has a X--

Proof:

iii)* ii) i s obvious. i i ) + i):L e t

F( X)

(f,)

J.

in

F(X)

. By

Lemma 1.2.6 t h e r e i s a s t a t e

1.1

of

with

I

i n f supX(fm) = i n f p ( f m ) 5 i n f fm d-c 5 s u p X ( i n f f m ) m m m X m where

T

i s a r e p r e s e n t i n g measure f o r

1.1

on

X ,which e x i s t s according

t o ii).The l a s t i n e q u a l i t y f o l l o w s f r o m t h e Monotone Convergence P r o p e r t y . On t h e o t h e r hand, Hence

s u p X ( i n f fm)5 i n f supX(fm) i s t r i v i a l . m m

s u p X ( i n f fm)= i n f s u p X ( f m ) which means t h a t m m

F(X)

i s a D i n i cone.

21 1

Represen ring Measures

i)

=$

o

Put

u : F(X)

iii): L e t =

I f

I

E VF(X)

with

r? be l i n e a r w i t h

bounded)

f

o

a v e c t o r l a t t i c e and be l i n e a r

+

i s a D i n i cone (Lemma 1.3.4).

ii Isupx

p IGIF(X)and

11

maximal, i . e . i f

v I sup x, v

I

. Assume

.

- o

E = o

and d e f i n e

.

supx

p 5

Then

E

ii is

furthermore, t h a t

l i n e a r , then

.

= u

is

: VF(X) -.fi

Let

G exists

Such

i n view o f Z o r n ' s Lemma t o g e t h e r w i t h t h e Dominating E x t e n s i o n Theorem 1.1.3.1 which a l s o shows t h a t

C must be monotone. v

can be extended u n i q u e l y t o a l i n e a r

@I'

ensures t h a t

>

;(f)

-

f E o

for all

a0

-

m

i

if

a E R

with

*

1 = u ( * lX) c G ( lX) 5 supX(* l X )=

l X )5 G(f).

< a = ;(a

1.3.6).

v

We c l a i m ,

G

*

n

and

then

alX I f

f

.

1 we have

has t h e decomposition p r o p e r t y .

X

. Then

with

hn t 0

there are states

m

m

un 5 supy

f E o

has t h e decomposition p r o p e r t y (Theorem

(Yn) be a c o u n t a b l e c o v e r i n g o f

Indeed, l e t

s i n c e t h e monotony o f

. Indeed,

i s bounded and t h e r e f o r e t h e r e i s a c o n s t a n t Since

E

on

1 A n = 1 and n= 1

I

n = l 'n 'n

*

use t h e Sandwich Theorem). Then t h e Dominating E x t e n s i o n Theorem 1.1.3.1 y i e l d s l i n e a r vn : E + R w i t h (To o b t a i n monotone

pn

'n l o lv n I o

vn

and

5

supy

for all

n

n. m

Then t h e m a x i m a l i t y o f all

n E N

.

=

Z

n= 1

and

An pn

Now t h e uniqueness o f t h e e x t e n s i o n s from

m

v =

implies

1 hn vn n=l

.

I n p a r t i c u l a r t h i s means t h a t

v

o

v

= unl

nl@

to

for

E yields

has t h e decomposition

property. E

i s a v e c t o r l a t t i c e and

applied t o EE- measure

E T

v

i s a s t a t e on

E

. Now,

P r o p o s i t i o n 2.1.6

and

u

shows iii).Indeed, P r o p o s i t i o n 2.1.6

on

X

with

v(f)

=

J f d-r f o r a l l f X

E E

yields a

, in

particular

212

Representing Measurns

f E 0. Note, t h a t , i f

for all

T h i s means

i n f fn = f .

n

u(f)

5

for all

2.2.2

;(f)

fn = max(f,-n)

zF = zD= zE . Therefore,

I i n f v(fn)

= inf

n

f E F(X)

f E F(X), t h e n

, which

1 fndr

n X

=

X

E @

and

we o b t a i n

f d.r

concludes t h e p r o o f .

Corollary:

be a -D i n i cone c o n s i s t i n g o f upper semicontinuous f u n c t i o n s on a t o p o l o g i c a l space X. Then every l i n e a r f u n c t i o n a l v : F(X) .* R w i t h p s supx has a r e p r e s e n t i n g zF-measure w h i c h can be u -on X --Let -

F(X)

extended t o t h e s m a l l e s t

u- a l g e b r a 1

c l o s e d compact subsets o f

Proof: Put

G(X) = F(X) t

U

X

c o n t a i n i n g IF -and a l l

X.

CL ( X ) .

Apply Theorem 2.2.1 t o

on

G(X)

Then (p

G(X)

i s a D i n i cone (Lemma 1.3.5).

can be extended t o a l i n e a r f u n c t i o n a l

on G(X) b y t h e Dominating Extension Theorem I 1.3.1). IIG measures -

T

.

This y i e l d s

xG c o n t a i n s a l l compact subsets o f X.

0

213

Representing Measures

2.3

WEAK D I N 1 CONES AND SIGNED REPRESENTING MEASURES

Throughout t h i s s e c t i o n f : X +R

F(X)

denotes a cone o f bounded f u n c t i o n s

n o t n e c e s s a r i l y c o n t a i n i n g t h e c o n s t a n t s . The aim o f t h i s

c h a p t e r i s t h e i n v e s t i g a t i o n o f s i g n e d measures on p r e s e n t i n g measures f o r

r(?)

t h e cone

and s i g n e d r e -

X

F ( X ) . The main t o o l s f o r t h i s i n v e s t i g a t i o n a r e

i n t r o d u c t e d i n 1.4 and a s i m p l e t r i c k c o n s i s t i n g i n t h e

t r a n s f e r between p o s i t i v e f i n i t e measures on ? = (X x (03) u ( X x { I ) ) and signed f i n i t e measures on X . We r e c a l l (Appendix Theorem A 5.2 o r Jordan Decomposition, s e c t i o n I 1.5.6.)

a s i g n e d measure

on

T

X

can

be w r i t t e n as

w i t h p o s i t i v e measures measures,

T+

variation

(TI

and

and

T-

T+

and

2.3.1

. S i n c e we

deal w i t h r e a l - v a l u e d

can be chosen t o be m u t u a l l y s i n g u l a r . The t o t a l

i s then

i s s a i d t o be f i n i t e i f

T

T-

Isl(X)


.

Definition:

L e t IF be a g a i n t h e s m a l l e s t

U-

: F(X)

IJ

normed, i f

f E F(X).

p(f) I supX(lfl)

f o r every

A s i g n e d measure T on xF w i t h t o t a l v a r i a t i o n a s i g n e d r e p r e s e n t i n g measure f o r p i f p(f) I

I

f d-r

J

f dr

X

f E P(X)

a l g e b r a such t h a t a l l

measurable. Consider a l i n e a r f u n c t i o n a l

+

R,

ITI(X)

5

1

i s called

F(X).

for all

f E

for all

f E F(X)

If i n particular

u(f) = then

T

X

,

i s s a i d t o be s t r i c t l y r e p r e s e n t i n g

p

.

are

i s called

IJ

,

214

Representing Measures

Recall t h a t in 1 . 4 we i d e n t i f i e d X with t h e subset X x {Ol of By a n t i X = ( X x { O l ) u ( X x 1 1 1 ) consisting of two d i s j o i n t copies of X symmetrization of functions f € F ( X ) we obtained t h e cone ^F(^x). Let us r e c a l l t h i s procedure: I f f E F ( X ) then we define i : 2 -* R by

.

If(x,O) and

F(X)

=

f(x)

and

If(x,l)

-

=

f(x)

i s t h e cone

-

g(x,O) = &

Clearly the r e s t r i c t i o n map f v(f) =

;(?I

for all

XI

.

i s l i n e a r and b i j e c t i v e . Hence

f

+

g(x,l) for all

f E F(X)

defines a l i n e a r b i j e c t i o n between the l i n e a r functionals v on F ( X ) S on ^F(^x). And the following lemma i s completely obvious: 2.3.2 If -

p

Lemma: : F(X)

I f ^v : -

+

fi & a normed l i n e a r f u n c t i o n a l ,

^F(x) - t r ?

i s l i n e a r with normed 1 inear functional. --

S

2 sup,

X

F

a r e measurable.

then

But a difficulty arises:

{O})

X x I03

I

B E t~ }

IF on X

7 E F(i)

X x I01 c

^x by

.

need not be t- - measurable in F

To overcome t h i s d i f f i c u l t y we prove:

i .

Lemma:

For every signed tF-measure -z- - p r o b a b i l i t y measure ? F

i; I supX u : F(X) -,i is a

such t h a t a l l

t- i s c e r t a i n l y r e l a t e d t o F

tF= I B n ( X x

.

then

NOW, l e t t- be t h e smallest a- algebra on

2.3.3

and

T

0" X

with

w i t h IT I

( X ) I1 there i s g

RepresentingMeasures

I

(*)

d?

If

=

X

if

Conversely,

for all

q-

7 is g

s i g n e d IIF - measure

d.r

X

0"

T

21 5

f E F(X)

2 , ---then there

probability _ measure _ _ _ -on

w i t h I T ~ ( X ) I 1 such t h a t ( * ) -

X

is a

holds.

Proof:

'i -,^x

L e t an i n v o l u t i o n j : j ( x , l ) = (x,o)

for all

-

1

=

1

7

at

b(x,o)

(x,o),

+

x E X

.

If

T

= 0

j(x,o)

= (x,l)

t h e n f i x some

where

6(x,1)

- measure w i t h 0

- T~ , ] T I

T = T

x- F

=

< I T ~ ( X )5 1

. The

NOW, assume, Bn

c

I-

?

,

T~

t

?

is a Bn

.

T~

Let

T ~ , T ~w i t h

and d e f i n e

B E 1F

c

Xx

7

and o f

E

n ;(n)

m

=

u Bn

n= 1

. Then

I01

= sup{?(B)

n

IBc

? ( i )immediately

- p r o b a b i l i t y measure on

IIF

such t h a t

1 ?(Bn) 2 (1 - -) SUP{?(B) I B c X n Put

and p u t

Jordan Decomposition o f

measure and we have

The d e f i n i t i o n s o f

Let

x E X

6

(see s e c t i o n I 1.5.6) y i e l d s p o s i t i v e IF measures -

T

and

a r e t h e D i r a c measures (XYO)' 6 ( x y 1 ) r e s p e c t i v e l y . C l e a r l y , we have ( * ) . Now, assume, 'I i s

9

(x,l),

a s i g n e d xF

be d e f i n e d b y

c

X

x

X

x

x

{O},

B E Z-1 F

101 and

103, B

E II-

F

1

.

.

-X

imply (*).

.

216

Representing Measures

Define 'I*(B)

= ?(B

whenever B y

n n)

E 1-

F

-

and

?(j(B)\n), B n(X

B E

lo})=

x

6

r- . We claim F n(X

x

'I*(B)

=

'1*(8)

IOI).

A t f i r s t , we have ? ( B n n) = ? ( B n n) . Let B A be the symmetric difference, i . e . B A = ( B U E), ( B n g ) . Then B A 8 c X \ ( X x {Ol) and hence j ( B ) A j ( g ) = j ( B A 8 ) c X x {Ol. Thus ;(j(B),n) = ? ( j ( B ) L n) which proves our claim. W e obtain that T , with T ( B n ( x x {Ol))= 'I*(B) for a l l B E E- , i s a signed zF- measure F

on X with

I'II(X) 5 1.

For f

E F(X)

we obtain

Now, the existence of signed representing measures i s settled as a simple application of Theorem 2.2.1: 2.3.4

Theorem:

The following are equivalent:

& a weak Dini cone

i)

F(X)

ii)

Every normed linear +- measure T

.

p

: F(X)

-+

6 -has - -a signed representing

Proof:

? ( i t) R

i s a Dini cone. So by 2.2.1 F(X) i s a weak Dini cone i f and only i f every linear functional i; : F(2) + R + 6 with C 5 suphas a representing z-- measure ? By 1.4.2 i ) holds i f and only i f

on

^x

. Since every

X

linear functional

-F ( X-)

i; :

?(a)

F

+

IR with

1; 5 supX

can

be extended uniquely t o t R such t h a t the extension i s s t i l l dominated by sup- , we infer from 2.3.2 and 2.3.3 t h a t F ( X ) i s a weak Dini X cone i f and only i f every normed linear p : F(X) -* fi has a signed representing rF-measure 'I . 0

217

Represenring Measures

Finally, we want t o mention t h a t there are indeed weak Dini cones with linear functionals which do n o t have representing measures. In f a c t , consider the cone of continuous real valued functions C(-1,l) on [-1,11 and let v(x)

Put

=

v C(-1,1)

Let

p

x > o

{ -11 =

x < o Ivfl

T

b1

and

represents 6-1

Of course,

lemma since [-1,11.

f €C(-l,l),

1

LI I

p

x E [-1,11

2 f(1)

be the linear functional with

f E C(-1,l). Hence If

for a l l

t

1

'2 f ( - 1 )

. I f(0))

1

~ ( v f )= '2 f ( 1 ) +

. f ( - 1 ) for a l l

sup

1-1311 * then we have necessarily

are the Dirac measures a t

1 and

T =

-1

b1

- '21 6-1

where

, respectively.

v C ( - 1 , l ) s a t i s f i e s the weak Dini condition in view of Dini's I v f I f o r a l l f E C(-1,l) i s a continuous function on

218

Representing Measures

2.4

REPRESENTING MEASURES ON WEIGHTED CONES

We can e a s i l y r e f o r m u l a t e t h e main r e s u l t s o f o u r p r e v i o u s s e c t i o n s f o r weighted cones. The main argument f o r p r o v i n g these r e f o r m u l a t i o n s c o n s i s t s o f t h e f a c t t h a t t h e Sandwich Theorem a l l o w s t o t r a n s f e r l i n e a r f u n c t i o n a l s v i a w e i g h t f u n c t i o n s . By a w e i g h t f u n c t i o n s we understand a f u n c t i o n w : X

R

--t

and, f o r a cone

F = F(X)

we mean by

WF

t h e cone g i v e n b y

W F = { w f I f E F3

2.4.1

Theorem: be a non-negative w e i g h t f u n c t i o n on X

Let w

k- valued f u n c t i o n s on X -such t h a t supx(w f ) the following are equivalent: i)

. Consider -a -cone <

QJ

for all f --

F

E F.

of Then

p : F + 6 with p ( f ) I supx(w f ) has a xWF- p r o b a b i l i t y measure T -such t h a t

Every l i n e a r functional

for all

f E F

-_.

p(f) I

X

ii) w F t R

_._

Wfd-r

for all

f E F.

i s a D i n i cone.

Proof: : w Ft R

Let p

: F

--i

6

--t

fi

p ( f ) = ;(wf)

defined by

p ( f ) I supx(w f ) Conversely, i f

p

. Then X i s a linear functional w i t h

be a l i n e a r f u n c t i o n a l w i t h

for a l l

f E F.

: F

i s linear with

+

I sup

p ( f ) I supX(w f )

,

f E F

then

Put 6 ( w f ) = sup{p(g) 6

1

i s s u p e r l i n e a r and

ii : W F

+

6

with

p ( f ) I;(wf)

6 I

g E

F with w f = w g l , f

6 I sup

ii I

X'

supx

5 supX(wf)

E F.

Hence t h e Sandwich Theorem y i e l d s a l i n e a r

, that for all

is f E F

.

219

RepresentingMeasures

li i s extended t o w F + R by p u t t i n g ;(1)

= 1. Now t h e theorem f o l l o w s

from t h e Representation Theorem 2.2.1.

2.4.2

0

Theorem: a cone o f r e a l v a l u e d ( n o t n e c e s s a r i l y bounded) f u n c t i o n s on X

Let F

and c o n s i d e r a (n o t necessarily p o s i t i v e ) weight f u n c t i o n such t h a t w f -equivalent:

i)

i s_ bounded r every _ _f o_

For every l i n e a r --___

F

1-1 :

-,

6

f E F , there i s a signed

,,(f)

ii)

WF

I

Iw

X

f dr

.

f E F

with

for all

+ R

Then t h e f o l l o w i n g are --

p ( f ) I supx( I w f I )

xwF- measure

w :X

T

,for

all

on X ~such t h a t -

f E F

i s a weak D i n i cone.

Proof:

;1 : W F

Every l i n e a r functional for a l l

: F

p

f E F

Conversely

let

.

p

k

by

: F

-,

-,

b ( w f ) 5 supx( I w f I )

with

-,

p ( f ) = <(w f )

6

.

We have

defines a l i n e a r p ( f ) 2 supx( I w f I )

u ( f ) 5 supx( Iw f I )

be l i n e a r w i t h

for all

f E F.

As i n t h e p r e c e d i n g p r o o f p u t

I

6(w f ) = s u p { p ( g ) 6

i s s u p e r l i n e a r on

yields a linear Theorem 2.3.4

b

g E F with

W F and

: WF

--f

6

6 I supx

with

6 5

w f = wgl

. Hence I supx

t o obtain the desired r e s u l t .

.

t h e Sandwich Theorem

.

NOW, we can a p p l y

Representing Measures

220

2.5

DIRICHLET STATES

I n the preceding sections we investigated the situation where a l l states on a cone F(X) admitted representing measures. A complete characterization was given in terms of order properties, namely by the notion of Dini cone. In case a cone under consideration f a i l s t o be a Dini cone then some states may have representing measures, whereas others certainly do n o t admit to representing measures (Theorem 2 . 2 . 1 ) . So, the probl em arises characterize those s t a t e s with representing measures in case t h a t the underlying cone does n o t have the Dini property. Such characterizations without any further assumptions Seem technically complicated. On the other hand, in special cases, i t i s easy t o give a description of those states having representing measures, for example when the cone i s a vector l a t t i c e (Daniell-Stone Theorem). We shall give in this section a slight generalization of this result.

Throughout this section l e t F(X) be a concrete order unit cone of bounded real-valued functions on some nonempty s e t X . Again, VF(X) denotes the max-stable cone generated by F(X). We deal in t h i s section with states of F(X) having unique extensions t o states of V F ( X ) . First we characterize these s t a t e s by a monotony-property and an algebraic property of the faces generated by them. Then we show that for these states a n individual Dini condition i s necessary and sufficient for the existence of representing measures. Let us f i r s t gather the necessary notions: 2.5.1

i)

Definition: A s t a t e p of F ( X ) i s said t o be a Dirichlet s t a t e i f i t has a unique extension t o a s t a t e of VF(X). Y c X , then a linear functional p on F(X) i s said t o Y- monotone i f p ( f ) I p ( g ) whenever f , g E F(X) such t h a t

ii)

be f(y)

Ig ( y )

for a l l y E Y

i i i ) A convex subset i f vl, v 2 E Q A v1

+ (l-x)v2

Q

.

of the s t a t e space of F(X) i s called a face whenever v1 and v2 are states such t h a t

E a

for some 0

c A c

1

.

221

Representing Measures

2.5.2

Remark:

Let us denote by Face(p) t h e f a c e generated by t h e s t a t e ?.I s e c t i o n of a l l faces containing u ) . I t i s very easy t o show Face(p) i s exactly the s e t of a l l those s t a t e s v such t h a t s t a t e vo and some 0 .c A I 1 w i t h u = x v + (1-l)vo. This

(i.e. interthat there i s a s e t i s auto-

matically convex. Furthermore, we have some k i n d of t r a n s i t i v i t y property f o r Face(u) , namely v E Face(p) implies Face(v) c Face(p) (compare with t h e part-order [2,p.l22I). Since Face(p) i s convex i t can be considered as t h e base of a convex cone, i . e . Cone ( u )

=

I x v I v E Face(p) ,

x

E R+1

i s a convex cone. A t t h i s point i t i s helpful t o r e c a l l a r e s u l t which was i m p l i c i t l y already proved i n s e c t i o n I . 1.5.5.

2.5.3

Lemma:

Assume t h a t -where X i

?.I

i s a s t a t e of

F(X)

and l e t X --

=

X1

U

X2

U...U

Xn

X for a l l i = 1, ..., n . Then n z x. u , of u into Xi- monotone s t a t e s i=l i i n

a r-e -non empty __ _ - subsets of

t h e r e i s a decomposition u u i , i = l , . . . , n , where

=

3 2 0

z x1. = 1 .

i=l

Proof: Let E = I f - g / f , g E F X ) } . Then E i s a vector space containing the constants. Recall , a1 1 f E F(X) a r e bounded. Since 1-1 i s a s t a t e , we have ~ ( f >) - m f o r a l l f F ( X ) . Hence can be extended t o a s t a t e LI on E by putting b ( f - g ) = u ( f ) - d g ) i f f , g E F(X) We have i; 5 max(supX ,. , .,supx ) , hence by t h e F i n i t e Decomposition 1 n n Theorem t h e r e a r e x i 2 0 , w i t h x A 1. = 1 and l i n e a r f u n c t i o n a l s i=l n p i : E + R with iiI supx , i = l , . . . , n , a n d I z X i Gi . Since E i s i =1 l n a vector space we have i; = Z x.i i and G 1. a r e X i - monotone s t a t e s i=l of E . P u t p i = ;. for all i = l,...,n.

.

'IF(X)

222

Represenring Measures

Recall t h a t we have a l s o shown i n t h e preceding p r o o f t h a t e v e r y s t a t e of

F(X)

can be extended t o a s t a t e o f

-

F(X)

Dominating Extension Theorem and t h e f a c t t h a t F(X)

space, we o b t a i n t h a t e v e r y s t a t e o f VF(X) - VF( X )

.

2.5.4

1.1

F ( X ) . Hence, a p p l y i n g t h e F(X)

-

i s a vector

F(X)

can be extended t o a s t a t e o f

Theorem:

L e t p : F(X)

i)

1.1

ii)

1.1

-,R

are e q u i v a l e n t .

be a s t a t e . -Then t h e f o l l o w i n g

-i s-a D i r i c h l e t s t a t e

can be extended -VF(X) - VF(X)

uniquely t o a state o f the vector l a t t i c e

iii)Cone(p) s a t i s f i e s t h e FSP. Furthermore, whenever is -

Y- monotone, -then a l l

a r e Y- monotone.

E Face(u)

p

u E Face(1.I)

Proof: (i) of

(ii): VF(X)

g i v e n by

By d e f i n i t i o n

and t h e n t h e unique e x t e n s i o n t o

b(f -9)

= G(f)

-

;(g),

o f bounded f u n c t i o n s , hence (ii) a

*

1.1 =

to

(iii):

Consider

A v t 6 uo

b(f)

f o r some 6

- VF(X) A u E Cone(u)

T =

VF(X)

.

f,g E VF(X) >-m

2 0

.

Since

the extension o f

u

T

-,?

i s l i n e a r . From t h e d e f i n i t i o n o f

that

T

-+

i s a b i j e c t i o n between

Cone(p)

(Theorem I 2.6.8).

to Cone(1.I)

and

sions t o

p

E Face(u) c Face(p).

E(X). Because

u 5 supy

sion t o

E(X)

b e i n g 5 supy.

equal t o

S

3

. Hence

E(X)

FSP,

is

Let

.

Hence,

Further-

we t h e n d e r i v e

Cone(;). E(X)

I t i s easy t o see t h a t t h e n Cone(;)

We r e c a l l has t h e

FSP

a l s o has t h e FSP.

so Cone(1.I) must have t h e

F o r t h e second p a r t we c o n s i d e r an a r b i t r a r y an element

such t h a t

must be unique,too.

t h a t t h e p o s i t i v e dual cone o f t h e v e c t o r l a t t i c e

NOW, l i n e a r b i j e c t i o n s preserve t h e

consists

has a unique e x t e n s i o n

has a unique e x t e n s i o n

more

i s obviously

VF(X)

f E VF(X)).

u,uo

1.1

- VF(X)

(Recall,

for all

a,X > 0 and s t a t e s

E(X) = VF(X)

every

can be extended u n i q u e l y t o a s t a t e

1.1

Y- monotone

FSP.

u E Face(p) and

S and i; be t h e unique exten-

t h e Sandwich Theorem g i v e s us an exten-

Because o f t h e uniqueness t h i s e x t e n s i o n i s

Y - monotone. I t s u f f i c e s t o prove t h a t

223

Representing Meaures

i; E Face(;)

i s again

f(y) 5 g(y)

for all

and some s t a t e

Y- monotone. Take a r b i t r a r y y E Y

such t h a t

f,g E E ( X )

0 < x < 1

and r e c a l l t h a t t h e r e i s some 3=

x P+(l-x)p.

with

S i n c e s t a t e s a r e monotone

we g e t

( * I P(max(g,f)) But

<(max(g,f))

2 ii(g)

= V(g)

and

p(max(g,f))

V

since

= v ( g ) = v(max(f,g))

is

=

2 C(g).

Y- monotone. T h i s y i e l d s

(A 6 +

(max(g,f)).

(1-1);)

T h i s i m p l i e s t h a t we have e q u a l i t y i n ( * ) , hence

b ( g ) 2 ;(f).

(iii)

IJ

=,

( i ) We can f i n d p r o p e r e x t e n s i o n s

hence t o Let

fly

C1

VF(X). Assume

...,fmE

ji2 a r e s t a t e s o f

F(X). We show

C,(max(f

= {x

I fi(x)

,... ,fm)},i=l, ...,m.

2 max(fl

Therefore, by Lemma 2.5.3,we 1

'I,

where

?'l,i'

i( l x )

may decompose 1 %,i ( 'x)

VF(X). T h i s means i n p a r t i c u l a r Let

=

p.

JYi

j=1,2,

"' IF( X ) m

'

and

m p1,i

i=l

there are p2yi

'i,j

=

=

v.

i=1

1 ,j

m

p2,i

E Cone(p)

.

F(X)

Xi-

and

X

j

- monotone. Hence

F(X) p.

ti2(maX(f1,...,fm)).

which a r e non-empty

m E Cl,i

ii, =

i=l

are

Then t h e

Xi-

p

Cone(v) -

in

z

pl,i - j=l

.X,

X = X1 U...U

We have

, C2

m

X ii2,i i=l

=

monotone s t a t e s o f

,..., f,,,)), j,i

are

i=1,2. monotone,

Xi-

satisfies the v

j,i

FSP,

'

E V . . The second assumption o f iii)i m p l i e s t h a t j=1 ~ Y J

are

-

which extend

= jik,i(max(fl

. Since with

VF(X)

-'2,i

iik,i(fi)

, ill,...,m,j=1,2.

Ci. .

, to

,,..., f,))=

Consider those s e t s o f t h e f o l l o w i n g form Xi

of

!J

v.

J,i

and

224

Representins Measures

and j

. We have

;l(max(fly...,fm)) m

rn

=

2.5.5

=

m

m

m I j=l

m

= ii2(max(fl,...,fm)).

Remark

A c t u a l l y we have proved more t h a n s t a t e d i n Theorem 2.5.4. if

i s a D i r i c h l e t s t a t e on

p

u n i q u e l y extended t o a s t a t e p +

G

F(X)

G

on

t h e n every

-

VF(X)

p E

VF(X)

.

p I supy

Face(p)

can be

and t h a t t h i s map

d e f i n e s a l i n e a r monotone b i j e c t i o n between

Furthermore, t h i s b i j e c t i o n has t h e p r o p e r t y t h a t

Namely t h a t

Face(p) and Face($) 6 I supy whenever

I t seems o f some i n t e r e s t t h a t a c t u a l l y we can

.

strengthen

t h e r e s u l t o f t h e Decomposition Theorem.

2.5.6 Let -

Lemma: p

(i)

be a maximal s t a t e o f

F(X).

Then t h e f o l l o w i n g a r e e q u i v a l e n t : -Every v E Face(u) has t h e p a r ti a l decomposition i,e. ~p r o p e r t y , f-o r e v e r y c o u n t a b l e covering, Yn,n E N, of X i s partially decomposable w i t h r e s p e c t t o t h e

(ii) Every

v

Proof: Only ( i )

E Face(p)

*

(ii)

I

supy

n

.

has -_t h e decomposition p r o p e r t y .

needs a p r o o f and t h i s p r o o f c o n s i s t s o f completely

t h e same arguments we used i n t h e p r o o f o f t h e Decomposition Theorem. L e t Ynyn E N , be any c o u n t a b l e covering. F o r sequence

It1

=

t tn

n€N

order. I f v

0

tn t 0

with

. Assume

v 5

z tn supy

nEN

t

E Face(p)

( 1 - l t l ) supx

consider a

, where

t h e sequence t o be maximal w i t h r e s p e c t t o p o i n t w i s e

I t 1 c 1 t h e n we decompose v

E Face(p).

n

v

=

t tnvn t ( l - l t l ) v 0

nEN

I t should be observed t h a t m a x i m a l i t y o f

p

, hence

(hence o f

v)

Representing Measures

225

i s assumed t o ensure e q u a l i t y i n t h i s decomposition. Now, representing i n t h e same way and adding the corresponding c o e f f i c i e n t s t o those of v 0 we g e t a contradiction t o the maximality of t h e t n and v i s decomposable. 0 v

. Hence

It1

=

1

This lemma then y i e l d s a representation theorem f o r maximal D i r i c h l e t states: 2.5.7 Theorem: s t a t e . -Then t h e following are Let - p : F ( X ) -,R be a maximal D i r i c h l e t equi Val e n t :

_has _ _a representing IF-measure

(i)

p

(ii)

Every v E Face(p)

& s t r i c t l y represented by a-

(iii) For a l l v E Face(u) --we have t h a t 0 2 f n E F(X)

(iv)

such t h a t

I nEN

fn(x)

IF-measure

I v(fn ) = -

n€N =

-

co

whenever

-for all x E X

.

Every v E Face(p) has t h e decomposition property.

Proof: ( i ) * ( i i ) : Since the extension 6 of p t o t h e vector l a t t i c e must have a representing measure. Hence E ( X ) = VF(X) - VF(X) i s unique the convergence property required i n t h e Daniel1 Stone Theorem is v a l i d f o r ii . I t i s q u i t e easy t o s e e t h a t i f ii has t h i s convergence property then a l l 3 E Face(;) have t h e same convergence property. Hence a l l t h e have IF-representing measures. B u t t h e r e s t r i c t i o n map from Face(;) onto Face(p) i s s u r j e c t i v e . So ( i i ) i s proved.

<

( i i ) * ( i i i ) i s an immediate consequence of t h e Monotone Convergence property. ( i i i ) * ( i v ) : By t h e P a r t i a l Decomposition Theorem every v E Face(p) has the p a r t i a l decomposition property. Remark 2.5.5 y i e l d s t h a t v E Face(p) has the p a r t i a l decomposition property i f and only i f t h e corresponding V E Face(;) has t h i s property. From Lemma 2.5.6 we obtain t h a t a l l 3 then have the decomposition property and ( i v ) holds obviously f o r the

RepresentingMeasures

226

restrictions

.

( i v ) * ( i ) : Again from Remark 2 . 5 . 5 we i n f e r t h a t v E Face(p) has t h e decomposition property i f and only i f the corresponding < has t h i s property. The existence of t h e representing measure f o r ; then follows from Proposition 2 . 1 . 6 0

2 . 5 . 8 Remark:

There i s a natural generalization of t h e notion of face: Let ( F ( X ) , s ) * be endowed with a preorder < compatible with t h e cone operations. Then a subset Q, of the s t a t e space i s c a l l e d f a c e with respect t o < i f v l Y u 2 E Q, whenever vl,v2 a r e s t a t e s such t h a t A v1 t ( l - x ) v 2 > v

for some 0 < 1 < 1 and

v E Q,.In 2 . 5 . 1 i i i ) we

considered faces of t h i s k i n d with respect t o the preorder given by equal i ty Clearly, we have: v i s an extreme point of the s t a t e space with respect t o 4 i f and only i f CVI i s a face ( s e e p. 1 4 2 ) .

.

Similarly, the face generated by a s t a t e such t h a t t h e r e i s a s t a t e v0 and 0 < kl

< A V

t

(l-X)uo

p

x

i s t h e s e t of a l l s t a t e s s 1 with

v

.

This generalization of f a c e i s of p a r t i c u l a r i n t e r e s t , when wise order i n ( F ( X ) , s ) *

.

Then we obtain: I f p i s a maximal s t a t e , then Face(p) with respect t o = w i t h respect t o < coincide. In p a r t i c u l a r Lemma 2.5.6 and Theorem 2.5.7 remain t r u e i f replaced by Face(p) w i t h respect t o 4 , This observation will be u s e d . i n Chapter 11.5.

i s the point-

and Face(p) Face(p) i s

227

Representins Measures

2.6

ELEMENTARY EXAMPLES AND APPLICATIONS

2.6.1

The Riesz Representation Theorem for non-Hausdorff Sets

For good reason the Riesz Representation Theorem i s usually stated only for compact Hausdorff s e t s . A1 though compact non-Hausdorff sets behave very similarly t o Hausdorff sets with respect t o functions attaining their maxima there i s quite some difference with respect t o Borel measures living on them. Especially arguments which depend on the support of a measure, or on regularity, do not go over unchanged. B u t i f one drops the regularity assertion in the Riesz Representation Theorem t h i s result in fact can be extended t o the non-Hausdorff situation. Here, we immediately present the more general version of the Riesz-Konig Theorem.

2.6.1 Riesz-Konia Theorem: Consider an abelian semigroup S with neutral element and 5 (not necessarily Hausdorff) compact topological space n --Assume t h a t by s + ; a for a l l w E R , -the map function S + U S C ( n ) i s given such t h a t , -s + ;(w) i s subadditive 0" S Moreover, let p : S + be additive t h a t such --

.

.

p(s) I

sup,(;)

for a l l

s E

Borel probability measure -Then _ _there _ is 5 p(s) I

j S d-r for a l l s E

R

s .

0"

T

S

R with

.

Proof: We proceed almost exactly as in the proof of I 1.6.1. We define a sublinear functional p and a superadditive functional on U S C ( n ) as follows:

6

228

Representing Measures

for all

h E USC(n). U n f o r t u n a t e l y

6

i s not superlinear since i t i s not

homogeneous. B u t u s i n g t h e Sandwich Theorem i n i t s o r i g i n a l semigroup v e r s i o n we f i n d a monotone a d d i t i v e

v

with

d l v C p . P r o p o s i t i o n I 1.2.4 t e l l s us t h a t

*

i s l i n e a r . Hence i t i s a s t a t e because

1. B u t USC(n) i s a Dini cone ( D i n i ' s Lemma o r F(X) = R ) . Hence b y t h e Representation Theorem 2.2.1 t h e r e i s a r e p r e s e n t i n g measure T f o r v . T h i s measure 0 tri v i a1 l y f u l f i11 s t h e d e s i r e d inequal ity. of

v ( * 1) I p ( i 1) =

v

1.3.5 a p p l i e d t o t h e t r i v i a l case

Corollary: Let -

n be-a compact t o p o l o g i c a l -space and F(n) --_-__ a s e t of upper semicontinuous f u n c t i o n s on n w i t h F(n) t F(n) c F(n). --Then f o r e v e r y w i t h u 5 sup, t h e r e i s a Bore1 probabia d d i t i v e f u n c t i o n u : F(n) -,6 -----

l i t y measure T on -~

u(f) 5

2.6.2

with -

R

f dr

n

for all f --

E

F(n)

.

R e g u l a r i t y and Support

We a l r e a d y know t h a t B a i r e measures on a compact space have a c e r t a i n r e g u l a r i t y p r o p e r t y . I n t h i s c o n t e x t one may ask i f t h e r e i s s t i l l some k i n d o f r e g u l a r i t y p r o p e r t y f o r those measures which appear i n t h e Representation Theorem 2.2.1.

I n fact, there i s a corresponding property

-

we s h a l l s k e t c h a p r o o f as a p p l i c a t i o n o f t h e decomposition p r o p e r t y . Lemma 1 [

27 , p.1981

Let -

a B a i r e p r o b a b i l i t y --_ measure on a t o p o l o g i c a l space

every

m

B E Bo(X) -we have m(B) = i n f I m ( A )

IA

an -open _

Fo - -set with

X

3

A

2

BI

X. -Then f o r

.

229

Representing Measures

Proof: Consider X

cp +

CB(X) (bounded r e a l continuous f u n c t i o n s on

BX

the set

X ) and map

i n t o i t s Stone Czech c o m p a c t i f i c a t i o n . By t h e Riesz

X

Representation Theorem t h e r e i s a unique B a i r e p r o b a b i l i t y measure

X

on

such t h a t p(f)

where

T

^f

* x

f dm

^f

=

BX

dr

A

i s t h e obvious e x t e n s i o n o f

f o r a l l open

Fo- subsets

f

to

U Kn, Kn nEN

A =

,

f E CB(X)

for all

p(7)

We c l a i m

3! X .

T(A) = 1

B X c o n t a i n i n g Q(X).

compact, o f

I n f a c t t h e decomposition p r o p e r t y ( 2 . 1 . 6 ) o f v ( . ) i m p l i e s t h a t t h e r e a r e A,

2 0

Hence

with p I 1

E xn = 1 such t h a t

xn supK

and

n

2 . 1 . 1 ( i i i ) we g e t t h a t S E B0((3X))

f E C B(X)

T

p I

(defined by

i s a B a i r e measure on

. Hence

m =

T

. Now,

cp

a s s e r t i o n o f t h e lemma.

n

, where

must be supported by

'I

cp

supx

E

X

If

dm =

using t h e r e g u l a r i t y of

.

From remark

= T(S)

~,(cp-'(S))

with

.

A

Xn = X r l K,

If T

for all

dTQ f o r a l l one g e t s t h e

0

C o r o l l a r y 2: E(X) __--____be a v e c t o r l a t t i c e o f bounded r e a l f u n c t i o n s -on some X . Assume t h a t E ( X ) c o n t a i n s t h e c o n s t a n t s and l e t m be a pro-- - - 1E(X) - ___, -we have b a b i l i t y ___ measure. __ Then, _.for all B E E E(X) Let -

m(B) = inf{m(A)

where a s e t

IA

A -i s called

is

E(X)-open

3

131

,

E(X)-open if i t s c h a r a c t e r i s t i c f u n c t i o n -i s the --._

supremum _o f_ a p o i n t w i s e i n c r e a s i n g sequence

fn t 0

i n E(X)

.

Representing Measures

230

Proof: Endow X

w i t h t h e c o a r s e s t topology such t h a t a l l elements o f

continuous and t r y t o convince y o u r s e l f t h a t t h e

E(X)

-

E(X) are

open s e t s a r e t h e

open Fu- s e t s o f t h i s t o p o l o g y . Then an a p p l i c a t i o n o f Lemma 1 g i v e s t h e 0 result. An obvious consequence o f Lemma 1 i s t h a t t h e f i n i t e B a i r e measures a r e r e g u l a r . A measure on a t o p o l o g i c a l space ( w i t h r e s p e c t t o an a r b i t r a r y u- a l g e b r a ) i s s a i d t o be r e g u l a r i f f o r any measurable

m(B)

= supIm(A)

1A

c

B we have:

B , A c l o s e d measurable}.

T h i s p r o p e r t y i s i m p o r t a n t f o r t h e d e f i n i t i o n o f a support. We s h a l l do t h i s immediately f o r L i n d e l o f spaces. Recall t h a t a t o p o l o g i c a l space i s s a i d t o be L i n d e l o f i f t h e c l o s e d s e t s have t h e c o u n t a b l e i n t e r s e c t i o n

F i s a f a m i l y o f c l o s e d s e t s such t h a t e v e r y countable s u b f a m i l i y has a nonempty i n t e r s e c t i o n t h e n F has nonempty p r o p e r t y , i . e . whenever

i n t e r s e c t i o n . The L i n d e l o f theorem s t a t e s t h a t e v e r y t o p o l o g i c a l space w i t h a c o u n t a b l e base i s such a space. Now, f o r a p r o b a b i l i t y measure support o f

m on a t o p o l o g i c a l space we d e f i n e t h e

m t o be

supp(m) = n { A c X I A

c l o s e d measurable w i t h

m(A) = m ( X ) = 11.

Of course i n general t h i s s e t can be empty, b u t n o t i n t h e case t h a t

X

i s Lindelof: Lemma 2 : Let X be-a L i n d e l o f space. ---Then f o r e v e r y c l o s e d measurable set B with B n supp(m) = 0 we have m(B) = 0 . I n p a r t i c u l a r , supp(m) i s n o t empty. L--

Proof: If

B n supp(m)

=

0 then, by t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y , t h e r e i s

a c o u n t a b l e sequence

An

o f measurable c l o s e d s e t s w i t h

m(An) = 1 and

Representing Measures

( n An) n B = nEN

0

.

Hence, we g e t the a s s e r t i o n from the cr- a d d i t i v i t y of

the measure m.

Corollary 3

231

0

1317 ,p.1751

Let X _ be -a Lindelof -space and m _be_ a_ _regular p r o b a b i l i t y measure _ ( w i t h respect t o a s u i t a b l e U - algebra Z ) . Then m(B)

=

0 -for all B

E

z _-w i t h

.

B n supp(m) = 0

An immediate consequence of this c o r o l l a r y i s t h a t every regular measure ( e s p e c i a l l y every Baire measure) m on a Lindelof space can be extended and t h a t i n this t o the U- algebra generated by II and supp(m) u- algebra supp(m) i s t h e smallestclosed s e t with mass 1, i . e . i n this case i t r e a l l y deserves t h e name support. In this context we should mention another notion. A Borel measure m on a topological space X i s c a l l e d t i g h t i f we have m ( K ) < f o r every closed compact K and i f f o r every measurable B c X 00

m(B)

=

sup{m(K)

IK

c

B , K closed compact}.

In some cases Borel measures a r e automatically t i g h t . For example, every f i n i t e Borel measure on a polish space i s t i g h t [ 27 ,p.201] ( X i s c a l l e d polish i f i t i s completely metrizable and has a countable base). The proof of t h a t f a c t i s q u i t e obvious. Because of t h e countable base we can find a dense sequence xn, n E N , of points i n X. Take f o r a r b i t r a r y P

> 0 t h e closed b a l l s

Then

X = U B(pyxn)

index Nn

nEN

B(p,xn)

. So,

f o r every

with p(X\Bn)

5

&

-

2n

w i t h radius E

p

> 0 and

around xn n E N

.

t h e r e i s some

Representing Measures

232

where Bn

=

U ' . B (-1

n

Ilk1

, xm ). Hence

p(X\K)

5

E

if

K =

n Bn

n€N

B u t K i s closed and obviously precompact, hence compact,since assumed t o be complete.

2.6.3

. X was

Representation by Unbounded Measures

Up t o now we have only considered f i n i t e representing measures. B u t i t i s well known the the Riesz Representation Theorem as well as the Daniel1 Stone Theorem carry over (under certain restrictions) t o situations where the representing measures are non-finite.

We show how t o recover these (slightly more general) versions from the f i n i t e case. The trick one has t o work with i s quite obvious: one has t o use suitable weight functions. We consider the following situations: Let E ( X ) be a vector l a t t i c e of real valued functions ( n o t necessarily bounded) on some s e t X For f € E with f L 0 we use the following notation:

.

Xf

=

{x

E

x I f(x) >

Ef = {g E E ( X )

xf

0)

I there

is

x

= u- algebra generated in

xE =

CI-

2 0

Xf

with by

algebra generated in X by

l g l S Af3

f-'Ef E(X)

=

{f-lhlh E E f l

.

.

For short we call a monotone linear functional p : E ( X ) -, R on a vector l a t t i c e E ( X ) Dini continuous i f p ( f n ) -, 0 whenever f n i s a n -,OD sequence in E ( X ) with f n J. 0 ( i . e . f n decreases pointwise t o zero).

Lemma 1: Let p : E ( X ) + R be a Dini continuous monotone linear functional. Then for every 0 5 f E E ( X ) there i s a Ifprobability measure mf such that --- I -

p(h) =

p(f) J

xf

( h f-')

dmf

for a l l

h

E

Ef

233

Representing Measures

Moreover, mf

is

unique i f

p(f)

0

Proof: I f p ( f ) = 0 then the assertion i s t r i v i a l . So assume p ( f ) > 0 and consider the vector l a t t i c e E; of a l l real valued linear functions on Xf given by the restrictions of f - 1 Ef . E; i s a n order unit vector

space (with 1 as order u n i t ) . And on xf

we define a s t a t e

E;

pf

by

where, of course, f a g stands f o r the extension t o X given by zero outside Xf pf i s Dini-continuous, since p was Dini continuous.

.

Hence by the Daniell-Stone Theorem there i s a unique probability measure representing the s t a t e pf NOW, by going back from E; t o Ef one gets

.

the desired result.

0

Let us turn our attention t o the u- completion (denoted by E u ( X ) ) o f E ( X ) . By that we mean the smallest u- complete vector l a t t i c e (with respect t o pointwise operations) which contains E ( X ) We recall t h a t u- complete means t h a t for every countable bounded s e t {fnln E N I the function dx) = sup f n ( x ) i s again an element of the vector l a t t i c e . n A s e t F i s called bounded i f there i s a n element g E E u ( X ) such t h a t If1 5 g for a l l f E F. E u ( X ) i s the intersection of a l l u- complete vector l a t t i c e s 3 E ( X ) I t i s a sublattice of the u- complete vector 1a t t i ce

.

.

E,(X)

where

IRX ‘E( X )

=

If E

X

R 1

i s the

E(X)

U-

I

there i s

g

E

EJX)

which means t h a t for every

such that If1 5 l g l l ,

complete vector l a t t i c e of a l l real

measurable functions. I n general we have E u ( X ) exercise t o show t h a t

E(X)

2 E,(X).

E,(X)

i f and only i f

f E Eu(X)

the function

=

EE(X)-

I t i s an easy Eu(X)

i s truncated,

Representing Measures

234

(f i s again in

A

l ) ( x ) = minCf(x),l)

Of course, i f

E,(X).

E(X)

i s truncated then

Eu(X)

is

truncated. Theorem 2:

Let

l~ :

Then -

p

: Eu(X)

E(X) R be a Dini continuous monotone linear functional. has t o-a- Dini continuous monotone linear - a unique extension -+

-+

IR

.

Proof: We f i r s t remark t h a t whenever u : E ( X ) -+R ( E ( X ) an arbitrary vector l a t t i c e ) i s monotone linear and Dini continuous then i t s Dini continuous extension t o E u ( X ) must be unique. I n f a c t , take two extensions v1 , v2 , then {f E

i,(x) I v l ( f )

hence equal to

=

v 2 ( f ) 1 i s a u- complete vector l a t t i c e

:

(Efl0

i(x),

Eu(X).

Now, we apply Lemma 1. If lJf

3

0 5 f E E ( X ) then we define the functional

+R

Then t h i s functional i s Dini continuous (Monotone Convergence Property) Therefore, i f 0 5 f l , f2 E E ( X ) , then and an extension of p IEf are Dini continuous extensions of p Hence Ufl, lJf2 IErnin(f1 ,f2) and LI coincide on { g E E,(X)I lgl 5 min(fl,f2)l llfl f2 Therefore, i f we take for g E E J X ) a n arbitrary f E E ( X ) with

.

.

.

l g l 5 f , we may define

235

Representing Measures

This functional does not depend on the choice of desired p r o p e r t i e s . 0

f

and i t has a l l t h e

Corollary 3 : Let -

E,(X) the

IJ

: E(X)

truncated. Then t h e r e i s a positive measure m ( w i t h r e s p e c t t o algebra generated by E ( X ) on X ) -such t h a t

is U-

-,R be monotone ~ l i n e a r and Dini continuous and assume ~ that -

v(f)

=

I X

f dm

for all

.

f E E(X)

Proof: of IJ t o E a ( X ) and consider Take the Dini continuous extension 2 = AEZ I t h e r e i s f E E & X ) such t h a t lA 5 f} E(X) Since E a ( X ) i s truncated we have i = t A E Z I1 E E u ( X ) } . E(X) A NOW, define f o r B E II E(X)

.

m(B)

=

sup{i;(lA) 1 A

E

with

Then m has a l l t h e desired p r o p e r t i e s .

A

= B3

.

0

This c o r o l l a r y e a s i l y leads t o t h e well known representation theorems f o r topological s i t u a t i o n s . Let us show this. On a Hausdorff topological space X we study t h e space C o ( X ) c o n s i s t i n g of a l l continuous functions f : X -,IR such t h a t If1 vanishes a t i n f i n i t y , i . e . the s e t s t x E X I I f ( x ) l L & I a r e compact f o r a l l E > 0 . Clearly, C o ( X ) i s a truncated vector l a t t i c e whose p o s i t i v e p a r t i s a subcone o f U C,+(X) .

-

Representing Measures

236

Theorem 4:

For every monotone linear v : C o ( X ) --

+

there i s a measure m (with -

R

respect t o the by C o ( X ) ) ~ - 0- algebra - generated p(f)

J f

=

dm -for a l l f E C o ( X )

-such t h a t

.

Proof: We only need t o prove t h a t suitable weight functions.

p

i s Dini continuous. This i s done by using

Consider a sequence f n in C o ( X )

with

f n +O.

Define W E C o ( X )

cp = f l

and a sequence

t

wn

E

W,(X)

=

1

n min(fl, -3 ) n=l n 1

with

Co(X)

cpn

+0

by:

otherwise Note,

cpn

w,,(xt)

+

Since

p

B u t the

i s indeed continuous since, by definition o f 0 for every net

xt

cp,

converginq t o a point x with

we have cp(x)

=

i s monotone we have

cpn

are uniformly decreasing t o zero since U C L ( X )

Dini cone (Lemma 1.3.5).

0

t R

is a

0

.

237

Representing Measures

Adapted Cones

2.6.4

Mokobozki - Sibony [2321 arld others (see also [721 and [255])introduced the concept of adapted cones. These cones came u p in potential theory. They have the property t h a t positive linear functionals always admit representing measures. Adapted cones are cones of nonnegative continuous functions on a locally compact topological space f u l f i l l i n g a suitable weight condition a t i n f i n i t y . We are going t o generalize the concept in such a way t h a t also nontopological situations and t h e i r representing measures are covered as we1 1 . be a truncated vector l a t t i c e of real-valued functions on some Let E ( X nonempty s e t X . E ( X ) i s said t o be adapted i f for every sequence f n E E ( X with f n J 0 there i s a 0 I Y E E ( X ) such t h a t f o r every there i s a 0 I cp E E ( X ) having the property t h a t cp-le,(x) converges uniformly t o zero, where E

> 0

cn(x)

As usual we p u t

f n ( x ) when f n ( x )

{

=

2

E

~(x)

otherwise and

= 0

0- (+-)

-1 (x) = +-

when ~ ( x =) 0.

cp

Lemma : Every monotone linear u : E ( X ) has - a representing measure.

+ R

on an adapted vector l a t t i c e E ( X ) -----

Proof: Consider arbitrary inf

nEN

E

> 0

u ( f n ) I E r (where r

fn E E(X)

and

with

fn

i s independent of

E R,

.

If we can prove

J

0

E

), then the Daniell-

Stone Theorem (section 2.6.3) yields the existence of a unique representing measure. I n order t o prove t h i s inequality we take the c p , ~ from the definition above, we define T = max(fl,cp,w) and we consider the subspace

i The functions

= {g E

c p , ~

E(X)

I

191 I A T

and a l l the

for some

x

E R,

I

.

f n are in t h i s subspace. Let

u

be the

Representing Measures

238

restriction o f

u

to

,?

and c o n s i d e r t h e s u b l i n e a r f u n c t i o n a l s on

g i v e n by:

Ix Ix

pn(g) = sup{T(x)-'g(x) qn(g) = s u p I T ( x ) - l g ( x )

i; i s monotone we have i;

Since

E X with

fn ( x ) I

E X with

fn ( x )

5 p max(pn,qn),

5 i n t o l i n e a r vn,nn w i t h

functional

I n particular then

6,

+

0.

(x)}

2 E Y (x)}

where

By t h e F i n i t e Decomposition Theorem we can decompose

E Y

p =

;(T)

fi = vn t

qn

the

From p n ( f n

-

E Y ) I

0

and nn I

and q,(f,-dn

(p)

we o b t a i n :

I 0

(adaptedness) we o b t a i n t h e d e s i r e d i n e q u a l i t y

Now, l e t

be a cone o f

F(X)

= p(T).

p 9., -1 vn and nn a r e monotone. P u t 6, = sup{(p (x)E,(x) I x E X I ,

vn I p pn

Since dn J. 0

adapted

.

real

f u n c t i o n s on

X

.

We c a l l

F(X)

i f t h e following c o n d i t i o n s a r e s a t i s f i e d :

i)

f o r every

f E F(X)

t h e r e i s some

ii) t h e t r u n c a t e d v e c t o r l a t t i c e

E

adapted, where generated means vector l a t t i c e o f functions

g E F(X)

generated by E

w i t h If1 I g F(X) i s

i s the smallest truncated,

X which c o n t a i n s F(X).

Theorem [ 1261 : Every monotone l i n e a r r e p r e s e n t i n g measure.

p

: F(X)

-,R ---on an adapted

cone

F(X)

-

has - a stri

RepresentingMeasures

239

Proof:

u can be extended t o a monotone l i n e a r

A l l we have t o show i s t h a t functional

on t h e t r u n c a t e d v e c t o r l a t t i c e

E(X)

generated by

F(X).

B u t t h i s i s easy. Because o f i ) i n t h e d e f i n i t i o n above

Ig

p(g) = i n f I u ( f ) defines a s u b l i n e a r f u n c t i o n a l on can be extended t o a l i n e a r g Io

since

If E F ( X ) I

E(X). Furthermore 2 p

on

E(X)

ii(g) 5 p(g) I~ ( 0 ) = 0

implies

.

. 6

u

p

Hence IF(X)' must be monotone =

0

Example 1 ( H e w i t t [1631 see a l s o [3171): ~

Let X

C(X) be t h e space o f continuous -r e a l valued f u n c t i o n s on X , where i s a normal t o p o l o g i c a l ___ space. _ Then e v e r y monotone l i n e a r functional _-

u :

-,R -can_ be

C(X)

r e p r e s e n t e d by

g

B a i r e measure. -____

Proof:

I t s u f f i c e s t o show t h a t C(X) fn J. 0, fn E C(X)

let

Yn = I x I f n ( X ) I T

n E C(X)

with

m

Z

n=l

every

, Zn

n

-2

x E X

~

E

> 0

and c o n s i d e r t h e d i s j o i n t c l o s e d s e t s

I x Ifn(x) 2

=

0 5 r n I 1 and

.

~~

g =

s

and

i s an adapted v e c t o r l a t t i c e . To show t h a t ,

T

1

E

. Take

Urysohn f u n c t i o n s = 0

nlYn

. The

function

-1

~+ f n( ) 1 i s o b v i o u s l y continuous and nowhere z e r o s i n c e i s i n some

Yn.

Put

cp = g

-1

and

Y

= 1. These f u n c t i o n s

have t h e d e s i r e d p r o p e r t i e s because o f

-1 cp

Example 2 (Mokobozky Let

X

-

E n = E n ' g I

f 5 c q

X

. We

such t h a t f o r e v e r y outside

0

-7

K

.

f ( x ) > 0. We c l a i m t h a t

P(X)

assume t h a t f o r e v e r y E

Furthermore we assume t h a t f o r every with

1

be a l o c a l l y compact H a u s d o r f f space and

q E P(X)

with

1

Sibony [2321 see a l s o [ 7 2 , p. 2831):

nonnegative f u n c t i o n s on a

"

m=n m

P(X)

> 0

a cone o f continuous f E P(X) t h e r e i s

t h e r e i s a compact s e t

x E X

there i s a function

i s an adapted cone i n t h e

K c X fEP(X)

Representing Measures

240

sense defined above. Clearly the property given above f o r P ( X ) a l s o holds f o r the p o s i t i v e cone of the truncated vector l a t t i c e generated by P ( X ) . For the proof of adaptedness we may t h e r e f o r e assume t h a t P ( X ) - P ( X ) = E ( X ) i s a truncated vector l a t t i c e . Now, take a sequence f n E P ( X ) with f n J. 0 . Then there i s some q E P ( X ) with

fl 5

E

q

o u t s i d e a s u i t a b l e compact

set (depending on E ) . P u t cp = f l t q and Y = q , then because of D i n i ' s Lemma on compact s e t s the functions cp and Y have t h e desired properties. Example 3:

Let us discuss a special case of example 2. Again X i s a l o c a l l y compact topological Hausdorff space and we denote by C a o ( X ) and C o ( X ) the complex-Val ued and real -Val ued continuous functions on X which vanish a t i n f i n i t y . We endow t h e s e cones w i t h t h e sup-norm. The cone of p o s i t i v e elements in C o ( X ) is c e r t a i n l y adapted. Hence, monotone f u n c t i o n a l s do have representing measures (Theorem 4 of section 2.6.3): W i t h Theorem 2.3.4 we obtain a s i m i l a r r e s u l t f o r bounded functions i n case t h a t p i s not monotone.

u For every l i n e a r functional -bounded ~ -

: C Co(X)

+

bounded complex-valued t i g h t measure such t h a t ~ - - v-p(f) =

I X

f dv f o r a l l

f E C Ro(X)

a t h-e r-e-i s a unique

.

Proof:

Because C

ao(X)

=

Co(X) + i Co(X)

functional on C eo(X)

from i t s r e s t r i c t i o n t o

r e s t r i c t our arguments t o C o ( X ) . functionals p r , p i by p(f) =

we e a s i l y recover every l i n e a r

pr(f)

+i

Co(X)

. Hence,

we can

Define bounded real-valued l i n e a r

pi(f)

for all

f E Co(X)

.

Since C o ( X ) i s a weak Dini cone these functionals have signed Baire representing measures mr and mi , respectively.

241

Representing Measures

Since, f o r example, r e p l a c i n g

mr

mr(A) = supImr(K)

by t h e measure

I K c A,

compact}

K

does n o t change t h e i n t e g r a l s f o r t h e f u n c t i o n s v a n i s h i n g a t i n f i n i t y we mr

can assume

mi

and

t o be t i g h t . Uniqueness i s easy t o see: E i t h e r

v i a t h e uniqueness a s s e r t i o n i n t h e D a n i e l 1

-

Stone Theorem o r f r o m t h e

f a c t t h a t d i f f e r e n t t i g h t measures l e a d t o d i f f e r e n t i n t e g r a l s .

2.6.5

0

R e p r e s e n t a t i o n o f F u n c t i o n a l s w i t h Zero Mass

We have seen t h a t o u r fundamental R e p r e s e n t a t i o n Theorem y i e l d s r e p r e s e n t i n g measures even i n f o r m a l l y more general s i t u a t i o n s (e.g.

signed

measures, measures on w e i g h t e d cones, unbounded measures). L e t us g i v e a n o t h e r exampl e o f t h i s k i n d .

L e t us s t a r t w i t h a s i m p l e o b s e r v a t i o n . We c o n s i d e r two cones

X

o f upper bounded f u n c t i o n s on cp : F(X)

{cp(f)

--t

If

G(Y).

E F(X)}

Denote by

.

and

cp(F)

F(X), G ( Y ) Y , r e s p e c t i v e l y , a n d a l i n e a r map

t h e subcone o f

G(Y)

given by

Lemma :

The f o l l o w i n g

are equivalent:

F? _ such _ -t h a t p ( f )

i )--For every l i n e a r

p

f E F(X)

p r o b a b i l i t y measure

U-

there i s

_ I -

2

: F(X)

a l g e b r a generated by cp(F)) ,(f)

ii) cp(F)

+

R

I

I Y

-+

T

4 supy(cp(f))

on Y

for all

( w i t h respect t o the

such t h a t

cp(f) d-r

for a l l

f E F(X)

i s a D i n i cone.

Proof:

i )=t. ii): Take an a r b i t r a r y s t a t e

w

of

cp(F)

t h e r e i s a s u i t a b l e r e p r e s e n t i n g measure

T

on

Hence,

v

and

T

i s a r e p r e s e n t i n g measure f o r

+ Y

R

.

Then, a c c o r d i n g t o i),

for

cp(F)

p =

+

R

v

o cp

.

must be a D i n i

242

Representing Measures

cone ( R e p r e s e n t a t i o n Theorem 2.2.1). ii)

3

i ) :L e t

6 : q(F)

+

assumption on state

on

v

q(F)

+

p r e s e n t i n g measure respect t o

6 5 supy

on

E F, cp(f) = g)

.

Because o f t h e

and t h e Sandwich Theorem g i v e s us a

v 2 6. Because o f ii)t h i s s t a t e has a r e -

with

R T

.

1-1

1f

6(g) = sup{p(f)

we have

1-1

be as i n i)and c o n s i d e r t h e s u p e r l i n e a r

F(X) - + R

1-1 :

g i v e n by

R

.

Y

And

T

has t h e r e q u i r e d p r o p e r t i e s w i t h

0

We g i v e two examples f o r s i t u a t i o n s where t h e lemma a p p l i e s .

Example 1: L e t now

F(X)

X which c o n t a i n s t h e

be a cone o f bounded f u n c t i o n s on

X

c o n s t a n t r e a l - v a l u e d f u n c t i o n s on

A linear

~ ( 1 =~ 0.) We r e c a l l t h a t

have mass zero i f -u ( f ) 5 supXlfl

.

for all

1-1

p : F(X) + R

i s said to

i s s a i d t o be normed i f

f E F(X).

Theorem: The f o l 1owing a r e equi Val e n t : i )

p : F(X) + R w_ i t h_mass - - - z e r o has a r e p r e s e n t i n g w i t h t o t a l v a r i a t i o n I T ~ I 1. -__

Every normed l i n e a r measure

on

T

X

For every normed l i n e a r

ii)

p o s i t i v e measure

v

0”

b r a generated by F(X))

p

: F(X)

X

x

X

with

-+

(rn,fn)

decreases -f o r a l l x,y inf nEN

-w_i t_h _mass _ _ _zero __

there i s a u-

(w i t h r e s p e c t -t o- _ t h_ e _p _ roduct such t h a t --

alge-

m( l X x ) 5

( f ( x ) -f(y))dm(x,y)

i i i ) -F o r e v e r y sequence

R

E R

x

for all --

f E F(X).

F(X) -such t h a t r n + f n ( X )

-

fn(Y)

E X we have --

sup (rn +fn(x) -fn(y)) x,y E X

= sup

inf (rn +fn(x) x,yEX nEN

-

fn(y)).

Proof: We c o n s i d e r

on Y

by

Y = X ^f(x,y)

x

X

= f(x)

and we c o n s t r u c t , f o r

-

f(y).

Let

G(Y) =

, a f u n c t i o n ?. F} , t h e n t h e map

f E F(X)

{? 1 f

E

243

RepresentingMeasures

f

-+

^f

i s l i n e a r from

F(X)

G(Y)

to

o n t o t h e same f u n c t i o n on

and two elements o f

F(X)

a r e mapped

i f and o n l y i f t h e y d i f f e r by a c o n s t a n t

Y

f u n c t i on. Hence, every

w i t h mass z e r o a n h i h i l a t e s t h e k e r n e l o f t h e map

IJ

and t h e r e must be a unique l i n e a r map

~ ( f= )u ( f ) c =

for all

(supX(f)

-

f E F(X).

If

G(Y)

v :

.+

such t h a t

R

+ i n f x ( f ) ) we g e t 1

Thus t h e Sandwich Theorem g i v e s us a s t a t e

. And

1

'IG(Y)

the equivalence ( i i )

lemma s i n c e ( i i i ) s t a t e s t h a t

(i)

=$

G(Y)

( i i ) : Take a norrned s t a t e

measure

a c c o r d i n g t o i)

T

and n e g a t i v e p a r t . hence,

-,^f

i s i n a d d i t i o n normed,then f o r

p

~ ( f =) u ( f ) = p ( f - c ) 5 ? I s u p X ( f ) - i n f X ( f ) } =

"'

f

. Then

'+(lX) = r - ( l X ) = A I

1

R

m = - ( ~ +

B

T

/2

G(Y)

+

supy(f) with

R

( i i i ) i s a consequence o f t h e

Q

has t o be a D i n i cone.

w i t h mass z e r o and a r e p r e s e n t i n g

decompose

must have mass z e r o

T

A

p

+

of

p

1

.

T

=

T+

-

T-

i n i t s positive

(F(X) contains t h e constants),

Then

( p r o d u c t measure)

-)

has t h e r e q u i r e d p r o p e r t y . (ii) + (i):

Take t h e marginalmeasures o f

p a r t s o f a measure on T ( A ) = m(A

x

X)

-

m(X

X x

m

as p o s i t i v e and n e g a t i v e

t o o b t a i n t h e r e p r e s e n t i n g measure A). 0

T

,

i.e.

Example 2: Take a t o p o l o g i c a l space on

X

.

We r e c a l l [ 2 4 l ] t h a t an o r d e r r e l a t i o n I

i s s a i d t o be continuous i f , whenever

neighbourhoods for all

X

U ( x ) , U(y)

of

x

and y

x 9y

, then there are

, respectively,such t h a t u 9 v

u E U ( x ) , v E U ( y ) . Such continuous o r d e r r e l a t i o n s a r i s e q u i t e

o f t e n . F o r example t a k e a p o i n t s e p a r a t i n g f a m i l y f u n c t i o n s on

X

and d e f i n e

F(X)

o f continuous

I t o be t h e s m a l l e s t o r d e r r e l a t i o n such

244

Representing Measures

t h a t the elements of f o r a l l f E F(X).

F(X) a r e monotone, i . e . x 5 y

f(x)

0

4

f(y)

In h i s pioneering work 12411 Nachbin has shown t h a t f o r completely regular spaces every continuous order is i n f a c t of t h i s kind.

.

NOW, l e t

X be compact with continuous order 5 Then by t h e fundamental Katetov - Nachbin Theorem ( a Tong - Katetov Theorem f o r normally ordered spaces, s e e [2411 o r [ 661) one can f i n d a monotone continuous function g with cp 2 g L Y whenever cp and Y a r e monotone and real valued w i t h cp lower semicontinuous and Y upper semicontinuous. This f a c t has the following consequence:

2 continuous real valued function 0" X , then t h e r e i s a mono-

Let f -

tone continuous function (*)

g

such t h a t --

1 I f + g I 5 7 supIf(x)

-

f ( y ) I x,y E

Proof: Define a monotone upper semicontinuous function semicontinuous function - ? by :

Put

A =

1

f ( x ) = sup{- f ( y ) I y

I XI

Ix

IZ I

Y(x)

= V

supX(f -

infi- f ( z )

7)

. Then -

V

f

t A L

-

^f - x

-

x

-f

Y

y

and a monotone lower

and by t h e Katetov

-

Nachbin Theorem there i s a monotone continuous function V Because - f 2 - f t -f we g e t f o r ( f t g ) : - x 5 ^ f t g s f t g

IY

IXI.

-

g i n between.

+ g s x .

1 Hence If t g l Ih supCf(x) - f ( y ) I x,y E X,y follows from t h e construction of ^f and y ) .

IX I

(the l a s t equality 0

As a consequence of t h i s i n e q u a l i t y and t h e lemma we obtain t h e following

i n t e r e s t i n g i n t e g r a l representation theorem ( a l s o due t o Nachbin [ 2 4 i l ) .

245

Representing Measures

Theorem : Let -

X

be a compact space w i t h continuous --o r d e r and l e t

l i n e a r functional on C(X)

such t h a t p ( g ) 2 0 --

LI

be-a_bounded __

f o r e v e r y monotone --

continuous f u n c t i o n . -Then t h e r e i s a p o s i t i v e -~ B o r e l measure v on

H = t ( x , y ) I y I XI with v(H) =

T IIp

II

and such t h a t ---

Proof: Observe t h a t

H

i s compact, map C ( X )

From t h e p o s i t i v i t y of

( * ) we o b t a i n

p(f) I

-,

C(H)

by

cp(f)(x,y)

=f(x) -f(y).

u on t h e monotone f u n c t i o n s and from t h e i n e q u a l i t y l l p II supH cp(f) f o r a l l f E C(X). S i n c e C(H)

1 7

i s o b v i o u s l y a D i n i cone (compactness o f

H ) t h e lemma g i v e s us a B a i r e

measure w i t h t h e d e s i r e d p r o p e r t i e s .

T h i s B a i r e measure can b e extended D t o a B o r e l measure by t h e usual procedure.

F o r a g e n e r a l i z a t i o n o f t h e p r e c e d i n g theorem t h e r e a d e r s h o u l d c o n s u l t [ 1661.

2.7

REMARKS AND COMMENTS

S e c t i o n 2.1:

The i n t e r r e l a t i o n between decomposition p r o p e r t i e s and

ineasure theory, d e s c r i b e d i n t h i s s e c t i o n , a r e t a k e n f r o m [1201 (see a l s o [1191,[1231,[1241).

An e s s e n t i a l t o o l i s P r o p o s i t i o n 2.1.4;

we l i k e t o

observe t h a t t h e u n d e r l y i n g s i t u a t i o n o f t e n occurs. F o r example, when F(X)

s a t i s f i e s a m i l d l a t t i c e c o n d i t i o n and

LI

i s maximal ( t h u s y i e l d i n g

b ) as a consequence o f t h e F i n i t e Sum Theorem). The p r o o f o f P r o p o s i t i o n 2.1.6

l o o k s r a t h e r complicated. I n f a c t , one can

g i v e a s i m p l e p r o o f b y use o f t h e D a n i e l l - S t o n e Theorem (as i t was done i n [1203). B u t we wanted t o g i v e a p r o o f o f t h e D a n i e l l - S t o n e Theorem as w e l l , so, we had t o proceed d i f f e r e n t l y . O f course, 2.1.7

i s somewhat weaker

t h a n t h e usual v e r s i o n o f t h e D a n i e l l - S t o n e Theorem s i n c e we assumed t h e functions i n

E(X)

t o be bounded. T h i s r e s t r i c t i o n w i l l be a b o l i s h e d i n

246

Represenring Measures

s e c t i o n 2.6.3 where the r e s u l t i s extended t o the case of unbounded functions. In f a c t , t h e theorem we a r e presenting t h e r e i s s l i g h t l y stronger than the usual Daniell-Stone Theorem, i n s o f a r as we obtain umonotone extensions without S t o n e ' s condition. The Daniell-Stone Theorem, in i t s o r i g i n s , goes back t o D a n i e l l ' s 1917 paper [ 831 ( s e e a l s o [3031).

I t seems t o be an i n t e r e s t i n g question, whether we can base a proof o f t h e Daniell-Stone Theorem e n t i r e l y on decomposition techniques, thus avoiding the use o f t h e Riesz Representation Theorem completely. In f a c t , t h i s i s possible; we sketch the procedure: Let E ( X ) be, with respect t o pointwise operations, a vector l a t t i c e and assume t h a t l X i s an order u n i t . Consider a decomposable s t a t e LI Define f o r A

c

.

X:

Call a sequence A n , n E N , o f pairwise d i s j o i n t s e t s a d i s j o i n t covering of A i f u An = A Define

.

nEN

m(A)

=

inf{

50

iii(An)

E

n=l

Then pick o u t subsets o f =

{A

c

X

I m(A)

Now, one can prove t h a t

1

X

by:

t

m(X\A)

5

is a

(A,)

=

d i s j o i n t covering of A 3

11

.

.

a- algebra

2

Z

E(X)

and t h a t m

is a

a d d i t i v e measure on which represents ii . ( I f t h e reader t r i e s t o prove these a s s e r t i o n s he should make extensive use of t h e FSP f o r (E(X),s)*).

a-

Represenring Measures

247

Section 2.2: Again, the material i s taken from [1201. A different proof of the Representation Theorem 2.2.1 was given by M. Neumann. His essential tools were Choquet's Theorem and Simons' Convergence Lemma. In [1201 one finds a condition for the existence of s t r i c t representing measures. This condition i s f a l s e and does n o t ensure the existence of s t r i c t represent i ng measures. Section 2.3: This material appeared in [1211. I n section 11.4 we are transferring the Representation Theorem t o a vector-Val ued s i tation, and i t seems interesting t o remark t h a t this procedure f a i l s for Theorem 2.3.4 ( a t least in the non-weakly-u-distributive case). Another open problem i s the uniqueness of signed representing measures. Whereas, in the case of positive representing measures, one can easily obtain results 1 i ke the Choquet-Meyer uniqueness theorem, no similar results for the signed case are knwon to the authors. Probably, one has t o work with a decomposition property being similar t o the one f u l f i l l e d by the preduals of abstracts L- spaces ( [2101,[2111). The Weighted Representation Theorem appeared, as a corollary, Section 2.4: in [120]. Frequently weight functions are considered in modern analysis. For example, see the extensive work a b o u t the weighted spaces C V o ( X ) and CV(X), where V i s a so-called Nachbin family ( [ 371,[ 771,[1441 [1961,[2421 [257 1 and 13061). Observe t h a t Lemma 2.5.3 i s a special case o f the abstract Section 2.5 Di sintegrat o n Theorem I 1.7.1. We included a separate and simple proof in order t o save the reader from going t h r o u g h the details of section I 1.7. The material i s taken from [1241. Theorem 2.5.7 can be considered as an integral representation theorem for those s t a t e s behaving locally simplicial. Of course, i n this case the representing measure i s unique. This section was mainly written t o demonstrate the wide Section 2.6: range of applications for the Representation Theorem. For this reason some of the proofs are rather brief. For example, the reader should observe that, although the proof of the uniqueness in Theorem 2 (section 2.6.3) i s correct, one needs some additional work t o verify the crucial assertion t h a t If E c , ( X ) I v l ( f ) = v2(f)l i s a vector l a t t i c e .

248

Representing Measures

The u- completeness i s no problem, b u t t o prove t h a t t h i s space i s a l a t t i c e one certainly needs transfinite induction (which, of course, proceeds in the obvious fashion). The intrinsic reason for that space t o be a l a t t i c e i s t h a t i ( X ) already i s a l a t t i c e . I n Lemma 1 (section 2.6.2) we work with the Stone-Czech compactification although we do n o t assume t h a t the space under consideration i s completely regular. Since some textbooks, unfortunately, define the Stone-Czech compactification only f o r completely regular spaces an additional explanation seems appropriate. Of course, by i3 X we mean the linear functionals on C B ( X ) which are given by the u l t r a f i l t e r s on X . Then everything which i s known for the Stone-Czech compactification goes through, except, t h a t X + B X need n o t be injective. (Another way o f defining B X i s via the structure space of C B ( X ) given by the KakutaniKrein-Stone-Yosida Theorem).

Section 2.6 deals w i t h several versions, and generalizations, of the fundamental Riesz Representation Theorem (which was originally proved by F. Riesz 1262 3 in 1909 f o r the case X = [0,11, see also [2651). This theorem has caught the attention of many outstanding mathematicians; they have genralized the result and reformulated the proof countless times. The case of compact subsets of Rn was settled by J. Radon [ 2591, for general metrizable compact sets the theorem i s due t o S . Saks I2821 and S. Banach (1937). For the general case of structure spaces of M- spaces Kakutani [1811 gave the proof. For non-Hausdorff compact s e t s , see A. Markov [2261 and A.D. Alexandrov t 1 1 (compare our Theorem 2.6.1). The C o ( X ) - case (see our Theorem 4 in section 2.6.3) i s due to several authors, independently. This case i s related t o the integral representation of the dual spaces with respect t o the s t r i c t topology (introduced by R.C. Buck [ 6 0 1, see also, for example, [1651 and t2901). For pseudocompact X the integral representation of monotone 1 inear functionals C ( X ) + R was f i r s t given by I . Glicksberg 11411. Hewitt's result [1631,[3171 ( o u r example 1 in section 2.6.4) i s usually stated in a more general form: Every bounded linear functional : C ( X ) -+ R can be represented by an integral, where bounded means t h a t for every 0 5 f E C ( X ) the s e t { p ( g ) I - f I g I f , g E C ( X ) I has t o be bounded. Of course, monotone functionals are bounded and a bounded functional can

248

Represenring Measures

be decomposed i n t o t h e d i f f e r e n c e o f two monotone ones. I f t h e r e a d e r i s i n t e r e s t e d i n e l e g a n t and s h o r t p r o o f s o f t h e Riesz R e p r e s e n t a t i o n Theorem he s h o u l d c o n s u l t [3161 o r [1381. G e n e r a l i z a t i o n s t o s e t - v a l u e d measures a r e t r e a t e d i n [2781. F o r more i n f o r m a t i o n about t h e s i t u a t i o n cansidered i n Example 2 ( s e c t i o n 2.6.5)

t h e r e a d e r s h o u l d have a l o o k i n t o Nachbin's b e a u t i f u l

book on t o p o l o g y and o r d e r [2411. Choquet's Theorem, which i n a c e r t a i n way i s a l s o a g e n e r a l i z a t i o n o f t h e Riesz R e p r e s e n t a t i o n Theorem,will

be t r e a t e d e x t e n s i v e l y i n t h e f o l l o w i n g

sections. There a r e many analoga o f t h e Riesz R e p r e s e n t a t i o n Theorem f o r t h e v e c t o r valued case. A p a r t f r o m t h e r e s u l t we g i v e i n s e c t i o n 11.4 t h i s a s p e c t

w i l l n o t be t r e a t e d i n t h i s book. For t h e s e m a t t e r s t h e i n t e r e s t e d r e a d e r s h o u l d c o n s u l t t h e e x c e l l e n t monograph o f D i e s t e l and Uhl [ 861.

Other R e p r e s e n t a t i o n Theorems:

L e t us b r i e f l y mention some o t h e r i n t e -

g r a l r e p r e s e n t a t i o n theorems f o r non-compact s e t s . A most i m p o r t a n t development i n t h i s f i e l d was i n i t i a t e d b y E d g a r ' s theorem. I n [ 9 3 1 he proved t h a t e v e r y p o i n t x i n a s e p a r a b l e RNP- s e t X ( s e e d e f i n i t i o n on page 198) i s t h e b a r y c e n t e r o f a p r o b a b i l i t y measure mx o n t h e u n i v e r s a l l y measurable s e t s i n X such t h a t t h e extreme p o i n t s o f X have

mx- measure 1. The p r o o f . amazingly s i m p l y , works v i a C h a t t e r j i ' s m a r t i n g a l e convergence theorem and von Neumann's s e l e c t i o n theorem. The use o f von Neumann's s e l e c t i o n theorem i s t h e reason t h a t t h e p r o o f cannot be t r a n s f e r r e d t o non-separable RNP- s e t s (where t h e theorem f a i l s ) . A f o r e r u n n e r o f t h i s theorem was g i v e n by Choquet [ 7 0 1. A non-separable analogue i s a l s o g i v e n b y Edgar . I n [ 9 4 ] he shows t h a t e v e r y element x

i n an

RNP- s e t

X

can be r e p r e s e n t e d by a t i g h t p r o b a b i l i t y measure

mx, which i s maximal w i t h r e s p e c t t o t h e d i l a t i o n o r d e r ( e q u i v a l e n t [951

t o b e i n g maximal, among t h e t i g h t measures, w i t h r e s p e c t t o t h e Choquet o r d e r ) . I n t h e s e p a r a b l e case these maximal t i g h t measures a r e supported be t h e extreme p o i n t s - which does n o t h o l d i n t h e non-separable case

-

i n f a c t t h e n t h e y o n l y vanish on t h e s e t s which a r e movable b y d i l a t i o n s .

250

Representing Measures

An equivalent non-separable version of Edgar's theorem i s g i v e n by P. Mankiewicz [225]. The c h a r a c t e r i z a t i o n of RNP- s e t s which a r e simplexes by means o f uniqueness of maximal measures is the work of R.D. Bourgin and G.A. Edgar [571. These r e s u l t s have been extendedfurther by the impressive work of E . G . F . Thomas ([307]to [311] on conuclear cones. Let us mention a special case of h i s r e s u l t s : Let S be a complete bounded closed convex Souslin subset of a l o c a l l y convex Hausdorff space. (Recall t h a t a s e t i s Souslin i f i t i s t h e continuous image of a separable completely metrizable space). Then, i f S has the Radon-Nikodym property, every point in S i s t h e barycenter of a Radon probability measure concetrated on t h e extreme points ( i . e . S has t h e i n t e g r a l representation property). This author shows furthermore t h a t the i n t e g r a l representation property characterizes t h e Radon-Nikodymproperty i n the following way: S has t h e RNP i f and only i f a l l closed convex subsets of SN do have the integral representation property. Actually a1 1 these r e s u l t s carry o v e r [ g i i l t o the case when t h e cone generated by S i s replaced by a socalled conuclear cone r with RNP which has t h e property t h a t closed convex h u l l s of compact subsets of r a r e again compact: (Another, b u t very r e l a t e d , 1 ine of non-compact Choquet theorems f o r bounded, weakely-closed, convex subsets of t i g h t measures was given by von Weizsacker[3251, [326] and von Weizsacker-Wink1 e r [327] ( s e e a l s o [331] ) . They show t h a t , roughly speaking, these s e t s do have t h e i n t e g r a l repres e n t a t i o n property.

SECTION 11.3 BOUNDARIES

This section i s dedicated t o the investigation of the so-called geometric situation. We show t h a t , in case of a compact convex s e t , the restrictions o f the affine continuous functions, as well as the restrictions of the upper semicontinuous convex functions, t o the extreme points are constituting a Dini cone. (Actually, we present this fact slightly more general since we replace the compact convex s e t by an arbitrary s t a t e space of an order unit cone, b u t t h i s generalization i s not essential). For this case the Representation Theorem then yields Choquet's celebrated theorem in the generalization of Bishop - de Leeuw. We s t a r t by an investigation of boundaries. We show t h a t there i s a minimal f i x p o i n t boundary for the so-called exposed maps. This immediately leads t o Bauer's Maximum Principle (and, of course, t o the Krein-Milman Theorem). I n section 3.2 we characterize the minimal fixpoint boundary in terms of separation properties; i t turns o u t t h a t t h i s boundary can actually be smaller than the Choquet boundary. This leads us t o the definition of five different boundary notions, and we investigate the relations between these boundaries. Since we did n o t need the notion of maximal measure for the proof of Choquet's Theorem (section 3.2) we append some information a b o u t the Choquet order in section 3.4. I n section 3.5 we t r e a t the Choquet-Meyer uniqueness theorem as well as the characterization of Bauer simplices ( t h i s i s done as an application of the material gathered in chapter 1 . 2 ) . Finally, in 3.6 we investigate in what kind of situation a Maximum Principle forces a cone t o be a Dini cone. Relevant results are obtained via Simons' Convergence Lemma. As a byproduct we get information a b o u t situations when the Choquet boundary i s the minimal max-boundary. This aspect of minimal boundaries will be further treated in chapter 11.5. 251

252

3.1

RepresentingMeasures

FIXPOINT BOUNDARIES, BAUER'S MAXIMUM PRINCIPLE AND THE KREINMILMAN THEOREM

I n t h e f o l l o w i n g we c o n s i d e r a f a m i l y o f upperbounded f u n c t i o n s on some nonempty s e t

X

and we s t u d y subsets

of

Y

X

i n which a l l f u n c t i o n s

X- suprema. I n p a r t i c u l a r we a r e

under c o n s i d e r a t i o n a l r e a d y a t t a i n t h e i r

i n t e r e s t e d i n " s m a l l " Y. The reason f o r t h i s i s , t h a t i n most cases t h e r e 1evant p r o p e r t i e s o f t h e f u n c t i o n s under c o n s i d e r a t i o n a r e c o m p l e t e l y

determined by t h e i r r e s t r i c t i o n s on Y , hence t h e r e i s some economical advantage i n making Y as small as p o s s i b l e . T h i s i d e a i s used w i t h g r e a t success i n d e a l i n g w i t h d i f f e r e n t i a l equations a d m i t t i n g a maximum p r i n c i p l e (as we have seen i n c h a p t e r 1.1.6). We s t a r t w i t h t h e d e f i n i t i o n o f t h e c r u c i a l n o t i o n Def in i t i o n :

3.1.1 Let

X

be a nonempty s e t and l e t F(X)

functions

f :X

f o r every

f E F

I n general even

+

6 .

A subset

, there X

Y

c

X

be a f a m i l y o f upperbounded i s c a l l e d a boundary f o r

i s a corresponding

i s n o t a boundary f o r

f e a t u r e s o f Dini cones i s t h a t i n t h i s case As we have seen i n 1.1.6.3 u n i t c i r c l e i n R2

with

y E Y

F(X) if,

f(y) = supX(f)

.

F(X). One o f t h e i m p o r t a n t happens t o be a boundary.

X

the, l e t us say, t o p o l o g i c a l boundary o f t h e

i s a boundary f o r t h e c o r r e s p o n d i n g harmonic f u n c t i o n s .

Another example f o r a boundary i s

K

i n case o f

C(K)

.

K ,

f o r compact

a r e s u l t which a l s o h o l d s t r u e f o r pseudocompact K T h i s l i s t c o u l d be c o n t i n u e d a l m o s t i n d e f i n i t e l y . We s h a l l deal w i t h some o t h e r i m p o r t a n t examples i n t h e n e x t s e c t i o n s . I n t h i s s e c t i o n we a r e g o i n g t o i n v e s t i a a t e a p a r t i c u l a r boundary b y means o f exposed selfmaps

3.1.2

i) L e t called

y : X

-t

X

.

Definition:

r r-

be a f a m i l y o f f u n c t i o n s i n v a r i a n t i f y(Y) c Y

y : X

for all

+

X

.A

Y c X.

i s a s i n g l e t o n we speak o f y- i n v a r i a n t i n s t e a d o f

subset

Y c X

I n case t h a t

is

r = Iyl

{yl- i n v a r i a n t .

253

Boundaries

ii) L e t

f : X

nuous f u n c t i o n s y : X -, X

= Ix

-,

r? .Then we c a l l a f a m i l y

F(X)- exposed

v a r i a n t subset

af K

F ( X ) c o n s i s t o f upper s e m i c o n t i -

be a compact space and l e t

X

E K

K c X

I f(x)

t h e s e t where

f

r

o f functions

i f f o r e v e r y nonempty c l o s e d compact

and e v e r y

= supK(f)3

f E F(X)

,is

r-

again

r-in-

the set i n v a r i a n t . Note t h a t

af K ,

K- supremum, i s nonempty s i n c e e v e r y upper

attains i t s

semicontinuous f u n c t i o n a t t a i n s i t s maximum on a compact s e t . L e t us g i v e an i m p o r t a n t example f o r t h i s s i t u a t i o n . 3.1.3

Example (Geometric S i t u a t i o n ) :

K o f a l o c a l l y convex Haus-

Consider a nonempty compact convex s u b s e t d o r f f v e c t o r space

x x+

(1-x)y E K

-,6

E

.

R e c a l l t h a t ,K

whenever

i s s a i d t o be convex i f

x

x,y E K, 0 5

5 1

,

and t h a t a f u n c t i o n

f ( x x + ( 1 - x ) y ) 5 x f ( x ) + (1-x)f(y) Conv(K) we denote t h e cone o f upper semicontinuous convex f u n c t i o n s f : K -+ r? . NOW, f o r z E E , d e f i n e f : K

whenever

i s c a l l e d convex i f

x,y E K, 0 5

5 1

We c l a i m t h a t

r

x+z, x - z E K

x+z

if

x

otherwise

= IyzIz

E E3

r- i n v a r i a n t compact k

We have t o show y z ( x ) E yz(x) = x

. By

by

yz : K j K

a

x

is

c K

afk

Conv(K)

-

and a r b i t r a r y

. If

x +z

or

exposed. To see t h i s we t a k e f E Conv(K), z E E, x E afK. x-z

and t h i s i s by assumption an element o f

i s not i n

af

k

K

then

. I n case

that I

x+z since that

as w e l l as x - z

a r e elements of

K

we know t h a t

yz(x) = x + z E K

and k i s r- i n v a r i a n t . Now t h e c o n v e x i t y o f f 1 1 f ( x ) I f(x+z) + f ( x - z ) , hence x + z E a f K because x E

k

f ( x ) = supl((f).

I n any case y z ( x ) E a f K

.

implies

Representing Measures

254

P r o p o s i t i o n [129]:

3.1.4

Consider a f a m i l y

F(K)

on a c l o s e d compact __--Then t h e s e t --of -

r

o f p o i n t s e p a r a t i n g upper semicontinuous f u n c t i o n s

set

, -and l e t r be an x ~for all y E r)

K

{x E K I y ( x )

F(K)

-

exposedfamily.

I=

.

i s-a boundary f o r F(X) -

Proof: Fix

fo E F(K)

. Since

r

is

F(K)- exposed

a

f0

K

must be

r-

invariant.

Furthermore t h i s s e t i s closed, hence compact. Thus

K =

{i c

a K 10

i s c l o s e d and compact and r - i n v a r i a n t )

f0

must be nonempty. By Z o r n ' s lemma t h e r e i s a minimal element Again, s i n c e

KO i s

r-

Hence, by m i n i m a l i h y o f I n o t h e r words,all KO = I x o l

Obviously, a

r-

maximum (because

K

.

.

af K O = KO f o r a l l f E F ( K ) .

KO we g e t

a r e c o n s t a n t on

i s a singleton since

a common f i x p o i n t

f E F(K), af K O € K

invariant,we have, f o r

f E F(K)

KO i n

F(K)

.

KO

This y i e l d s t h a t

was assumed t o be p o i n t s e p a r a t i n g .

i n v a r i a n t s i n g l e t o n must be a f i x p o i n t . So we have found xo

, where

Ixol c a

f0

t h e a r b i t r a r i l y chosen

K).

fo a t t a i n s i t s

0

Using t h i s r e s u l t i t i s now obvious how t o make boundaries s m a l l e r , j u s t by making

r

larger.

To t h i s end we n o t e t h a t t h e u n i o n o f an a r b i t r a r y

c o l l e c t i o n o f F(K) - exposed f a m i l i e s i s a g a i n F(K)- exposed. Hence t h e r e i s a unique maximal F(K)- exposed f a m i l y , namely rmax = UIr Replacing

r

Ir

is

i n P r o p o s i t i o n 3.1.4

F(K)- exposed) by

.

rmaxwe g e t a minimal f i x p o i n t

boundary. The f o l l o w i n g theorem g i v e s a complete c h a r a c t e r i z a t i o n of t h i s minimal f i x p o i n t boundary i n terms o f p r o p e r t i e s w i t h r e s p e c t t o

F(K).

255

Boundaries

3.1.5

Theorem [1291:

L e t F(K) be a f a m i l y o f p o i n t - s e p a r a t i n g upper semicontinuous f u n c t i o n s on a _ compact c l o s_ e d s_ et K . 5 max(K) -be t h e c o l l e c t i o n -o f those _ ~ _ such t h a t f o r each compact,closed w i t h {XI 5 R c K x E K ----______ t h e r e i s some

f E F(K)

with -

f ( x ) = sup-(f) > i n f - ( f ) . K K

(*)

Then max(K) -----i s t h e s e t o f a l l common f i x p o i n t s o f rmax. I n particular f o r F(K). max(K) i s-a boundary Proof: x E max(K) and p u t

Let

KO = Assume since is

n {k

c K

compact closed, rmax - invariant, x E

{ x ) t: KO. Then t h e r e i s x E max(K).

rmax-i n v a r i a n t s i n c e

2

mality o f rmax

KO

.

Hence

f E F(K)

with

rmaxi s

{XI = KO and x

KO

k~ .

( f ) > infK (f) 0

t h e o t h e r hand,

F ( X ) - exposed and

we have

KO

f ( x ) = sup

. On

x E af KO 8 KO

We i n f e r

v a r i a n t . S i n c e . B KO ~

Y

Ik

KO

is

af KO rmaxin-

a c o n t r a d i c t i o n t o t h e minimust be a f i x p o i n t f o r a l l

*

Now c o n s i d e r x

E K

with y(x) = x

for all

Then t h e r e i s a compact, c l o s e d subset such t h a t f o r each be c o n s t a n t on

KO

f E F(K)

.

KO c K

we have e i t h e r

F o r each

y E KO

y

define

E rmax.Assume x B max(K). containing

x

f ( x ) < supK ( f ) 0

yy

: K

--t

K

.

.

with

{XI t: KO,

or f must

by y y ( x ) = y

Then r i s L e t r = {yyl y E KO} and y ( z ) = z i f z E K \ { x I Y F(K)- exposed. Indeed, l e t k c K be compact, c l o s e d and r- i n v a r i a n t . f E F(K).

Let Hence all

KO

c

y E KO

k

Assume, x E af

k.

This implies,

.

Hence

-

af K

f

is

. Then

y = y ( x ) E k f o r a l l y E KO. Y i s c o n s t a n t on KO , i . e . y E af k f o r

c ?i

r-

invariant. If

-,

x B af K then, t r i v i a l l y

256

Representing Measures

( a I?) = af I? f o r a l l y E KO . We have r c rmax s i n c e rmax i s t h e Y f maximal F(K)- exposed f a m i l y . Because y ( x ) = y x f o r y E KO\ {XI , Y x cannot be f i x p o i n t f o r a l l y E rmax, a c o n t r a d i c t i o n .

y

*

We conclude

x E max(K).

0

Two o t h e r consequences o f 3.1.4 a r e t h e c e l e b r a t e d Krein-Milman Theorem and Bauer's Maximum P r i n c i p l e . L e t us come back t o t h e Geometric S i t u a t i o n o f Example 3.1.3.

Again

H a u s d o r f f v e c t o r space x = y = z

whenever

K

E

.

i s a compact convex s e t i n a l o c a l l y convex Recall t h a t

x = x y t (1-x)z

x E K

with

we denote t h e s e t o f a l l extreme p o i n t s o f rates the

PO

nts o f

K

since f o r

a continuous l i n e a r f u n c t i o n a l obviously

i s c a l l e d extreme p o i n t i f

y,z E K, 0 < K

.

Note t h a t

x,y E K w i t h

x

+

u on E w i t h p ( x )

x

< 1

.

ex(K)

By

Conv(K)

sepa-

t h e r e i s always

y

~ ( y ), and

E Conv(K) (compare Remark 2.5.8).

p

H. Bauer's Maximum P r i n c i p l e :

3.1.6

f o r Conv(K) , i . e . -e v e r y upper semicontinuous ex(K) -i s_ a boundary a t_ t a_i n_s _i t s K- supremum on ex(K) convex f u n c t i o n on K _

.

Proof: Take t h e s e t

r

is

r

= Iy,lz

E

El. which we considered i n 3.1.3. We know t h a t

Conv(K)- exposed, hence i t s common f i x p o i n t s a r e a boundary (3.1.4).

But, o b v i o u s l y ,

ex(K)

.

r

i s e x a c t l y t h e s e t o f a l l common f i x p o i n t s o f 0

Let

n

z

i=1 n

X

be a subset o f a v e c t o r space

Xi xi

with

n E N

, xl,

... ,xn

E X

E

, then and

a v e c t o r o f t h e form

x1

,..., xn

2 0

such t h a t

xi = 1 i s c a l l e d , as u s u a l l y , a convex combination o f X . i=1 By co(X) we denote t h e s e t o f convex combinations and , i n case t h a t

E

257

Boundaries

-

i s a topological vector space , co(X) stands for the closure o f co(X). O f course, co(X) and are the convex hull o f X and the closed convex hull respectively.

z(X)

3.1.7

Krein-Milman Theorem:

convex -be a -compact ~ convex ~ subset _ of_a locally _ - Hausdorff ~ -vector K = =(ex K ) , i . e . K i s the closed convex hull o f its space. Then -------extreme points. ~Let K

Proof:

.-.

.

K = =(ex K ) 3 ex(K) i s certainly a compact convex subset o f K The s e t k must be nonempty since every f E Conv(K) attains i t s K- supremum on i?, according to Bauer's Maximum Principle. Assume there i s some xo E K \ K . Since K x k i s open in K and E i s locally convex, there

i s an open convex A c K x k containing xo

. Then Hahn-Banach

Separation

Theorem (1.5.8) applied t o A and B = k yields a continuous linear functional u such t h a t u ( A ) n p ( B ) = 0 . Without loss o f generality assume p ( a ) > u ( b ) for a l l a E A, b E B (otherwise take - p ) Hence supK(u) 2 supA(p) > supB(p) since p a t t a i n s i t s supremum on

.

the compact s e t B v l K E Conv(K)

.

. This

IB

contradicts 3.1.6 since ex 0

K c k and

Represenring Measures

258

3.2

MORE BOUNDARIES

In the l a s t section we have mainly considered the minimal fixpoint boundary and we have characterized t h i s boundary in terms of a separation property with respect t o compact subsets. This separation property ( i n the definition of max(K)) can be modified in several ways thus leading t o different subsets of a compact s e t . In the important cases these modified sets are again boundaries and, under additional assumptions, there are many relations between these boundaries. Another way of picking out special subsets as boundaries can be given in terms of characterizations o f the support o f representing measures. This concept i s mainly due to Heinz Bauer and has become a very useful tool in several parts of analysis

.

We s t a r t by giving a l i s t of the different subsets under consideration. Let K denote a compact Hausdorff space and l e t F(K) consist of upper semicontinuous 6- valued functions on K We assume (for convenience) that the elements of F ( K ) do separate the points of K , b u t i t i s n o t assumed, a t least n o t for the moment, t h a t F(K) i s a cone of functions. The u- algebra under consideration will be the family of Borel sets and by a representing measure for xo E X we mean a -,regular Borel probability measure T such t h a t

.

f(xo) 5 By

S U ~ ~ ( Tw )e

I

f

X

dT

for a l l

f E F(X)

.

denote the support of the measure

T

.

For completeness we also include the definition o f max(K)

3.2.1

i)

.

Definition max(K) i s the s e t of those x E K such that for each closed k c K with {XIc k there i s an f E F(K) with

*

f ( x ) = sup-(f) > i n f - ( f ) K K

.

strong maximum max(K) i s called the s e t of ~ _ _ _ _points. -

259

Boundaries

ii)

Max(K) (called the s e t of ~maximum points) i s the s e t of those x E K such that for each nonempty compact I? c K with x 13 k there i s some f E F(K) with

w

f ( x ) > sup-(f). K

i i i ) Max(K) i s the s e t of those x E K such that for every compact, nonempty k c K with x B k there i s some f E F(K) with f ( x ) 2 sup-(f) and K iv)

Let

0 > B > a

. Then

Max

a,B

f(x) > inf-(f) K (K)

.

i s the s e t of those x E K

such t h a t for every compact, nonempty k c K with there i s some f E F(K) with f I 0 and

x B k ,

f ( x ) 2 B > a 2 sup-(f) K

v)

C h ( K ) (called the Choquet boundary) i s the s e t of those x E K such t h a t the Dirac measure 5, a t x i s the only representing measure for x.

vi )

N

C h ( K ) (called the -weak Choquet boundary) i s the s e t of those x E K such that for every representing measure T of x we have x E S U ~ ~ ( T )

.

3.2.2

(i)

ProPosi tion:

MaxaYB(K) c Max(K)

c

N

N

Max(K) and C h ( K ) c C h ( K )

( i i ) max(K) c C h ( K ) n $x(K) ( i i i ) Ch(K)

U %(K)

N

t

Ch(K)

( i v ) Max(K) contained in every closed boundary in the closure of max(K). ----

of

K

, inparticular

Represen ring Measures

260

Proof: ( i ) i s an immediate consequence o f t h e d e f i n i t i o n s . ( i i ) Consider x E max(K). Let

k

be as i n D e f i n i t i o n 3.2.1 iii)and p u t

t h e i n e q u a l i t y o f D e f i n i t i o n 3.2.1 i ) AJ

to

k

^K

=

ku

{XI Then a p p l y

t o obtain the inequality

o f 3.2.1 iii).Hence, x E Max(K). Now, l e t

be a r e p r e s e n t i n g measure f o r

T

We have t o prove

s u p p ( ~ )= Cxl

x.

which c l e a r l y i m p l i e s

T =

6

X

. Assume

{XI and c o n s i d e r k = S U ~ ~ ( Tu ) Cx3 (which cannot be a s i n g l e f s a t i s f y i n g t h e i n e q u a l i t y o f 3.2.1 i). Put ^K I y E s u p p ( ~ )I f ( y ) = f ( x ) l , t h e n t h i s i n e q u a l i t y i m p l i e s t h e contradiction: SUPP('I)

t o n ) . Then t a k e an

Hence s u p p ( ~ ) = {XI and (iii)

I 4

Ch(K) c Ch(K)

Now, l e t

N

x E Max(K)

x E Ch(K).

i s a g a i n an immediate consequence o f D e f i n i t i o n 3.2.1. and l e t

'I

be a r e p r e s e n t i n g measure f o r

We have t o prove x E s u p p ( ~ ) . So, assume

-

K = supp(~)

.

By 3.2.1 iii)t h e r e i s some

f(x

t sup-(f) K

and

( i v ) Take

x E Max(K)

t h e n 3.2.1 i i ) contradiction to

and l e t

g i v e s some

k

f E F(K)

f(x) >

with

.

J f dr (a c o n t r a d c t i o n ) . R

be a c l o s e d boundary o f

f E F(K)

K.

If x C K

f ( x ) z sup-(f). This i s i n K b e i n g a boundary. The second p a r t o f t h e a s s e r t i o n

f o l l o w s t h e n from Theorem 3.1.5.

with

.

x C s u p p ( ~ ) and p u t

f(x) > inf-(f) K

These two i n e q u a l t i e s c l e a r l y i m p l y

x

0

26 1

Boundaries

3.2.3

Theorem t1311:

L e t F(K) + F(K)

Then

c F(K),

i.e.

f

+ g E F(K) whenever f,g E F(K).

CWK) c Max(K).

Proof:

x d Max(K). Then b y d e f i n i t i o n t h e r e is a compact c X which does x such t h a t f ( x ) I s u p - ( f ) f o r a l l f E F(K). By t h e K Riesz-Konig Theorem I 1 . 6 . 1 t h e r e i s a p r o b a b i l i t y measure T on such Let

not contain that

f(x) I

K

f dr

for all

f E F ( K ) . We can assume t h a t

because o t h e r w i s e we can r e p l a c e

T

x

3.2.4

and

N

x d Ch(K)

because

x

i s regular,

b y a r e g u l a r measure h a v i n g t h e same

i n t e g r a l s f o r continuous f u n c t i o n s . Hence, for

T

B k

3

T

i s a r e p r e s e n t i n g measure

supp(~).

0

Corollary:

L e t F(K)

t F(K) c F ( K ) .

i) There i s

fi s m a l l e s t

c l o s e d boundary of K

boundarv.

ii) --A l l the sets

d

, --called the Shilov N

max(K), Max(K), Max(K), Ch(K), Ch(K)

t h e i r closures a r-e-a_ l l_equal to_ t h e S h i l o v boundary ___ IL

a r e boundaries,

.

r\r

i i i ) max(K) c Ch(K) c Max(K) = Max(K) = Ch(K).

Proof: ( i i i ) f o l l o w s from t h e i n c l u s i o n s g i v e n i n 3.2.2 and 3.2.3. Since max(K) i s a boundary (Theorem 3.1.5) a l l t h e o t h e r s e t s must t h e n be boundaries, and from 3.2.2 ( i v ) we g e t t h a t t h e c l o s u r e o f Max(K) i s t h e s m a l l e s t c l o s e d boundary. E v e r y t h i n g e l s e f o l l o w s t h e n f r o m t h e f a c t t h a t t h e s e t s under c o n s i d e r a t i o n a r e boundaries and subsets o f

Max(K).

262

Representing Measures

Before we a r e going t o c h a r a c t e r i z e t h e Choquet boundary in terms of separation properties we remark t h a t conditions on t h e representing measures in t h e d e f i n i t i o n of the Choquet boundary can be formally weakened. Remark:

3.2.5

Let M,

denote the s e t of p r o b a b i l i t y measures on

Then

Ch(K) = i x E K

I TIXI

for all

> 0

K representing T

x.

E Mx3

Proof: I t i s t r i v i a l t h a t the Choquet boundary is contained i n t h e s e t on t h e r i g h t s i d e . NOW, assume t h a t f o r every T E Mx we have TIXI > 0 . We claim t h a t then

-

-( T - T

T =

l-T{XI

~ 1 x 1= 1 f o r a l l {XI

E M,

M,

i s an element of

)

Hence a c o n t r a d i c t on.

3.2.6

T

.

Indeed, i f

with

~ 1 x 1< 1 then

;{XI = 0.

0

Theorem:

be-a-cone containing a l l constant functions -and l e t a < p Let F(K) Then C h ( K ) = Max ( K ) .

< 0

.

a,!3

Proof: Let x E Max

(K)

x d

k

.

If

T

E

M,

T(K)

with

of

K with

f s 0 and f ( x ) 2 p > a t s u p - ( f ) .

K

then B

Hence

k

and consider a nonempty compact subset

a,D Let f E F(K)

5

< f(x)

aB

< 1

5

K

f dr

r(k) a

f o r every compact

. k

c K

with

x B

.

By regula-

Boundaries

r i t y of

we get then

T

Now, assume x Assume

E

Ch(K)

263

~ 1 x 12 1 - B/a > 0

.

for a l l

f E F ( K ) . The Sum Theorem yields linear K

and f(X)

p2 I supK

aB

5

pl,v2

: F(K)

B a) v2(f)

u l ( f ) + (1 -

respectively such that

K

Then B/a

T~

t

then

>

(1 - B/a)

sup-(g) K

T~

-,

with

pi(f)

I

f

i=1,2,

dTi,

i s a n element in Mx

T~

, T ~

on i;,

for a l l

f E F(K).

which i s n o t equal t o g

E

F(K)

with

x

E

Max

(K)

B ' ( l - a) s u P K ( g ) * Put

f E F(K) , f

and therefore

x B R.

.

for a l l f E F(K)

the Dirac measure 6, , a contradiction. Hence there i s g(')

Ch(K).

such t h a t

By the Riesz-Konig Theorem there are probability measures and

E

;supi;,(f)+ (1 - B/,)supK(f).

f(x)

p1 I sup-

x

and l e t k c K be non empty compact with

(*)

I

yields

And 3.2.5

Max

I0

a,B

, f(x)

2

B > a 1 supl((f)

( K ) 3 Ch(K)

.

. Hence

a,P

We conclude t h i s section with a discussion of three examples. If F(K) i s N N a cone we have proved (3.2.4) max(K) c C h ( K ) c C h ( K ) = Max(K) = Max(K) c closure of (max(K)). We show that i n general a l l preceding inclusions are proper.

264

Representing Measures

Exampl es :

I 1I n

i ) Let K

Let F(K)

If : K + R

=

.

Clearly, 1 = lim f(Ti) n +-

E N) U {O}

I f(0)

=

i s a vectorspace of continuous functions. We have 0 B Ch(K) s i n c e , "X 1- 6 , 6, Dirac measure a t Ti 1 , we obtain 'I = n = l 2"

F(K) with

f ( 0 ) = "X 1-n f(,)1 = f dr n=l 2 K

.

measure on K with

f dp

for all

Xi 2 0

with

I

f(0) =

K

On t h e other hand, l e t

m

P =

.X X i di , f o r s u i t a b l e

1 =o

Dirac measure a t 0 )

. Fix

m ,N E

by

1

fm,N (E)

and f

m,N

Ch(K)

6,

*

1

2m-N)

x xi

i =o

=

1

be a p r o b a b i l i t y

W e have

.

, and define f

c N

, n

=

is t h e

( 6,

m,N

E

F(K)

m

0

for m * n s N

1

for n > N

n+ m

Hence, i f = A.

=

(zm -

I

,m

N

.

f E F(K) m

p

( 0 ) = lim f m , N);( 1 = 1. W e obtain

1=

P

K i s compact. "1 1-n f ( 1 x)} . n=l 2

N

I fm y N dp + m ,

=

x0

t

m

(2m-2m-N)~mt 1 i=Nt1

for all

Am = 2 - m ( l - ~ 0 )

.

(1 - Xo)r and 0 E supp(p) ry rv C h ( K ) = Max(K) = Max(K) t

m E N

Thus 0

xi

.

.

We conclude, rJ

E Ch(K)

and

.

i i ) Let K be as i n i ) . Define now F(K) = I f : K

T r i v i a l l y , we have t h a t

+

R

1 f(o)

I 1I n

=

lim f ( n1) = 1 ( f ( 1 ) w-

E N)

t

1 f(-2))3

,

is the smallest boundary of

K with

265

Boundaries

respect t o But

F(K). Hence

f4

0 d Ch(X)

since

Let

0 C supp(

I an E R

iii) L e t co = {(a,) co

must be t h e c l o s u r e o f

K

1 + '2 b2)

b1

,n

.

, with l i m

E N

1 1 . II w i t h

be t h e sequence whose

an = 01

n+

i s a Banach space i f we c o n s i d e r en E co

max(K).

n'th

.

II ( a n ) l l = sup l a n l nEN

1

component i s equal t o

II enII

and a l l o t h e r components a r e zero. C l e a r l y ,

1 for all n

n

= -

.

. Note

m

t h a t we may w r i t e a l l elements o f

co

as

(a,)

=

z n an en where t h e

n=l

s e r i e s converges i n t h e norm t o p o l o g y . D e f i n e t h e f u n c t i o n m

Pm : co

+

P (

by

R

z n an e n )

= am

n=l

. Clearly,

i s continuous w i t h

Pm

r e s p e c t t o t h e rIOrTntoplOgy . L e t us c o n s i d e r t h e f u n c t i o n defined by

f(n) = n

[rl

=

- [ 2I-

sup{n E N U I03 [ n 5 r3

For every with

mo, N E

f ( m ) = mo

N

Let

1

n E N3

F(K)

functions on

, there

r 2 0

. n > 1

for all

m E N ,m

is

,n

E N

, and:

2 N

, and

- n-" ( x ~ ( +~ en) )

IPN(xn)I I 1 f o r a l l

.

for

f(n) < n

x1 = el

xn = n-'(nn+t)el

K = ixn

N

.

L e t us d e f i n e by i n d u c t i o n

We have c l e a r l y

+

, where

By elementary a n a l y s i s we have

(*I

f :N

We have

xn

+

x1

for all

N > 1 and n E

and consequently,

n = 2,3, N

.

K

...

Consider i s compact.

c o n s i s t o f the r e s t r i c t i o n s o f a l l realvalued l i n e a r continuous co. C l e a r l y

f

a p r o b a b i l i t y measure on

K

P

nl K f u n c t i o n s separate t h e p o i n t s o f

E F(K)

with

K

. We f(xl)

n E N , and t h e s e

for all claim, =

K

x1 E Ch(K) : L e t

f dT

for all

T

f E F(K).

be

266

Representing Measures m

Hence

T =

1

i=1 measures a t xi

di,

Xi

1

. Assume

be an index such t h a t

0 = f(Xl)

,

A. 2 0

m

Z

i=1

AN = sup Xi

Ai

N-(Ntl)~N

-

i=N

a c o n t r a d i c t i o n . Thus

there i s

g E F(K) N

such t h a t

m

t e o E U

2 N

such t h a t

f(Xi)

where

f = -P

=

1 -N N AN t

di

NIK

are the Dirac

xi

* 0 . Let

N > 1

E F(K). Hence

W

k=Nt1

m

1 lk k-k 2 N-(Ntl)~N k=Nt1

x1 E Ch(K).

with

i s a neighbourhood of mO

,

.Define

i>1

f dT = Z

2

x

1

=

X

mo E

1

i > 1 such t h a t

there i s

m

is

A. =

g(xl)

x

for all

mO

.

) en

sup Xi > 0 irN+1

Hence

-t

I n o t h e r words, t h e r e

U = {y E colg(xl)

> g(y))

0 we have

m t N , N s u f f i c i e n t l y large. (*) yields

f ( m ) = mo

and t h e r e f o r e we a r r i v e a t a c o n t r a d i c t i o n :

(Here we used t h e same n o t a t i o n f o r

,

x1 B max(K). Indeed, o t h e r w i s e

But

. Since

mO

- (Ntl)-N.N-l

= supK(g) > i n f K ( g ) .

> g(x

g(xl)

A k k-k PN(Xf(k))

g E F(K)

and i t s e x t e n s i o n t o

*

So, we have indeed max(K) c Ch(K).

The same argument as b e f o r e shows t h a t i n f a c t

Ch(K) = K

.

co).

267

Boundaries

3.3 CHOQUET'S THEOREM I n t h i s section we deal with representation of points of compact convex sets by means of integrals over the extreme points. We do t h a t a t several levels of abstraction, although a l l these results are - more or less consequences of Choquet ' s fundamental representation theorem in case of the geometric situation (described in 3.1).

I n f a c t a l l theorems of this section can be obtained as corollaries of Theorem 3.3.6. B u t in order t o i l l u s t r a t e different viewpoint we give for a1 1 theorems different approaches.

We recall the geometric situation: K i s a nonempty compact convex subset of a locally convex Hausdorff vector space E . Again Conv(K) denotes the i- valued upper semicontinuous convex functions on K and A ( K ) stands for the subcone of affine continuous functions. A function cp i s said t o be affine i f

x,y E K , 0 5 X 5 1

for all of K

.

.

And by

ex(K) we denote the extreme points

The importance of Bauer's Maximum Principle (or the Krein Milman Theorem) comes from the f a c t that i t allows t o identify A ( K ) and A ( K ) lex(K) (restrictions t o ex(K)): 3.3.1

Lemma:

For every --

h E A ( K ) lex(K) there i s a unique

The map h

+

-_.

6 ----i s linear and we have supK 6

6 E A ( K ) with =

~up,,(~)h

.

lex(K)

= h.

Proof: Assume there are h l , h 2 E A ( K )

such t h a t their restrictions t o ex(K)

are equal. By Bauer's Maximum Principle we get supK(hl - h 2 ) = ~ ~ p ~ ~- h(2 ) ~= )0 .( The h ~same holds for supK(h2- h l ) . Hence h l

= h2

. So,

the extension t o

K

i s unique and the map h

+

6

268

Representing Measures

must be l i n e a r s i n c e i t s i n v e r s e ( r e s t r i c t i o n map) i s l i n e a r . The r e s t o f t h e a s s e r t i o n f o l l o w s a g a i n f r o m t h e Maximum P r i n c i p l e .

0

The n e x t o b s e r v a t i o n i s i m p o r t a n t f o r t h e subsequent theorems: 3.3.2

Lemma:

Conv(K)lex(K)

a D i n i cone.

Proof: P r o v i n g t h e D i n i c o n d i t i o n d i r e c t l y i s n o t t h a t easy. B u t i n Lemma 1.3.4 we have a f o r m a l l y weaker c o n d i t i o n . L e t Conv( K )

l W K ) . Then

take

(pn E Conv(K)

From Bauer's Maximum P r i n c i p l e we g e t of

%

be a sequence i n

having the

(pn

K we g e t ( f r o m D i n i ' s Lemma) some m ...

5 0

as r e s t r i c t i o n s .

'pn

. Because o f

6 0

t h e compactness

x E K with W

.

Y(x) = i n f supK( I: gn) where Y = I: The f u n c t i o n Y i s a g a i n m n=l n=l an element o f Conv(K) hence ( B a u e r ' s Maximum P r i n c i p l e ) x can be assumed t o be an element o f Lemma 1.3.4

3.3.3

ex(K). T h i s shows t h a t c o n d i t i o n ( i i i ) o f

i s fulfilled.

Choquet's Theorem:

F o r e v e r y x E K t h e r e i s a p r o b a b i l i t y measure mx --

on ex(K) -

(with

a l g e b_ r a making a l l elements o f A(K) r e s p e c t t o t h e s m a l l e s t u- ~ --_ _ lex(K) measurablel such t h a t h(x) =

I

ex(K)

h dmx -f o r a l l h E A(K)

.

Proof: l e t 6 be t h e e x t e n s i o n t o K (Lemma 3.3.1). I N K ) The map h + 6 ( x ) i s o b v i o u s l y a s t a t e o f A(K) B u t t h i s must be lex(K) a D i n i cone s i n c e i t i s a subcone o f a D i n i cone (Lemma 3.3.2). By t h e For

h E A(K)

.

R e p r e s e n t a t i o n Theorem (2.2.1)

t h e r e i s t h e n a measure mx

with

269

Boundaries

for all

( o r for all

h E A(K)lex(K)

Since A ( K ) i s a vector space this measure s t r i c t l y represents x Inserting for h the function 1 one gets t h a t mx(ex(K)) = 1 .

1

h E A(K)).

. 0

.

Choquet's original theorem was stated for metrizable K I n t h i s case the Baire and the Borel sets coincide and - since ex(K) i s then a Borel subset of K 12501- the measure mx can then be considered as a Borel measure supported by the extreme points. The non-metrizable case was f i r s t proved by Bishop and de Leeuw ([441 . Using the Sandwich Theorem we immediately can transfer the result t o Conv(K) instead of A( K ) : 3.3.4

Theorem:

x

For every --

E K

there i s a probability measure mx

on

ex(K) (with -

respect to the u- ~~algebra making a l l elements o f Conv(K) ~ -smallest measurable) -such that

4x1

I

I

cp dmx

ex(K)

for a l l

cp E

Conv(K)

.

Proof: Define a superlinear 6 : C o n v ( K ) J e x ( K ) 6(cp) =

supI;p(x) I % E Conv(K) with

(p

lex(K)

From Bauer's Maximum Principle we get 6

Theorem gives us a s t a t e

2 6

on

6

+

= cp)

sup ex( K ) Conv(K) lex(K)

. and the Sandwich

I

Conv(K)lex(K) i s a Dini cone (3.3.2),hence the Representation Theorem (2.2.1) yields a representing measure mx for v By definition of 6 this measure

.

f u l f i l l s the desired inequality.

0

*

270

Representing Measures

Now, i t i s quite natural t o ask i f these theorems can be transferred t o s t a t e spaces of order unit cones. F i r s t we have t o find a suitable boundary which can replace the extreme points of a compact convex s e t . Fortunately, the extreme points of the s t a t e space can take this role. This i s not completely t r i v i a l since we cannot always consider the s t a t e space as a compact convex subset of a locally convex Hausdorff vector space. This i s so, because s t a t e s are allowed t o attain the value .

-

3.3.5

Theorem:

Let ( F , < , I ) ----be an order unit cone. Consider a sequence f n @ F such that u ( f n ) i s decreasing --for every s t a t e p . --_Then. there i s a character v with inf v ( f n ) = inf S I ( f n ) nEN

nEN

where SI i s the -order unit functional. Proof: Consider on the compact s t a t e space n the cone of continuous functions F. For W,P E n we define functions n + R by

^F given by the Gelfand transforms of yP

Yw

The s e t r

P

i f there i s a

u

otherwise

I

= { y p y wp,w E

0

n l i s obviously

x

I 1 with p s x p + ( 1 - x ) ~

?- exposed.

The s e t K of states ( n o t necessarily characters) v f u l f i l l i n g the equality of 3 . 3 . 5 i s nonempty (Lemma 1.2.6) and compact since i t i s the s e t where the upper semicontinuous function inf ^fn attains i t s maximum. K

i s r- invariant since

Tn

nEN

i s decreasing. Therefore

.

r l K is FIK -

exposed. Hence, there i s a r- fixpoint in K By definition of the t h i s must be an extreme point and the assertion i s proved since ex-

yP ,w

treme points are characters ( I .2.10.6).

0

27 1

Boundaries

3.3.6

Theorem:

Let ( F , < , I ) be an order u n i t cone, ~denote by char(F) the characters of F . ---Then for every s t a t e P of F there i s a probability measure -

m

P

on char(F)

-

(with respect t o the

U-

algebra generated -by the

restrictions t o char(F) of the Gelfand transforms of F ) -such t h a t for all --

f E F.

Proof: Theorem 3 . 3 . 5 states t h a t char(F) i s a boundary of the s t a t e space for Exactly as in the proof Of 3 . 3 . 4 we then show the Gelfand transforms t h a t for a s t a t e p of F there i s a s t a t e i; o f ? [char(F ) such t h a t F(p) I for a l l T E ^F . Another immediate consequence of Theorem 3 . 3 . 5 i s t h a t F l c h a r ( F ) i s a

.

-

Dini cone. Hence by the Representation Theorem we find a representing measure m for This measure does the j o b . 0

.

3.3.7

Theorem:

Let K _ be -a_ compact space and F(K) a_ family _ _of- point separating _ _ Hausdorff -upper semicontinuous functions on K Then for every x E K there i s-a~ _ __ probability measure T on the Choquet boundary C h ( K ) (with to ~ _ respect _ _ _ the u- algebra generated by the restrictions -of the elements of F ( K ) to -~ C h ( K ) ) -such that

.

_ .

-_.

f(x) 5

I

Ch(K)

f d-r -for all f E F(K).

Reprssenting Measurns

272

Proof: Obviously we can replace

F(K)

by the cone generated by

F(K) and the

constant functions, t h a t changes n e i t h e r the Choquet boundary nor the inequality f o r

T

We embed the s e t p(x)(f) = f(x) p(Ch(K)) from 3.3.6.

. So,

K

l e t us assume t h a t

F(K)

i s an order u n i t cone.

n o f F(K) by

i n t h e s t a t e space

: K

p

-,n

with

f E F(K) (canonical embedding). We c l a i m t h a t

for all

i s equal t o t h e s e t o f characters. Then the a s s e r t i o n f o l l o w s

Proof o f the claim: L e t 1.1

be a character. By the Riesz-Konig Theorem there i s a r e g u l a r

representing measure

for

T

and s p l i t up the support

1.1

on

s u p p ( ~ )o f

1.1 5

T ( S 1 ) U l t T(s2)1.I2

u 2 ( f ) < u ( f ) . Hence p

+

1.1~

and

T

. We p

f E F(K)

:

v1,u2

If T(s2) > 0 then we have s t a t e s

such t h a t

. Take an a r b i t r a r y

K

given by

have, by d e f i n i t i o n o f

p2

,

cannot be an extreme p o i n t o f t h e

s t a t e space, which i s a c o n t r a d i c t i o n t o t h e assumption t h a t character. Therefore T ( S ~ =) 0 and T i s supported by S1

p

is a

. Since

f

was a r b i t r a r y we have

and t h i s i n t e r s e c t i o n must be nonempty and f o r every P(X) 2 x E k

1.1

.

.

x E

k

we have

Since a character i s maximal we then g e t p ( x ) = p f o r a l l can only c o n s i s t o f one p o i n t because t h e F(K) were assumed

273

Boundaries

t o be p o i n t s e p a r a t i n g . And f o r t h i s p o i n t we have proved t h a t i t i s t h e s u p p o r t o f any r e p r e s e n t i n g measure, hence

F o r t h e second p a r t t a k e an p1 3p2

and

0 <

x

< 1 such t h a t

r e p r e s e n t i n g measures Then T =

6,

x T~

t (l-X)T2

. This

gives

x E Ch(K)

T~

=

T

and

T~

x E Ch(K).

and assume t h e r e a r e s t a t e s

p(x)

I

for

p1

Apl

and

t (l-x)p2

p2

. Take

respectively.

x

i s a r e p r e s e n t i n g measure f o r

T~ = T~ =

bX

or

p1 = p2 =

~ ( x ) . So,

an extreme p o i n t o f t h e s t a t e space, hence (1.2.10.6)

regular

, hence P(X)

must be

a character.

We s t a t e t h e e s s e n t i a l p a r t o f t h e p r o o f s e p a r a t e l y :

3.3.8

Lemma:

be a compact H a u s d o r f f -space and F(K) ---a f a m i l y o f p o i n t separaLet K _ L e t p : K -,---s t a t e space o f t h e t i n g upper semicontinuous _ f u n_ c t_ i o n_ s -on K . -cone 2 R, generated by F(K), be t h e c a n o n i c a l embedding. Then p(Ch(K)) are the characters --

3.3.9

o f t h e s t a t e space.

Corollary:

L e t (FYI,<)

be an order unit cone. Then the characters -

a r e t h e Choquet

-_.

boundary o f t h e s t a t e space ( w i t h r e s p e c t t o t h e Gelfand t r a n s f o r m s o f F).

274

Represenring Measures

MAXIMAL MEASURES

3.4

I n t h i s s e c t i o n we g i v e another c h a r a c t e r i z a t i o n o f t h e p r o b a b i l i t y (F,< , I )

measures on t h e s t a t e space o f an o r d e r u n i t cone

which a r e

supported by t h e c h a r a c t e r s ( i n t h e sense o f Theorem 3.3.6). (F,< , I ) i s an o r d e r u n i t cone, n i t s s t a t e space which i s compact w i t h r e s p e c t t o t h e c o a r s e s t t o p o l o g y

L e t us s t a r t by f i x i n g t h e n o t a t i o n : such t h a t a l l Gelfand t r a n s f o r m s

?

forms a r e t h e f u n c t i o n s ?(w) = w ( f ) f u n c t i o n s on

By

VflB

for all

n

. By Vp

we mean t h e l o w e r bounded f u n c t i o n s i n V F

VF

We f i r s t observe t h a t

or

VflB

.

VflB)

separates t h e p o i n t s i n

i s a canonical correspondence between

VFB

.

3.4.1

of

F

,

i.e.

Prob(n)

.

Prob(n)

denotes t h e

u- a l g e b r a generated b y

R with respect t o the

( o r e q u i v a l e n t l y by

f

we denote t h e max-stable cone o f

, i.e.

generated by

p r o b a b i l i t y measures on

F

on n g i v e n by t h e elements

n

w E

a r e continuous. The Gelfand t r a n s -

cp E

Prob(n)

and t h a t t h e r e

and t h e s t a t e space o f

Observation:

(i)VflB

- Vfl,

i s , b y t h e Stone-Weierstrass theorem, sup-norm dense i n

C(n). I n p a r t i c u l a r we have f o r

J

cpd~=

R (ii)F o r e v e r y s t a t e

cpdp

T,P

for all

R

LI

of

E Prob(n)

VFB

that

T = p

whenever

-

c p E VFB

I

t h e r e i s a unique

T

E Prob(n)

such t h a t

275

Boundaries

The map u -,T i s a n affine bijective map between the s t a t e space of V F B and Prob(n). We really don't need a proof. Assertion ( i ) i s a t r i v i a l application of the Stone-Weierstrass theorem. The uniqueness in ( i i ) follows from ( i ) and the existence in ( i i ) i s easily seen as follows: F i r s t extend p t o the space V f B - VFB then apply the Riesz-Konig Theorem. The ordering

< in

-

given by the pointwise order on VFB i s

Prob(n)

called the Choquet-order , i . e . for

R

c p d ~I

n

cpdp

~ , pE

for a l l

Prob(n) we write

cpE

T

<

p

if

-

VFB

Because of 3.4.1 i ) this i s a n order relation ( n o t only a preorder). Of course, instead of V^FB one can take V F t o define the same order relation. A measure Choquet order.

Prob(n)

€

T

i s called maximal i f i t i s maximal in the

are bounded from below the Cartier-FellMeyer Theorem (1.2.7.7) immediately yields a complete characterization of the Choquet ordering: I n case that the elements of

3.4.2

Theorem:

Let the functions -equivalent:

in -

F

Then the following are be bounded from below. --

-forall

(i)

j c p d ~I J q d p

(ii)

Whenever there are positive measures

n

n

=

T

T~

R

T~,...,T,,

yitJ

+...+ T n __-then there are positive measures

with p = p l

J

cpEVF

'i d T i

I

pl,...,pn

+...+ p n -such t h a t J ^f n

dpi

for a l l i --

=

1 , ..., n -and a l l f E F

.

Representing Measures

276

Proof: Note t h a t

VFB =

Vf

since the elements o f

f

a r e bounded from below,

Therefore (Vf): corresponds t o t h e p o s i t i v e measures on the s t a t e space n (Observation 3.4.1). Now, the Cartier-Fell-Meyer Theorem (1.2.7.7) s t a t e s t h a t t h e Vf- pointwise order (Choquet o r d e r ) i s equal t o the ^Fdecomposition order, which i s e x a c t l y t h e a s s e r t i o n o f t h e theorem.

0

Now, we t u r n our a t t e n t i o n t o maximal measures. 3.4.3

Remark:

For every -

T

dominates

T

E Prob(n)

t h e r e i s a maximal measure

E Prob(n) which

i n the Choquet order.

Proof: Zorn's l e m a gives us a maximal l i n e a r

p :

VFB

-t

R

with

-

By t h e R i e s z - K h i g Theorem there i s a E Prob(n) which represents Obviously, T has the r e q u i r e d p r o p e r t i e s . 0

p

.

This r e s u l t y i e l d s another d e s c r i p t i o n o f characters. 3.4.4

Lemma:

-A point x E i s maximal. --

n -i s-a character -i f and only ---i f t h e Dirac measure 6,

Proof: -

.

at

x E n Then x i s an element o f the Choquet boundary o f the s t a t e space w i t h respect t o ^F ( C o r o l l a r y 3.3.9). By d e f i n i t i o n o f t h e Choquet boundary then 6, i s t h e o n l y representing measure w i t h

Take a character

respect t o

F , hence

a l s o w i t h respect t o

that

i s maximal and take a

Vf

. So,

6,

i s clearly

maximal. Now,assume

6,

T

E Prob(n)

with

x

277

Boundaries

x ( f ) = T(x) I

I ^f d

T

for a l l

f E F

(T

n

-F .

represents

x with

?I.

respect t o

T h i s i n e q u a l i t y e a s i l y goes over t o p o i n t w i s e suprema o f f i n i t e subsets o f Hence w(x)

cp d r

5

for all

cpE

62

and

n

6

T =

. Thus

X

since

VF ,

dx was maximal. So

x

i s i n t h e Choquet-boundary o f

i t i s a character ( C o r o l l a r y 3.3.9).

O

Maximal measures can e a s i l y be c h a r a c t e r i z e d by envelopes and by t h e f a c t t h a t they a r e supported i n a c e r t a i n sense by t h e extreme p o i n t s o f t h e s t a t e space. L e t us f i r s t e x p l a i n what k i n d o f envelopes we have i n mind. i s an upperbounded f u n c t i o n on t h e s t a t e space n then we d e f i n e o be t h e pointwise infimum of those c(cp) = i n f { -

i! I f

E F

cp

~(cp)

+

3.4.5

Let

cp

~ ( c p ) = cp

Lemma: 'I

E Prob(n), then the f o l l o w i n g

T

(ii)

I &(h)dT = 1 h dT

are equivalent:

i s maximal i n the Choquet order. ------

n

R

for all

h E C(n)

.

Proof:

*

(ii):

Define a s u b l i n e a r

F, w i t h

. One e a s i l y observes

i s sublinear. Furthermore we have

(i)

(i)

f E

7t

~p5

o :

7 I- cpl

with

s c a l l e d t h e F- envelope o f

- 7,

p : C(n) + R

by

if

-

t h a t t h e map cp E

vFB.

270

RepresentingMeasures

Now, c o n s i d e r an a r b i t r a r y cpo E C(n) and c o n s i d e r t h e l i n e a r v : R cpo + R g i v e n b y v(rcpo) = - r &(-cpo)d~ f o r a l l r E R

.

From

0= J we conclude

v 6 p

on R qo

i s a s t a t e since

v ( - h ) 5 p(-h) =

J

&(V)dT +

. Hence we

.

p 5 sup,

For an a r b i t r a r y element

since

J

C(n) dominated b y p

f u n c t i o n a l on v

w)dT 6

E ( W -

h E V^FB

&(-cp)d~ can extend

to a linear

v

(which w i l l be c a l l e d

v

again).

we have

J &(-h)dT

= J(-h)dT

& ( - h ) = -h. T h i s i m p l i e s v(h) 2

hdr

for all

h E

VFB .

By t h e Riesz Representation Theorem t h e r e i s a p r o b a b i l i t y measure v(w) = This implies

T

v(w) = since

J

cp

=

T

d;

for all

q E C(n)

for all

c p E C(n)

with

.

or

1 cp d.r

was assumed t o be maximal. Now, r e c a l l t h a t we s t a r t e d w i t h

T

v(-cpo) =

E

(-(Po)&

f o r t h e a r b i t r a r i l y chosen cpo E

C(Q).

Thus ( i i ) i s

proved.

(ii) * (i):

Take a maximal

t h e form - E (cp) i s t h e VFB , t h e r e f o r e we have

J

(*)

-&(cp)

dT 6

J

i

E Prob(n)

with

. Every

T <

function o f

p o i n t w i s e supremum o f an upwards d i r e c t e d n e t i n

-&(V)d;

for all

W E

C(0)

.

From ( i i ) and t h e p a r t (i) es (ii) which we a l r e a d y proved we o b t a i n f o r a l l

w E VFB

:

278

Boundaries

Combining t h i s with ( * ) we have

for a l l

And

cp E

VFB , hence

for a l l

must be maximal since

T

cp E

C ( R ) (Observation 3.4.1 ( i ) ) .

T =

0

Combination of the l a s t r e s u l t with Lemma 3.4.4 gives 3.4.6

Corollarv:

Let x -

E

n

.Then

x i s-a character if and only if

d x ) for all

E((P(x)) =

cpE C ( n ) .

Another consequence i s : 3.4.7

Theorem:

Let -

T E

(i)

T

(ii) (iii)

I

Prob(n).

Then the following are equivalent: --

i s maximal in the Choquet order E

R

If

(h)dT = d T

=

Ig

I

h dT

dT

for all

_._.

h E C(n)

-f o r a l l f , g E C ( n ) --such t h a t the r e s t r i c t i o n s

t o the characters a r e equal. -( i v ) For every Baire s e t B c n which contains the characters of the s t a t e space n we have r(B) = 1 . Proof: ( i ) +=+ ( i i ) we already know (3.4.5). ( i i i ) * ( i i ) i s an immediate consequence of Corollary 3.4.6. (iv) * ( i i i ) is trivial.

280

Representing Measures

( i ) * ( i v ) : We f i r s t remark t h a t t h e Choquet boundaries o f n w i t h respect t o and V? a r e equal (hence equal t o t h e s e t o f c h a r a c t e r s , Corollary 3.3.9) since t h e Choquet boundary does not change i f a cone of functions is replaced by t h e max-stable cone i t generates. Prob(n). From Theorem 3.3.7 we obtain a prob a b i l i t y measure m on t h e characters (equal t o Choquet boundary) w i t h

Now consider a maximal

J

sl

cpd~5

T

E

qdm

Char(n)

for all

cpE

VF

.

For Baire s e t s B we p u t ;(B) = m ( B n char(s2)). This defines a Baire s i n c e T was maximal. probability measure with ? > T . Hence T = And T has obviously the required p r o p e r t i e s . 0

281

Boundaries

3.5

THE CHOQUET

-

MEYER THEOREM

The central result of t h i s section will be the Choquet (see [ 2 1 or [2501) in the following form:

-

Meyer Theorem

An order unit vector space ( E , < , I ) i s simplicia1 i f and only i f every s t a t e has a unique representing measure on the extreme points of the s t a t e space ( i n the sense of Theorem 3 . 3 . 6 ) . Essentially t h i s result i s a simple combination of the characterization of probability measures on the extreme points as maximal measures (Theorem 3 . 4 . 7 ) and the characterization of simplicia1 cones by the FSP of the positive dual cone (Theorems 1.2.9.5 and 1.2.9.7). In the proof of the central result we make a l i t t l e detour in order t o present some results of the same flavourfor order u n i t cones instead of vector spaces. I t should be remarked t h a t the result as i t i s stated above does carry over to cones only for maximal s t a t e s (which certainly comprises the result for vector spaces because every s t a t e i s maximal on a vector space). Let (F,< ,I) again be an order unit cone. A s t a t e p i s said to be maximal i f for everylinear monotone v with v 2 p we must have v = !J This i s the same as saying t h a t for every linear monotone v with S I -> v 1 !J we have v = p On a vector space every s t a t e ( i n f a c t every linear functional) i s t r i v i a l l y maximal.

.

.

We know (Theorem 3 . 3 . 6 ) t h a t for every s t a t e p there i s a representing measure m f o r !J which i s carried by the characters of the s t a t e space. P

Representing measure means t h a t m p(f) I

I 'i dmll

for a l l

P

i s a probability measure with

f E F

,

And "carried by the characters" means t h a t i t i s a measure on the characters

endowed with the smallest U- algebra such t h a t a l l the f E are Gelfand transforms of F) ; t h i s i s equivalent t o saying measurable (? t h a t there i s a Baire probability measure T~ on the s t a t e space n which gives, with respect to ? , the same integrals as m and which has the !J property that i t vanishes on any Baire s e t having empty intersection with

282

Representing Measures

t h e s e t o f c h a r a c t e r s (Remark 2.1.1). Furthermore we know (Theorem 3.4.7) on t h e s t a t e space

t h a t a B a i r e p r o b a b i l i t y measure

'I

i s maximal w i t h r e s p e c t t o t h e Choquet o r d e r i n g

R

(meaning maximal as a l i n e a r f u n c t i o n a l w i t h r e s p e c t t o t h e

VFB

-

point-

w i s e o r d e r ) i f and o n l y i f t h e r e i s a measure c a r r i e d b y t h e c h a r a c t e r s which has t h e same i n t e g r a l s w i t h r e s p e c t t o t h e f u n c t i o n s i n

VFB

(or

because o f Observation 3.4.1,

or

C(R) V? s i n c e t h i s c o n t a i n s VFB)

.

Now, l e t us combine a l l t h i s knowlegde w i t h Theorem I . 2.9.11. 3.5.1

Theorem:

Let F be-a s i m p l i c i a 1 ---o r d e r u n i t cone and c o n s i d e r an -

R- valued maximal

o f F. Then t h e r e i s a unique p r o b a b i l i t y measure -

state

P

by t h e --

c h a r a c t e r s --o f the state

J

P(f) =

7

dmu

m

P -

carried

space -such t h a t -for all

f E F

.

Proof:

.

F: Now, c o n s i d e r as G t h e F i n G v i a t h e Gelfand t r a n s form. Then a l l t h e requirements o f S i t u a t i o n I . 2.9.9 a r e o b v i o u s l y f u l -

Since

i s R- valued i t i s an element o f

1.1

and map b y T : F

V':

cone

-,G

t h e cone

f i l l e d . Theorem I . 2.9.11 g i v e s a unique l i n e a r map u(;)

o

(*) for a l l

T =

i

* -,GT

with

and

u(;)(g)

G

u : F,

E Ff

= sup(v(g)

. Hence

I

u(U)

v E G,* is

with

voT =

V? = G- p o i n t w i s e maximal i f

is

maximal. T h e r e f o r e i t s s t r i c t r e p r e s e n t i n g measure on t h e s t a t e space must be maximal i n t h e Choquet o r d e r i n g and t h e e x i s t e n c e o f t h e s t r i c t r e p r e s e n t i n g measure in

'c1

(*)

f o l l o w s from t h e p r e c e d i n g remarks.

a l s o y i e l d s t h e uniqueness o f

would be dominated b y m

P

m

because o f

u

The p r o p e r t y

s i n c e any o t h e r such measure

(*); hence i t must be equal t o

because i t corresponds t o a maximal measure

.

fi

P

m

u

Boundaries

283

u

A c t u a l l y we have proved a l i t t l e b i t more s i n c e t h e map

we used i n t h e

proof i s additive: 3.5.2

Remark:

Consider t h e same s i t u a t i o n as i n 3.5.1 and assume t h a t t h e s e t o f maximal R- valued s t a t e s i s a convex s e t . Then t h e map

When

F

u

+

m

U

i s affine, i.e.

R- v a l u e d and

i s a v e c t o r space t h e n e v e r y s t a t e i s a u t o m a t i c a l l y

maximal; thus t h i s s i t u a t i o n i s much more t r a n s p a r e n t : 3.5.3

Theorem: are equibe an order unit vector space. Then t h e f o l l o w i n g --

L e t (E, < , I )

valent:

i)

E i s simplicia1

ii) Ef

-has t h e FSP o r , equivalently,

( E T - E*,)

i s 2 vector l a t t i c e ~~

w i t h~ r e s p ect t o t he o rder g i v en b y t h e p o s i t i v e cone ET -

iii) --F o r e v e r y two p r o b a b i l i t y measures there i s ---

iv)

v)

an

ml,m2

.

the characters c a r r i e d by -

f E E such that

Every s t a t e u of --

E has a unique r e p r e~ s e n t i n g measure m v ca_ r r i e_ dby- t h e c h a r a c t e r s . _ There i s an a f f i n e

.___---

u

+

m v from t h e s t a t e space i n t o t h e

p r o b a b i l i t y measures _ c a r_ r i e_ db y- t h e c h a r a c t e r s such t h a t ~ ( f )= J f dmU

for all

f E F -and s t a t e s

P

.

Representing Measures

284

Proof: i s t h e c o n t e n t o f Theorems 1.2.9.5 and 1.2.9.7. (i) Y (ii)

--

(i) * (iv)

f o l l o w s f r o m o u r l a s t theorem.

(iv)

i s a consequence o f Remark 3.5.2.

(v)

(i): Consider t h e embedding T : E + G = C(R) g i v e n by t h e Gelfand (v) transform. Then ( v ) says t h a t t h e r e i s an a d d i t i v e map u : ET -,GT w i t h u(1-1)o T = p f o r a l l Theorem 1.2.9.8.

1-1 E

. Hence,

Er

E

i s simplicia1 by v i r t u e o f

Since e v e r y s t a t e has a r e p r e s e n t i n g measure ( s t r i c t s i n c e (iii) Y (iv): we a r e i n a v e c t o r space s i t u a t i o n ) uniqueness o f t h a t r e p r e s e n t i n g measure must o b v i o u s l y be e q u i v a l e n t t o t h e p r o p e r t y t h a t f separates these measures. n A d d i t i o n a l i n f o r m a t i o n i s o b t a i n e d when t h e s i m p l i c i a 1 map i s c l o s e d . We s t a r t b y making a few remarks on elementary topology. We have endowed t h e s t a t e space

nF

o f an o r d e r u n i t cone w i t h t h e c o a r s e s t t o p o l o g y such

t h a t , f o r e v e r y f E F , t h e Gelfand t r a n s f o r m i s continuous. T h i s means t h a t a f u n c t i o n @ : nF1 + nF2 i s continuous i f and o n l y i f 7 o is continuous f o r e v e r y f E F2. can c o n s i d e r t h e cone R

nF

O f course, i n s t e a d o f t h e s t a t e space one = i r w l r 1 0 , w E nF3 generated by t h e s t a t e

space. F o r a v e c t o r space t h i s cone i s equal t o a l s o h o l d s i f t h e elements o f

f E F t h e r e i s some a E R

(F,< )

* . This

F a r e bounded f r o m below, i . e . such t h a t

aI

< f. We endow

same topology. I t should be observed t h a t a f f i n e maps T :

R

equality f o r every

nF w i t h t h e

nF

1

+

n

F2

between s t a t e spaces correspond u n i q u e l y t o l i n e a r maps + R nF between t h e generated cones ( t h i s e x t e n s i o n , o r 1 2 r e s t r i c t i o n , we denote b y t h e same symbol). A l i n e a r map T :

T : R

nF

i s continuous i f and o n l y i f i t s r e s t r i c t i o n T

RRF1

lnFl

t o t h e base R

-+ F1

i s continuous. F i n a l l y we remark t h a t we endow t h e bounded measures ( o r t h e base g i v e n b y t h e p r o b a b i l i t y measures) on

nF 1

w i t h t h e weak-star

F2

285

Boundaries

-

C(nF ) 1 all

VFB

i s sup-norm dense i n

t h i s t o p o l o g y i s t h e same as t h e c o a r s e s t t o p o l o g y such t h a t , f o r

f E F

3.5.4

-

-

C(nF ) . S i n c e VFB 1

topology w i t h respect t o

, the

functions

m

+

1f

dm

a r e continuous.

Theorem:

Consider -a g a i n a s i m p l i c i a l ---o r d e r u n i t v e c t o r space E are equivalent:

. -Then t h e

following

i s c l~ o s e d in the s t at e space. (i) -The _ _s -e t o f c h a r a c t e r s (ii) The sup-norm c l o s u r e o f

a t t i ce w i t h respect to -i s a -v e c-t o r l~

t h e p o i n t w i s e o r d e r on t h e c h a r a c t e r s . (iii)

The 9 -

u

a s s i g n i n g --t o each s t a t e

p

i t s unique maximal measure

_ .

(------maximal i n t h e Choquet o r d e r ) i s continuous.

(iv)

The s i m p l i c i a l

map

1 : E:

+

g

(L E )

continuous.

Proof:

a n be t h e c h a r a c t e r s o f E . T h i s i s a compact s u b s e t o f t h e s t a t e space. The map p : 2 + c(a a ) g i v e n by t h e r e s t r i c t i o n s t o a R i s monotone. And because o f (i) * (ii):

Let

t h i s map i s i n j e c t i v e and we o n l y have t o p r o v e t h a t p ( E ) i s a sup-norm dense subspace o f c ( a s2 ) . We know t h a t p ( E ) separates t h e p r o b a b i l i t y measures on

as1

(Theorem 3.5.3 ( i i i ) ) .

c(a

norm continuous l i n e a r f u n c t i o n a l s on i n C(a n )

(ii) ( i v ) :

Hence

by t h e Hahn-Banach Theorem

0 ) . Thus

(I

^E.

E*, t h e r e i s a unique such t h a t p(p)(?) = p(f)

p E

p(i)

must be dense

e : E + Clos(E) o f

E

into

An easy e x e r c i s e shows t h a t f o r each s t a t e in for all

~(1.1)

LE

(2);

Clos f E E

i s continuous and l i n e a r . Now, if Clos o f t h e f r e e l a t t i c e cone

separates t h e sup-

1.5.8).

Consider t h e canonical embedding

t h e sup-norm c l o s u r e o f

p(2)

(E)

and t h a t t h i s map

p

i s a l a t t i c e then by d e f i n i t i o n

t h e r e i s a unique l a t t i c e cone homomorphism

286

E*

Representing Measures

: LE

+

Clos(E)

such t h a t

commutes. Then c o n s i d e r t h e map z : E: T h i s map i s l i n e a r w i t h

X(p)

0

L =

for all

p

g i v e n by

(LE)

+

. Hence

p

p

+

p(p)o c*.

IT must be t h e

s i m p l i c i a l map. F o r i t s c o n t i n u i t y we have t o show t h a t f o r a r b i t r a r y

W E LE p

t h e map

-,p ( p ) ( t * ( q ) )

(iv)

*

(iii):

E*

: LE

E*

o

L

-,V

= E

p + z ( p ) ( ~ )

Consider t h e embedding

: E

+

V

2

and l e t

i s a f f i n e (Remark 3.5.2 and Theorem 3.4.7) we have p; ) and u must be continuous s i n c e I: i s continuous. a

(iii) * (i):R e c a l l t h a t i f c h a r a c t e r (Lemma 3.4.4). c l o s e d because a Theorems 3.5.3

E

i s continuous.

p

be t h e unique l a t t i c e cone homomorphism such t h a t

. Since

u ( p ) o e* = ~ (

i s continuous. B u t t h i s i s equal t o

and i s c l e a r l y continuous s i n c e

~ ( p )=

dV t h e n

Hence a-1{6ylv E

i s continuous.

QI

p =

u

and

p

must be a

i s t h e s e t o f c h a r a c t e r s and

0

and 3.5.4 are, o f course, minor m o d i f i c a t i o n s o f r e s u l t s

well-known f o r t h e geometric s i t u a t i o n when A(K) a r e t h e a f f i n e c o n t i n uous R- valued f u n c t i o n s on a compact convex subset K o f a l o c a l l y convex H a u s d o r f f v e c t o r space. The f u n c t i m

t h e n serves as o r d e r u n i t lK and an easy e x e r c i s e (Hahn-Banach Theorem) t h e n shows t h a t t h e s t a t e s of A(K) are exactly given by t h e p o i n t evaluations i n K K then i s c a l l e d

.

a siniplex ( o r r a t h e r a Choquet-simplex) i f A(K) i s s i m p l i c i a l . Because of t h e f a c t t h a t t h e n t h e cone generated by K i s a l a t t i c e cone t h i s

n o t i o n r e a l l y i s an a p p r o p r i a t e g e n e r a l i z a t i o n o f t h e w e l l known d e f i n i t i o n of a simplex i n R"

.

I n t h i s s i u t a t i o n the characters are obviously given

by t h e extreme p o i n t s of

K

. When t h e

extreme p o i n t s a r e c l o s e d t h e n

K

i s c a l l e d a Bauer-simplex (because o f t h e p i o n e e r i n g work o f Heinz Bauer i n combining p o t e n t i a l t h e o r y and t h e t h e o r y o f compact convex s e t s ) .

287

Boundaries

3.6

DIN1 BOUNDARIES

I n t h i s section we investigate two problems. The f i r s t one i s motivated by the observation (Lemma 1.3.3) t h a t X i s a boundary whenever F(X) i s a Dini cone. We look for conditions such t h a t F(X) becomes automatically a Dini cone whenever i t has X as a boundary. An essential tool for finding meaningful conditions i s Simons Convergence Lemma (see 12971 and [2441 ,[2681). This lemma i s very powerful in several areas of functional analysis, i t i s particularly useful in context with the proof of James 'Theorem (see [298 I and [1771 12583 ,[lo61 ). The second problem looks very different from the f i r s t one, b u t they are related with respect t o the techniques which are involved. We have already seen (section 3.2) t h a t a compact convex subset may have a boundary for the affine functions smaller t h a n the s e t of extreme points (Choquet boundary). We ask under w h a t kind of conditions such a behaviour i s prohibited, in other words: When are the characters the smallest boundary of the s t a t e space (with respect t o the Gelfand transforms) of an order unit cone? W e s t a r t with Simons Convergence Lemma ( i n the version of M. Neumann [2441). 1

Consider C, , the s e t of sequences y all

n

= (y,,y2,...)

m

E

and I ~ I =1 <+m,Let (f,), n n=l n

N

E

m

m

where c

I: Y n f n = c lyl + inf

n= 1

. By

< y , f > we

t y n f n . This s e r i e s always converges in

n=l

we can write: (1)

m

m 1

n=l

yn(fn - c ) ,

i s a common upper bound f o r the

fn.

for

N , be a sequence of uni-

formly upper bounded functions on some nonempty s e t X denote the function

with yn 2 0

since

288

3.6.1

Representing Measures

Simons Convergence Lemma:

Assume t h a t -supx< y,f

>

= lim

m

(supx

E ynfn) n=l

W m

for a l l --

1

y E Lt

. Then,if

Y c X

_i s _a

> I

boundary f o r I < y,f

y E

L,),1

we have -s u p y ( l i m sup fn) 2

n

(**I 2

-)a

m

infIsupX( z

n= 1

Xnfn)

I

m

J t i y

E N , An t 0

m

An =

II

n=l

1)

Remark: C o n d i t i o n ( * ) i s q u i t e o f t e n a u t o m a t i c a l l y f u l f i l l e d . F o r example when t h e

.

This follows a r e upper semicontinuous f u n c t i o n s on a compact s e t X fn immediately w i t h (1) f r o m D i n i ' s Lemma. Also,when t h e fn a r e u n i f o r m l y bounded ,then ( * ) h o l d s . P r o o f o f 3.6.1.: L e t a be t h e i n f on t h e r i g h t s i d e o f ( * * ) . 6 > 0 there are y E Y

can show t h a t f o r each ntm

g, E Km = I such t h a t

Z

i=m g,(y)

and thus, s i n c e

I

Xifi

ntm

z

i=m

t a - 2 6. 6

Xi

= 1, Xi

t 0, i=m,mtl,

n

.

g,

E

(a- 6(lth)

...,ntm,

-

+ m

E N,

n

E NI

-

2 6

. We can c l e a r l y

. Let

p be an upper bound f o r t h e 6 > 0 a r b i t r a r i l y and l e t

px)(l-A)-'

Km by i n d u c t i o n , such t h a t

m

l i m sup f n ( y ) 2 a n+m

s u p y ( l i m sup fn) 2 a

r e s t r i c t o u r s e l v e s t o t h e case a > - m fn. We t h e n have - m < a I p < m F i x

Define

and , f o r a l l

Because,then we have

was a r b i t r a r y ,

1 > h > 0 be so t h a t

The theorem f o l l o w s i f we

t a - 2 6

.

289

Boundaries

.

+ 6 2-m A m We c l a i m t h a t t h e r e i s y E Y

such t h a t

g,(y)

ta

-2

6 for a l l

(1t

x ) -1(g,tx

m 2 1.

Proof o f t h e claim: Define h, =

m

E xi-' i=l

gi

and p u t

ho = 0

. Since

m E N , we have

for a l l

I f we m u l t i p l y

the l a t t e r i n e q u a l i t y by

I

x

(Note, t h a t always supX(hm) > We d e r i v e

+ supX(hmtl)

supX(hm-l)

x-m(supX(h,l)

-

-

since

supX(hm)) 2

Because o f ( * ) we have s u p X ( l i m hm) = l i m supX(hm) m+m

.

(ltx)

we o b t a i n

+ 6 2-"(1+ h ) A". a >

- -).

gmtl)E

I(,,

Representing Measures

290

The boundary property of We conclude

Y yields

y E Y

such t h a t h(y) = s u p X ( h ) .

hence

t (l-x)-'(a-6(1tA)

m E

for a l l

N

.

-

AD) 2 a

-

2 6

0

For our considerations we only need a special version of the Convergence Lemma,dealing with the situation when the f n are decreasing. I n t h i s case the inequality has a simple form: 3.6.2

Corollary:

Let the --

f n be as in 3.6.1

pointwise decreasing.

Then

and assume furthermore that the f n are

supy inf f n t inf supx f n n n This inequality shows t h a t the Convergence Lemma must have nice applications for Dini cones. Let us f i r s t briefly recall the notation we adopted in the context of Dini cones. We assume that F = F(X) i s a cone o f upper bounded functions on a nonempty s e t X containing a l l constant functions. F i s endowed with the pointwise order on X and the function lX serves as order unit. VF i s

Boundaries

the max-stable cone generated by and

functions in VF(X), sup- norm.

5

29 1

VFB consists of the bounded

F(X).

denotes i t s closure with respect t o the

Let us introduce two additional closures. By F ( X ) we denote the cone consisting o f a l l

< where

y

m

Y,cp>

and

E L:

=

t Ynfn

n= 1

cp = (f,)

i s a uniformly upper bounded sequence in

F ( X ) . And by F(X) we denote the cone of a l l uniformly converging sums 1 gn n€N

, where the

g n are elements of

F v R

F(X)

Imax(f,r) I f E F, r E R3

=

def

.

was called a Dini cone if inf supX(fn)= supx inf (f,)

n a

nEN

for a l l

(f,)

+

in F(X) , where

+

(f,)

means t h a t the sequence i s

pointwise decreasing. We are going t o call a subset Y

c

X

a Dini boundary for F(X)

if

inf supX(fn)= supy inf (f,) nEN

for a l l sequences say that

p

nEN

(f,)

+

in F ( X ) .

If

has a representing measure u(f) I

I

Y

f d-r

for a l l

p T

: F(X) -ti i s linear,then we

0”

Y

f E F(X) ,

if

292

where

Representing Meagtres

must be a p o s i t i v e measure w i t h respect t o the

T

generated by the r e s t r i c t i o n s t o

Y

o f the elements o f

u- algebra F(X).

An example o f a D i n i boundary i s the s e t o f characters f o r the Gelfand transform o f an order u n i t cone (Theorem 3.3.5).

One o f our problems w i l l

be t o f i n d o u t i f a l l D i n i boundaries are o f t h a t kind. This example shows, furthermore ,that D i n i boundaries and subsets c a r r y i n g representing measures f o r a l l s t a t e s are i n t i m a t e l y connected.

3.6.3

Improved Representation Theorem t1291:

Consider

Y cX

F(X)

Y

, then

&a

-i s-a

t h e f o l l o w i n g a r e equivalent:

D i n i cone and

boundary f o r

_ .

F(X)

6

i s a D i n i boundary f o r F(X)

For every s t a t e ----

of

For every s t a t e ---

1.1

l i m sup n +m whenever Let -

F(X)

t he r-e i s a representing -measure on Y -

o f F(X) we have --

p ( f n ) I sup

fiY

( l i m sup f n ( y ) ) n+m

is an upper -

(f,)

bounded sequence i n F(X).

fn E F(X) w i t h fn I 0 m

supy(

z fn) >

n=l

-

E p(fn) > n=l

If i n f supx(

mEN

IJ

-- . m I fn) > -

n=l

.Then

03

whenever ---there i s a s t a t e

(viii)

-i s-a

boundary f o r F(X)

Y i s-a boundary for Y

Y

m

o f F(X) w i t h -

293

Boundaries

then supy( Z. f n ) > n= 1 whenever

(f,)

&2

m

sequence

fi

with

F(X)

fn

5 0

-for a l l

n E N .

Proof: ( i ) * ( i v ) : Since F(X) i s a Dini cone,formula (1) (preceding 3.6.1) shows t h a t ( * ) of 3.6.1 i s a tomatically f u l f i l l e d . Now, take a decreasing sequence (f,) + in F(X) , then from 3.6.2 we obtain supy inf f n 5 nf supx f n . n n Since the inequality in the other direction i s t r i v i a l the cone must have Y as a Dini boundary. ( i v ) 4 ( v ) : When Y i s a Dini boundary i t must be a boundary f or F(X) (exactly the same argument as in Lemma 1.3.3). B u t then we can use a routine argument: Take a s t a t e IJ of F ( X ) . Define a superlinear 6 on F ( X ) l y ( r e s t r i c t i o n s t o Y ) by:

Since Y

i s a boundary we have 6

get a s t a t e

G on F ( X ) l y with G p(f) 5

Now, since Y

I

; ( f l y ) for a l l

i s a Dini boundary of

supy t 6.

. By

the Sandwich Theorem we then

B u t t h i s means

f E F(X). F(X)

the cone F ( X ) J y must be a

i ) ) . From the Representation Theorem 2.2.1 Dini cone (Lemma 1.3.4 i i i ) we then get a representing measure on Y for n This clearly i s a representing measure for LI

.

.

( v ) * ( v i ) : i s an immediate consequence o f the Monotone Convergence

Property ( i n the version of Fatou's Lemma) applied t o the representing measures on Y .

294

Representing Measures

-

(vi)* ( v i i ) + ( v i i i ) i s t r i v i a l .

( v i i i ) ( v ) : Apply ( v i i i ) t o the constant sequence t o s e e t h a t , f o r a l l f , we have

fn

=

(f

-

supX(f))

Then take an a r b i t r a r y s t a t e u of F(X) and consider on F ( X ) l y ( r e s t r i c t i o n s t o Y ) the superlinear functional

From ( 2 ) we g e t 6

of

I supy.

Hence the Sandwich Theorem gives us a s t a t e ;I

F(X)\y j(f

Iy

) > p(f) -

for all

f E F(X).

We claim t h a t ( v i i i ) implies t h a t F(X)

i s a Dini cone. Hence

1Y

has a

representing measure (Representation Theorem) which i s then a l s o a representing measure on Y f o r u Thus ( v ) i s proved s i n c e p was a r b i t r a r y .

.

Proof o f the claim: We use an argument we already used in t h e proof of Lemma 1.3.4. Take an a r b i t r a r y sequence cpn I 0 i n F(X) and p u t m m a = i n f supy ( z cp,) and 0 = supy ( t cpn) m n= 1 n= 1

.

We need t o prove a IB s i n c e a t 0 i s t r i v i a l . I n order t o do t h a t , we take numbers N n such t h a t Nn 1h e r e g = t a I s u p Y( g n) $ a + - , w cpn. n k=l 1 1 Then we consider 0 t f n = n ( g n - a - -). Since cpn 5 0 , we o b t a i n , by n d e f i n i t i o n of a , for every m E N and every E > 0 some y, E Y with 1 f n ( y m ) 2 - -2 - E f o r a l l n 5 m. Hence n m i n f s u p y ( t f n ) t - "t 1-2 > - m m n=l n=l n

"

.

295

Boundaries

Now, ( v i i i ) g i v e s us some y

z fn(y ) >

n=1

-

E Y

.

m

i n f gn(y ) 5 p , n

This inequality,together w i t h

forall

n1 ( p + t - a ) > - w

Z

with

implies

E > O .

n=l Hence (v) for

B 2 a

.

* (i):Actually,we prove t h e statement t h a t Y F ( X ) . T h i s statement i s , f o r m a l l y , s t r o n g e r t h a n

state o f

F(X)

and l e t

a r e p r e s e n t i n g measure 1-1

( i ) . Let

be i t s r e s t r i c t i o n t o t h e subcone

F(X). Then

ji i s c l e a r l y a r e p r e s e n t i n g measure f o r

for

T

u be a

(Monotone Convergence P r o p e r t y and f o r m u l a (1) b e f o r e 3.6.1).

state o f

F(X) has a r e p r e s e n t i n g measure on

R e p r e s e n t a t i o n Theorem ( h e r e ( i i ) +. ( i ) )

Y

vergence P r o p e r t y t h a t (iii)

=S

(ii)

(ii )

*

(iv);

Y

.

So e v e r y

And e x a c t l y as i n t h e

one shows w i t h t h e Monotone Con-

F(X).

must be a D i n i boundary f o r

is trivial Let

be a r b i t r a r y i n

(f,) J.

a = i n f supx fn n

We have t o prove

p 2 a

= max(fn,a-I).

Then

'pn

i s a D i n i boundary

p

and

,for a < y,(p > E

=

F(X). P u t

supy i n f f n ' n -

w

F(X)

.

I n o r d e r t o do t h a t ,define whenever

y

E Li

since

the

'pn

a r e c o n s t i t u t i n g a sequence which i s u n i f o r m l y bounded. NOW, t h e i n e q u a l i t y coming o u t o f Simons' Convergence Lemma has t h e f o l l o w i n g f o r m supy i n f n since the

'pn

r i g h t hand s i d e

'pn

2

i n f supx n

'pn

a r e decreasing. The l e f t hand s i d e i s L a

, hence

p

L a

.

5 max(p,a-I)

and t h e

296

Represenring Measures

( v ) * ( i i i ) : Assume t h a t every s t a t e of F(X) has a representing measure on Y . We then know t h a t every s t a t e of V F has a representing measure

on Y (2.2.1 and 1.3.4). We claim that every s t a t e of has a representing measure. Then ( i i i ) i s an immediate consequence of 2.2.1 and 1.3.3 (applied t o Proof of the claim: Take an arbitrary s t a t e i; of VFB,

q).

take a maximal s t a t e l.~ 2 i; , r e s t r i c t i t t o VFB and extend i t t o a maximal s t a t e i,i on VF Then the corresponding representing measure for G on Y i s clearly a representing measure for l.~ Since both p and

.

I vFB .

the integral are sup-norm continuous we immediately have a representing measure on Y for IJ 0

.

3.6.4

Remark:

I t i s quite obvious that whenever one of the conditions of Theorem 3.6.3 i s replaced by the corresponding condition for VF, VFB or VFB (instead

-

of

F(X))

then we obtain an equivalent condition. The reasons for that are

i)

the Dini property remains valid i f we replace F(X) by VF,

ii)

any element of i n VFB ,

iii)

states of

VF

i s the infimum of a decreasing sequence

VFB are sup-norm continuous and therefore they

(as we1 1 as integrals with respect t o probabil i t y measures) have unique extensions t o the sup-norm closure.

Now some examples. 3.6.5

Examples:

( i ) Let X be a pseudo-compact topological space, i.e. X i s completely regular such that every bounded continous real valued function attains i t s maximum on X (see 1921 or [140]). Then every s t a t e on the (bounded) continuous real-valued functions has a Baire representing measure on X. (One should observethat every continuous R- valued function on a pseudocompact space i s automatically bounded 1320, p.341). This example i s due t o G1 icksberg [ 1411 see a1 so [3171 ,[1181.

297

Boundaries

(ii)

X

Let

be a convex closed and bounded ( n o t necassarily compact)

subset o f a l o c a l l y convex t o p o l o g i c a l Hausdorff vector space and consider t h e continuous, r e a l valued bounded and convex functions ConvB(X) on X. Then, i f Y c X i s a boundary f o r ConvB(X),there i s , f o r every, x E X a representing measure on (iii)

Let

Y.

be a convex bounded p o l i s h subset o f a l o c a l l y convex t o -

X

pological Hausdorff vector space

and consider the weakly

E

r e a l valued,bounded and convex f u n c t i o n s

*

t p Conv ~

o(E¶E )

(X)

* (E) Conv u(E,E )

a t t a i n s i t s maximum on X

X

then

on

continuous,

X

.

I f every

i s weakly

compact. Proof: ( i ) and ( i i ) are,obviously,immediate

consequences o f Theorem 3.6.4.

Only t h e p r o o f o f ( i i i ) needs a d d i t i o n a l remarks. Let

F(X)

be t h e r e s t r i c t i o n s o f

. All

n o f F(X)

space

(E*+R)

to

we have t o prove i s

and consider the s t a t e

X

n

since

= X

x E X with

i n i t s canonical topology. Clearly, here we i d e n t i f y a l l 6, E n

, where

extend

p

t o a state o f

representing measure

Convo(E,E*) on

T

generated by Convu(E,E*) to

Now, l e t p E n ¶ t h e n we can

= f ( x ) , f E F(X).

6,(f)

(X)

and from Theorem 3.6.4 we g e t a

( w i t h respect t o

X

n i s compact

(X) ,or,equivalently,by

1

, the

u- algebra

the r e s t r i c t i o n s o f

X ) . By t h e Hahn-Banach Theoremyand t h e f a c t t h a t

E*

has a countable

X

base ( s i n c e i t i s polish),we obtain,that every closed convex subset o f belongs t o

.

1

Hence every open convex subset belongs t o t and 1

must t h e r e f o r e be t h e ure

( s e c t i o n 2.6.2). T =

where

xn

2 0

convex h u l l s

<

1 An

n E N.

>c X

T

T

i s a t i g h t meas-

as

Tn

1 An = 1 and where

Kn, Kn

u- algebra o f Bore1 sets. Thus

This means,that we can w r i t e

n=l

with

compact supports

X

Since

o f the

X

Kn

T,

are p r o b a b i l i t y measures w i t h

i s complete and contains t h e closed

, these

closed convex h u l l s must be

298

Representing Measures

E**

a g a i n compact. A p p l i c a t i o n o f t h e Hahn-Banach Theorem i n r e s p e c t t o t h e weak*topology) y i e l d s t h a t t h e s t a t e by t h e i n t e g r a l w i t h r e s p e c t t o

i s an element

T,

(with

on

11,

xn

F(X)

-

<

of

Kn

given

> .

[2501

( T h i s i s e x a c t l y t h e argument which was used i n t h e p r o o f o f P r o p o s i t i o n 1 . 2 ) . Hence m

lJ =

where t h e since

p

n=l

xn E X.

AnXn

3

The s e t

was a r b i t r a r y .

X

i s complete, thus

11

E X

and

X = R

0

L e t us now i n v e s t i g a t e under what k i n d o f c o n d i t i o n s t h e Choquet boundary i s a minimal boundary. L e t F ( X ) , F(X) t h a t a subset

Y c X

3.6.6

be as b e f o r e . R e c a l l

o f a t o p o l o g i c a l space i s c a l l e d L i n d e l o f i f e v e r y

Y

open c o v e r i n g o f

and

c o n t a i n s a c o u n t a b l e sub-covering.

Theorem:

L e t X be a compact nonempty H a u s d o r f f --space and l e t F(X) be a cone o f uppersemicontinuous 6- valued f u n c t i o n s which ~contains a l l constant t h e f o l l o w i n g asserfunctions and separates t h e p o i n t s o f X . Consider t i o_ ns f or a -

subset

Y c X

.

For t h e Choquet boundary CH(X) ( --w i t h respect t o (i) --Ch(X) c Y (ii)

Y

f o r F(X) -i s-a boundary -

(iii)Every s t a t e of (iv)

F(X)

F ( X ) ) we have --

(o r equivalently f o r F(X)

has measure on - a r e p r e s e n t i n g ~-

or -

Y

s u p X ( i n f fn) = supy ( i n f fn) f o r every decreasing sequence n a nEN

i n F(X) Z

(v)

Ch(X)

(vi)

Ch(X) c Z

-We have:

c

(i)

4

f o r every --

Fo- subset

-

Z

c

-

X

with -

Y cZ

f o r e v e r y L i n d e l o f set Z c X with Y --

(ii) o (iii)

(iv)

4

(v)

(vi).

c

q)

Z.

(f,)

299

Boundaries

Proof: ( i ) + (ii):I t i s c l e a r from t h e d e f i n i t i o n s o f l i k e w i s e t h e Choquet boundary o f

X

Ch(X)

that

F(X),

w i t h respect t o

Ch(X)

F(X) o r

is

5

Hence (ii)f o l l o w s i n view o f C o r o l l a r y 3.2.4. ( i v ) i n c o n n e c t i o n w i t h D i n i ' s Lemma and Lemma 1.3.4, t e l l s us t h a t o ( i v ) f o l l o w from Theorem 3.6.3. i s a D i n i cone. Thus ( i i ) * (iii)

-

(iv)

(v):

Assume t h e r e i s

x B Z.

,

and a l l

U Kn, Kn 5 Kn+l n= 1

Z =

We have

x B Kn

Hence

x E Ch(X) such t h a t

m

for all

for all

n E N

IY

a r e compact.

Kn

n.

By Theorem 3.2.6 t h e r e a r e

'pn

E F(X)

such t h a t

3 2 supK (wn) n Put

F(X)

'pn

and

I0

for all

.

n E N

n

1 '4( ,then we have i n c o n t r a d i c t i o n t o ( i v ) : k=l

fn =

m

supx i n f fn 2 n

-1

Z

n=l

n

> - 3 2 supy i n f fn n

.

Hence ( v ) must be t r u e . (v)

-

(vi):

x ECh(X) with

x

Let Z c X

B U(z)

such t h a t

x

be a L i n d e l o f s e t such t h a t

B

f o r each

Z

Y

z E Z.

zn E Z

Fo- s e t c o n t a i n i n g

course , c o n t r a d i c t s ( v ) . ( v i ) + v) i s t r i v i a l since sets.

0

.

Z

I f t h e r e were

U(z)

By t h e L i n d e l o f p r o p e r t y we c o u l d s e l e c t m

a c o u n t a b l e number o f elements s e t wou d be an

c

we would o b t a i n open neighbourhoods

Y

with

Z

c

U n=l

n(Z,) . The

but not containing

Fo- subsets o f t h e compact

x X

latter

. This,

of

are Lindelof

300

Representing Measures

(ii) ( i ) i n the preceding theorem does n o t h o l d i n general. The f o l l o wing example, due t o Simons [ 297,p. 7061 ,may i l l u s t r a t e t h i s . This example shows furthermore t h a t t h e r e are i n f a c t boundaries which are d i s j o i n t from the Choquet boundary.

3.6.7

Example:

.e""(n) = { f l f : R

L e t n be an uncountable s e t and consider Endow

k?(a)

-,R , f

bounded).

w i t h the weak-star topology w i t h respect t o

k?(n) = I d r p : n -,R such t h a t

E I v ( w ) l < -1.

That i s the coarsest

WE61

topology such t h a t a l l t h e functions

r(n) 3 f

+

<

lP,f

>

1 cp(W)f(W), l p E e'(n)

=

WEQ

a r e continuous.

i s then compact. Consider F = Convc(X) , the continuous convex f u n c t i o n s on X I t i s easy t o see t h a t the extreme p o i n t s o f X are

.

ex(X) = { g E X I g ( W ) =

*

1 for all

w E

nl

.

And t h i s i s o f course the Choquet boundary Ch(X) ( w i t h respect t o F = Convc(X)). F i x x E X , then we f i n d , f o r every h E Convc(X) ( i n f a c t f o r every continuous f u n c t i o n ) , countably many

%

E

1

l (n) such t h a t f o r every y

E X

( J u s t consider the compact subsets For each

n we f i n d f i n i t e l y many

< Ij yx > '4( i n

=

<

( * ) ) a

Y.,Z

J

we have

Iz

Kn =

1'

*

.

*

E X

"mn

> , f o r a l l j, i m p l i e s z d Kn

I Ih(z) - h ( x ) l E

ll(n)

. This

2

Ti1 l

.

such t h a t produces the desired

30 1

Boundaries

Since f o r every 'pn

'pn

t h e r e i s a c o u n t a b l e s u b s e t nn c n such t h a t

n we can f i n d a c o u n t a b l e s e t

outside

= 0

the

'pn

vanish outside

6

outside

x E Ch(X) ( a boundary f o r

1 g(w)

= i 1 or

i s a g a i n a boundary f o r

0

n such t h a t a l l

means t h a t changing t h e values o f

does n o t change t h e v a l u e o f

servation t o the

Y = {g E X

. This

6

c

,g

x

h ( x ) . A p p l i c a t i o n o f t h i s obConvc(X))

yields that

vanishes o u t s i d e a c o u n t a b l e s e t )

Y n ex(X) = $

Convc(X). B u t c l e a r l y

.

L e t us now t u r n o u r a t t e n t i o n t o cases where t h e " p a t h o l o g i c a l b e h a v i o u r " as d e s c r i b e d i n t h e l a s t example cannot o c c u r

3.6.8

.

Corollary:

If, i n Theorem 3.6.6., Y -( i ) - (vi) are equivalent.

i s- a L i n d e l o f -subset o f -

X

T h i s i s c l e a r s i n c e i n t h i s case t h e i m p l i c a t i o n ( v i ) 3.6.9

, -then a l l e

assertions

(i) holds t r i v i a l l y .

Corollary:

Suppose, each

f E F(X)

i s B a i r e measurable. Then f o r a r b i t r a r y Y --

t h e a s s e r t i o n s (ii) - (vi)

are e q u i v a l e n t .

c X

Proof: I t remains t o p r o v e ( v ) e ( i i ) . L e t

f E F(X)

continuous and B a i r e measurable. The s e t a compact B a i r e s e t and t h e r e f o r e a Z = {x E X

I

f ( x ) < supX(f)}

i s an

i t s maximum a t some p o i n t i n Y Ch(X) c Z. B u t Ch(X) Z. T h i s proves ( i i ) .

. Here

I f(x)

{x E X

f

i s upper semiis

t supX(f))

Gu- s e t . We conclude t h a t

Fu- s e t . Assume,

. Then we have

Y

c

Z

f

does n o t a t t a i n and hence,by ( v ) ,

i s a boundary which c o n t r a d i c t s t h e d e f i n i t i o n o f 0

Representing Measures

302

3.6.10 Corol 1a r v : Suppose t h a t each Bore1 s u b s e t o f

X

,& a

X

i s metrizable.

B a i r e s e t . Then f o r a r b i t r a r y

are e q u i v a l e n t .

Y c X a l l assertions ( i ) - ( v i ) Note, t h i s h o l d s i n p a r t i c u l a r i f Proof:

*

Only ( v )

( i ) r e q u i r e s a p r o o f . Assume t h a t t h e r e i s some

x E Ch(X)

{XI . Since {XI i s compact , hence a compact B a i r e set, i t follows t h a t Z = X {XI i s an Fo- s e t . From ( v ) we conclude

such t h a t

Y c X

x E Z = X \

3.6.11

\

{XI which i s impossible.

o

Corollary:

Suppose t h a t assertions ( i )

X

i s f i r s t countable. -Then f o r a r b i t r a r y

-

(vi)

s m a l l e s t boundary (-o r max(x,F(x)) 1. By maX(X,F(X))

for

all -

Y cX

a r e equivalent. I n p a r t i c u l a r , Ch(X) -i s the F(X) ( o r F(X)) and c o i n c i d e s w i t h max(X,lF(X))

we meanyof course,the boundary which we discussed i n

s e c t i o n 3 . l ( w i t h r e s p e c t t o t h e cone F ( X ) ) . R e c a l l , t h a t t r i v i a l l y t h e Choquet boundaries o f

X

w i t h respect t o

F(X)

and IF(X)

and

F(X)

coincide. Proof: Again, we prove ( v ) + (i). F o r any is a

Go- s e t and

p r o o f we conclude

X

\

X

particular

x E Ch(X), {XI

whenever

Y

i s a boundary f o r

Y = max(X,F(X)). On t h e o t h e r hand, b y P r o p o s i t i o n

max(X,F(X)) c Ch(X)

The assumption t h a t

, in

{XIi s an Fo- s e t . E x a c t l y as i n t h e p r e c e d i n g

x E Y. Hence Ch(X) c Y

F(X), i n p a r t i c u l a r i f 3.2.2.,

x E X

,

i.e.

max(X,F(X)) = Ch(X).

0

i s f i r s t c o u n t a b l e i n C o r o l l a r y 3.6.11 i s e s s e n t i a l

i n view o f t h e Example 3.6.7.

Boundaries

3.7

303

REMARKS AND COMMENTS

Section 3.1: The f i x p o i n t boundary was introduced i n [1291. Much e a r l i e r [1131 i t was discovered t h a t t h e s e t w i t h t h e separation property s t a t e d i n Theorem 3.1.5 i s a max-boundary which i s contained i n the Choquet boundary ( s e e a l s o [1311). The Krein-Milman Theorem f i r s t appeared [2061 in 1940 f o r compact convex subsets of dual spaces. The importance of the Maximum P r i n c i p l e was discovered i n 1960 by Heinz Bauer [19 3 . Section 3.2: For function algebras t h e Shilov boundary was i n t h e e a r l y f o u r t i e s introduced by Shilov (published [I391 i n 1946). For more general s i t u a t i o n s i t s existence was l a t e r shown by several authors ( [ 9 1,11671 [ 411,[29 I ) . The most general assumptions seem t o be those s t a t e d i n Corollary 3.2.4, t h i s r e s u l t , again, is due t o H . Bauer [20 1 ( s e e a l s o the e x c e l l e n t survey [ 221). The boundary Max(K) was introduced i n [1131, ry the boundaries Max(K), Maxa,p(K) and t h e weak Choquet boundary were considered i n [1311. Of course,

MaxccdK) generalizes a well known concept

59, p.901 from the theory of function algebras, namely E . Bishop's [40 I c h a r a c t e r i z a t i o n of t h e weak peak p o i n t s . Pioneering work about t h e Choquet boundary, f o r t h e general case of cones of USC- functions was done by H. Bauer, he a l s o c l a r i f i e d the r e l a t i o n between Choquet and Shilov boundary [ 171 ( s e e a l s o [ 411). C r i t e r i o n s f o r the e q u a l i t y Max(K) = C h ( K ) a r e hidden i n [1221 , most of t h e examples t o show t h a t , i n g e n e r a l , t h e inclusions between the boundaries a r e proper a r e taken from [1131 and [1291. The weak Choquet boundary has some i n t e r e s t f o r function algebras s i n c e i t always supports s u i t a b l e Jensen measures [1151, this is not t r u e f o r t h e Choquet boundary. (For general information about Jensen measures s e e [1371). [

Section 3.3: The b e s t sources f o r general information about Choquet's Theorem a r e [2501 and [ 2 I . For t h e case of a metrizable compact convex s e t t h e theorem was f i r s t proved i n 1956 by G . Choquet [67 1 . Later on, s h o r t e r proofs were given, independently, by M. Hervi! [ 1611 and F.F. Bons a l l [ 531. They use t h e f a c t t h a t a compact convex s e t i s metrizable i f f t h e r e i s a continuous s t r i c t l y convex functions f . Hence , f = 'i only ) c l e a r l y be on the extreme points, and a measure with p ( f ) = ~ ( 7 must

304

Representins Measures

c a r r i e d by t h e extreme p o i n t s . F o r n o n - m e t r i z a b l e compact convex s e t s t h e theorem was proved by E. Bishop and K. de Leeuw [ 411. F o r a d i f f e r e n t proof see [114]. I f one extends t h e B a i r e measures c a r r i e d by t h e extreme points o f

t o Borel measures on

K

K

one has t o be v e r y c a r e f u l . T h i s i s

i l l u s t r a t e d by an example [ 731 due t o G. Mokobodzki. He shows t h a t t h e r e i s a simplex

K

such t h a t t h e extreme p o i n t s a r e a B o r e l subset o f

such t h a t t h e r e i s a B o r e l p r o b a b i l i t y measure m on f o r every B a i r e subset

B

3

ex(K)

, and

m(ex(K)) = 0

K with

K and

m(B) = 1

,

(see a l s o [25Oyp.721).

The g e n e r a l i z a t i o n o f Choquet t h e o r y t o t h e case o f cones o f continuous f u n c t i o n s i s t h e work o f many authors. The most i m p o r t a n t c o n t r i b u t i o n s a r e due t o H. Bauer, D.A. Edwards ( f o r example [ 9 8 1 ) , E.B. and

-

N. Boboc

A(K)lex(K)

Davies 184

1

A. Cornea 1471. F i n a l l y , l e t us observe t h a t t h e f a c t t h a t

i s a D i n i cone i s c l o s e l y r e l a t e d t o R a i n w a t e r ' s theorem [2601.

S e c t i o n 3.4:

Most r e s u l t s o f t h i s s e c t i o n a r e s l i g h t m o d i f i c a t i o n s o f

r e s u l t s which a r e w e l l known i n case o f t h e geometric s i t u a t i o n (as general

2 1 ) . Compare, f o r example C o r o l l a r y 3.4.6 w i t h M. Hervh [1611, and Lemma 3.4.5 w i t h G. Mokobozki 12311. The Choquet o r d e r was i n t r o d u c e d i n 1960 by G. Choquet 169 1.

r e f e r e n c e see

S e c t i o n 3.5: The uniqueness theorem i s t h e work o f G. Choquet and P.A. Meyer ([2301,[71 1 , [ 6 9 I and [ 7 3 I ) . Simp1 i c e s w i t h c l o s e d extreme p o i n t s e t s were c o m p l e t e l y c h a r a c t e r i z e d by

H. Bauer [20 I . Many more d i f f e r e n t c h a r a c t e r i z a t i o n s o f Bauer-simplices can be found i n t h e l i t e r a t u r e 2 1 , most o f these r e s u l t s can e a s i l y be o b t a i n e d by a s k i l l f u l a p p l i c a t i o n o f t h e m a t e r i a l i n s e c t i o n I 2.

S e c t i o n 3.6:

A l l t h e m a t e r i a l i s t a k e n from [1291. O f course, by s i m i l a r

arguments as i n C o r o l l a r y 3.6.11, we o b t a i n f o r t h e s i t u a t i o n considered i n

3.6.6 t h a t t h e Choquet boundary Ch(X) i s t h e mimimal boundary i f every x E Ch(X) i s a G6- s e t . I n case t h a t F(X) i s a f u n c t i o n a l g e b r a t h i s was a l r e a d y observed by E. Bishop and K. de Leeuw [41 I . A s i m i l a r r e s u l t was o b t a i n e d by J . S i c i a k [2931. A p a r t from these o b s e r v a t i o n s we do n o t know o f many r e s u l t s about minimal ( n o t n e c e s s a r i l y closed) boundaries.

SECTION 11.4 INTEGRAL REPRESENTATION OF OPERATORS TAKING VALUES I N AN ORDER COMPLETE VECTOR LATTICE

The problem we c o n s i d e r i n t h i s s e c t i o n i s e a s i l y formulated. L e t F(X) a D i n i cone and c o n s i d e r an a r b i t r a r y o r d e r complete v e c t o r l a t t i c e R

.

be

Then we ask i f e v e r y monotone l i n e a r o p e r a t o r

R- v a l u e d measure T on u- a l g e b r a generated by F(X)), w i t h

has a r e p r e s e n t i n g measure, t h a t i s an respect t o

Z

F(X) p(f)

the

5

J f dT X

for all

f E F(X)

X

(with

.

A somewhat more s p e c i a l i z e d q u e s t i o n would be whether o r n o t Choquet's fundamental theorem c a r r i e s o v e r t o t h e case when t h e r e a l s a r e r e p l a c e d by t h e o r d e r complete v e c t o r l a t t i c e R . F i r s t one m i g h t g e t t h e i d e a t h a t i t i s p o s s i b l e t o adapt a l l t h e d e t a i l s of t h e p r o o f o f t h e R e p r e s e n t a t i o n Theorem t o t h e p r e s e n t s i t u a t i o n . T h i s i d e a i s e s p e c i a l l y i n t r i g u i n g s i n c e one o f o u r main t o o l s ( t h e Sandwich Theorem) c a r r i e s o v e r c o m p l e t e l y t o t h e p r e s e n t s i t u a t i o n . B u t soon one d i s c o v e r s t h a t an e s s e n t i a l step, namely t h e P a r t i a l Decompos i t i o n Theorem 1.2.4,

cannot be t r a n s f e r r e d t o t h e s i t u a t i o n where

R

takes t h e p l a c e o f t h e r e a l s , a t l e a s t n o t w i t h o u t assuming a d d i t i o n a l p r o p e r t i e s for

R

,l i k e

weak

u- d i s t r i b u t i v i t y (see Appendix A 5 ) .

Another r e s u l t which cannot be e s t a b l i s h e d f o r t h e general case i s t h e Daniel1

-

Stone Theorem (see[3391,[338Iand[3411), a l s o more o r l e s s a

consequence o f t h e P a r t i a l Decomposition Theorem. 305

306

Representing Measures

B u t f o r t u n a t e l y t h e s i t u a t i o n i s n o t as hopeless as i t seems a t t h e f i r s t glance. I t t u r n s o u t t h a t a s i m p l e t r i c k a l l o w s t o deduct from t h e Rep r e s e n t a t i o n Theorem i t s analogy f o r t h e v e c t o r - l a t t i c e s i t u a t i o n under

R

c o n s i d e r a t i o n . For t h e case when

C(S)

=

,S

compact and e x t r e m a l l y

disconnected, t h e t r i c k c o n s i s t s i n a p p l y i n g t h e Representation Theorem

uy s E S

u

for

6,

the

i n f i m a o f bounded subsets o f

0

instead o f

and t h e n u s i n g t h e f a c t t h a t

C(S) a r e n o t t o o d i f f e r e n t f r o m t h e

S- p o i n t w i s e i n f i m a . Although t h i s l a s t argument w i l l be d i s g u i s e d i n a

lemma about maximal s t a t e s i t i s n o t t o o d i f f e r e n t from t h e argument we used i n t h e c o n s t r u c t i o n of t h e Loomis-Sikorski homomorphism (Theorem I.2.3.3).

We a r e going t o f o l l o w v e r y c l o s e l y t h e arguments o f [1341.

R

L e t us b e g i n w i t h t h e d e t a i l s . We s t a r t w i t h t h e s i t u a t i o n when Again

F(X)

empty

X

F(X)

=

C(S).

i s a cone o f upperbounded R- valued f u n c t i o n s on some non-

. We

endow

F(X)

w i t h t h e p o i n t w i s e o r d e r on

c o n t a i n s a l l c o n s t a n t R-. valued f u n c t i o n s .

-

always compact and e x t r e m a l l y disconnected.

-

function being

i d e n t i c a l l y on

S

-

X

-

-

I n the following i s

The element

and

and assume t h a t

-

S

means t h e

i s d e f i n e d t o be t h e

C(S) which i s n o t bounded from below. The

infimum o f e v e r y subset o f

a l g e b r a i c o p e r a t i o n s a r e d e f i n e d as i n 1.1.

A l i n e a r monotone u : F(X)

-,C ( S )

u ( f ) 5 Is s u p X ( f )

u

As b e f o r e we say t h a t with

I.!

i s a maximal

we must have u = u

5 v

l i s h e s f o r every

S- s t a t e

V

w i l l be c a l l e d an S

for a l l

lS i s t h e i n d i c a t o r f u n c t i o n on

where

u

u I- -1

.

-

state i f

f E F(X),

i.e.

S

lS(s) = 1 f o r a l l s

S- s t a t e i f f o r every

E S.

S- s t a t e

Again by Z o r n ' s Lemma one estab-

t h e e x i s t e n c e o f a maximal

S- s t a t e

u 2

G.

Now t h e c r u c i a l argument: 4.1

Lemma:

L e t u : F(X) -

the value --

-

+

Q)

C(S)

.

U

I- -1

F o r each --

be some maximal s E S

define

us

S- s t a t e which never a t t a i n s ---I___

t o be t h e r e a l valued s t a t e

307

Integral Representation of Operators

p c J let

ps(f) = p(f)(s)

g i v e n by --

s t a t e s w i t h us --

S

Y ~ , SE

, be maximal r e a l v a l u e d

f E F(X), the set

I pS. Then, for each

i s meagre. Proof: Denote b y

: F(X) + R s

Y

s E S, f E F ( X ) .

t h e map g i v e n b y

And l e t

A(f) c C(S)

A ( f ) = { W E C(S)

14s)

Y(f)(s) = YS(f)

for all

be t h e s e t for all

t Ys(f)

s E Sl

.

As i n P r o p o s i t i o n I 2.3.2 we denote w i t h i n f and I n f t h e i n f i m a i n and Rs

r e s p e c t i v e l y . One e a s i l y notes

contains

supX(f)lS)

? : F(X)

a functional checks t h a t

that

and bounded f r o m below +

C(S)

by

A ( f ) i s nonempty ( i t p ( f ) E C(S)). We d e f i n e

(by

i ( f ) = i n f A ( f ) , f E F(X).

i s s u b l i n e a r , f u r t h e r m o r e i t i s monotone s i n c e

monotone. From t h e obvious i n e q u a l i t y p o s i t i o n I 2.3.2

C(S)

Y(f) 5 I n f A(f)

One e a s i l y Y

was

we g e t w i t h Pro-

that

Is E s I G ( f ) ( S ) < Y S ( f ) l f E F(X). We c l a i m t h a t

i s meagre,for a l l

p =

\r

which t h e n , c l e a r l y ,

proves t h e 1emma. Proof o f the claim: F i r s t we d e f i n e a " H i l f s f u n k t i o n a l " ( a u x i l i a r y f u n c t i o n a l ) bY Y (f) = infIi(f+g)

Then

p Iy I Y

t h e r e i s some funktional

.

Now, we prove

fo E F(X)

6 : F(X)

+

with

C(S)

by

-

p(g)

p = y

Ig

E F(X)I

. I n order

y :

F(X) - r C ( S )

. t o do t h a t l e t us assume

p ( f o ) < y ( f o ) . D e f i n e a second H i l f s -

308

Representing Measures

Then

i s s u p e r l i n e a r and

6

t h e n an S- s t a t e maximality o f

u

.

;I .

!.I I 6 I

w i t h 6 Ip I @ Hence !.I = y . p

By t h e Sandwich Theorem t h e r e i s

, clearly

i n contradiction t o the

Now t h e p r o o f i s completed by l o o k i n g a t t h e f o l l o w i n g i n e q u a l i t i e s :

The V e c t o r Valued Representation Theorem[l341:

4.2

Let

'

F(X)

be 5 -cone o f

upper bounded

6-

valued f u n c t i o n s

contains a l l c o n s t a n t -r e a l - v a l u e d f u n c t i o n s -and l e t R

--

a r e. e q u i v a l e n t : t e v e c t o r l a t t i c e . Then t h e f o l l o w i n g _ ---

(i) F(X) (ii)

0" X

which

be an o r d e r ----

comple-

i s a Dini cone ----

Every monotone l i n e a r !.I

has an

_.-

i-

: F(X)

-+

valued

k

= R U 1- -1

r e p r e s e n t i n g measure.

Proof:

(i) * (ii):L e t e = ~ ( 1 ,~t h e) n e t 0. By R(e) R g i v e n by

we denote t h e o r d e r

complete s u b - v e c t o r - l a t t i c e o f R(e) = { a E R

I there

is a

x

E R+

with

-x e

I a It

xe I .

Since R(e) i s an o r d e r complete o r d e r - u n i t v e c t o r l a t t i c e we know from I 2.2.3 and I 2.3.1 (Kakutani-Krein-Stone-Yosida) t h a t R(e) i s

309

Integral Representation of Operators

order-unit and vector-lattice isomorphic t o some C(S) , S compact Hausdorff and extremally disconnected. For convenience we j u s t write R(e) = C ( S ) . Now, l e t

VF and VFB again denote the max-stable cone generated by F(X) and i t s subcone of bounded functions, respectively. We know t h a t both are Dini cones (Lemma 1 . 3 . 4 ) . By p we denote the sublinear functional on VF

given by

Then

p Ip

IF(X) linear G : VF -t

and the Dominating Extension Theorem gives us a monotone

u 5 iiI F ( x ) and i; I p.

with

t o VFB , then

restriction of

G

G

we denote the

i s an S- s t a t e of

Zorn's Lemma we get a maximal S- s t a t e t h a t there i s an R(e)- valued measure v ( g ) = J g d.r X

By

for a l l

v T

VFB

on VFB with with

g E VFB

C

. From

Iv

. We

claim

.

This ,clearly ,proves the assertion since we have p(f)

I inf v ( f

nEN

v ( -n e ) )

5 inf nEN

X

(f

v ( - n e ) ) d.r

=

X

f dr

.

Proof of t h i s claim: Choose for every s E S a maximal real-valued s t a t e v ( g ) ( s ) 5 YS(g)

for a l l

g E VFB

yS

on VFB such that

.

From Lemma 4 . 1

we obtain t h a t , for every g E VFB , the s e t

(*I

I v(g)(s)

{s E S

\ys(g)}

i s meagre.

The Representation Theorem 2 . 2 . 1 gives us real-valued probability measures p s on X such t h a t

Represn tins Measures

310

for all (equality since

i s maximal). W i t h these measures we can d e f i n e a

yS

-,Rs

by

I h d ps

,

l i n e a r monotone Y : M y(h)(s) =

vFB

X

h E M y s E S,

where M stands f o r t h e v e c t o r l a t t i c e o f a l l bounded, r e a l valued and

k)- measurable f u n c t i o n s L

= {h E M

I

on

X

. NOW,

t h e r e i s some and y ( h )

From ( * ) we know t h a t

VFB c M

l e t us c o n s i d e r t h e v e c t o r space

n ( h ) E C(S)

. As

i n t h e p r o o f o f Theorem I 2.3.3 h E L

.

Sl

,

t h e continuous f u n c t i o n

i s unique.

L

One e a s i l y v e r f i e s t h a t L

n(h)

d i f f e r o n l y on a meagre subset o f

B a i r e ' s c a t e g o r y theorem t e l l s us t h a t f o r n(h)

such t h a t

-,C(S).

t h e cone

i s a v e c t o r l a t t i c e and t h a t

n

n and

v

Furthermore ( * ) t e l l s us t h a t t h e maps VFB

.

i s l i n e a r from c o i n c i d e on

L

Since c o u n t a b l e unions o f meagre s e t s a r e a g a i n meagre one o b t a i n s t h a t is

U-

complete.

bounded sequence i n L functions

(f,)

To be more p r e c i s e , l e t

I n f n(fn) n

, with and

f = i n f f,.

n

be a decreasing l o w e r

Then b y P r o p o s i t i o n 1.2.3.2

the

d i f f e r o n l y on a meagre subset, t h e r e -

i n f q(fn) n

f o r e ( s i n c e t h e meagre s e t s a r e s t a b l e a g a i n s t c o u n t a b l e unions and because o f t h e Monotone Convergence P r o p e r t y o f t h e measures and Y ( f )

agree except on a meagre s u b s e t o f ~ ( f =) i n f q ( f n ) n

Hence

L i s u- complete and

since

M

i s the smallest

which c o n t a i n s

n

(i.e. q

S

inf n

(f,)

f E L

and

ps)

. Thus

i s u- monotone).

i s u - monotone. Because o f

L

3

VFB and

u- complete v e c t o r l a t t i c e o f bounded f u n c t i o n s

VFB we have

L

= M.

Since

q

is

U-

monotone we know

31 1

Integral Representation of Operators

from the Appendix (section A 5 ) t h a t respect t o a C(S)- valued measure 'I i s proved. (ii)

4

( i ) i s rather t r i v i a l . Let

F(X) and choose some e s t a t e p with

5

-

o in R

q

.

(f,)

i s given by an integral with Because of rl = u the claim I vFB be a decreasing sequence in

J.

. Lemma

I 1.2.6gives us a 6- valued

inf u ( f n ) = inf supx f n . n n Then

if p(f) > -

m

else

- m

defines an R- valued monotone linear functional which has by ( i i ) a representing measure T . And application of the Monotone Convergence Property ( i . e . the f a c t t h a t the integral d.r i s u- monotone) X yields easily

-

supx inf f n n

2

inf n

p(fn)

.

Hence F(X) must be a Dini cone since the decreasing sequence was chosen arbitrarily 0

.

This vector valued representation theorem has some well known results as consequences. First of a l l an analogue of the Riesz Representation Theorem due t o J.D. Maitland Wright: 4.3 Corollary [340]: Consider C ( K ) , the continuous real valued functions on some compact space K -and l e t R be an order complete vector l a t t i c e . Then every monotone : C ( K ) -,R has a n R- valued representing measure (with respect linear

-Proof: C(K)

i s a Dini cone and

'C( K )

=

Bo(K)

(Baire s e t s ) .

0

312

Note t h a t

Representing Measures

K

i s n o t required t o be Hausdorff.

One should observe that from t his theorem one easily recovers the existence of the Loomis-Sikorski homomorphism (Theorem I 2.3.3) by p u t t i n g C ( K ) = C(S) = R, S extremally disconnected. Another consequence i s a vector valued Choquet Theorem:

4.4 Corollary [1341: K be-a compact --______convex subset of a locally convex Hausdorff vector space, A ( K ) the realvalued continuous affine functions on K and let R be an order complete vector l a t t i c e . Then for every monotone linear ---~ ---

Let -

I.I

: A(K)

-+

R

there e x i s t s an R- -~ valued measure T (with - respect -t o the --ex(K) n Bo(K)) ---on the extreme points ex(K) -such t h a t

u- algebra

Proof:

A ( K ) I ex( K ) i s isometrically order isomorphic t o A ( K ) ( c . f . Lemma 3.3.1) 0 and furthermore i t i s a Dini cone (Lemma 3.3.2). For weakly u- dist ri buti ve R t h i s re sult was already obtained by G.F. Vincent-Smith [3191.

SECTION 11.5 GENERALIZED HEWITT - NACHBIN SPACES

In t h i s chapter we t r e a t y a s a n application of the preceding methods, some topological aspects of the theory of continuous functions in a generalized way.Using the notions of characters and Dini cones,which we have discussed in preceding sections,it i s possible t o transfer classical notions such as Stone - Czech - compactification, realcompactifi cation and pseudocompactness from the situation of a completely regular topological space X t o a general s e t where a given order unit cone F ,consisting of bounded functions, takes overthe place of the continuous, or bounded continuous,functions on X in the classical situation. I n the f i r s t section we introduce the notion of F- compactification and F- realcompactification. We show t h a t , indeed, these definitions include the classical definitions of Stone - Czech - and realcompactification. Our

generalizations allow us t o describe these compactifications easily e.g. by means of decomposition properties of s t a t e s or integral representations of s t a t e s and characters. So, many theorems in the following a r e j u s t recollections of what we have proved so f a r , which, in t h i s context, i s seen from a different angle and obtains a new meaning. There i s one new aspect: Here we also discuss f i l t e r convergence properties. Finally, we consider some applications. Forexample, a generalization of G1 icksberg's integral representation theorem ( o r Alexandrov - G1 icksberg theorem [317]) ,for pseudocompact spaces which i s , ultimately, contained already in Theorem I1 2 . 2 . 1 and Theorem I1 3.3.6 and, therefore, sheds a new l i g h t on these theorems. 313

314

5.1

Representing Measures

BASIC DEFINITIONS AND THEIR MEANING IN THE CLASSICAL SITUATION

Throughout t h i s chapter l e t X a nonempty --s e t and l e t F be an order unit cone consisting of bounded realvalued functions on X , which separate -always denotes the s t a t e space of F . the points of X . R -----2 . 6 . 3 ) t o general

We want t o extend the notion of Dini continuity (see s t a t e s u on F . 5.1.1 Let

Definition: p

be a s t a t e on F.

p

i s called Dini continuous

whenever f n E F i s pointwise decreasing such t h a t

if

inf f n n

inf u ( f n ) I 0 n I 0

.

This definition indeed includes the former definition of Dini continuity since we have 5.1.2 i)

Lemma: p

i s Dini continuous ---i f and only i f -inf p ( f n ) n

5

supX(inf f n ) n

for a l l pointwise decreasing sequences f n

_.-

ii)

If

F i s max-stable,then p p ( f n ) -,0 for a l l sequences

E F.

i s Dini continuous -i f and

fn E

on’ly if F pointwise decreasing -t o zero.

Proof: i ) The sufficiency i s obvious. To prove the necessity assume inf p ( f n ) > a > supX(inf f n ) for some a E R and some decreasing n n sequence f n E F. Then, with g n = f n - a , we have, g n i s decreasing, inf g n I n

0

but

inf n

p(g,)

> 0

, a contradiction.

i i ) We only need t o prove the sufficiency of t h i s condition. Indeed, l e t f n E F be a r b i t r a r i l y decreasing such t h a t inf f n I 0 . n

315

Generalized Hewitt-NachbinSpaces

Then p(max(f,,O))

+

0 by assumption.

A Dini continuous character of F

5.1.3

i)

is called Dini character.

Definition:

Put p XF v XF p XF

= {p

E n

= Ip E

n

I I

p

character) and

p

Dini character3

.

is called the F- compactification of X ,

v

XF is called the

F- realcompactification (or F- Hewitt-Nachbin-completion). Moreover, X is defined to be F- compact, if p XF c X . Similarly, X is defined to be F- realcompact (and sometimes called F- Hewitt-Nachbinspace) if v XF c X . ii)

X is called F- pseudocompact if every element of the sup-norm closure VF of VF attains its maximum on X . (Recall VF = {max(fl ,..., fn) I fl,...,fn E F, n E N1).

-

Rema rk :

If we take F to be the continuous, bounded realvalued functions'on a completely regular topological space X , denoted by CB(X) , then p XF is just the classical Stone-Czech-compactification. Indeed, recall that the Stone-Czech-compactification p X of a completely regular space X is the set of all multiplicative nonzero linear functionals on CB(X) endowed with the weak-star topology. This means that p X coincides with the set of all lattice homomorphisms S on CB(X) (see 1 . 2 . 2 ) . Indeed, by the Kakutani-Krein-Stone-Yosida Theorem the sup-norm closed vector lattice CB(X) is isometrically isomorphic to C(S). So any multiplicative nonzero linear functional p on CB(X) corresponds to a multiplicative linear functional on C(S) which must be the Dirac functional at some element in S . Therefore p E S . Conversely, by the same theorem, every p E S is multiplicative on CB(X). Clearly, S is the extreme point set by Theorem of the state space of CB(X). Therefore, B X = B X CB(X) I 2.10.6 ).

316

Represenring Measures

S i m i l a r l y , the realcompactification o f a completely regular Hausdorff space

X,

v X

a l s on CB(X) on

i s t h e s e t o f a l l m u l t i p l i c a t i v e nonzero l i n e a r f u n c t i o n which can be extended t o m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l s

C(X), t h e space o f a l l continuous r e a l v a l u e d f u n c t i o n s on

general we have,

*

vX c

BX

X.

In

(examples see below). Note, t h a t if

E p X w i t h p ( f ) E f ( X ) f o r a l l f E CB(X) t h e n p can be represented by a zero-one measure m on X , i . e . m(A) E 1 0 , l ) f o r a l l Bore1 s e t s A . T h i s i s a consequence o f z e r o - s e t f i l t e r t h e o r y ([1401).Thereforey i n t h i s case, p can be extended t o a m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l on C(X), thus p E v X . Conversely, i t i s w e l l known t h a t i f p i s a m u l t i p

p l i c a t i v e l i n e a r f u n c t i o n a l on

C(X)

then

p ( f ) E f(X)

f E CB(X)

for all

(see Theorem 5.3.1 below). As general r e f e r e n c e s f o r t h e c l a s s i c a l s i t u a t i o n we r e f e r t o [ 1 4 0 ] ,[3201and [3241. 5.1.4 Let X true: -

Proposition:

-be_ a

completely r e g u l a r H a u s d o r f f space. Then t h e f o l l o w i n g

are

ii)

Proof:

i) f o l l o w s d i r e c t l y f r o m t h e preceding remarks. ii)Consider

p

E v X cB(x)

c pXcB(x) = p X

and a l a t t i c e cone homomorphism on

. Hence

p

i s multiplicative

CB(X). We c l a i m t h a t

p(f) E f(X)

a l l f E CB(X) from which, b y t h e p r e c e d i n g remarks, we o b t a i n and t h e n v X CB(X) *

p

for

E vX

f E CB(X) a r b i t r a r i l y and assume p ( f ) B f ( X ) . Consider t h e

So t a k e sequence

fn = max(1 - n ( f - p ( f ) ) 2 ,0)

which i s t h e n p o i n t w i s e decreasing t o

zero. Hence we o b t a i n t h e c o n t r a d i c t i o n

since

p

was D i n i continuous. Here we used t h e f a c t t h a t

p

i s a multi-

317

Generalized Hewitt-Machbin Spaces

plicative l a t t i c e cone homomorphism on C B ( X ) . Therefore p ( f ) E f(X). Conversely, l e t

p E

uX

. Consider

a sequence

decreasing t o zero,and assume inf u ( f n ) n Kn = I T E B X

I

T ( f n ) 5 pa 1.

=

f n E C B ( X ) , pointwise

a > 0

i s compact and

Kn

consists of the Dirac functionals

bX

at x

E X.

. ?

Consider c

U Kn nEN

Furthermore

, where p

B

U Kn. nEN

Therefore, by Urysohn'slema, we find a function g E C ( B X ) with g(p) > g ( T ) for a l l T E U Kn Indeed, consider g n E C ( p X ) with nEN

.

m

.

1 9,. ' 'nIKn = o for a l l n , a n d p u t 9 = n x= l 2n Consequently, with g ( x ) = b x ( g ) for a l l x E X , we have

0

5

gn

~ ( 6 =)

1

, gn(ll)

g(p)

B g(2)

5

fact t h a t

p

= 1

g(X).

By the preceding remarks t h i s contradicts the i s extendable t o a multiplicative linear functional on C ( X ) . =

The notion of F- pseudocompactness clearly coincides with the classical one i f F= C B ( X ) . Recall t h a t there are indeed non-compact completely regular Hausdorff spaces X which are nevertheless C B ( X ) - pseudocompact. An easy example ([1401) i s X = {a I a ordinal number with a < w13 endowed with the order topology; here

w1

i s the f i r s t uncountable ordinal. The pseudo-

compactness here follows easily from the fact t h a t every sequence in X has a cluster point. B u t for general F we have already more examples of F- pseudocompact s e t s a t hand. I t follows e.g. from 1.3.3 t h a t X i s F- pseudocompact whenever F i s a sup-norm closed max-stable Dini cone. We have already seen that F being a Dini cone i s equivalent t o X being F- pseudocompact (Improved Representation Theorem 3.6.3 with Y = X ) . Let K be a compact convex subset of a locally convex Hausdorff space and denote by ex(K) i t s extreme p o i n t s e t . Another interpretation of Bauer's Maximum Principle (11. 3.1.6) i s t h a t ex K i s A(K),exK-and ConvB(K)lex Kpseudocompact. Here A ( K ) = {f:K + R I f affine, continuous), ConvB(K) = tf:K + R [ f convex, uppersemicontinuous, boundedl.This i s a n easy consequence o f Theorem 5 . 4 . 1 below.

fact

318

Representing Measures

5.2 THE F- COMPACTIFICATION We want t o s t u d y f i l t e r p r o p e r t i e s o f t h e s e t s s u p p o r t i n g c e r t a i n l i n e a r f u n c t i o n a l s . L e t us f i r s t i n t r o d u c e d i f f e r e n t f a m i l i e s o f subsets o f

X

which we s h a l l discuss i n t h i s c o n t e x t . Let

be a s t a t e o f

p

subsets

Z(p)

i s defined t o be t h e f a m i l y o f a l l

such t h a t f o r e v e r y a < p < 0

Z c X

, and

f I0

F. Then

p(f)

2 p > a 2 sup

there i s

(X L Z ) ( 0

i s c a l l e d t h e s e t o f s t r o n g domination f o r

Z(p)

f E F with

p

.

Furthermore, we c o n s i d e r

x I

D(p) =

CY c

N(u)

{Y c X

I

supy]

P 5

p $ SUP

(XLY)}

U(P) i s t h e s e t of domination, for

u

.

*

i s c a l l e d t h e complementary s e t

N(p)

Recall, by Face(p) we mean t h e c o l l e c t i o n o f a l l t h o s e s t a t e s such t h a t t h e r e a r e 0 < l,l I x v + ( l - A ) v o . We c l e a r l y have

i f and only i f

I 1 and a s t a t e

Face(v) c Face(p)

Furthermore, o b v i o u s l y , F

x

p

whenever

of

v0

v E Face(p)

Let i)

n be a r b i t r a r y . --Then we have:

Z(u) c N ( I J ) n D ( v )

ii) Z ( P ) c Z ( V ) iii)

F

R

of

Face(p) = { p l (see Remarks 2.5.2 and 2.5.8).

Lemma: p E

of

(see 11. 2.5.2).

i s an extreme p o i n t o f t h e s t a t e space

A t f i r s t we prove some t e c h n i c a l f a c t s concerning

5.2.1

u

F with

for all

Z1 n Z2 E N(v)

v E Face(u)

for all

Z1,Z2 E

Z(p)

P ( u ) , N(u) and Z(v).

319

Generalized Hewitt-NachbinSpaces

Proof: i)

follows from the definition. We claim that

Z ( p ) c N(p)

whenever Z E

This proves

Z(p).

Indeed, l e t Z E there are s t a t e s

Z(p) c D ( p ) .

By the Finite Decomposition Theorem I , p 2 and 0 I x I 1 with and

(1-x)p2

p I A p1 t

According to the definition of Z(p),with a obtain f E F , f I 0 such t h a t

Hence, in view of

Therefore

0 I

p I

i i ) Let v

A

Hence iii)

I

A p <

sup

x

1 n which means

v(f)

Z1,Z2 E p I

x

I

E

1

t

Z(p)

- x

I

x

- 1> - n

2

= - 1

we

sup ( X \ Z ) ( f )

=

. Consider f E F, f

x

z,))

=

I

(l-X)vo

0 , with

p > a t sup ( X \ Z ) ( f )

Z

E

Z(V).

we would have

N(p)

"'Pyl

w t

some real numbers a < B

( f ) which implies

, b u t Z1 n Z2 6

sup ( x \ ( z l n

since n was arbit.rary.

0

Then we have N 5 X

(l-h)vo(f) 2 p(f) t (X\Z)

n.

D(u).

Z(p).

and there i s

v ( f ) 2 p > a t sup If

(f)

(X\Z)

and 0 <

0

- n , n > 1, p

,

Face(u) and Z E

E

v(f) t

x

A

=

(l-A)supZ(f) 2 p ( f ) t

supz and hence Z

f o r some v0 E n

Then a

f I 0

1 s p(f) 5

This implies

+

(f)

A sup(x\z)

-

1.4.4

Z(p). pl

supz

p I

u

y2

-

0.

Representing Measures

320

with

Y i = XxZi,

we would o b t a i n

0 Ix I1 w i t h

p I

Then c o n s i d e r

By t h e F i n i t e Decomposition Theorem 1.1.4.4

i=1,2.

x

a,P

supy with

1

+ (l-x)supy

2

.

a c 2 p < p < 0 and

flyf2 E F, fl I 0, f 2 I 0,

such t h a t p ( f i ) 1 p > a 2 sup

'i

(fi)

,i=

1,2

.

We a r r i v e a t t h e c o n t r a d i c t i o n 2B I

p(fl

+ f 2 ) I x supy (fl) 1

+ (l-x)supy (f2) 2

5 a

.

Z1 n Z2 E N ( p )

Hence

NOW, we f o r m u l a t e t h e main c h a r a c t e r i z a t i o n o f t h e F- c o m p a c t i f i c a t i o n which i s , p a r t i a l l y , a r e c o l l e c t i o n o f former theorems.

5.2.2

Theorem:

The f o l l o w i n g are equivalent:

_.

i)

P

ii)

p

E P XF

i s a character of F

iii) p has t o VF ---and f o r every f i n i t e - -a unique dominated e x t e n s i o n c o v e r i n g Y1,...,Yn o f X t h e r e i s some k I n such t h a t ----

iv)

p

can be extended t o-a c h a r a c t e r o f VF

v)

p

can be extended ---t o a l a t t i c e cone homomorphism VF --

maximal s t a t e -vi)

p

vii)

P

viii) p

+R

being a

E ex n can be extended t o- an nt_ o f- t h e s t a t e space aVF of VF -- _extreme _ _ _ p_o i_ i s maximal and Z(P) = N ( p )

321

GeneralizedHewitr-Nachbin Spaces

If

ix)

X

Z(p)

&

identified, b-y evaluation, with a subset of n , then i s a f i l t e r base of a _ f i l_ t e_ r -in n converging 2 p .

Proof: i)

o

ii)

i s j u s t the definition of i s Theorem I

i i ) o iv) ii)

o

-

v)

.

2.10.10 i )

i s Corollary I 2.10.4 together with Theorem I

2.10.10.

i s Theorem I 2.10.6

i i ) o vi) ii)

D XF

v i i ) i s Theorem I

2.10.6 together with Theorem I

2.10.10.

i i ) =$ i i i ) : The uniqueness of the dominated extension follows d i r e c t l y from the definition of a character (Definition I 2.10.1). Let Y 1 , . . . , Y n be a f i n i t e covering of X . From the Finite Decomposition Theorem I

n z

i=l

A. = 1

1.4.4 we obtain s t a t e s n 1 such t h a t p I x. i=l

Since, by v i ) , p 5 supy . k

p

E ex n

pi xipi

, there i s

k

and real numbers and In

p . I supy 1

i

such t h a t

xi

L

0

for all

p =

with i

.

u k , hence

i i i ) =$ v ) : Let fi be the unique dominated extension t o VF . Then fi clearly i s maximal. Another consequence of i i i ) , the uniqueness, the covering property and the Dominating Extension Theorem I 1.3.1 i s t h a t ;5 supy f o r some k I n , whenever Y1, ...,Y n i s a f i n i t e covering of k X . Finally, because p can be extended t o the vector space plF = F - F which i s a subspace of VF-VF , which means there i s an extension

<

of p and then, by uniqueness, = fi . (Recall, t h a t a l l f E F are bounded, hence p i s realvalued.) TO complete t h i s p a r t of the proof we show t h a t ; i s a l a t t i c e cone homomorphism. Take f l , . . . , f n E VF and consider the covering

U : VF

Y Let

+

W

=

{x

k In

E X

I f .1( x ) L max(fl(x),..., f n ( x ) ) l , i = l , ..., n .

be such t h a t

p

I supy

k

.

322

Representing Measures

By t h e Sandwich Theorem t h e r e i s a l i n e a r , with

i; I v 5 supy

f(x)

Ig ( x )

i.e.

for a l l

v ( f ) Iv(g)

whenever

f,g E VF

and

x E Y k . From t h e uniqueness o f t h e e x t e n s i o n we

. I n particular

i n f e r i; = v i;(fk)

..., f,))

2 C(max(fl,

. .,:(fn))

Hence max(?(f,),. Thus

,

k

Yk- o r d e r - p r e s e r v i n g s t a t e v

2 C(fi),

i=1,.. .,n

.

. . ,fn)).

= i;(max(fl,.

i; i s a l a t t i c e cone homomorphism.

.

i s maximal s i n c e i t i s an extreme p o i n t o f n F u r t h e r more, we know by Lemma 5.2.1 t h a t Z ( p ) c N ( p ) . Consider Z € N ( p ) . Then we vi)

4

viii):

claim,

p

x

0 > p > a

and p u t

t (1-x)supx (X\Z) would o b t a i n s t a t e s u1,p2 w i t h p I x p l t ( l - x ) p 2

have

u1 5

p I

sup

SUP(^,^),

p 2 I supx

x

p a-l s i n c e by t h e Sum Theorem I

Indeed, f i x

Z E Z(p).

. T h i s would

imply

pl

= p

=

,

f o r some

Then

f

N(p).

1.4.1 we

because

p

p I sup

was (X\Z)

Thus we have

g E F. Put

i s w e l l defined,

u(f) > Therefore, viii) base.

Z E

cannot

where

assumed t o be an extreme p o i n t . Hence we would a r r i v e a t which c o n t r a d i c t s

. We

> a

Z E Z(p)

*

ix): v i i i ) L e t G 1 Z(u)

From Z(p) =

N(p)

=

f E F, f I 0, and

sup ( X \ Z ) ( f )

and hence

N(p)

*

= Z(u).

and Lemma 5.2.1 i i i ) show t h a t Z(p) i s a f i l t e r be an u l t r a f i l t e r converging t o some i; E n

we d e r i v e

.

GeneralizedHewitt-Nachbin Spaces

p I

supy f o r a l l

323

Y E G

s i n c e otherwise t h e r e i s Y E G with p Ir s u p y and hence X Y E N ( p ) = Z ( p ) c G which c o n t r a d i c t s t h e f i l t e r p r o p e r t y o f G . Therefore p 5 , and , by maximal i t y of p , p = 11 . Hence Z(p) has only one accumulation p o i n t , namely p . That i s , Z ( p ) converges t o p s i n c e n i s compact. i x ) * vi):

Let u Z(U)

Hence

v I SUPy

E

Face(p). Then we have, by Lemma 5.2.1,

= Z(V)

for all

v I p

c Wv).

Y E

Z(p)

for all

and thus

v 5 lim Z ( p ) = p

. We obtain

v E Face(p)

and t h e r e f o r e , by d e f i n i t i o n of Face(p), v = p which means Hence p E ex n . This completes t h e proof of Theorem 5.2.2.

Face(u)

=

{PI.

324

5.3

RepresentingMeasures

THE F- REALCOMPACTI FICATION

I n t h i s section we turn t o various characterizations of the F- realcompactification u XF Recall t h a t ZF i s the smallest u- algebra on X such

.

that a l l elements in F are measurable. Let measure m i s said t o represent p i f p ( f )

p

5

E n. Recall, a IF f d m for a l l f E F. X

If equality holds for a l l f E F , then m i s called s t r i c t l y representing. Let E be the sup-norm closure of VF - VF. Note, that every s t a t e of VF has a unique extension t o a s t a t e of VF - VF a n d , furthermore, t o a s t a t e of E . (Recall, VF consists of bounded functions on X ) . By definition, a l l states are continuous with respect t o the sup-norm. Let p be a character of F . Then by Theorem 5.2.2 and the preceding remark, p has a unique extension t o a character of E Occasionally we denote this extension again by p , because no confusion can a r i s e . Since E i s a vector l a t t i c e , p on E necessarily i s a l a t t i c e homomorphism (see I 2 . 2 . )

.

Let Y c X and assume 1-1 I supy on dominated extension of the character we obtain p 5 supy on E .

F. I n view of the uniqueness of the p

t o E and the Sandwich Theorem

5.3.1 Theorem 11273: Let be-a character o f F. -Then the following --a r e equivalent: - p -

i)

p

i s Oini continuous

ii)

For every sequence f n E F with - fn there i s x E X with r f n ( x ) > -nEN

iii)

p

iv)

p

has -

a ( s t r i c t l y ) representing

I 0

and

1 p(fn) > nEN

-

OJ

- O D

ZF-

measure

has fi ( s t r i c t l y ) representing zF- measure m which i s a 10,l)- measure, i . e . m ( A ) = 1 or m ( A ) = 0 -for a l l A E IF

325

Generalized Hewitt-Nachbin Spaces

with -

Far

vi)

p I supy

Z

any sequence no nemp ty

the i n t e r s e c t i o n

E Z(p)

n

fl

Czn I

f E E

vii)

u(f) E f(X)

viii)

F o r e v e r y f E E -t h e r e i s x E X with u ( f ) If ( x ) -u , as d c h a r a c t e r o f E , -i s D i n i continuous.

ix)

n E NI

whenever

Proof:

i)=9 ii)i s an easy consequence o f t h e D i n i c o n t i n u i t y o f u 2.5.7 iii) ii)t o g e t h e r w i t h 2.5.8. ii)a iii) i s Theorem (Recall,

iii)* i v ) : If

p

as a c h a r a c t e r i s an extreme p o i n t o f Let

m

and

R

. i s maximal).

p

be t h e r e p r e s e n t i n g measure o f iii)and l e t

A E IF.

0 < m ( A ) < 1 t h e n we have

m = m ( A ) ml where

ml

+

(1-m(A))

i s t h e measure w i t h

m2

ml(B)

and

=

m2

i s d e f i n e d by

m(A) m2(B) = p

iv)

m((x'A)n B, m(X A)

i s an extreme p o i n t o f 4

v):

B E IF . This contradicts the f a t t that

for all

By Theorem

t h e s t a t e space o f 2.5.7.

E = VF

we know t h a t

- VF

.

has t h e decomposition

p

m

property, i . e . there are such t h a t

A

with

n 2 0

= 1

t A, n=l

and s t a t e s

m

p I

Since

p

t xn n=l

and

pn

i s an extreme p o i n t of R p

=

un

I supy

n

pn

5 supy

we have f o r some

n

.

n

for all

n.

pn

of

F

326

v)

Representing Measures

c;)

f o l l o w s , by c o n s i d e r i n g complements, d i r e c t l y from t h e d e f i n i t i o n

vi)

of

N(p)

v)

* vii):

and t h e f a c t t h a t Put

Yn = { x E X

I f(x)

Zn = I x E X

N(p) =

I p(f)>. I f

t

supy

n

2

1

p(f)

(Theorem 5.2.2). p(f)3

and

B f(X)

then

t

X =

U (YnU Zn). nEN

n such t h a t

Hence, by v), t h e r e i s some p 5

I f(x)

Z(U)

or

p I supz

n

.

The second i n e q u a l i t y l e a d s t o t h e c o n t r a d i c t i o n supz ( f ) 4- n 5 p ( f ) I supz ( f ) n n 1

Hence we have

LI I sup

maximality o f

p

p(f)

1

vii) viii)

1

there i s

Hence Thus

is

Y-,

monotone, 1.e.

ix):

Let

a

with

m I: n=l

fn E E

f 2

f

+

on

p(f)

u(f) E f(X).

+

be p o i n t w i s e decreasing t o zero and assume,

< a = i n f v(fn) n

1 min(fn,a)

E E

2"

fn(x) 2 a f o r a l l p(fn)

, since

is trivial.

o We have

and consequently, by t h e Sandwich Theorem and

'n

u ( f ) , a c o n t r a d i c t i o n . Thus

.t

viii)

*

,p

*

0

n

,a

.

i x ) * i) i s t r i v i a l .

D

. and by v i i i ) t h e r e i s

contradiction since

x E X

with

i n f f, = 0. n

Y,

Generalized Hewin-NachbinSpaces

327

Example: Theorem 5.3.1 and Proposition 5.1.4 provide us with examples for realcompact Hausdorff spaces X which are n o t compact. E . g . take X = N . Then the multiplicative li ne a r functionals on C ( X ) differ ent from zero ( i . e . the Dini characters on C B ( X ) ) a re , according t o Theorem 5.3.1, j u s t the characters of C B ( X ) which a re s t r i c t l y represented by a { O , l l - measure on N . These measures, of course, a re the Dirac measures of the elements in N . Hence , we obtain the well-known f a c t N = v N . Let us return t o Theorem 5.3.1. As remarked previously every character of F can be extended uniquely t o a character of E = sup-norm closure of VF - VF. Furthermore, by Theorem 5.3.1 , i f p i s a character of F then p i s a Dini character i f and only i f i t s unique extension t o E i s a Dini character of E . On the other hand, there may e xi st a Dini continuous character T of E such t h a t T i s n o t maximal while T i s always maximal on the vector space

E.

I e. we have

v XE

3

v

XF

in general

(here we have identi-

fied the Din characters of F with t h e i r unique extensions t o For the same reason we obtain B XE 2 B XF in general. We can identify the elements of

E).

X , by evaluation, with elements of

Hence, as a subset of the compact Hausdorff space p XE

the s e t X

P XE. is

completely regular. Nevertheless, the classical realcompactification v X of X i s d i f f ere nt from w X E . Indeed, by Theorem I 2.2.3 there i s a n order unit isomorphism E C(B XE).

+

C ( p X E ) , hence we may identify E

with

C ( p X E ) , in general, i s only isometrically l a t t i c e isomorphic t o

a subspace of

C B ( X ) . Hence v x = w xC B ( X )

2

v

XE

in general.

Representing Measures

328

5.4

F- PSEUDOCOMPACTNESS

I n this section we give characterizations of F- pseudocompact sets in terms of Dini-cones and representing measures. Then we study the connection between F- pseudo, F- real- and F- compactness. As special cases we obtain several cl assi cal resul ts concerning pseudocompactness and the real -and Stone-Czech- compactification of completely regular Hausdorff spaces.

5.4.1

Theorem [1271:

The following are equivalent:

g

F- pseudocompact

i)

X

ii)

p XF = v XF

iii) F

& 5 -Dini cone

xF- ~measure on

Every s t a t e o f F has - a representing

iv)

X

.

Recall, F i s a Dini cone i f and only i f supX(inf f n ) n

=

for every decreasing sequence

inf supX(fn) n (f,)

in F

.

Proof: i i i ) co iv) i s Theorem 2.2.1. iv) + i i ) follows a t once from Theorem 5.3.1 i i i ) i ) . i i ) i ) : According t o o u r remark in the introduction o f section 5.3, everystate o f VF has a unique extension t o a s t a t e of Therefore we can identify P,

.

Since we know by Theorem 5.3.1 t h a t every character continuous i f and only if i t s unique extension t o E we obtain with the obvious identifications v XF = v XVF -- u x w

.

p

of F i s Dini i s Dini continuous

329

Generalized Hewitt-Nachbin Spaces

Now, by assumption, 3.3.5

there i s

v

X z

=

x~ .

@

= VXE

LI E BXvT

Consider

by Theorem

with

= vXF

By Theorem 5.3.1 v i i ) there i s x E X Hence X i s F- pseudocompact.

. Then

f E

with

f(x) = u(f) = supX(f).

i ) + i i i ) : I n view of Lemma 1.3.4 i t suffices t o prove t h a t VF cone. Let h n be a decreasing sequence in VF. We always have

i s a Dini

inf s u p X ( h n ) 2 supX(inf h n ) .

n

n

Hence we have t o prove the converse inequality. To t h i s end we may clearly assume a = inf s u p ( h ) > - 03 .

X n

n

Let 6 < a be arbitrary and p u t

g n = max(hn,6). Then g n

i s decreasing

-

and uniformly bounded. X i s a boundary f o r VF by assumption Therefore we may apply Simons' Convergence Lemma 3.6.1 to prove

m

( * ) inf{supX( I.

n= 1

I

A,,

gn)

supX(lim sup 9,)

n

I xi

.

m

z xn

t 0 , i = l, . . . , m ,

n=l

i).

= 11 I

'

+-

Here we consider the pointwise lim sup, i . e lim sup g n ( x ) = inf sup{g,(x)

n

for

x E X

since the

n

-,-

. Hence

m t n3

the right-hand side of ( * ) i s equal t o

g n are decreasing. The left-hand

inf supx(gn) = a since 6 < a

n

We obtain

I

supX(inf g n ) 2 a

n

side

0-f

supX(inf 9,)

n

( * ) i s equal t o

.

. This

implies, since 6 < a

was a r b i t r a r y ,

Representing Measures

330

that s u p X ( i n f hn) 2 a = i n f supX(hn). n n Hence V F

i s a D i n i cone.

o

Theorem 5.4.1 c o n t a i n s t h e well-known Alexandrov ([317l),which

G l i c k s b e r g Theorem X

,

has an i n t e g r a l r e p r e s e n t a t i o n by a measure on

X

s t a t e s t h a t , f o r a c o m p l e t e l y r e g u l a r H a u s d o r f f space

e v e r y s t a t e on

C(BX)

i f and o n l y i f

X

Moreover,if

-

i s pseudocompact.

K

i s a compact convex subsetof a l o c a l l y convex H a u s d o r f f space, t h e n t h e s e t o f a l l extreme p o i n t s o f K, ex K , i s Convg(K) , e x Kand A(K)

I

(Recall

Convg( K

ex

-

pseudocompact.

I

ex K =

K

1

f : K

+

, f upper semicontinuous, convex, bounded3

A(K) l e x K = { f

ex K

If : K

+

R

,

,

a f f i n e continuous))

T h i s i s a consequence o f Theorem 5.4.1

iii)* i), Lemma 3.3.2

and t h e

f a c t t h a t a subcone of a D i n i - c o n e i s a Dini-cone. We conclude t h i s s e c t i o n w i t h a c o r o l l a r y which c l a r i f i e s t h e c o n n e c t i o n between

F- r e a l -

and

F- pseudocompactness and

F- compactness. I n t h e

c l a s s i c a l s i t u a t i o n i t i s a well-known t o p o l o g i c a l r e s u l t ([1401): 5.4.2

Corollary:

The f o l l o w i n g are equivalent:

_.

i) X & F- pseudocompact ii) X i s F- compact

and

F- realcompact

Proof: i)

4

ii):

compact ii)

=,

.

i):

p XF c X

By Theorem 5.4.1 we have

By d e f i n i t i o n ,

, every

X

c h a r a c t e r of

a

XF = v XF c X. Hence

X

is

F-

i s F- realcompact. Furthermore, here, s i n c e F

is the Dirac functional o f a p o i n t o f

hence i t i s D i n i continuous. Therefore,

13 XF

= v XF

.

0

X,

GeneralizedHewitt-NachbinSpaces

33 1

SOME CONSEQUENCES

5.5

n ,

I n t h i s s e c t i o n we t u r n o u r a t t e n t i o n t o subsets o f t h e s t a t e space

elements o f

X

X

. Here,

u pXF

X

i n p a r t i c u l a r t o subsets o f

as u s u a l l y , we i d e n t i f y t h e

w i t h t h e i r corresponding D i r a c f u n c t i o n a l s , t h a t i s t o say,

n

i s regarded as a s u b s e t o f

. As

such, c l e a r l y

X

i s a topological

space c a r r y i n g t h e t o p o l o g y i n h e r i t e d f r o m t h e t o p o l o g y o f t h e s t a t e space.

A t f i r s t we want t o s t u d y subsets F l y - realcompact whenever ly

. Where,

in

c l e a r l y ,F

.

to Y

F

X

X

is

of

X

which a r e

F- compact o r

-

Fly Or F- realcompact, r e s p e c t i v e -

i s t h e cone o f t h e r e s t r i c t i o n s o f a l l elements

IY

Note, t h a t , b y Theorem 5.2.2 that i f

Y

X

ix),

i s dense i n

X U p XF

. We

point out

F- compact i t i s n o t n e c e s s a r i l y compact o r even c l o s e d as

is

a t o p o l o g i c a l subspace o f

R

.

Example: Take

X = N

and d e f i n e

F = If : X + R I f

-

lim n-t

bounded w i t h

f(n) =

n , since

bn, n E N

, i.e.

p XF = X

t

.

f(2)l

n c l e a r l y are a l l Dirac

Here t h e extreme p o i n t s o f t h e s t a t e space functionals

$ f(1)

. However,

p XF

i s n o t closed i n

1i m n+ 0

T h i s example a l s o shows t h a t

A subset

Y c X

X U p XF i s , i n g e n e r a l , n o t c l o s e d i n

i s c a l l e d sup-boundary i f we have

supy(f) = supX(f)

for all

f E F

Note t h a t sup-boundary n o t n e c e s s a r i l y means t h a t

Y

i s a boundary.

Remark: Let

Y c X

Define

cp :

be a sup-boundary and

aY be t h e s t a t e space o f

nY + R by

rp(p)(f) = p ( f

IY

)

for all

p

E

fiY

and

f E F.

F

IY

*

n

.

Representing Measures

332 tp

i s continuous and i n j e c t i v e . We c l a i m :

(1)

I f v E n then there i s

(2)

q ( P yF

IY

P xF

)

p E

sly

with

~ ( p 2 ) v

.

f E F,y

.

*

P r o o f o f (1): D e f i n e 6 5 supy

by

6 ( f ) = supIv(g) (To o b t a i n

6 5 supy

Ig

E

F, gIy

5 fl

for all

we need t h e assumption t h a t

Y

be a sup-boundary).

6 i s s u p e r l i n e r r and hence t h e Sandwich Theorem p r o v i d e s us w i t h t h e d e s i r e d s t a t e p E n w i t h S 5 p l s u p y whence rp(!.~) 2 v . T h i s proves (1). Y ( 2 ) , ( 3 ) a r e easy consequences of Theorems 5.2.2 iii)and 5.3.1 v)

5.5.1

Proposition:

be a sup-boundary of

Let Y -

i)

If

i s closed i n X

Y

Fly

-

ii) If Y

is

X

and

. --Then we have X

F- compact t h e n Y

compact. is an Fa- -subset o f X and X i s F- realcompact, -

Fly

( R e c a l l , an

-

Fa- subset i s a c o u n t a b l e union o f c l o s e d subsets o f X )

i):We have Y = q-l(i(Y)) i : Y

then

Y

realcompact.

Proof: -

where

is

+

X

,

i s t h e embedding, and, b y ( 2 ) ,

.

333

Generalized Hewitt-NachbinSpaces

Cp(YUPYF Since, b y Theorem 5.2.2 Y u p YF

) c x u g x F = x .

ix),

i s dense i n

Y

we o b t a i n

Y U 13 Y

IY

. Hence

= Y

IY

IY

Y

ii):Take t h e f u n c t i o n

is

F

compact.

I y-

as b e f o r e . S i n c e

cp

cp :

Y u w YF

subsets o f there i s

u

Y

w YF

.

IY

no E N w i t h

definitions o f

Z(p)

p 5

and

u

Let

5.5.2

w YF

.

that the restriction o f

IY

(Here we closed

be a r b i t r a r y . By Theorem 5.3.1 v ) IY T h i s means by Theorem 5.2.2 and t h e

f i l t e r b a s e o f a f i l t e r converging t o

. Hence we have

.

is

E w YF

supy

N(p)

X

-*

IY continuous we o b t a i n t h a t Y i s a Fa- subset o f Y U w YF OD IY used t h a t X i s F- realcompact). So assume Y = U Yn , Yn n=l

c Y

p

.

. Since

Z(V) t o Y

is a

i s c l o s e d we o b t a i n

Y

D

ProDos it ion :

The f o l l o w i n g

equivalent:

i) p E p X F ' w X F

ii)There i s an

Fa- s u b s e t

Y

o f X u p XF with

Y

3

X

,but

p

B Y.

Proof: ii) i ) : The same argument as i n t h e l a s t p a r t o f t h e p r o o f o f P r o p o s i t i o n 5.5.1 i i ) ( i . e . u l t i m a t e l y Theorem 5.3.1 v))shows t h a t Y 3 X U w XF . T h i s y i e l d s i). i)+ ii): By Theorem 5.3.1 v ) t h e r e i s a c o u n t a b l e c o v e r i n g such t h a t

p $

supy

OD

Hence

p

B

U n=l

n

for all

n E N

Ync X

of

X

.

in where in denotes

the closure o f

Yn

in

X U p XF. 0

334

Representing Measures

Reformulation of Proposition 5.5.1 and 5.5.2 well known r e s u l t s : 5.5.3

Theorem:

The following i)

y i e l d s generalizations of

X

&

are equivalent:

F- realcompact.

i i ) F o r every with X --

iii) X

p

c Y

i s the --

--t h e r e is

E 13 X F \ X

and -

p

B Y

an Fo- subset Y

. Fo- ~subsets of

intersection o f a l l

X

u

of 3! XF

X

u

p XF

containing X.

We c l o s e t h i s section by proving t h a t some topological spaces a r e automatically F- realcompact. Recall, a topological space X i s c a l l e d a Lindelof space i f f o r any open covering of X t h e r e i s always a countable subcovering. 5.5.4

Theorem:

Let X be endowed with a topology such t h a t X & 5 Lindelof space and i s F- realcompact. a l l f E F a r e upper semicontinuous. Then X I

Proof: Let

v

p

E v XF

and assume

1-1

B X

. Consider

F = I

7 IY

E Z(p)l

, where

is the closure of Y under t h e given topology. In view of t h e d e f i n i tions of Z(1-1) and N(u) and Theorem 5.2.2, F i s a f i l t e r base of Z(p). (Here we use t h a t a l l f E F are uppersemicontinuous). 1-1 B X implies n F = 0 t h e r e f o r e { X \ ! 1 Y E Z ( p ) > is an open covering of X which contains a countable subcovering { X \ Y n I n E N) i n view of t h e Lindelof property. We a r i i v e a t Theorem 5.3.1 v i ) .

n yn

n€N

= 4

vn E

Z

, which c o n t r a d i c t s 0

335

GeneralizedHewitt-NachbinSpaces

We conclude t h i s s e c t i o n w i t h a s l i g h t g e n e r a l i z a t i o n o f a w e l l known r e s u l t due t o Choquet [ 72 1, p. 146: Again, l e t

K

be a compact convex

s u b s e t o f a l o c a l l y convex H a u s d o r f f space. Then, under t h e g i v e n t o p o l o g y , i s a B a i r e space. F o r g e n e r a l i z a t i o n s t o cones, see D.A.

ex K

R e c a l l , a B a i r e space

X

i s a t o p o l o g i c a l space w i t h t h e p r o p e r t y t h a t

t h e i n t e r s e c t i o n o f every sequence dense i n

X

Edwards [991.

(P,)

.

X

o f dense open subsets o f

is

The B a i r e space p r o p e r t y indeed remains t r u e f o r a l l F- compact s e t s . To t h i s end l e t

X

be a g a i n an a r b i t r a r y s e t and c o n s i d e r a cone

F

as

above, c o n s i s t i n g o f bounded r e a l v a l u e d f u n c t i o n s on X h a v i n g lX as o r d e r unit. R e c a l l , by Theorem 5.2.2. , D X F = e x 0 , where R i s t h e s t a t e space o f F. 5.5.5

Theorem:

If a x ,

i s endowed the ~w i t h s t a t e space o f F , t h e n BXF ---

r es t r i c t i o n -o f t h e topology on n is a -_--

,the

B a i r e space.

Proof: The p r o o f i s e s s e n t i a l l y t h a t o f [ 8 8

, p.3951.

We can, by e v a l u a t i o n , r e g a r d t h e elements o f

F

T h e r e f o r e i t i s no l o s s o f g e n e r a l i t y t o assume t h a t p E

9XF c R = X

t h e elements

Indeed, by d e f i n i t i o n o f f I 0 such t h a t

Put

Uf

= -JC(

I x E X I f(x)

Y E

Z(p)

>c(

.

I . Then we o b t a i n

i s a neighbourhood o f

p

which converges t o

(Theorem 5.2.2

p

a neighbourhood base o f

compact subset o f

R

and

1~.

Since

.

~1

there i s

p

E Uf,

.

X = n. F o r any

a r e neighbourhoods o f

0 2 B >

z ( 1 - 1 ) ~f o r

n

as f u n c t i o n s on 1-1

.

f E F with

C1c Y

. Hence

Y

i s a filterbasis o f a f i l t e r

Z(p)

ix))

t h e elements o f

(closure o f

Z(p)

form

Representing Measures

336

Since 2 ( p ) i s a neighbourhood base o f 0 t fl,f2 E F, E R , such t h a t

V E U ~

lYa?

g

there are

E IR, f 3 E F

oxF

To show t h a t Pn

subsets

nu

I . !

f242

with

i s a B a i r e space c o n s i d e r a sequence o f dense open

o f 9XF.

Note t h a t d e n s i t y of

Pn

means t h a t

Pn

c o n s t r u c t a decreasing sequence

n

n + l * an+l

pxF c

u

F fn,%’

n pXF

fn,S,

0 , whenever

n U

nonempty and r e l a t i v e l y open. F i x such a s e t

0 * F

we deduce

E BXF,

. We

U

Ff

n y*n

We c l a i m t h a t

i s compact we o b t a i n

W

n 3XF

8

P1 n... n Pnn U

c

(

n

i=1

W

convex and

n

n pXF n\W

is

3.3.5

n...n

O X F c P1

.

n U

Pn

for a l l

n

, i.e

03

n

i=1

P

i s dense i n 8XF

0 n o t e t h a t , by c o n s t r u c t i o n o f W , W i s compact i s convex. L e t

L e t nW be t h e s t a t e space o f

W

* 0 .

a

n’ n

n.

U was a r b i t r a r y . Thus ?XF i s a B a i r e space.

To show

Since

n=l

n U i s non-empty. T h i s i m p l i e s t h a t

Pi)

since

n Ff

W:=

for all

0

Once we know this,we have W 03

use i n d u c t i o n t o

such that

m

Since

U cpXF i s

nw D

, by

p

is

F

Iw

p

FIw

.

F

-

W I

ex fiw

.

F

i s a c l o s e d sup-boundary o f

compact, i . e .

( a p p l i e d t o a c o n s t a n t sequence) pE

+

.

-compact and W

P r o p o s i t i o n 5.5.1,

an extreme p o i n t

be t h e r e s t r i c t i o n map. IW Then we have W c o w o P c n

: F

,

ex ow

.

ex fiw c W

* 0 . Hence

W

F

IW’

By Theorem

possesses

337

Generalized Hewitt-NachbinSpaces

W

has, by c o n s t r u c t i o n , t h e p r o p e r t y t h a t , w i t h

n , we have

p2 E

p2 E

maximal s t a t e o f F

p

=

x

+ (l-x)v2 ,

v1

Assume

p !f

ex R

and one e n d p o i n t Let

=

fixF

, vl,

. Then

v1 1.1 =

A v2

+

n ,

dn n

I

there i s

x

0 5

x

v1

5 1

E W

> 0

i s not i n W

or

v2

such t h a t

1.1 E

i n t ( L n n)

.

G2 E W w i t h

p =

a 3

2

.

t (l-a)vl}

and, b y compactness, t h e r e i s

E W

v2

.

( l - ~ ) v ~

be an extreme p o i n t o f p E

*

E W whenever

A two-dimensional computation shows t h a t , b y m i n i m a l i t y o f that

FIw

.

t h e r e i s a 1i n e d

of

d W we o b t a i n

is a

1.1

i s maximal regarded as a s t a t e of

1.1

v2 E

vl,

p2 L p l y

i m p l i e s t h a t t h e extreme p o i n t

i s convex, we have

A = i n f { a E [O,l]

Since with

, since

n\W

Note t h a t , s i n c e

. This

W

and

E W

p1

n

ex nw and t h a t

. Essential W

x , v2

must

arguments f o r t h i s computation a r e

n\W

as w e l l as

a r e convex. 0

5.6

REMARKS AND COMMENTS

A l l t h e m a t e r i a l , e x c e p t 5.5.5,

of t h i s s e c t i o n i s t a k e n f r o m 11271. We

l i k e t o emphasize t h a t t h e analogy between t h e t h e o r y o f c o n t i n u o u s f u n c t i o n s and, more o r l e s s , Choquet t h e o r y , which we have presented h e r e goes f a r beyond t h e few examples we gave. Most o f t h e m a t e r i a l concerning t h e s i t u a t i o n which we c a l l e d t h e c l a s s i c a l s i t u a t i o n goes back t o H e w i t t ' s fundamental paper 11621 o f 1948. By t h e same t i m e s i m i l a r r e s u l t s were found b y

L. Nachbin, a l t h o u g h h i s i n t e r e s t

i n t h e m a t t e r was a c o m p l e t e l y d i f f e r e n t one. He was m a i n l y i n t e r e s t e d i n spaces which admit complete u n i f o r m s t r u c t u r e s . Nachbin's r e s u l t s 12391 were n o t p u b l i s h e d b e f o r e 1954. Because o f these o u t s t a n d i n g c o n t r i b u t i o n s realcompact spaces a r e sometimes c a l l e d H e w i t t

- Nachbin spaces. The i n t e r -

r e l a t i o n between complete u n i f o r m s t r u c t u r e s and realcompact spaces a r e expressed by t h e f o l l o w i n g a s s e r t i o n ( compare S h i r o t a l2921) :

338

Representing Measures

A completely regular space such t h a t every closed d i s c r e t e subspace has

nonmeasurable c a r d i n a l i t y admits a complete uniform s t r u c t u r e i f and only i f i t is realcompact. The original d e f i n i t i o n o f realcompact spaces is d i f f e r e n t from t h e one we gave f o r t h e generalized s i t u a t i o n . S t a r t w i t h a completely regular TI- space X and a continuous map f : X + R and denote by f * : 13 X + R U I-) the canonical extension of f t o a map from t h e StoneCzech compactification to t h e A l e x a n d r o v - c o m p a c t i f i c a c t i o n of R . Denote by v f X the s e t p XxI y I f * ( y ) = -I. Then v X was o r i g i n a l l y defined as

UX =

n

{vf X If E

c(x)}.

Since we d i d not admit unbounded functions i t i s not completely obvious how t h i s d e f i n i t i o n c a r r i e s o v e r t o t h e generalized s i t u a t i o n . B u t i t can I f does not a t t a i n i t s be done. Consider f o r example- v"F = i f E maximum on X I and define f o r f E v"F v f X = {x E p XF I f ( x ) < supx f 1. Then, in f a c t , v XF =

n Ivf X If E VFI N

This i s l e f t as an exercise f o r the reader. Hewitt proved t h a t X i s realcompact i f and only i f every real zero-set u l t r a f i l t e r i s f i x e d . Recall, t h a t a zero-set f i l t e r i s a f i l t e r having a f i l t e r base consisting of zero s e t s of continuous functions, and such a f i l t e r i s s a i d t o be real i f every countable i n t e r s e c t i o n i s nonempty. This c h a r a c t e r i z a t i o n corresponds t o t h e f a c t t h a t X 3 v X F i f and only i f every u l t r a f i l t e r being contained i n some

Z(p),

p E

B X F , and which

has t h e countable i n t e r s e c t i o n property, has t o be fixed. A condition which can be easilyobtained from Theorem 5.3.1. As a general observation one finds t h a t t h e properties of zero-set f i l t e r s a r e r e f l e c t e d in t h e corresponding properties of t h e f i l t e r s Z ( p ) , p E p X F

.

Other characterizations o f realcompact spaces a r e t h e following 13241 due t o Katetov [1831, Mrowka [2341, Frolik [1121, Wenjen [3281 and, of course, imp1 i ci t e l y t o Hewi tt [ 1621 :

339

Generalized Hewitr-Nachbin Spaces

Let

X

be a c o m p l e t e l y r e g u l a r

T1- space. Then t h e f o l l o w i n g a r e e q u i -

Val e n t :

1)

x

2)

For every

i s realcompact

and 3)

E BX\X

f(x) > 0

For every

t h e r e i s some

for all

For every

x E 8X\X

t h e r e i s some

5)

X =

6)

X

2 E 3X\X

E A

that

n{BlpX i s the

2

B

X

3

X

f(2) = 0

,B

intersection o f

f E C(X)

which has no

u {XI.

t h e r e i s some

X n A =

and

with

x E X.

continuous e x t e n s i o n t o

4)

f E CB(X)

0

.

i s an

G6- s u b s e t A o f pX

such

Fu- s e t } .

u- compact subspaces o f gX

.

The r e a d e r i s c e r t a i n l y a b l e t o p r o v e these equivalences and t o t r a n s f e r them t o t h e general s i t u a t i o n . Then he can o b t a i n p r o o f s f o r t h e g e n e r a l i z e d statement

v i a an a p p l i c a t i o n o f t h e r e s u l t s i n s e c t i o n 5.

Condition 4) i s p a r t i c u l a r l y i n t e r e s t i n g since i t i l l u s t r a t e s t h e i n t e r r e l a t i o n between realcompact spaces and L i n d e l o f spaces. A space

X

is

L i n d e l o f i f and o n l y statement 4 ) h o l d s f o r e v e r y c o m p a c t i f i c a c t i o n ( i n s t e a d o f gX )

. This

r e s u l t i s due t o Mrowka

[2351.

Another d e f i n i t i o n o f realcompact spaces i s t h e f o l l o w i n g : . X compact i f and o n l y i f

m

X

i s real-

i s homeomorphic t o a c l o s e d s u b s e t o f

Rm

some c a r d i n a l number.

F o r more m a t e r i a l about u n i v e r s a l p r o p e r t i e s and p r o d u c t s o f g e n e r a l i z e d r e a l compact

spaces t h e r e a d e r i s r e f e r r e d t o [ 127 I .

F o r i n f o r m a t i o n about pseudocompact spaces ( c l a s s i c a l s i t u a t i o n ) t h e r e a d e r i s r e f e r r e d t o H e w i t t ' s o r i g i n a l paper [162] and subsequent work

([1401~[1411,[3171~[324I).

SECTION 11.6 EXAMPLES AND APPLICATIONS

Most " r e a l " a p p l i c a t i o n s o f i n t e g r a l r e p r e s e n t a t i o n theorems deal w i t h s i t u a t i o n s where t h e extreme p o i n t s o f t h e s t a t e space a r e e i t h e r compact o r n o t f a r from b e i n g compact. I n f a c t , one o f t h e few e x c e p t i o n s i s t h e Poulsen simplex, which p l a y s a r o l e i n s t a t i s t i c a l mechanics ([376] ,[377]

,

[ 581). T h i s simplex (which i s unique i n t h e m e t r i z a b l e case [2131,[2211)

has a dense extreme p o i n t s e t . S i n c e we c o u l d n o t go deeper i n t o s t a t i s t i c a l mechanics, which i n i t s e l f i s convex-cone-techniques,

an i m p o r t a n t area f o r a p p l i c a t i o n s o f

we o m i t t h i s example. I n s t e a d o f t h a t , we a r e pre-

s e n t i n g s e v e r a l examples which i n p r i n c i p l e c o u l d be t r e a t e d w i t h t h e RieszKonig Theorem, t o g e t h e r w i t h some a d d i t i o n a l techniques. Nevertheless, we hope t h a t by these examples we can evoke t h e i m p r e s s i o n t h a t i n t e g r a l r e p r e s e n t a t i o n s a r e an i m p o r t a n t t o o l i n several areas o f mathematics. The examples are: B e r n s t e i n ' s i n t e g r a l r e p r e s e n t a t i o n o f c o m p l e t e l y monot o n i c f u n c t i o n s , K e n d a l l ' s theorem on i n f i n i t e l y d i v i s i b l e c o m p l e t e l y monotonic f u n c t i o n s , t h e m u l t i p l i c a t i v i t y o f extreme s t a t e s i n case of mu1 t i p l i c a t i v e cones and t h e a p p l i c a t i o n s t o t h e Gelfand Representation and t h e GelfanrkNaimark Theorem, t h e Bochner-Weil Theorem i n harmonic a n a l y s i s and, f i n a l l y , t h e Levy-Khintchine formula. 340

341

Examples and Applications

6.1

COMPLETELY MONOTONIC FUNCTIONS

Let E be t h e vector space of a l l i n f i n i t e l y o f t e n d i f f e r e n t i a b l e functions f : I 0,+ m [ + R on t h e open h a l f - l i n e . We endow E with t h e topology of uniform convergence o f a1 1 d e r i v a t i v e s ( including t h e zero order d e r i v a t i v e ) on compact subsets of t h e open half l i n e . With t h i s topology E becomes a l o c a l l y convex Hausdorff space. Clearly the given topology i s generated by t h e following family of seminorms Ip,,, I m E N , n E No} , where

A function f E E i s c a l l e d completely monotonic i f ( - l ) nf ( n ) 2 0 f o r a l l n = O,l, ... . Examples a r e the functions e x p ( - a x ) , xqa , where a 2 0 . Note t h a t w i t h t h e usual pointwise operations, the completely monotonic functions form a cone. The function x - ~ shows t h a t completely monotonic functions a r e not n e c e s s a r i l y bounded. B u t i f such a function i s bounded then c l e a r l y i t s l i m i t lim f ( x ) e x i s t s . We want t o prove an

i n t e g r a l representation f o r Indeed, roughly speaking, f exp(-ax) for different a X = { f E E I f completely X i s convex. W e must show,

x+o+

a r b i t r a r y completely monotonic functions f . i s t h e "sum" of functions of t h e type . A t f i r s t , we t r e a t t h e bounded case. P u t monotonic, f ( x ) I1 for a l l x >'O} . C l e a r l y , t h a t X i s compact under the given topology.

Lemma : X

subset of _i s_a compact --

E.

Proof: Let R be the Alexandrov compactification o f R be mutually d i s j o i n t copies of I O,m [ . Introduce a topology i n

and l e t

Rn, n=O,l,

u Rn by defining a subset t o be open

nEN

only i f a l l i t s i n t e r s e c t i o n s w i t h t h e Rn

a r e open. The union

is then a l o c a l l y compact space. We may i d e n t i f y t h e functions

...

,

i f and U

nENo

Rn

f E X with

342

Representing Measures

:

u

Rn

i=

{

?. I

^f

+

R

'.

d e f i n e d by

f(n)

=

I Rn

.. .

n = 0,1,2,

I f we endow

w i t h t h e topology o f u n i f o r m convergence on a l l compact

f E XI

u Rn , t h i s i d e n t i f i c a t i o n becomes a homeomorphism. We show,

subsets o f

i

i s equicontinuous. Then, by A s c o l i ' s Theorem,

must be c o n d i t i o n a l l y

compact w i t h r e s p e c t t o t h e topology o f compact convergence. C l e a r l y , i s c l o s e d i n t h e space o f a l l f u n c t i o n s from t h e topology o f compact convergence. Hence To show t h e e q u i c o n t i n u i t y o f

r > 0

For each that

and

i

= 1for all

r > 0

n and a l l

f o r some

;? r < ro < r w i t h

Since

(-1)

x 2 r

for all

to

and

X

?

R w i t h respect t o must be compact.

there i s a constant

for all

T h i s f o l l o w s e a s i l y by i n d u c t i 0 n . n = 0 U(r,O)

Rn

,it s u f f i c e s t o p r o v e t h e f o l l o w i n g :

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

(-l)n f ( ' ) ( x ) I U(r,n)

2

u

-X

. Assume

x 2 r

and

U(r,n)

such

f E X.

i s c l e a r s i n c e we may t a k e

t h a t we have a l r e a d y found

U(r,n)

r > 0. Then by t h e mean v a l u e theorem we o b t a i n

(ro) = 2 (f(n) (r)

f(n+l)

f(n+l)

-

An)({))

I

i s non-increasing we have

and

. And

f E X

we can d e f i n e

2

U ( r , n t l ) = F(U(r,n)tU($,n))

Pro pos it i on : F o r each --

t h e r e i s & p o s i t i v e Bore1 measure

f E X

T

on

with

R,

T(R+) I 1 such t h a t

j

f(x) =

exp(-ax) dr(a)

for a l l x > o

.

R+

Proof: Put X

n

Denote by

Conv(X)

= { exp(-ax) l a E

, hence

t h e continuous convex f u n c t i o n s

R+I U

compact. We show,

and d e f i n e yr : X

+

X

{

01.

r > 0

+

R.

n i s a c l o s e d subset o f

Clearly,

n i s a boundary f o r

as f o l l o w s : I f

X

and

Conv(X). L e t f E X\{Ol

r E b{Ol put

343

Examples and Applications

yr(f)(x)

= f(r)-'f(r+x)

r < 0

x > 0. Define

for all

x > 0.

If

f E X.

Note, t h a t even i n t h e l a t t e r case

define yr(f)(x) = f(x)

-

yr(0)(x) = 0

f(x-r)

+

for all

f(x)f(-r)

for all

i s again completely

yr(f)

monotonic. L e t us c o n s i d e r t h e f i x p o i n t boundary i n t r o d u c e d i n c h a p t e r

I 1 3.1. K c X

be a compact

. Let

5

say

r

The s e t

r

T h i s shows t h a t

r-

Conv(X)

-

exposed. Indeed, l e t

i n v a r i a n t s e t and f i x some element i n

E a K 5

ylrl(f), y - l r l ( f )

r

of

f

Conv(X),

f E aE K . Then we have

r

and t h e r e f o r e

necessarily s a t i s f y

u

exp(-a x ), a E R U I - -1

functions

exp(-ax)

boundary f o r with

is

is

Conv(X)-

f(x+y) = f ( x ) f ( y )

x,y > 0. I t i s w e l l known t h a t , among t h e c o n t i n u o u s f u n c t i o n s ,

for all only

01

be a r b i t r a r y and c o n s i d e r

0

exposed. The f i x p o i n t s

with

+k

functional w i t h

x > 0.

p f I supx

{m}

bX E

Conv(X)

f E X. Then, o f course,

7 on n w i t h on

is a

-bX E Conv(X). And

E Conv(X)

5

uf(f bX) I

i s a linear

I * b x ( w ) d ;(u)

for

wER

n n a t u r a l l y defines a p r o b a b i l i t y

whose r e s t r i c t i o n t o

i n t e g r a l i n e q u a l i t i e s now i m p l y

X. Hence R

be t h e f u n c t i o n s

f E X. By C o r o l l a r y I 1.612 we o b t a i n .

for all

The measure R+ U

let

p f ( s ) = ~ ( f )f o r a l l

with

t h i s e q u a l i t y . Only t h o s e

a E [O,m] a r e elements o f

for all

a p r o b a b i l i t y measure

measure on

, satisfy

{a)

x > 0

Conv(X). F o r

bx(f) = f(x)

uf : Conv(X)

all

Ir E R,r

= {yr

f(x) =

j

R+

we c a l l

. The

T

preceding

exp(-ax)dT(a) f o r a l l

x

0.

7

R+ 0

Theorem (Berns t e i n [ 36

I,

A

+

function

-i s_

f :

3 O,=[

see a1 so [ 2501 ) : c o m p l e t e l y monotonic i f and o n l y i f t h e r e

R

a p o s i t i v e Bore1 measure

f(x) =

J R+

exp(-ax)dT(a)

T

0"

for _ all _

R+

such t h a t

x >

o . The measure

T

i s unique.

Representing Measures

344

Proof: The s u f f i c i e n c y i s s t r a i g h t f o r w a r d , i t remains t o show t h e n e c e s s i t y . I f i s completely monotonic and for a l l R,

. Then

x > 0

1>

fc E X

E

then put

> 0

. There

f

fE(x) = f ( e ) - l f ( x + c )

i s a p o s i t i v e B o r e l measure

T~

on

with

(*)

J

f(x+e) = f ( e )

exp(- a ( x t e ) ) e x p ( a e ) d-rE(a)

R+

for all

x > 0, i n view o f t h e p r e c e d i n g P r o p o s i t i o n . L e t

measure on

dT

=

Rt

i.e. T

I g(a)f(e)

exp(a&)d-re(a)

for all

g E Co(Rt)

Rt

f o r a l l continuous i s independent o f

I

be t h e B o r e l

d e f i n e d by

R,

Ig

T

E

g : R+

+R

. Indeed,

with

t a k e any

l i m g ( r ) = 0 . Because o f ( * ) r1 > el > 0 Then we have

e x p ( - a n ) f ( & )e x p ( a & ) d T E ( a ) =

.

f(n) =

R+

I

R+

exp(-an)

f(E1)

exp(a&, )dTe ( a ) 1

The subspace generated b y t h e f u n c t i o n s

for all

exp(-an),

n E N

.

n E N , i s dense i n

w i t h r e s p e c t t o t h e sup-norm which i s a consequence o f W e i e r s t r a s s ' theorem. T h i s shows, T i s independent o f e , f ( x ) = e x p ( - a x ) d r ( a )

Co(Rt)

I

R

for all

x > 0

and

'I

i s t h e unique measure w i t h t h i s p r o p e r t y .

0

345

Examples and Applications

6.2

KENDALL’S THEOREM ON INFINITELY DIVISIBLE COMPLETELY MONOTONIC FUNCTION

L e t cp : 1 0,-

be a bounded c o m p l e t e l y monotonic f u n c t i o n .

R

+

c a l l e d i n f i n i t e l y d i v i s i b l e if f o r e v e r y

Y with

monotonic f u n c t i o n

y n = cp

n E N

. Later

cp

is

there i s a completely

on we s h a l l see, i n t h e c o n t e x t

o f t h e L6vy-Khintchine formula, t h e reason f o r t h e importance o f t h i s n o t i o n . A t t h e moment we a r e g o i n g t o g i v e a c h a r a c t e r i z a t i o n o f t h e i n f i n i t e l y d i v i s i b l e c o m p l e t e l y monotonic f u n c t i o n s . T h i s c h a r a c t e r i z a t i o n Kendall [1931 ( s e e a l s o [ 2 5 0 1 ) .

i s due t o D.G. Theorem: Let -

cp

be a bounded e l y d i v i s i b l e c o m p l e t e l y monotonic f u n c t i o n . - i n f i n i t~

Then t h e r e i s a p o s i t i v e -Bore1 measure ----

on [O,ml such t h a t -

T

1 - exp(-a x 1 d r ( a ) [o,-] 1 - exp(-a)

}

for all --

x E lo,-[

.

B e f o r e we p r o v e t h e theorem we d i s c u s s some b a s i c p r o p e r t i e s o f i n f i n i t e l y d i v i s i b l e c o m p l e t e l y monotonic f u n c t i o n s . L e t monotonic f u n c t i o n s that

X

P,,~,

m E N

with

cp

cp(x) 5 1

.

X

be t h e s e t o f c o m p l e t e l y

I n t h e l a s t s e c t i o n we proved

i s compact w i t h r e s p e c t t o t h e t o p o l o g y generated b y t h e seminorms

elements o f

,n

.

E No

By

K

we denote t h e s e t of i n f i n i t e l y d i v i s i b l e

.

X

Lemma :

i)

K

iii)

I f cp E K -

iv)

If

_i s _a

cp

E

compact s u_ bset o f

x and

element o f ~v)

I f cp E K to -

cp

.

and -

K

.

a > 0 p

2 0

X

then

. cpa E K.

then t h e f u n c t i o n --

then t h e function --

1/ n exp(n(cp ( x )

exp(p(cp(x)

- 1)) is an

- 1)) converges p o i n t w i s e

346

Representing Measures

Proof:

i)- i v ) a r e s t r a i g h t f o r w a r d v e r i f i c a t i o n s .

F o r ( v ) we use t h e f a c t t h a t

exp(n(a

l/n

- 1)) -, a as n

for all

a > O . o r cp(x) = 1, f o r some

I t remains t o prove v i ) . I f cp(x) = 0

,

x > 0

then by complete m o n o t o n i c i t y , t o g e t h e r w i t h a s i m p l e c a l c u l a t i o n o r w i t h B e r n s t e i n ' s Theorem, we have cp cases j u s t t a k e

= cp(x) cp(xta)

al(x)

= n(l

-

= y e = cp

o r cp

. Assume

cp(a)

l/n

)

/ cp(a)

for all

x

, respectively.

For these

l/n

-

x > 0

.

K by

. I n order

1 - cp(a)

Then cpn E

1

y1 E

x > 0 cp(x)

and cpn(x) =

=

now, 0 < cp(x) < 1 f o r a l l

e = 1. Then we can d e f i n e

A t f i r s t , take

'n

5,

= 0

cp(xta) 1/n

to find

put

51

l/n

and

exp(n(cp(x)

l/n

- l))exp(-pn)

= exp(n(cp(x+a)

for all The f i r s t t h r e e e x p o n e n t i a l s converge t o

11n

n E N

-

l))exp(pn(cpn(x) and

x > 0

-

1))

.

cp(x), @ ( a ) and cp(xta). Hence

t h e f o r t h e x p o n e n t i a l a l s o converges p o i n t w i s e t o a f u n c t i o n

E~

which i s

e a s i l y seen t o be c o m p l e t e l y monotonic. (We may go o v e r t o a s u i t a b l e subsequence which converges i n t h e g i v e n t o p o l o g y on compactness o f with

e

5, = 5,

=

1 cp'-'

K).

. Now

51

assume

and

K

i n view o f t h e

i s c l e a r l y i n f i n i t e l y d i v i s i b l e and s a t i s f i e s v i ) 0 <

Y & = Y:

E

< 1 -&

. Consider 0

c1

and

y1

and t a k e

347

Examples and Applications

P r o o f o f t h e Theorem:

Iw E K,

C = {- log(w)

Define t h e compact s e t

s i d e r t h e t o p o l o g y generated b y t h e

w ( 1 ) = e-’} m E N , n € No)

p m,n,

.

, con-

(again

C

i s even

of t h e lemma. I n view o f i v ) i n t h e lemma convex i n view o f i i ) and i-i)

( I -exp(-a))-l

= ( 1- e x p ( - a x ) )

Q,(X)

a € [O,c0].(0f

course

R = { Q aI a E [ O p ] }

functions,

c

a r e elements o f

for all

= 1). We prove t h a t

Q o ( x ) = x, Q,(x)

i s a boundary f o r t h e continuous convex

c C

Conv(C), on

Once t h i s i s e s t a b l i s h e d we can proceed

C.

s i m i l a r l y as i n t h e P r o p o s i t i o n o f t h e l a s t s e c t i o n : L e t dx,x > 0, be t h e function w i t h

Q E C

Fix

Since

= Q(x)

for a l l

for all

LI

f E Conv(C). We have

-

for all

on

x > 0.

with

R

p

uQ 5 supc

( * bX) 5

Q

The measure

p r o b a b i l i t y measure Q(x) = uQ(dX) =

on

T

dX d?

n

n

r

= {T

a,@

I

Qa(x)dT(a)

[ O ,-I

I a E lO,-[,p Q(x)

Ta,,(Q)(x)

1

I

‘I

=

+ +

E R}

T

common f i x p o i n t s o f By d e f i n i t i o n of t h e

s o l v e t h e equations:

r

is

r

( * 6,)d;

p(Q(l+a)

-

Q(a))

-

Q(a))

x > 0.

n i s a boundary. D e f i n e T

a,a

: C

if lB(Q(l+a)

-,

-

C

by Q(a))l < 1

otherwise.

a,@( Q ) E C

e a s i l y checks t h a t

Q E C.

for all

t o obtain a

for all

o f functions

p(Q(x+a)

Q(X)

I n f a c t , we have

by

n defines n a t u r a l l y a

T h i s proves t h e Theorem. I t remains t o show t h a t a family

6

-,

such t h a t we have

[Op] =

on

hX E Conv(C).

Conv(C)

Q ’

n i s compact we can a p p l y C o r o l l a r y I 1.6.2

p r o b a b i l i t y measure

*

We have

Q E C.

and d e f i n e t h e l i n e a r f u n c t i o n a l

LI ( f ) = f ( Q )

Q

6,(Q)

i n view o f iii)and v i ) o f t h e lemma. One

Con(C)

-

exposed. T h e r e f o r e t h e s e t o f a l l

must be a boundary ( P r o p o s i t i o n 3.1.4). T

a,@

the fixpoints are the functions

Q

in

C

which

348

Representing Measures

(1)

Q(x)(Q(lta)

Define

- Q ( a ) f o r a l l a,x

-

1 for all

Q ( x ) R(a) = Q ( x t a )

-

Q(a)

-

x > 0

.

Q ( x ) R ( l ) = R(x)

Then (1) i m p l i e s

Q(x) f o r a l l

.

a,x > 0

for all

.

x > 0

Define

H(x) =

.

> 0

a = 1

From ( 2 ) we i n f e r w i t h

(3)

= Q(xta)

- Q(x)

R(x) = Q ( 1 t x )

(2)

- Q(a))

-Q(x) =

Q(Xt1)

l t R ( X )

.

= 1 t R(l)Q(X)

We o b t a i n H ( x t a ) = 1t R ( 1 ) Q ( x t a ) = 1t R ( 1 ) Q ( x ) t R ( 1 ) Q ( a ) t R ( l ) * Q ( x ) Q ( a ) = H ( x ) H ( a ) for a l l

a,x > 0. Here t h e second e q u a l i t y f o l l o w s from ( 2 ) and ( 3 ) .

Hence H(x) = exp(8 x )

,B

E

[-my-[

, and

therefore

~ ( x )= ( e x p ( B x ) - 1) ~ ( 1 1 - l f o r a l l With

Q ( l ) = 1 we o b t a i n

t h a t both, R(l) > 0

R(l)

and

B

R(l)

, are

t

x > 0.

1 = exp(B). The monotony o f

Q

p o s i t i v e o r negative. I f B > 0

implies and

t h e n cp w i t h

cp(x) = exp(- Q ( x ) ) = exp{

-

exp(g x ) - 1 exp(8) 1

for all

-

x > 0

i s n o t completely monotonic. Hence we o b t a i n t h e f u n c t i o n s =

-1 -1

exp(-ax ) exp(- a)

for all

Q,

with

x > 0, a E [ o p ] ,

Since these f u n c t i o n s i n f a c t s o l v e (1) t h e y a r e t h e o n l y s o l u t i o n s ( i n c )

o f ( 1 ) . T h i s means

n i s a boundary.

0

349

Examples and Applications

6.3

MULTIPLICATIVE CONES

Throughout t h i s s e c t i o n we assume t h a t ( F , + I ) i s an order u n i t cone such t h a t F = F, + R I , i . e . f o r every f E F t h e r e i s some h 2 0 such f + A I . The cone i s s a i d t o be m u l t i p l i c a t i v e i f we have defined that 0 on F, such t h a t an a s s o c i a t i v e d i s t r i b u t i v e mu t i p l i c a t i o n

-

f = xf

f .(AI) = (XI)

for all

f E F,

.

This ensures t h a t t h e r e l a t i o n s one usually expects from an algebra do hold ( w i t h R, instead of R ) .

The m u l t i p l i c a t i o n i s s a i d t o be monotone i f , f o r a l l f

4

h, g

iI ,

we have f g < h g

and

f , g , h E F,

with

f + h g < h + f g .

And t h e m u l t i p l i c a t i o n i s s a i d t o be weakly monotone i f , f o r a l l

f , g , h E F,

with

f
,g

4

I

and a l l c h a r a c t e r s u

Of course, monotone implies weakly monotone. A typical example f o r a cone with a monotone m u l t i p l i c a t i o n is t h e cone of bounded upper-semicontinuous functions on some X endowed with t h e point-

wise s t r u c t u r e . Other examples f o r m u l t i p l i c a t i v e cones a r e furnished by the s e l f a d j o i n t p a r t s of commutative Banach algebras w i t h involution. (We discuss these examples i n s e c t i o n 6.4) Remark: i ) I f a m u l t i p l i c a t i v e cone ( F , < , I , * ) i s a vector space then t h e m u l t i p l i c u t i o n on F, can uniquely by extended t o an algebra on a l l of F. This e a s i l y seen: For f , g E F, and h , 6 2 0 one obviously has t o define: (f

-

h

I)

-

(g - d I )

=

def

f . g -Ag - 6f

+ A6I.

RepresentingMeasures

350

ii) T

i n t o some a l g e b r a A. i s s a i d t o be m u l t i p l i c a t i v e if i t i s m u l t i p l i c a t i v e on F, , i . e .

T(f

Consider a l i n e a r map T : F + A

.

. T(g)

g) = T ( f )

for all

mapping

f,g E F,

F

. Then,if

F

i s a v e c t o r space,

such a map i s n e c e s s a r i l y an a l g e b r a homomorphism ( w i t h r e s p e c t t o t h e algebra g i v e n by i ) ) . Lemma : be- a m u l t i p l i c a t i v e o r d e r u n i t cone such t h a t t h e m u l t i p l i c a t i o n i s weakly monotone. -Then e v e r y c h a r a c t e r ( i . e . extreme p o i n t o f

L e t (F,<,I,-) -

_ the state

space)

g

multiplicative.

Proof: Let

be a c h a r a c t e r . F i x a r b i t r a r y

p

l i n e a r monotone

v(g) = p ( f - g ) We c l a i m Since p

.

f

with

g E F,

.

f

I and d e f i n e a

for all

'

, i . e . p ( f g) = u ( f ) u ( g ) f o r a l l g E F, . Ft I was a r b i t r a r i l y chosen t h i s proves t h e m u l t i p l i c a t i v i t y o f

v = ~ ( 1 )p 4

f E F,

by

v : F + R,

(The c o n d i t i o n

f < I can be dropped s i n c e

u i s l i n e a r ! ) . I n order

t o prove t h e c l a i m we observe t h a t 0 5 v ( g ) s p ( g ) f o r a l l g E F, s i n c e p i s monotone and s i n c e we assumed t h e m u l t i p l i c a t i o n t o be weakly monotone. I n p a r t i c u l a r ,

x

= ~ ( 1 ) must be i n

x

> 0

[0,13.

I n addition this

- v i s a monotone l i n e a r f u n c t i o n a l on F, . Hence IF, we can f i n d monotone l i n e a r f u n c t i o n a l s C,G on F, w i t h 1 = 3(I)=(p(I)

yields that

cp = 11

such t h a t

Just take

-1 (p = (1-1) cp

- v,cp

=

x -1 v if x <

if 1

and $

extend u n i q u e l y t o s t a t e s

and = p

1 F,

S,G

<

= p

I Ft

for

otherwise of

F.

A = 0

. Since

And we have

, and F = F, t R I 1-1 =

x S t (1-x)G

Taking i n t o account t h a t p i s an extreme p o i n t o f t h e s t a t e space (Theorem1 2.10.6) we have i n any case p = S o r p = G, i . e . p = IF, 0 = (1-x)-'(plFt - v ) . T h i s immediately i m p l i e s t h e c l a i m . u IF,

or

35 1

Examples and Applications

Theo rem : Assume t h a t --

& 2 m u l t i p l i c a t i v e ----o r d e r u n i t cone such t h a t

(F,<,I;)

the set o f % ---

m u l t i p l i c a t i o n _i s weakly monotone. -Denote by states. F o r every s t a t e --such t h a t --

i)

!.I

of F

t h e r e i s a p r o b a b i l i t y measure

Here, -o f course, we c o n s i d e r t h e s m a l l e s t U- -a l g e b r a on ___ t h e f u n c t i o n s v -,v ( f ) , f E F , a r e measurable. i i ) ---For the order u n i t functional This __

means --

iii If F

fi a

SI

we have

SI(f)

SI(f*g)

5 SI(f)

v e c t o r space such t h a t

d e f_ i n e_ s a -norm,then _ -

_i s_a

(F,.)

fo_ r all

SI(g)

ISIl(f)

T

%

on

% -such t h a t

= sup{v(f)lv E

i n p a r t i c u l a r ---that the order u n i t functional

plicative, i.e.

multiplicative

%I.

& submulti-

f,g E F .,

= max(SI(f),

SI(-f))

commutative a l g e b r a .

Proof: i)

immediate consequence o f t h e lemma and Theorem 3.3.6.

Consequence o f t h e lemma and t h e f a c t t h a t t h e c h a r a c t e r s a r e a ii) boundary of t h e statespace. iii)

B(%),

Consider

t h e bounded r e a l valued f u n c t i o n s on

D e f i n e t h e Gelfand t r a n s f o r m a t i o n i.e.

?(v) = v ( f )

for all

v

E

F 3 f

% . Then

-, f E B(%)

%.

i n t h e obvious way,

t h i s G e l f a n d t r a n s f o r m i s an

a l g e b r a homomorphism i n t o a commutative a l g e b r a . And i i ) y i e l d s t h a t ISIlV) =

171

for all

f E F.

Since

norm t h e Gelfand t r a n s f o r m must be i n j e c t i v e .

ISII

i s assumed t o be a 0

352

Representing Measures

6.4

BANACH ALGEBRAS AND SPECTRAL THEORY

The main o b s e r v a t i o n o f t h i s s e c t i o n i s t h a t i n case o f commutative Banach algebras t h e spectral r a d i u s norm can be understood as t h e o r d e r u n i t norm

o f an o r d e r u n i t cone such t h a t t h e Let N

m u l t i p l i c a t i o n i s weakly monotone.

be a normed algebra, i . e . a normed space which i s an a l g e b r a such

that

(1)

I1 x e y l l 5 II xi1 . l I y I I

for a l l

x,y E N.

The s c a l a r s a r e denoted by K ,they a r e e i t h e r equal t o a l g e b r a ) o r equal t o

R ( r e a l normed

(complex normed a l g e b r a ) . O f course, a complex

(t

normed a l g e b r a can always be considered as a r e a l normed a l g e b r a ( b u t t h e n x

and

ix

a r e l i n e a r i l y independent).

We assume t h a t

N

has a u n i t

I , i.e. x I

=

Ix = x

for all

x E N.

T h i s i s n o t very r e s t r i c t i v e s i n c e we can always a t t a c h a u n i t : I f N has no u n i t t h e n we embed fl a l g e b r a i c a l l y i n t o N = I ( x , r ) I x E N, r E K} endowed w i t h m u l t i p l i c a t i o n and norm g i v e n by:

II ( x , r ) I I = I r l + II x I I

-

(xl,rl)

( x 2 ,r2 ) = (x1x2 t rlx2

C l e a r l y , t h i s a l g e b r a has a u n i t , namely

The s p e c t r a l r a d i u s

p(x)

of

x E N

t r2 x1 ’ r1 r2 )

(0,l).

i s defined t o be

The l a s t e q u a l i t y i s an immediate consequence o f (1). T h i s d e f i n e s a seminorm w i t h

(2)

p(xk ) =

p ( ~ ) f o~ r

all

k E N, x E N

.

The importance o f t h e s p e c t r a l r a d i u s norm, i n case o f a Banach a l g e b r a

B

,

353

Examples and Applications a

comes from the elementary f a c t t h a t

a n xn

converges in

B

1 l a n l ( p ( x ) t c ) ~converges f o r some e > 0. Hence, f o r n=O following elements e x i s t in B :

x

1

n=O

m

(3)

JiT-7

(4)

ln(1

-

X

K{ z

=

n=O

def

-

) =

1

m

112

(

n

)(+

if > p(x),the

}

x n

n (7)

1

n=l

Of course, these functions f u l f i l l the usual relations, l i k e =

x I- x

and

exp(ln(1

X - x)) =

I -

X

,

where exp(z) i s defined t o be: exp(z) =

(5)

zn z n!

a

n=O

*

Now, we consider the fol owing Situation 1: that

Let N be a real normed algebra and P

i)

I E P , -I !k P

ii)

II xII I t x E P

iii)

P

*

for all

x

c N

be a cone such

E N

P = { x y Ix,y E P ) c P

Remarks : We introduce in

-

N

the order given by

x s y

y-XEP.

Then i ) and i i ) guarantee t h a t I unit functional by SI and I S I ]

P

, i.e.

i s an order unit

. We

denote the order

denotes the order unit norm

354

Representing Measures

I S I I = maxISI(x),

(6)

SI(-x)l

=infIx>O

IXI -x

and XI t x E P I .

E P

Because o f ii) we have

(7)

ISII(x)

I IIxII

Now, c o n s i d e r a r b i t r a r y

- X Z = (y-x)z

Then y.2

E P

x,y,z E P

x E N.

for all

and

with

x Iy

and

z II .

x t y z - ( y t x z ) = (x-y)(I-z)

E P.

Hence i i i ) i s e q u i v a l e n t t o t h e a s s e r t i o n t h a t t h e m u l t i p l i c a t i o n i s monotone. Define

%(N)

t o be t h e nonzero m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l s

v : N - r R .

Theorem I n case o f S i t u a t i o n 1 we have

i) SI(x)

Iv

= supIv(x)

ii) 6-'p(x)

I SI(x)

I . The sup i s a t t a i n e d a t some

E %(N

Ip ( x )

x E P, where

for a l l

v E %(N

6=supIp(y)lyEP,I-y

PI.

Proof: The lemma i n t h e p r e c e d i n g s e c t i o n shows t h a t e v e r y c h a r a c t e r i s m u l t i p l i c a t i v e on

P

, hence

contained i n +(N)

N

on

and

(&mark i n 6 . 3 ) . So t h e c h a r a c t e r s a r e

i ) i s an immediate consequence o f Theorem 3 . 3 . 5

( a p p l i e d t o c o n s t a n t sequences). A s s e r t i o n i )y i e l d s = (sI(x))2

S+X)

f o r x E P. F o r x E N we o b t a i n by ii)o f S i t u a t i o n 1 t h a t SI(X.X) 2 SI(X) 2 S i n c e we have I S I ] 6 II II we immediately o b t a i n

.

ISII I

p

( f r o m t h e c o n s t r u c t i o n o f t h e s p e c t r a l r a d i u s norm), I f

6 =

m

t h e theorem i s proved. So, l e t us assume Then f o r e v e r y p ( x ) I6 x

.

x

Since

6

< O D

> SI(x)

x

. Note,

that

we have

> SI(x)

P(X) 2 6 S I ( X ) .

0 I

6 t 1 since X

2

. Hence

I

p(1) = X

p(,)

1. Take I 6

was a r b i t r a r i l y chosen t h i s y i e l d s 0

or

x 2 0.

355

Examples and Applications

, we d i s t i n g u i s h between r e a l 1 in e a r i t y and complex 1 i n e a r i t y . R e c a l l a map u between

Since from now on we a l s o deal w i t h v e c t o r spaces o v e r

N1

two v e c t o r spaces

and

N2

t

,if

i s called real-linear

u(a x + B y ) = a u ( x ) + B ~ ( y )f o r a l l and r e a l numbers

.

p

a,B

.

a,p

and complex numbers

E N1

X,Y

if

i s c a l l e d complex-linear,

*

A l l o u r examples deal w i t h i n v o l u t i o n s . An i n v o l u t i o n

:

N

+

N

is a

r e a l -1 i n e a r map such t h a t

for a l l

A(N)

By

N

x,y E

(X*)*

= x

(xyy

= y* x*

(A xf

=

N

and

x x* A

,

x denotes

E K . (i.e.

t h e c o n j u g a t e complex number

K = R

or

K

t )

=

we denote t h e s e t o f nonzero m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l s

A ( N ) a r e supposed t o be r e a l - l i n e a r , i f K = R and c o m p l e x - l i n e a r i f K = 0 . I n case o f a normed a l g e b r a N w i t h i n v o l u t i o n an element I, E A(N) s a i d t o be symmetric i f +

K.The elements o f

-

v ( x * ) = I,(x)

for all

x E N.

By A ~ ( N ) we denote t h e symmetric elements o f Example 1 ( Consider

A

*-

A(N).

algebras)

t o be a

*

- algebra w i t h u n i t

I. A

*-

complex Banach a1 gebra w i t h continuous i n v o l u t i o n . By ,A, r e a l normed a1 gebra o f s e l f a d j o i n t elements

Asa = IX E A Obviously, we have

A = As,

I X*

=

@ i ,A,

algebra i s a we denote t h e

XI .

. Let

P denote t h e c l o s e d cone

of

x

356

Representing Measures

-

P = Ix

generated by

i s closed. Hence

Ix

with

E As,}.

S i n c e t h e i n v o l u t i o n i s continuous

As,

P c Asa. Let x 4

Observation 1: y E As,

2

.

x = y

II I - x I I < 1. Then t h e r e i s

be s e l f a d j o i n t w i t h

Proof: Recall t h a t f o r

I 1 a l l < 1 we d e f i n e

with

a E As,

by the

f o l l o w i n g a b s o l u t e l y converging s e r i e s

Clearly,

For

d

z = I

m i s s e l f a d j o i n t . From t h e t r i a n g l e i n e q u a l i t y we o b t a i n :

- dr-a’,

we have

must be s e l f a d j o i n t and assertion. Observation 2:

z

f o r some

E As,

Let

.

a E As,

and

E

Put a = I

. Then

> 0

Proof: Apply Observation 1 t o

x = I t ( ( Ia l l +

Now, d e f i n e an o r d e r r e l a t i o n i n As,

a Ib That means

a

E)-’

Ix

101

-

( I l a II +

.

i s an o r d e r u n i t of

c -

As,

x E P with

.

and we have SI I II II

Proof: O b s e r v a t i o n 2 shows t h a t , f o r a l l

E )

I +a = z 4

0

Lemma 1:

I

Hence, y =

t o obtain the

x

by

i f and o n l y i f t h e r e i s some

P = Ix E As,

.

I I z II < 1 and z E AS,

m= y2 .

a E Asa

, we

have

.

a t x = b.

357

Examples and Applications

-

a 5 ( I l a II + &)I .

T h i s c l e a r l y proves t h e c l a i m .

0

II II of

L e t us c a l l t h e norm

monotone if II II

A

i s monotone on

P.

Lemma 2:

The f o l l o w i n g are equivalent: II II

i)

ii) SI(x)

% monotone = 11 x l l

for all

x 2

o

Proof:

SI

Since

i s monotone w i t h r e s p e c t t o t h e g i v e n o r d e r ii)i m p l i e s i ) .

II II

Now, l e t

II II I p

be monotone, t h e n

(restriction t o the positive

cone) i s a monotone s u b l i n e a r f u n c t i o n a l . Hence, f o r e v e r y f i x e d 0

9

u : P

x E P, t h e r e i s a monotone l i n e a r f u n c t i o n a l

II

p 5 I1

~ ( x =) I I x I l (Theorem I 1.3.3).

and

, we

II I l l s 1

and

.

ASa = P- R, I

state o f

SI(X) The i n e q u a l i t y

2

R

Because o f

~ ( 1 )= I I I I I = 1. So,

obtain

+

p

with x 5 IIx I I I

can be extended t o a

From Lemma 2.1.2 we d e r i v e

U(X)

= IIXII

.

i s a consequence o f Lemma 1.

IIxII 2 SI(x)

c

Another u s e f u l o b s e r v a t i o n i s : Lemma 3:

fi

The m u l t i p l i c a t i o n -

& monotone ---i f and o n l y

,A,

if

A i s commutative.

Proof:

P

-

P

c

P

i s equivalent t o the assertion t h a t the multiplication i s i s commutative, t h e n x x* y a y * = ( x y ) . ( x y ) * .

-

monotone. I f A

T h i s c l e a r l y imples Since

,A,

ASa = P

,A,

c ,A,

-

I

Ix

if

P

{A

P

P

c

( I

E R,}

.

P

P . Hence, P

c

P

*

P c P.

i s an o r d e r u n i t ) we have

. Therefore

we g e t , f o r a r b i t r a r y

Representing Measures

358

, that

x,y E Asa

x-y

i s s e l f a d j o i n t . This implies

,

( x . y ) = (X.Y)* = y*x* = y * x and

x

must commute w i t h

Therefore

A

Example 2 Let

A

. Hence,

y

As,

i s a commutative algebra.

must be commutative. (C*- a l g e b r a s )

be a

algebra w i t h u n i t .

C*-

A

C*-

*

algebra i s a

- algebra

such t h a t

II X ~ * I I

=

II~II

for a l l

II x I I

I n particular t h i s implies

2

=

x E A.

2 I I x II and

I I x I1 = IIx* II

for all

x E A. Remark : Let A

be a commutative

C*-algebra then, f o r

a,b E As,

the following

are true: 2 2 2 I I a t b I l t I I a II

i)

I1 I- a 2 II I 1

ii) If II a2q1 5 1 then I I a 2 t b 2I I < : l

iii) I f

1>

Let

iv)

> 0

E

1 1 1 - a 2 - b2I I s l

then

, then

t h e r e i s y E As,

such t h a t

c

I t

a 2 t b 2 = y2.

Proof:

i) I I a2 t b 2 I I = II ( a + i b ) ( a - i b ) l I = I l a t i b l l I I a - i b l l = l l a t i b l l 2

.

From t h e t r i a n g l e i n e q u a l i t y we d e r i v e 2 IIatiblI= IIatibll+ IIa-ibllt 2 Ilall. Hence 2 2 2 2 I I a t b I12 I I a l I = I l a II

ii) L e t (ltc) I

t > 0

-

a

2

, t h e n from Observation 2 we g e t some c

= c2

( I t &= )

And

E+

0

.

. Hence, (1tc)llIii

E As,

such t h a t

we o b t a i n from i ) : =

I I2 t~c 2 112 I I C 2 11 = I I ( l + t ) I - a 2 I I .

proves t h e a s s e r t i o n

359

Examples and Applications

iii) L e t

E

> 0 , then f r o m O b s e r v a t i o n 2 we g e t some ( 1 + ~I)- a 2 - b 2 = c2

(1tE) t I I ( l + c ) I Again,

E

-+

-

.

such t h a t

From i)and ii)we o b t a i n :

a 2 I I = I I b2 t c 2 l l t I I c 2 I I = l I ( l + ~ ) I - 2a - b 2 I I .

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

0

iv)

S i n c e t h e norm i s homogeneous we can assume

Put

x =

E

c E Asa

I1 a 2 + b2 11s 1 - E

I + ( a 2 + b 2 ) , t h e n a c c o r d i n g t o iii)we have

And t h e a s s e r t i o n f o l l o w s from O b s e r v a t i o n 1

.

II I - x I I

.

I (1-E).

o

Lemma 4:

If A

C*- a l g e b r a . t h e n t h e norm i s monotone.

-i s_ a commutative

Proof: E As,

Consider al,...,am

Put

a =

x

,@

a' i

yields that for and

p

I + D = x2

and

= 1b

6,p

2 0

. Then

2

bn E As,

bly...,

. Assertion

6,p

-+

0

gives the desired r e s u l t .

Now, c o n s i d e r on

such t h a t

A

2

. 0

A ~ ( A ) t h e s t a t e space t o p o l o g y . Then

And f o r commutative

61+ a = y

i) o f t h e Remark y i e l d s

II ( 6 + p ) I + a + @ l l t II 6 I + a I I And

have t o show

( v ) o f t h e Remark and i n d u c t i o n

y,x E As,

there are assertion

. We

aS(A)

i s compact.

we o b t a i n t h e f o l l o w i n g i m p o r t a n t r e p r e s e n t a t i o n .

Gel fand-Naimark Theorem: Let -

A

-be_

a commutative

C*- ___-algebra w i t h u n i t . Let A

=

symmetric nonzero m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l s on A

x E A

+

i

E C a(A)

, -given

i s o m e t r i c a l g e b r a isomorphism

by

from

i ( ~= )~ A

onto

AS(A)

.Then

( x ) -for all u E A

C,(A).

be the

is an

Representing Measures

360

Remark: Because o f

A

=

,

x E A

A ~ ( A ) we have, f o r a l l

/\

z

x* = x

(where t h e b a r

I = 1A *

denotes complex c o n j u g a t i o n ) . Furthermore P r o o f o f t h e theorem: Consider

Asa

. We

have t o prove t h a t

isomorphism f r o m ,A,

onto

x

+

x

i s an i s o m e t r i c a l g e b r a

Consider i n ,A,

C R(A).

the order

i n t r o d u c e d i n Example 1. Then

i s an o r d e r u n i t cone (Lemma 1) (Asa,s,I) w i t h monotone m u l t i p l i c a t i o n (Lemma 3 ) . A s t r a i g h t f o r w a r d computation shows ( A s a ) = AS(A)

*

From Lemma 2 and Observation 3 we o b t a i n The theorem o f t h i s s e c t i o n y i e l d s t h e n

for all

. This

x 2 0

I I X II for a l l

x E As,

= II x II f o r a l l

SI(x)

x 2 0.

implies Sup{

=

1 i(u) I v

E A)

( t o see t h a t c o n s i d e r

x2 2 0).

X i s an isometry. Obviously t h i s i s o m e t r y must be an a l g e b r a isomorphism s i n c e t h e elements o f A a r e m u l t i p l i c a t i v e and l i n e a r .

So

x

+

That t h e i s o m e t r y i s o n t o i s an immediate consequence o f t h e StoneWeierstrass Theorem.

Example 3

0

(Gelfand r e p r e s e n t a t i o n o f commutative a l g e b r a s ) .

We s t a r t w i t h a commutative complex normed a l g e b r a

I . L e t us f i r s t embed A T h a t i s a commutative morphism J : A morphism

3

+

: a(A)

construction o f

B

t(A)

in

the free

C*-

A

w i t h u n i t element

a l g e b r a generated by

C*- a l g e b r a such t h a t f o r e v e r y nonzero i n t o a C*- a l g e b r a B t h e r e i s a unique

-,B

II 211

with

=

1.

**-

homohomo-

L e t us f i r s t d e s c r i b e t h e

a(A). Consider f o r a moment A

as a v e c t o r space w i t h

( i n s t e a d o f t ) and denote by A Q A t h e a l g e b r a i c t e n s o r p r o d u c t ( a g a i n w i t h r e s p e c t t o t h e r e a l numbers). r e s p e c t t o t h e r e a l numbers

R

A.

361

Examples and Applications

D e f i n e m u l t i p l i c a t i o n , norm and i n v o l u t i o n i n

x

(

i

A @ A

Then

ai 8 bi)*

=

f bi 8 ai 1

A 8 A

by:

.

i s a normed r e a l a l g e b r a w i t h i n v o l u t i o n and u n i t

I 8 I.

Take t h e q u o t i e n t a l g e b r a w i t h r e s p e c t t o t h e i d e a l generated by

.

{ (ai) 8 b t a 8 ( i b ) la,b E A1

Since t h i s i d e a l i s i n v a r i a n t under t h e i n v o l u t i o n we have a c a n o n i c a l l y d e f i n e d i n v o l u t i o n i n t h e q u o t i e n t . O f course, t h e d i f f e r e n c e between

A Q A

and t h e q u o t i e n t i s t h a t we can now i d e n t i f y

.

(ia) Q b = - a Q ( i b )

T h i s means t h a t we can c o n s i d e r t h e q u o t i e n t a l g e b r a as a complex a l g e b r a such t h a t t h e i n v o l u t i o n

*

satisfies

( x ( a 8 b ) ) * = x(b Q a ) .

NOW, t a k e as

t(A)

the completion o f t h i s algebra w i t h respect t o t h e

s p e c t r a l r a d i u s norm. By abuse o f n o t a t i o n we can now i d e n t i f y As(a(A)) = A(A)* Observe, t h a t t h e u n i v e r s a l p r o p e r t y o f w i t h t h e s p e c t r a l r a d i u s norm of

implies that

A

endowed

can be c o n s i d e r e d as a complex subalgebra t(A) yields:

And a p p l i c a t i o n o f t h e Gelfand-Naimark Theorem f o r

t(A).

-Gelfand

p

t(A)

Representation ( c f . [ 275

F o_ r e_ very

2 i f II a II =

a E A llalI2

-we have for all

m e t r i c a l g e b r a isomorphism -~

1)

p(a) = s u p { l v ( a ) l a E A between

, Jhe~ p A

IU E =

A(A)). & p a r t i c u l a r ,

II II and t h e r e i s an i s o -

and a subalgebra

of

C(A(A)).

362

6.5

Representing Measures

THE BOCHNER

-

WEIL THEOREM

We rephrase some o f t h e r e s u l t s o f s e c t i o n s 6.3 and 6.4 f o r t h e case o f without unit). commutative i n v o l u t i v e normed algebras ( perhaps Let

N

be such an algebra, i . e . a commutative complex normed a l g e b r a w i t h

involution x By

Nsa

+

x*

.

We assume t h e i n v o l u t i o n t o be continuous.

we denote t h e r e a l a l g e b r a o f s e l f a d j o i n t elements i n

Nsa = C X E N I X = x * }

u : N

A functional x E N

all

+

.

a i s s a i d t o be symmetric i f u ( x * )

. The symmetric

c o m p l e x - l i n e a r f u n c t i o n a l s on

u n i q u e l y t o t h e r e a l - l i n e a r f u n c t i o n a l s on symmetric m u l t i p l i c a t i v e

If

i s continuous w i t h

p

C(V)

Ns,

=

N

u(x)

correspond

N , i.e. t h e

.

u i s called positive i f

= 1 t h e n we c a l l i t a s t a t e .

f o r a p o s i t i v e f u n c t i o n a l we have

for

AS(N) a r e t h e non-zero

Nsa.

c o m p l e x - l i n e a r f u n c t i o n a l s on

m u l t i p l i c a t i v e r e a l - l i n e a r f u n c t i o n a l s on

A symmetric c o m p l e x - l i n e a r f u n c t i o n a l

N:

p(xx*) 2 0

for all

Note t h a t

x E N.

Theorem 1:

i)

If

measure

&

u T

continuous and p o s i t i v e t h e n t h e r e i s a p o s i t i v e f i n i t e

on A,(N) -

such t h a t --

ii) _ If

N -----i s a Banach a l g e b r a t h e n e v e r y p o s i t i v e f u n c t i o n a l _ i s -autom a t i c a l l y continuous. iii)

Every extreme p o i n t -~

of t h e s e t o f s t a t e s i s m u l t i p l i c a t i v e . -

-

_

-

I

_

_

_

363

Examples and Applications

Proof:

As described i n 6.3 we adjoin a u n i t I t o N and obtain t h e algebra NI. By B we denote the Banach algebra given by the completion of N I . The convolution i s extended t o N I by putting I* = I , and t o B by c o n t i n u i t y . From s e c t i o n 6.4 we know t h a t Bsa ( s e l f a d j o i n t elements of B ) c o n s t i t u t e s an order u n i t cone, with I as order u n i t . Since B i s commutative the m u l t i p l i c a t i o n i n BSa i s monotone. Now, extend

p t o N I by putting p(1) = C ( p ) . ( I f NI i s not a Banach algebra then extend i t f u r t h e r t o B by continuity).We claim t h a t p i s p o s i t i v e on B , i . e . ~ ( y y*) 2 0 f o r a l l y E B . Since p i s l i n e a r and continuous i t s u f f i c e s t o prove t h a t f o r y = I - x , x E N. In f a c t

-

Therefore p is monotone, hence continuous with respect t o t h e order u n i t norm. Since t h e order u n i t norm i s equal t o the s p e c t r a l radius norm ,IJ i s continuous with respect t o t h e norm. This proves i i ) because we needed c o n t i n u i t y only f o r t h e case when N was not a Banach algebra. a r r i v e d a t t h e s i t u a t i o n considered i n s e c t i o n 6.3 a s s e r t i o n Since we i ) follows from t h e theorem i n t h a t s e c t i o n and i i i ) i s an immediate 0 consequence of t h e preceding lemma. We i l l u s t r a t e this r e s u l t f o r a concrete s i t u a t i o n . Let ( G , - ) be a l o c a l l y compact abelian group. Locally compact a b e l i a n groups have a very rich s t r u c t u r e concerning d u a l i t y (Pontryagin d u a l i t y theorem) and i n v a r i a n t measures (Haar measure, s e e 12401). For a general introduction i n t o t h e theory of these groups t h e reader i s r e f e r r e d t o any textbook on harmonic a n a l y s i s ( 1 i ke [219 I , [148 1 , [184 1 o r [273 I ) . Let us r e c a l l some of t h e basic notions. A m u l t i p l i c a t i v e map 6 denotes the s e t 3 : G -+ t z E t I I z I = 1) i s c a l l e d a character on G o f a l l characters and i s endowed with t h e topology of uniform convergence i s then l o c a l l y compact and i t i s an abelian on compact subsets of G . group with respect t o pointwise m u l t i p l i c a t i o n . Take an element s E G , then s E --t i ( s ) defines a c h a r a c t e r on . Hence, t h e r e i s a

.

364

RepresentingMeasures

canonical homomorphism G

-

which i s continuous a c c o r d i n g t o A s c o l i ' s

+

1emma. One o f t h e most i m p o r t a n t p r o p e r t i e s o f

m

sure

, that

i s a p o s i t i v e nonzero, r e g u l a r Borel' measure m which i s

, i.e.

translation invariant

A c G. Consider

i s t h a t i t admits a -Haar mea-

G

C ao(G)

, the

m(A.s) = m(A)

s E G

for all

and B o r e l s e t s

space o f complex-valued continuous f u n c t i o n s on

G

which vanish a t i n f i n i t y . Endow C ao(G) w i t h t h e sup-norm t o p o l o g y and denote by MB(G) t h e space of continuous l i n e a r f u n c t i o n a l s on C a,(G). Example 3 i n s e c t i o n 2.6.4 sho\:s t h a t f o r every unique bounded complex-valued t i g h t measure iC

IG

x(f) = Thus we can i d e n t i f y

MB(G)

.

G

measures on

f di

S p e c i a l elements o f

MB(G)

For

L 1 (G)

x,y E MB(G)

I G

L 1(dm)

. The

define

we d e f i n e c o n v o l u t i o n

*

y) =

.

E C U,(G)

a r e o b t a i n e d when t h e Haar measure

. Similarly,we

f d(x

J

s ,tEG

f(S*t)

L 2 (G)

.

x*y

and i n v o l u t i o n x* by

L2( G )

even an i d e a l o f

. L1(G)

i s a H i l b e r t space.

dx(s)dy(t)

(MB(G),*,*)

i n v o l u t i v e complex Banach a l g e b r a w i t h u n i t element G)

m is

subspace o b t a i n e d ' i n t h i s way i s

where t h e b a r denotes complex c o n j u g a t i o n . t h e u n i t of

such t h a t

G

w i t h t h e space o f bounded complex-valued t i g h t

m u l t i p l i e d by a d e n s i t y cp E denoted by

,f

x E MB(G) t h e r e i s a on

i s a commutative be

( D i r a c measure a t

i s a complex Banach subalgebra ( w i t h o u t u n i t ) ,

(MB(G),*,*)

.

365

Examples and Applications

For

x E MB(G)

we d e f i n e bounded, continuous, complex-valued f u n c t i o n s

and

Fx

by:

6

on

(F,)(s)

=

sEG

,

S(S) dx(s)

E

G

These f u n c t i o n s a r e c a l l e d t h e F o u r i e r - t r a n s f o r m o f x , r e s p e c t i v e l y . When

cotransform o f

complex-valued f u n c t i o n s on

i) a r e

FX

x

and t h e F o u r i e r

-

C aB(E) ( t h e bounded continuous

endowed w i t h t h e sup-norm and t h e i n -

v o l u t i o n g i v e n b y complex c o n j u g a t i o n , t h e n

C LB(g)

c o n s t i t u t e s a complex

i n v o l u t i v e Banach a l g e b r a w i t h r e s p e c t t o p o i n t w i s e m u l t i p l i c a t i o n . The importance o f t h e F o u r i e r - t r a n s f o r m stems from t h e f a c t t h a t Banach a l g e b r a homomorphism f r o m

MB(G)

into

F

is a

C OB(6). T h i s homomorphism

i s symmetric w i t h r e s p e c t t o t h e i n v o l u t i o n s , i . e . F

u

-

*

= F

u

for all

p E

MB(G)

.

Observation 1: Every

3 E 6 ~-~ d e f i n e s an element (*)

And f o r e v e r y ---

v,(x)

v,

= (Fx)(S')

1

v E A ( L (G))

S

1 E A ~ ( L( G ) )

for a l l

t h e r e i s some

Hence

C

1

= A ( L (GI)

= A,(L

by

1 x E L (G) S E

.

G -such t h a t

v = v-

S

.

1( G ) ) .

Sketch o f t h e P r o o f : R e c a l l t h a t t h e F o u r i e r t r a n s f o r m i s a symmetric Banach a l g e b r a homomorphism. T h i s immediately i m p l i e s t h e f i r s t p a r t .

366

Representing Measures

Now t a k e

1

v E A ( L (G))

Im1-l v ( x

SO(S) =

So E

Then

(1)

s,

. I f one

x

*

y =

1 x E L (G)

and

*

~ ( x + ) 0

.

s E G

Define f o r

bS)

keeps i n mind t h a t

tEG

*

x

bt d y ( t )

v, = v

t h e n one e a s i l y sees

with

R e c a l l t h a t a Banach a l g e b r a separates t h e p o i n t s o f

A

.

.

for all 0

A

i s s a i d t o be semisimple [275 I

if A(A)

Lemma 1:

1 L (G)

i s semisimple

Proof: I n view o f Theorem 1 i ) i t s u f f i c e s t o show t h a t f o r a g i v e n p o s i t i v e 1 x * x * E L (G) t h e r e i s a p o s i t i v e l i n e a r f u n c t i o n a l p such

element that

~ ( *x x*) > 0

F o r y E L 1( G )

.

Such a f u n c t i o n a l i s e a s i l y found:

consider the l i n e a r operator

.

T : L 2 (G) + L 2 (G) g i v e n by Y then t h i s o p e r a t o r i s i n -

2 T cp = y Y cp f o r a l l cp E L (G) If y P 0 Y j e c t i v e . Take now cp + 0 and c o n s i d e r t h e l i n e a r f u n c t i o n a l

P

1

: L (G)

g i v e n by ~ ( y )=

< q,Tycp>

(

< ,>

scalar product i n

L'(G)).

T h i s f u n c t i o n a l i s indeed p o s i t i v e s i n c e

Observation 2:

i) E

separates t h e p o i n t s o f

G

, i . e . t h e canonical map G

+

is

+

t

Examples and Applications

367

in

G)

inj e c t i ve . ii)

6

The l i n e a r h u l l o f

C a ( G ) ( c o n t i n u o u s complex f u n c t i o n s on

i s , w i t h r e s p e c t t o u n i f o r m convergence on compact subsets, dense i n

c

D(G)

.

Proof: i ) i s a consequence o f Lemma 1 and t h e f a c t i i ) Stone-Weierstrass Theorem. 0

6

1

= A(L ( G ) )

(Observation 1 ) .

Another a p p l i c a t i o n o f Theorem 1 i s t h e c h a r a c t e r i z a t i o n o f F o u r i e r - t r a n s forms o f p o s i t i v e f i n i t e measures. L e t us b e g i n w i t h a d e f i n i t i o n :

A function

i s s a i d t o be o f p o s i t i v e t.ype on

f E C ag(G)

f i n i t e s e t o f numbers we have

E a

cl,...,cn n

G

i f f o r every

s ~ ~ . . .E ~G s ~

and elements

ci Ck f ( s i S i l ) 2 0

1

i,k = l O b s e r v a t i o n 3:

f E

i ) Every

6 _ is _ of

p o s i t i v e -t y p e on

G

ii)

. -finite tight

The F o u r i e r t r a n s f o r m of e very p o s i t i v e -~ G _ is _ o f positive t y pe on 6 . iii)If f _ is _ o f p o s i t i v e -t y p e on G then f ( e ) iv)

f E C aB(G)

is _ of _

(*) f o r every --

I

p o s i t i v e ----t y p e i f and o n l y i f f d(x

*

x*) 2 0

x E MB(G).

Sketch o f Proof: i)

direct verification.

ii)Take

cl,.

..,cn

E t

2 (f(s)l

and

fl,. ..,gn

- . Then

E G

measure

T

-for a l l s

on E G

.

Representing Measures

368

n

X

ck

Ci

i, k = l

FT(Ti

5;')

=

SEG

(

X i,k

ck si

Ci

5;')

J I t ci Ci\ 2 ( s ) d.r(s)

=

dT(S)

(S)

2 0

.

s€G

iii) Take

c1 = 1

, c 2 --

a and

. Then

s1 = e, s2 = s

the definition o f

p o s i ti ve t y p e y i e l ds (a

Taking i(f(s) type) Q =

6

t

1) f ( e ) t a f ( s )

a = 1 yields that

-

f(s-'))

- Ifol , i f

(Note, t h a t

f(s-'). f(s)

f(s)

af(s-l) z

+

o

.

i s r e a l , a = i shows t h a t

f(s) t f(s-l)

i s real.

. Hence fo =

t

f(e) 2 0

, since

f

i s o f positive

NOW, e x p l o i t i n g t h e same i n e q u a l i t y w i t h 0, proves t h a t f ( e ) 2 I f ( s ) I

.

i v ) Note t h a t f o r sums o f D i r a c measures ( * ) i s e q u i v a l e n t t o t h e c o n d i t i o n r e q u i r e d i n t h e d e f i n i t i o n o f p o s i t i v e type. Then a p p r o x i m a t i o n o f f i n i t e measures by sums o f D i r a c measures y i e l d s t h e equivalence o f p o s i t i v e t y p e and ( * )

.

0

The f o l l o w i n g theorem i s due t o Bochner 150 1 f o r t h e case t o H e r g l o t z [1601 f o r

G = Z

, and

G = R " , due

t o A. Weil [3221 f o r t h e general case.

Theorem 2:

. -Then t h e

Let f E

C aB(G)

i)

is o f p o s i t i v e type on -

f

following are equivalent: G

ii) There ---- i s a unique p o s i t i v e measure f(s) =

I, S ( s )

ZEG

d.r ( S )

T

E MB(G)

for a l l

-such t h a t

s E G.

369

Examples and Applications

Proof:

ii)* i )L e t x

be a sum o f D i r a c measures on

G

.

Elementary computation

yields:

I

*

f ( s ) d(x

I

x*)(s) =

G

S ( s ) S ( t ) d x ( s ) dx*(t)dT(S)

&G2

and t h i s q u a n t i t y i s Hence

f

since

1 0

i s assumed t o be p o s i t i v e .

T

i s o f p o s i t i v e type.

i)* ii) S i n c e f

6

functional

Denote by

p

on

i s bounded ( O b s e r v a t i o n 3 iii)we can d e f i n e a l i n e a r MB(G)

by

the r e s t r i c t i o n t o

1

L (G)

. We

claim t h a t

p

i s positive i n

t h e sense o f Theorem 1. I n f a c t w i t h O b s e r v a t i o n 3 i v ) we o b t a i n : U(X +

-

X*

x

*

x*) = 5

- i((6, ;(he)

-

X)

=

*

(de

f(e)

Hence, Theorem 1 p r o v i d e s a p o s i t i v e measure U(X)

=

I

V(X)

dr(v)

-

X)*)

6

1

= A(L ( G ) )

;(be

*

he)

. T

1 on A(L ( G ) ) such t h a t for all

VW1(G)) With

+

( O b s e r v a t i o n 1) we can r e w r i t e :

1 x E L (G)

.

37 0

Representing Measures

i s a B a i r e measure on

6

(since a l l

measurable). T h e r e f o r e we can extend

T

t o a r e g u l a r Bore1 measure, i . e .

Note, t h a t

T

S have t o be p we

a t i g h t measure. Combining t h e formula above w i t h t h e d e f i n i t i o n o f have:

Hence

1,

f(s) =

Uniqueness o f

T

?EG

S ( s ) dT(2)

for all

s E G.

i s an immediate consequence o f Observation 2 i i ) .

Note, t h a t Theorem 2 s t a t e s t h a t t h e f u n c t i o n s o f p o s i t i v e t y p e on the restrictions t o

G

( c o n s i d e r e d as a subgroup o f

.

6

)

G

are

o f the Fourier

Since t h e elements o f MB(6) a r e transforms o f p o s i t i v e measures on ^G d i f f e r e n c e s o f p o s i t i v e measures t h i s means t h a t t h e r e s t r i c t i o n map t o G o f t h e F o u r i e r transforms o f

i s b i j e c t i v e . NOW, c o n s i d e r

MB(g)

G

6 .

as subgroup o f

subgroup. Assume t h a t t h e r e i s an element x o E E\G symmetric neighbourhood xo U2 = I x o s t

I s,t

U

of

E U3 c Z \ G

e E

. Then

o f the characteristic functions o f CP E L2('G)

n L 1(t)

f: U

and

xo E E L G

. T h-i s

P o n t r y a g i n D u a l i t y Theorem: G = ~~

lX u2

p

E L2(g)

E.

proves

. Then

cp =

cp

+

lU * lxou2

0,

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

G.

d i c t s t h e f a c t t h a t t h e r e s t r i c t i o n map t o can be no element

a compact

take the convolution

vanishes on

must be t h e F o u r i e r transform o f some

. Take

such t h a t

0

and cp

This i s then a closed

G

n

L1(c)

. But

cp

t h i s contra-

i s b i j e c t i v e . Hence, t h e r e

37 1

Examples and Applications

6.6

- KHINTCHINE FORMULA

THE LEVY

Again

MB(G)

denotes t h e bounded complex-valued

l o c a l l y compact a b e l i a n group measures i n MB(G).

. By

G

MB,(G)

i g h t measures on t h e

we denote t h e p o s i t i v e

Q u i t e o f t e n ,in p r o b a b i l i t y t h e o r y ,one has t o deal

w i t h i n f i n i t e l y d i v i s i b l e elements of t h a t f o r every given

t h e r e i s some

n E N

*

LI = v

*. . .

v

MB+(G)

*v

v

i.e.

E MB,(G)

p

such

with

E MB,(G)

(n-times).

F o r example, c o n s i d e r t h e p r o b a b i l i t y d i s t r i b u t i o n o f a random v a r i a b l e which, f o r a r b i t r a r y

, can

n

be w r i t t e n as t h e sum o f

n

independent

random v a r i a b l e s . T h i s i s t h e s i t u a t i o n which leads t o t h e c e n t r a l l i m i t theorem. Other examples a r e g i ven by t r a n s 1 a t i on

- in v a r i a n t Markov-semi -

) . S i n c e t h e s e examples p l a y an i m p o r t a n t r o l e i n

groups ( s e e [1421,[27]

appl ied p r o b a b i 1 it y t h e o r y

t h e c h a r a c t e r i z a t i o n o f in f i n i t e l y d i v i s i b l e

measures i s o f general i n t e r e s t . I n o r d e r t o a v o i d t e c h n i c a l i t i e s we r e s t r i c t o u r c o n s i d e r a t i o n s t o t h e s p e c i a l case

G =(R,+).

B u t a l l t h e arguments can b e t r a n s f e r r e d t o a more

general s i t u a t i o n . Those readers who a r e i n t e r e s t e d i n t h e general t h e o r y of convol u t i o n semi groups a r e r e f e r r e d t o [158 1,1171 1

81 1 , see a1 so

[1781 and [1421. Recall where

if

G = R

x E R

transform

.

t h e n a1 1 c h a r a c t e r s a r e o f t h e form

Hence we can i d e n t i f y o f a measure

F

lJ

F,(x)

=

R

p €

MB(R)

fi w i t h

, we

ei t x d p ( x )

,

t

+

e

-i tx

and,for t h e F o u r i e r -

R

o b t a i n t h e w e l l known f o r m u l a x E R

=

6 .

Consider t h e f o l l o w i n g s i t u a t i o n : Fix

N E N~

and l e t

f,goygly...ygN~l

f u n c t i o n s such t h a t :

(1)

lim It1

f(t) = 1 + -

: IR + R

be bounded

continuous

372

Representing Measures

(2)

f(t) > 0

(3)

f

t

for all

+

0

N-times continuously d i f f e r e n t i a b l e a t 0 with

is

and f ( " ) ( O )

f")(O)

> 0

(4)

g0(O)

g1(0) =

(5)

t h e functions g k a r e (N-k)gi") ( 0 )

=

=

* * .

k

< m

for 0 5 m < N

0

gN-,(0)

1 -< m 5 N

0 for

lim S U P It g k ( t ) l It1 + -

(6)

=

=

1

times d i f f e r e n t i a b l e a t 0 with

-

k

for k = OY1s2,...,N-1.

Theorem: Let i)

: R +

Whenever p1

ii)

be continuous. -Then t h e following a r e equivalent:

R

f tF

p E MB(R)

P

2-pf

has compact ~support and pl,p2

-,

E K,

-such t h a t

then

There a r e a polynomial P ( t ) of degree < N a p o s i t i v e regular Bore1 measure T W\{Ol and a t 0 w i t h a + ~ ( b I 0 ) I ) 1 -representation --holds f o r a l l t E R: such t h a t t h e following integral ---

u(t)

=

A d +

P(t)

t

f(N)(O)

(For N = 0 t h e sum term of t h e integrand vanishes). Note t h a t the integrand i n t h e i n t e g r a l representation of u i s a bounded continuous function on R ( a consequence of T a y l o r ' s formula together with the assumptions on f and go,...,gN-l) .

Examples and Applications

373

B e f o r e we p r o v e t h e theorem we mention a n o t h e r o b s e r v a t i o n which can be proved by elementary c a l c u l a t i o n : Remark: Let

MB(R)

p E

have compact s u p p o r t . Then t h e F o u r i e r t r a n s f o r m

J

F (x) =

u

R

eixt

dp(t)

i s i n f i n i t e l y o f t e n d i f f e r e n t i a b l e and t h e moments o f

are given by

LI

P r o o f o f t h e Theorem:

(ii) + (i):L e t

x

> 0

. Since

must h o l d f o r

have compact s u p p o r t such t h a t

p

the f i r s t

(N-1) d e r i v a t i v e s o f

. Therefore

Fp

the f i r s t

(N-1)

IF I 5

u

xf

f o r some

v a n i s h a t zero, t h e same

f

moments o f

vanish

p

(remark above). A p p l i c a t i o n o f F u b i n i ‘ s theorem t o t h e i n t e g r a l representation yields:

where

= T

T~

+ a 6

0

and where

FU(c’) f ( E ) - l

i s extended c o n t i n u o u s l y

i n t o t h e o r i g i n ( 1 ’ H o p i t a l ’ s lemma). Th s f o r m u l a immed a t e l y y i e l d s t h e required inequalities.

(i) (ii): measures p

Consider t h e v e c t o r space such t h a t

(7)

p

(8)

F ( 0 ) = F (1)( 0 ) =

E c MB(R)

cons s t i n g o f those

has compact s u p p o r t and t h e F o u r i e r t r a n s f o r m

u

P

moments a r e z e r o

*.*

= F(N-l)(0)

u

= 0

F

v

i s r e a l valued

, i.e. the f i r s t

(N-1)

374

Representing Measures

(9)

.

FP vanishes a t i n f i n i t y

Recall, t h a t t h e F o u r i e r t r a n s f o r m q u i t e o f t e n vanishes a t i n f i n i t y , f o r example i f !J has a continuous densi t y ( R i emann-lebesgue Lemma). Observe t h a t i f f o r a continuous f u n c t i o n v we have t h a t

Iv

!R

then

v

0

dp =

for all

1-1 E

must be a polynomial o f degree

, the

Now consider

R*

each element

r o f IR*

E

< N

.

one p o i n t c o m p a c t i f i c a t i o n o f a linear functional

on

cpr

R

,

and a s s i g n t o

E :

r+O,m

r = m

1F !J( N ) ( 0 )

One e a s i l y checks t h a t

r

+

if

cpr(p)

r = O

i s continuous on R*

f o r every

!J

E

The c o n d i t i o n imposed i n ( i ) guarantees

I u d!J I

R

sup CP,(!J)

for a l l

rER*

!J

E E

.

The Riesz-Konig Theorem g i v e s us a r e g u l a r Bore1 p r o b a b i l i t y measure on IR*

such t h a t

I u d!J = I R*

IR Define p

a = 0(0)

and

E vanish and s i n c e

T = u

cpr(u) d u ( r )

IR\

cp,(!~)

I01

.

p E

E

.

Since t h e f i r s t (N-1) moments o f

i s equal t o

moment we o b t a i n v i a F u b i n i ' s theorem

for a l l

f(N)(0)-l

times t h e N-th

a

E

.

Examples and Applications

J u

R

du =

J v

R

dp

for all

p E

375

E ,

where

B u t v is a continuous function ( T a y l o r ' s formula). F i n a l l y , s i n c e J ( u - v ) d u = 0 f o r a l l p E E the functions

u and v

R

must be equal u p t o a polynomial of degree < N . The condition a + ~ ( R \ { 0 1 ) I 1 i s f u l f i l l e d because T was a p r o b a b i l i t y measure. Let us i l l u s t r a t e this r e s u l t f o r concrete cases. F i r s t observe t h a t we have proved again Bochner's Theorem i n t h e following fomulation: Bochner's Theorem: A continuous function u -

: R +R i s t h e Fourier of-a p o s i t i v e - transform ~ T on R w i t h T ( R ) 5 1 i f and only i f f o r every regular Borel measure ~-~ -----complex Borel measure p 0" R with compact support -and w i t h p1 2 F 2 - p 2 (p, , p 2 E R+) -we have: u

Proof: Take N

=

0 , f = 1, go = l .

0

I t i s not a t a l l d i f f i c u l t t o transform t h i s condition i n t o the positivetype-condi t i o n .

376

Representins Measures

Another s p e c i a l case i s : The Levy

-

Let u : R

Khintchine

hu

g continuous f u n c t i o n . Then,for every

A 2 0, e i s t h e F o u ri e r t r a n s f o~ rm _ o f -some p o s i t i v e f i n i t e r e g u l a r B o r e l measure on R -i f and only i f t h e r e a r e a 2 0 , E R and a p o s i t i v e -f i n i t e Borel measure T JIJ R \ I O ) such t h a t -+

R

Formula:

_

_

_

_

_

I

_

_

_

~

-

_.-

for all --

t E IR

Proof:

A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t f o r any u ( t ) , which has an i n t e g r a l r e p r e s e n t a t i o n as above, t h e f u n c t i o n s e x u(t), x 1 0 , a r e indeed F o u r i e r t r a n s f o r m s o f p o s i t i v e f i n i t e B o r e l measures o n R . N = 2

put

,

f ( t ) = ( l + t 2 ) -t2 I , g o ( t ) = 1 and

x

Assume t h a t f o r every

t 0

the function

exu

gl(t)

=

( 1 + t 2 -1

.

i s the Fourier transform

of some p o s i t i v e f i n i t e B o r e l measure on R. We c l a i m t h a t t h e r e i s some

0 <

p

< +

whenever

such t h a t

pl,p2

t 0

and

p E

MB(R)

has compact s u p p o r t w i t h

Then t h e theorem y i e l d s t h e i n t e g r a l r e p r e s e n t a t i o n f o r

(i~ measure.

since

p-lu

i). Note t h a t t h e c o e f f i c i e n t s o f t h e a r b i t r a r y p o l y ii) must be r e a l s i n c e eB1 and 2 ~ a r e) t h e ez e r o '~t h and~ f i r s t moment, r e s p e c t i v e l y , o f a p o s i t i v e

satisfies assertion nomial

u

P ( t ) = B~ + i B t appearing i n

Examples and Applications

377

Proof o f t h e claim: F i r s t , we show t h a t i f

(*I

Judv

R

and i f

M > 0

M f t Fv 2 0

such t h a t

v

E MB(R)

has compact s u p p o r t

then

t o .

Indeed Bochner's Theorem shows t h a t

J eAudv

Hence

1

I (eAU-l)dv

A

The l i m i t

A

X 2 0

.

- -A1 J d v

t

=

-

-1 F (0) A V

= 0.

of t h e l e f t s i d e y i e l d s ( * ) .

0

-*

for all

2 0

R

Now, c o n s i d e r t h e f o l l o w i n g measure 1 T = b0 - 2 m , where

m

i s t h e r e s t r i c t i o n o f Lebesgue measure t o

t r a n s f o r m i s equal t o M1,F2 > 0

such t h a t

.

M2 FT s f s M I F Now, c o n s i d e r plf

t F

1-1

3 = (pl MI

M 2 0

where

t - p2 f T

-

. Hence,

p

,p2

p1

1-1)

t 0

. Since

that

The F o u r i e r -

sin t 7 ).

( F T ) ( t ) = (1 We can f i n d

[-1,+1].

u

and p1

M1 FT 2 F

.

Fy t 0

E MB(R)

u

w i t h compact s u p p o r t such t h a t we o b t a i n f o r t h e measure

I n a d d i t i o n we have

F- 5 M f

f o r some

we o b t a i n f r o m ( * )

i s independent o f

= ( p 2 M 1 ~t u )

-

u

. For

t h e o t h e r i n e q u a l i t y we p u t

and o b t a i n w i t h ( * )

J u d u

So, t h e c l a i m i s proved.


J

U

~

< Tp 2 p

< m .

Representing Measures

378

6.7 REMARKS AND COMMENTS The s e t X , c o n s i s t i n g of those c o m p l e t e l y monotonic S e c t i o n 6.1: f u n c t i o n f w i t h f 5 1 , c l e a r l y i s a simplex, hence a Bauer simplex, s i n c e t h e extreme p o i n t s e t i s closed. T h i s i s a consequence of t h e uniqueness o f t h e i n t e g r a l r e p r e s e n t a t i o n . I f we c o n s i d e r a1 1 c o m p l e t e l y monotonic f u n c t i o n s t h e n we have an example f o r a s i m p l i c i a l cone w i t h o u t base. F o r i n f o r m a t i o n about c o m p l e t e l y monotonic f u n c t i o n s t h e r e a d e r i s r e f e r r e d t o [3291,[3301,[ 351 and [36

.

1

S e c t i o n 6.2: The l o g a r i t h m s o f t h e i n f i n i t e l y d i v i s i b l e completely monotonic f u n c t i o n s are, again, an example f o r s i m p l i c i a l cone. S e c t i o n 6.3:

In case t h e cone under c o n s i d e r a t i o n i s a v e c t o r space t h e

n o t i o n s o f "monotone mu1 t i p l i c a t i o n " and "weakly monotone mu1 t i p l i c a t i o n " c o i n c i d e . T h i s i s a consequence o f iii)i n t h e Theorem. S e c t i o n 6.4:

I n t h i s s e c t i o n we wanted t o a v o i d t h e symbolic c a l c u l u s

and Cauchy's formula. F o r t h a t reason we considered, f o r example, t h e function

o n l y i n case t h a t

x

has s p e c t r a l r a d i u s norm s t r i c t l y

l e s s t h a n 1. Another f a c t , which we d i d n o t use,is t h a t , i n case o f a C*-

d1

algebra, a s e l f a d j o i n t

m.

x

i s p o s i t i v e i f and o n l y i f

O f course, a l l t h i s can be o b t a i n e d f r o m t h e II x It - x' 5 Gelfand Representation o r t h e Gelfand-Naimark Theorem, r e s p e c t i v e l y . Most

textbooks (see I2751 which we h i g h recommend) use t h e symbolic c a l c u l u s i n o r d e r t o develop t h e elementary t h e o r y , which t h e n l e a d s t o t h e Gel fand Representation. S e c t i o n 6.5:

I n t h i s s e c t i o n some o f t h e arguments, and many o f t h e proofs,

a r e very b r i e f . We squeezed l a r g e p a r t s o f t h e general t h e o r y i n t o a few pages i n o r d e r t o be a b l e t o p r e s e n t t h e Bochner-Weil Theorem i n t h e prop e r c o n t e x t . We thought t h i s necessary i n o r d e r t o emphasize i t s importance as a b a s i c t o o l i n harmonic a n a l y s i s . S e c t i o n 6.6:

One e a s i l y remarks t h a t t h e theorem o f t h i s s e c t i o n c o n t a i n s

a whole h i e r a r c h y o f Levy-Khintchine t y p e formulas. Nevertheless we admit t h a t we do n o t know o f any p r o b a b i l i s t i c i n t e r p r e t a t i o n o f t h e i n t e g r a l r e p r e s e n t a t i o n formulas o f h i g h e r o r d e r .

Examples and Applications

379

The reader should be aware of the f a c t that the decomposition of e l u , given by the Levy-Khintchineformula, i s a decomposition o f this semigroup into a translation-semigroup, a Gaussian semigroup and a Poisson semigroup. Of course, one can use the general integral representations o f the theorem

t o obtain other well known results. For example, N = 2, f ( t ) = t2, g o ( t ) = g l ( t ) = 1 yields Kolmogoroffs formula characterizing those measures of f i n i t e variance [1421 .

APPEND IX MEASURES AND THE R I E S Z REPRESENTATION THEOREM

The fundamental Riesz Representation Theorem concerning t h e r e p r e s e n t a t i o n o f c e r t a i n l i n e a r f u n c t i o n a l s is(among t h e b a s i c theorems)one o f t h e most u s e f u l r e s u l t s i n modern a n a l y s i s . I t l i n k s a n a l y s i s and measure t h e o r y and a l l o w s t h e a p p l i c a t i o n o f measure t h e o r e t i c methods (and language) i n connection w i t h a n a l y t i c problems. F o r t h e convenience o f those readers, who a r e n o t t o o f a m i l i a r w i t h measure theory, we r e c a l l t h e b a s i c d e f i n i t i o n s and we g i v e a s h o r t o u t l i n e how t h i s b a s i c r e s u l t can be proved. For f u r t h e r r e f e r e n c e t h e reader should c o n s u l t

A 1

[ 2471.

ALGEBRAS

U-

We s t a r t w i t h a nonempty s e t

1.1

C 271, L 1511or

.

X

Definition:

A c o l l e c t i o n I: o f subsets o f the following hold : (i)

X E z

(ii)

if A E z

then

X

X\A E

is a

U-

X

if

x W

(iii) i f An E z, n E N, t h e n If z

i s s a i d t o be a u- a l g e b r a i n

U An E n=l

algebra then the p a i r

x

(XJ)

i s c a l l e d a measurable space.

From t h i s d e f i n i t i o n i t i s q u i t e obvious t h a t any i n t e r s e c t i o n (as subsets of

P(X))

of

u- algebras i n

X

i f we t a k e t h e i n t e r s e c t i o n o f a l l

f o r some f i x e d containing M

M c P(X)

. We

U-

u- a l g e b r a i n

algebras i n

X

.

U-

a l g e b r a by

380

X.

Hence,

which c o n t a i n M ,

t h e n t h i s must be t h e s m a l l e s t

denote t h i s

u- a l g e b r a generated by M

i s a again a

algebra i n EM and c a l l i t t h e U-

X

Measures and the Riesz Repremntation Theorem

381

Usually very special u- algebras, like the Borel sets or the Baire sets are studied. Let X be a normal topological space. Then the members of Borel s e t s . If in the u- algebra generated by the closed sets are called -this definition the closed sets are replaced by the open Fu- s e t s (countable unions of closed s e t s ) then we obtain the

U-

algebra of the

-Baire s e t s . We denote these u- algebras by B(X) (Borel sets in X ) and B o ( X ) (Baire sets in X). If X i s metrizable then every open s e t i s an Fu

-

s e t , hence for metrizable X we have:

Bo(X)

B(X) =

.

O f course, we always have

since 1.1 ( i i i ) implies that every Fa- s e t belongs t o the u- algebra generated by the closed s e t s . In general ( i f n o t mentioned otherwise) the u- algebra considered in R ,

6 or 6 U I + - ) , resp., i s the u- algebra of a l l Borel s e t s . 1.e. B(R), B(6) and B(6 u {+-=I) are generated by the following collection C

(--,a1 I a E R I

of intervals.

Now, consider a measurable space from a s e t ?, into X. Then

.

i s a u- algebra in X f : ?, + X i s said t o be if

f-’(z)

c

2 . Of

I f in

(X,z) and a function

X a a- algebra

*

f : X

+

X

-

i s given then ( i ,1)- measurable ( o r measurable in short)

course, i f z

i s generated by

Z

Mc

P(X) then for the

Appendix

382

f

measurability o f

Instead o f

U-

algebras generated by a g i v e n c o l l e c t i o n o f s e t s one can

u- a1 gebras generated by f u n c t i o n s .

consider

F be a c o l l e c t i o n of f u n c t i o n s

So, l e t

stand t h e s m a l l e s t

U-

algebra i n

i s exactly the

Bo(X)

continuous f u n c t i o n s on (upper

x

X)

U-

f € F

such t h a t a l l

a l g e b r a generated by

and t h a t

semicontinuous f u n c t i o n s on

X

Let

t h e n by IF we under-

+ li;

a r e measurable.

i s a normal t o p o l o g i c a l space, t h e n a s i m p l e c o n s i d e r a t i o n y i e l d s

that

Y :

X

f : X

u- a l g e b r a generated by

This i s the

If X

i t s u f f i c e s t o show t h a t

B(X)

C(X)

(realvalued

i s generated by

USC(X)

X).

be an a r b i t r a r y t o p o l o g i c a l space. We r e c a l l t h a t a f u n c t i o n +

6

i s upper semicontinuous i f

e v e r y a E IR

and t h a t

{x E X

a E

IR

( x ) < a3

i s open f o r

cp : X + IR U { + a }i s l o w e r semicontinuous

tx E X

i s upper semicontinuous, o r e q u i v a l e n t l y , i f for all

IY

.

I ~ ( x >) a1

if

-

cp

i s open

A r a t h e r good-natured o b j e c t i s Bd x ( X ) = (X,x)

{flf

: X +IR

i s bounded and

an a r b i t r a r y measurable space.

(I, Bor(lR))- measurable},

T h i s i s n o t o n l y a v e c t o r space, b u t

i n a d d i t i o n a v e c t o r l a t t i c e (under p o i n t w i s e suprema and i n f i m a ) which is

u- complete. Here

whenever

u- complete means t h a t

fn E B d I ( X ) , n E N, such t h a t

pleteness o f Instead o f respectively

BdI(X) BdB(X)

.

sup fn belongs t o nm

sup fn i s bounded. The nm

r e f l e c t s t h e p r o p e r t y r e q u i r e d i n 1.1 and

BdB, (X), we w r i t e

Bor(X)

BdI(X) u-com-

(iii).

and B a i r e ( X ) ,

Measures and the Riesz Representation Theorem

A 2

MEASURES

2.1

Definition:

Let (X,E) be a measurable space. A map m : X measure i f m(D) = 0 and i f (i) m(A U 8 ) t m ( A

(ii)

If

fi

B)

=

for all

m ( A ) t m(B)

An E X, n E N , such t h a t

+

An+1

2

An

383

i s called

uI+-)

IRt

A,B E

for a l l

x n E N

then we have always sup m ( A n )

n a

=

m(

U An) nEN

.

Sometimes the measures defined A measure i s c a l l e d f i n i t e i f m ( X ) < above a r e c a l l e d p o s i t i v e measures t o s e t them a p a r t from signed measures. 03.

Signed measures a r e differences of positive measures. The tripe1 (X,r,m) i s usually s a i d t o be a ~measure space. P o s i t i v e f i n i t e measures w i t h m ( X ) = 1 a r e c a l l e d p r o b a b i l i t y measures, and (X,Z,m) i s then s a i d t o be a p r o b a b i l i t y space. An obvious consequence of i .e.

2 . 1 ( i ) i s t h a t p o s i t i v e measures a r e monotone,

m ( A ) I m(B)

whenever

A,B E II w i t h

A

c

B.

And .:om 2 . ( i i ) t h e important property follows t h a t measures a r e 0- a d d i t i v e , t h a t means : m(U{An

which

In

E Nl)

I 1

n a

m(An),

when An E E, n E N

,.

implies e q u a l i t y i n c a s e t h a t t h e An a r e pairwise d i s j o i n t .

A s e t N E x with m ( N ) = 0 i s c a l l e d an m-nullset and two functions f, g on X having d i f f e r e n t values only on an m-nullset a r e s a i d t o be equal m- almost everywhere ( f = g m- a . e . ) .

Appendix

384

(X,x,m)

Let

Then a measurable f u n c t i o n measure m

?\

For

Then of

m

from

X

to

X

~p :

x

--t

. This

X

a measurable space.

can be used t o t r a n s f e r t h e

can be done i n t h e f o l l o w i n g way.

define

E

m

(i,i) be

be a measure space and l e t

. This

i s a measure on

under 10

.

measure i s c a l l e d t h e -~ image measure

The most i m p o r t a n t f e a t u r e of measures i s t h a t t h e y enable us t o d e f i n e i n t e g r a l s i n a v e r y n a t u r a l way. Roughly speaking, i n t e g r a l s a r e p o s i t i v e l i n e a r f u n c t i o n a l s w i t h s t r o n g convergence p r o p e r t i e s .

(X,r,m) be a measure space. For s i m p l i c i t y we assume t h a t m i s a f i n i t e measure. To d e f i n e t h e i n t e g r a l we s t a r t w i t h measurable step-

Let

f u n c t i o n s . These a r e f u n c t i o n s o f t h e f o l l o w i n g t y p e :

n s = E a k=l where

1 Ak

1 Ak

,

akEW,

AkEE,nEH,

denotes t h e c h a r a c t e r i s t i c f u n c t i o n o f

Ak, For a such a step-

f u n c t i o n an i n t e g r a l i s g i v e n by

X

s dm

n E ak m(Ak) k=l

=

One can e a s i l y show t h a t t h i s d e f i n i t i o n makes sense,i.e. s,

another r e p r e s e n t a t i o n o f

s

=

m

Now, f o r a r b i t r a r y

J

X

f dm =

m E B . 1 , B . E IR, 6 . E j=1 J Bj J J

x ,

then

n

f E Bd 1 (X)

sup[

i f there i s

we d e f i n e

J s dm I f X

2 s,s

measurable s t e p - f u n c t i o n } .

385

Measures and the R i e v Representation Theorem

The i n t e g r a l o f

f

B E II of

on a subset

J f d m = B

J f . l B d m X

X

i s d e f i n e d as

.

An easy c a l c u l a t i o n y i e l d s t h a t by

-,

f

j f d m X

a l i n e a r f u n c t i o n a l on

Bd Z ( X )

i s g i v e n . Furthermore t h i s f u n c t i o n a l i s

monotone, i.e.

1f X

dm 5

X

f

dm

for

f,? E B d I ( X ) w i t h

f I

f .

B u t i n c o n t r a s t t o o r d i n a r y l i n e a r f u n c t i o n a l s t h e i n t e g r a l has t h e f o l l o w i n g s t r o n g convergence p r o p e r t y : Monotone convergence p r o p e r t y : (i.e.

f n+l < for a l l n) - fn -

L e t fn, n E N, be-a d e c r e a s i n g sequence n a

bounded, t h e n ~inf n€N

j X

fn dm

is

i n B d z ( X ) -such t h a t i n f (f,) -

=

I

X

i n f (f,) n€N

T h i s convergence p r o p e r t y r e f l e c t s t h e

0-

dm.

a d d i t i v i t y o f t h e measure m.

Only f o r convenience we have s t a t e d t h i s p r o p e r t y f o r bounded f u n c t i o n s .

A s i m p l e e x e r c i s e shows t h a t i t goes o v e r unchanged t o unbounded f u n c t i o n s . V a r i a t i o n s o f t h e above r e s u l t can be found i n c o u n t l e s s forms i n t h e 1 i t e r a t u r e (Lebesgue Dominated Convergence Theorem, F a t o u ' s Lemma, Lemma o f Beppo -Levi

, etc. ) .

NOW, l e t us c o n s i d e r a monotone, r e a l v a l u e d l i n e a r f u n c t i o n a l B d Z (X)

h a v i n g t h e monotone convergence p r o p e r t y , i . e . i n f n€N

u on u(fn) =

386

Appendix

p ( i n f fn) whenever n€N

5 fn

f,+l

for a l l

n E N and

i n f fn i s bounded. nm

One c o u l d have t h e i d e a t o c a l l such a l i n e a r f u n c t i o n a l an a b s t r a c t i n t e g r a l . B u t t h i s does n o t l e a d t o a n y t h i n g new, s i n c e from an a b s t r a c t integral

for

p :

A E 1

Bdx(X) + R we can e a s i l y o b t a i n a measure by d e f i n i n g

. And -

If X

Finally, a t r i v i a l

- we

no wonder dm =

p(f)

- but useful

should be mentioned. L e t

have then for a l l

-

be a measure space and l e t

measurable space. Then g i v e n a measurable f u n c t i o n

J f dm

=

where

A 3

J

X

dm

focp

i s t h e image measure o f

--

f o r m u l a f o r i n t e g r a l s o f image measures

(X,f,m)

X

.

f E Bz(x)

forall

m under

cp

cp : X + X

(XJ)

be a

we have

f EBd;(i),

.

THE RIESZ REPRESENTATION THEOREM

w i l l be a nonempty compact Hausdorff space and p : C(X) + I R i s a monotone l i n e a r f u n c t i o n a l on t h e r e a l v a l u e d continuous f u n c t i o n s on X Here "monotone" i s always used w i t h r e s p e c t t o t h e p o i n t w i s e o r d e r o f C(X). For t h e e x t e n s i o n o f p t o an i n t e g r a l we need t h e knowledge o f t h e f o l l o w i n g s i m p l e f a c t ( p r o o f l e f t t o t h e r e a d e r ) :

Throughout t h i s s u b s e c t i o n X

.

3.1

Remark:

Let Y

semicontinuous. I f Y < i s a continuous

beupper semicontinuous on (o

g with

(i.e.

X

~ ( x < ) q ( x ) for a l l

Y 5 g < cp

.

and l e t cp be l o w e r x E X)

then there

C e r t a i n l y , t h e reader has observed immediately t h a t , because of t h e compact-

387

Measures and the Riesz Representation Theorem

ness of X , t h i s i s a s p e c i a l case o f t h e famous Tong - K a t e t o v Theorem. We s t a t e d i t i n t h i s s p e c i a l form o n l y t o enable everybody t o f i n d t h e simple proof f o r himself. Now, we extend

G

By d e f i n i t i o n

;:

t o a functional

p

USC(X)

6

-*

on t h e cone o f upper

X. T h i s i s done b y

semicontinuous f u n c t i o n s on

i s s u b l i n e a r and monotone ( w i t h r e s p e c t t o p o i n t w i s e

o r d e r ) . Furthermore,

c o i n c i d e s on

LI

d i f f i c u l t t o see i s t h e f a c t t h a t

C(X)

with

.

p

Somewhat more

G i s indeed l i n e a r .

3.2

Lemma:

(i)

G

i s linear

( i i)

i

has t h e monotone converqence p r o p e r t y USC(X) , i.e. when n E N , jsi p o i n t w i s e d e c r e a s i n g sequence in USC(X) then

,Y,

~

i i ( y n ) = G ( i n f yn) n a

inf na

Proof:

. i s s u b l i n e a r , so i t remains t o

(i) We a l r e a d y observed t h a t

i s s u p e r a d d i t i v e . F o r t h i s purpose we t a k e a r b i t r a r y

prove t h a t I Y ~ , YE ~ USC(X).

the function

Then f o r

cp =

g

Using Remark 3 . 1

-

yll

g E C(X)

+

E

with

g 2

yll

+

y2

and

i s l o w e r semicontinuous w i t h

we f i n d a continuous

h with

h2

cp>

t

> Y2

cp y12

> 0

. We

. put

h 1 -- g - h t t r ~ a ~n d g e t

Using t h e f a c t t h a t

LI

i s a d d i t i v e on

C(X),

t h i s yields:

NOW, t a k i n g on t h e r i g h t hand s i d e t h e infimum o v e r

g

and

t

we have

Appendix

388

Hence,

must be s u p e r a d d i t i v e .

(ii) The i n e q u a l i t y

i s t r i v i a l , since w i t h g 1 Y = i n f Y, n€N some

no E N

i s monotone. Now, we t a k e an a r b i t r a r y g E C(X) by compactness of X , we find, f o r E > 0

. Then

such t h a t

Taking t h e i n f over

gt

g and

E

> Y

E

.

,

Hence

on t h e r i g h t s i d e we g e t t h e d e s i r e d

inequal ity

I n t h e c o n s t r u c t i o n o f C we have approached t h e upper semicontinuous f u n c t i o n s f r o m above by continuous f u n c t i o n s w i t h o u t l o o s i n g a- a d d i t i v i t y . Now, we c o n t i n u e t h i s process by approaching t h e bounded f u n c t i o n s f r o m below by upper semicontinuous f u n c t i o n s . By d o i n g t h i s we l o o s e u- a d d i t i v i t y as w e l l as l i n e a r i t y . Hence we have t o r e s t r i c t t h e con-

s t r u c t e d f u n c t i o n a l t o a s u i t a b l e subspace where a1 1 d e s i r e d p r o p e r t i e s remain v a l i d . F o r t u n a t e l y , i t t u r n s o u t t h a t

Bor(X)

i s such a s u i t a b l e

subspace. L e t us d e s c r i b e t h i s procedure i n g r e a t e r d e t a i l . We denote t h e space o f bounded f u n c t i o n s on functionals f o r

b E Bd(X) ;(b)

X

by

by:

= SUP{;(Y)

I \y

E USC(X),

5 b)

Bd(X). We d e f i n e

389

Measures and rhe Riesz Representation Theorem

and

-

G(b) =

;(-

b ) = i n f { - ;(P)

Both f u n c t i o n a l s a r e monotone and d e f i n i t i o n s u p e r l i n e a r whereas G super1 in e a r i t y o f

;

IY

E USC(X),

-

\y

2 b}

.

( c a l l e d t h e l o w e r i n t e g r a l ) i s by

(upper i n t e g r a l ) i s s u b l i n e a r . The

imp1 i e s :

Hence we have -

(3.1)

v

ll2ll.

Obviously,

equals

; on

from t h e d e f i n i t i o n o f

.

USC(X). And, s i n c e

C ( X ) c USC(X), we o b t a i n

;(Y) s ;(Y) f o r a l l

that

Y E USC(X).

Hence, we have

;(Y) as a consequence o f

=

G(Y)

for all

Y E USC(X)

(3.2)

(3.1).

L e t us t u r n o u r a t t e n t i o n t o

U-

a d d i t i v i t y . The b e s t we can do i s gathered

i n t h e f o l l o w i n g obvious 3.3

Let

Remark: bn E Bd(X), n E N

supremum

, be

-

b

=

sup bn n a

(ii) ;(b)

s

sup L ( b n ) n a

a p o i n t w i s e i n c r e a s i n g sequence w i t h bounded

Then

.

I f we have i n a d d i t i o n

G(bn) = <(bn)

for all

n

390

Appendix

t h e n t h i s i m m e d i a t e l y i m p l i e s ( w i t h t h e h e l p of

(3.1))

T h i s g i v e s us some i m p o r t a n t i n f o r m a t i o n concerning t h e s e t E

F i r s t we c l a i m t h a t Sketch --_ (3.1)

I G(f)

I f E Bor(X)

=

E equals

Bor(X)

= {(f)}

. i; t o g e t h e r w i t h

o f a p r o o f --f o r t h i s c l a i m : S u p e r l i n e a r i t y of and t h e s u b l i n e a r i t y o f ;(b)

i t i s a v e c t o r space s i n c e

whenever

i; i m p l y t h a t E i s a cone. I n a d d i t i o n

-

=

G(-b)

,b

E Bd(X), shows t h a t

-

b E E

b E E.

The supremum o f two upper semic,ontinuous f u n c t i o n s i s a g a i n upper semicontinuous, t h i s e a s i l y leads t o t h e o b s e r v a t i o n t h a t E i s a v e c t o r l a t t i c e w i t h r e s p e c t t o p o i n t w i s e suprema. Remark 3.3 t h i s vector l a t t i c e i s

U-

complete. And, because of

c o n t a i n USC(X). Now, s i n c e

Bor(X)

i s the smallest E

l a t t i c e c o n t a i n i n g USC(X), we o b t a i n

i

The s u b l i n e a r

i;

must be l i n e a r on

. And

3

Bor(X).

E = Bor(X)

(iii) shows t h a t (3.2),

E must

u- complete v e c t o r 0

s i n c e i t i s equal t o t h e

(iii) shows t h a t i t i s i n a d d i t i o n Bor(X) i s an a b s t r a c t i n t e g r a l and so, t h e r e must be a Borel measure m (meaning a measure on t h e B o r e l s e t s ) with superlinear

U-

Remark 3.3

a d d i t i v e . Hence

X

Hence m

restricted to

b dm = i ( b )

for a l l

i s a r e p r e s e n t i n g measure f o r

we mean t h a t t h e i n t e g r a l w i t h r e s p e c t t o t h e given functional

X

p

b E Bor(X)

p

. Here, m

by r e p r e s e n t i n g measure

restricted to

, i.e.

f dm = v ( f )

for all

(3.3)

f E C(X)

.

C(X)

equals

39 1

Measures and the Riesz Representation Theorem

Our c o n s t r u c t i o n o f

m y i e l d s some more i n f o r m a t i o n . Namely,

m

is

r e g u l a r i n t h e f o l l o w i n g sense: m(B) = sup{m(K) for all

I

K c B y K compact}

(3.4)

B E B(X)

T h i s i s e a s i l y seen. Our c o n s t r u c t i o n o f

m

gives

where t h e i n e q u a l i t y i s d e r i v e d from t h e f a c t t h a t t h e f u n c t i o n s

lK, K c X pact s e t

compact, belong t o K,(Y)

USC(X).

NOW, f o r

Y E USC(X)

d e f i n e a com-

by

Then because of m o n o t o n i c i t y and SUP{~;(Y)

Y E

a d d i t i v i t y we o b t a i n :

USC(X),

5 sup{c(lK ( Y ) )

n

0-

1

Y IlB)

n E N,

Y E

U S C ( X ) , Y 5 lg) .

T h i s t o g e t h e r w i t h (3.5) g i v e s t h e d e s i r e d r e g u l a r i t y r e s u l t . I n f a c t , i f we i n s i s t on r e p r e s e n t i n g o u r f u n c t i o n a l by a r e g u l a r measure t h e n a l l t h e e x t e n s i o n s we have done i n t h e c o n s t r u c t i o n o f G a t h e r i n g these r e s u l t s we o b t a i n t h e c e l e b r a t e d :

rn a r e unique.

392

Appendix

Riesz ReDresentation Theorem:

Let

X

be a compact H a u s d o r f f space and l e t

C(X)

p:

-*

R

-be-a

monotone

linear functional. Then t h e r e i s a unique r e g u l a r B o r e l measure m such . _ _ _ _ - ~ - -

that -

X

f dm =

p(f)

for a l l --

f E C(X)

.

I n o u r above reasoning we considered B o r e l -measurabl e f u n c t i o n s . So t h e measure, which appeared i n t h e statement o f t h e Riesz Representation Theorem, was a B o r e l measure. We c o u l d have r e p l a c e d a l l t h e f u n c t i o n s b y Baire-measureable f u n c t i o n s i n t h e p r e c e d i n g c o n s t r u c t i o n , e s p e c i a l l y USC( X )

by Baire-measurable upper semicontinuous f u n c t i o n s . T h i s would

have enabled us t o s t a t e t h e Riesz Representation Theorem w i t h B a i r e measures i n s t e a d o f B o r e l measures. However, s i n c e B o r e l measures r e s t r i c t e d t o t h e B a i r e s e t s a r e B a i r e measures, t h e case o f B a i r e measures

i s a l r e a d y c o n t a i n e d i n t h e preceding f o r m u l a t i o n o f t h e Riesz Representat i o n Theorem. Nevertheless u s i n g B a i r e measurable f u n c t i o n s i n s t e a d o f B o r e l measurable f u n c t i o n s i n t h e p r e c e d i n g c o n s t r u c t i o n y i e l d s a n o t i o n

o f r e g u l a r i t y adapted t o t h e B a i r e s e t s ; i . e . t h e r o l e o f compact s e t s i s taken by t h e compact G6- s e t s . ( R e c a l l , a G6- s e t i s , by d e f i n i t i o n , a c o u n t a b l e i n t e r s e c t i o n o f open s e t s ) . On t h e o t h e r hand we g e t uniqueness

o f t h e r e p r e s e n t i n g B a i r e measure w i t h o u t any f u r t h e r r e s t r i c t i o n ( 1 ike r e g u l a r i t y ) since Baire(X)

i s the smallest

U-

complete v e c t o r l a t t i c e

c o n t a i n i n g C(X). Hence e v e r y B a i r e measure must have t h e adapted r e g u l a r i t y property :

3.5

Remark:

L e t m be a B a i r e measure on m(B) = s u p h ( K )

f o r a l l Baire sets

B

.

I K c B,

X K

. Then i s a compact

G6- s e t } ,

393

Measures and the Riesz Representation Theorem

A 4

THE RANDON

-

NIKODYM THEOREM

L e t u s r e c a l l some n o t i o n s . We s t a r t w i t h p o s i t i v e measures a measurable space

nuous w i t h r e s p e c t t o m-nullset when

T

. This

A E and

m

(denoted

by

i s s a i d t o be a b s o l u t e l y c o n t i -

T

if

<< m)

T

T(A) = 0

f o r every

s i t u a t i o n c o n t r a s t s w i t h t h e case

m a r e m u t u a l l y s i n g u l a r , where m u t u a l l y s i n g u l a r means t h a t

one can s p l i t up such t h a t

The measure

(X,X).

m on

and

‘I

X = A U B

T(A) = 0

and

A

i n t o d i s j o i n t measurable s e t s

.

m(B) = 0

I n o t h e r words,

c o n c e n t r a t e d on d i s j o i n t s e t s . T h i s i s denoted by

T

and

T

.

Im

B

and

m are

F o r t h e p r o o f o f t h e Radon-Nikodym Theorem we need t o r e c a l l t h e d e f i n i t i o n of t h e c l a s s i c a l

Lp- spaces. But, up t o now, we o n l y d e f i n e d i n t e g r a l s

f o r bounded f u n c t i o n c s , so we have t o s t a r t by e x t e n d i n g t h e i n t e g r a l t o unbounded measurable f u n c t i o n s

-,6

f : X

U

I + -1, where

measure space and where measurable means -I.

B o r e l - measurable. Using t h e

monotony o f t h e measure t h i s e x t e n s i o n i s , f o r

J f

dm = sup{

X

For

-f

J

X

g dm

If

i s some

(X,X,m)

f 2 0

, easily

2 g, g E Bd

z(X)>

obtained by

.

f I 0 we d e f i n e t h e i n t e g r a l t o b e t h e n e g a t i v e o f t h e i n t e g r a l o f and f o r a r b i t r a r y

f

the i n t e g r a l i s obtained by s p l i t t i n g

f

up

i n t o i t s p o s i t i v e and n e g a t i v e p a r t ( p r o v i d e d t h a t t h e i n t e g r a l s o v e r these

-

parts are not both i n f i n i t e ) . For

15 p <

t h e c l a s s i c a l space

measurable f u n c t i o n s and

Lm(m)

f : X

+

LP(m)

6 u I +-I

-

i s t h e n t h e space of a l l such t h a t

I

l f l P dm <

,

i s t h e space o f a l l measurable f u n c t i o n s which d i f f e r f r o m a

bounded f u n c t i o n o n l y on an m - n u l l s e t . As u s u a l l y , two f u n c t i o n s i n these spaces which d i f f e r o n l y on an m - n u l l s e t a r e i d e n t i f i e d .

Now, l e t us g i v e two i m p o r t a n t r e s u l t s o f measure t h e o r y . I n case o f f i n i t e measures

m

and

T

a b e a u t i f u l u n i f i e d approach

274 , p. 1231 ) . He considered t h e f i n i t e measure v = T t m and t h e i s due t o von Neumann (see [

394

Appendix

2 L ( v ) . I t i s q u i t e obvious t h a t t h e map f

H i l b e r t space

+

I X

bounded l i n e a r f u n c t i o n a l on

2 L ( v ) . Hence t h e r e must be some

such t h a t , f o r a l l

, one

f E L’(v)

T

is a

2 g E L (v)

has

.

1fd.r = I f g d v X X R e w r i t i n g t h i s one o b t a ns ( w i t h

f d

v =

T t

m)

dT = l f g dm. X Elementary c a l c u l a t i o n shows t h a t one can assume t h a t i n [0,11. So, we c o n s i d e r t h e d i s j o i n t s e t s

Ix E X

D = NOW,

1 g(x)

< 1) and

we d e f i n e measures

Then

T~

Because

<< T

m and =

T

a

t T

T~

s

1

g

E = X \ D = { x l g ( x ) = 1)

takes o n l y values

.

by

T ~ , T ~

m

.

we have decomposed

T

i n t o a b s o l u t e l y continuous and

s i n g u l a r p a r t s w i t h r e s p e c t t o m. Furthermore one e a s i l y shows t h a t such a decomposition i s unique. B u t von Neumann’s argument shows a l i t t l e b i t more. I n o r d e r t o see t h i s we i n s e r t , f o r a r b i t r a r y

A E t , the function

n fn = lA(l t z gk) k=l i n (*)

and we g e t fn ( 1 - g ) d T

A

= I (1-gntl)d-r A

= I g ( 1t

A

Z ‘

k=l

9k ) d m

.

395

Measures and the Riesz Representation Theorem

NOW, writing

h = g ( l + "Z gk ) k=l

and using the f a c t t h a t

D monotonically converges t o

1 and i s equal t o with the monotone convergence property T,(A)

I 1d r = Ih

=

DnA

This implies in particular h

(i)

E

L 1( m ) .

-

0 outside

gntl)

in

D we obtain

dm.

Summing a l l t h i s u p we get:

Theorem ( [ 274 , p.1241):

4.1 Let -

A

( 1

T

and m -be f i n i t e positive measures on There i s a unique decomposition of T~ 4: m and T~ 1 m __

.

---__.

with -

T

(X,E).

into

T = T~

+

rs

such t h a t , -for a l l ( i i ) There i s a function h E L1(m) (called density) -A E E , -we have ~~

ra(A) =

I

A

h dm.

The f i r s t part of 4 . 1 i s called the Lebesgue decomposition theoremandthe second part the Radon-Nikodym theorem. A particular consequence of ( i i ) i s that i f T i s already absolutely continuous with respect t o m then i t can be represented by a density. Both parts of Theorem 4.1 do n o t remain valid in case one drops the condition that the measures be f i n i t e . B u t the second p a r t i s true in a more general situation. All w h a t one has t o assume i s t h a t m i s U- f i n i t e . We recall t h a t m i s said t o be a- f i n i t e i f there i s a sequence An E I: such t h a t m ( A n ) < = and U A,, = X . nEN

Then, by exhausting s e t s with the sequence An , one can prove: Radon-Ni kodym Theorem:

m & positive measures on (X,I:) such t h a t m a- f i n i t e and assume that T i s absolutely continuous --with respect t o m . Then there -i s_ a measurable function (or : X +iR+ u { + = I (called the density of T ) such _ _t h a t T ( A ) = I (or dm -for a l l A E E . Let -

T

_.___-

---I

A

2 uniquely

determined up t o equality m-almost everywhere. If a- -----f i n i t e then we can assume t h a t (or takes only - values ~ in -IR,

(oT

.

T

is

396

A 5

Appendix

SIGNED AND LATTICE

-

VALUED MEASURES

Let us s t a r t with some remarks a b o u t measures having valued in some order complete vector l a t t i c e R I n order t o be able t o deal with i n f i n i t e measureswe adjoin t o Rt = { r E R I r t 01 the abstract element + and we define t h i s element t o be the supremum of every unbounded subset of R . Furthermore we extend the cone operations of R in the obvious way t o R U {+-I So in particular we require t h a t 0 . ( + O D ) = 0 . We remark t h a t multiplication with -1 i s a linear map from R UI+-I into a ( i f one puts - ( + m ) = -).

.

.

-

The theory of R- valued measures can be developed very much in the same way as the theory of real-valued measures, although a word of warning seems appropriate. Arguments depending on techniques related to densities or t o notions like regularity or localization can be transferred t o t h i s nice general situation only under strong additional hypothesis. Let ( X , 1 ) be a measurable space. Then a map an R- valued measure i f T($) = 0 and i f (i1

T

i s positive, i . e .

(ii)

T

i s additive, t h a t means T ( A U B)

(iii)

T

t

T ( A ) E R, U I + - } T ( A ) + T(B)

T(A n B)

i s u- monotone, i . e .

T

: I:

+

R

for a l l for a l l

u {+-I 4E

i s called

Z

A, B E 1

for every increasing sequence An

in

1 we have

lim inf T(A,) n 3-

= T(U An) nEN

O f course, such a measure i s said t o be a fi n i t e measure -if

T(X) < +

Now, l e t us be more precise a b o u t what we understand by a siqned

m.

R-

valued measure (signed measure in short). T h a t i s an additive, u- monotone map T : 1 + R with T($) = 0 such t h a t (iv)

T

i s bounded, i . e .

SUP{T(A)

IA

E I:} < +

m

.

We should remark t h a t in many cases the l a s t condition i s redundant, for

397

Measures and the Riesz Representation Theorem

example i f

5.1 If -

R =R ! :

Lemma: T

: t

+

k i s additive and

P r o o f : ( p r i v a t e communication by For

if

m -n -tone t h -

U-

bounded.

Anton Schep) :

.+(A) = S U ~ I T ( B I) B c A, B E tl . We c l a i m t h a t then, f o r e v e r y n , t h e r e i s B E z w i t h B c A,

A E t we d e f i n e .+(A)

= +

T(B) 2 n

and

T

+ (B)

. This

= +

proves t h e lemma: Because i f

t h e n t h e a s s e r t i o n o f t h e c l a i m p r o v i d e s a d e c r e a s i n g sequence with

T(X,)

t n. Since

T(x1)

T(X,)

=

+

=

in I

-

T

u- monotony and t h e f a c t t h a t

NOW, u s i n g t h e

Xn

i s a d d i t i v e we have

T

=

+ (X)

T

+

l i m i n f - r ( X1\ Xn) n +a

i s i n c r e a s i n g we g e t

Xl\Xn T(

n Xn)

n€N

This implies 0 =

-

l i m i n f T(X,) n +-

hence a c o n t r a d i c t i o n t o

T(X,)

-

P r o o f o f t h e c l a i m : Because T(C,)

that

2

+ T (C,)

n. I f

+ T (A1)

+

=

procedure. Take put

A2 = A1\

C2

-

C2

= +

T

L n

+(A)

put

=

n

Xn)

for all

n.

+

+

B = C1.

T(

n€N

we f i n d

,

C1 c A

I f not,we have f o r

(consequence o f t h e a d d i t i v i t y o f c

A1

with

T(C,)

2

such t h a t

n. Assume a g a i n

T).

A1 = A \ C 1

Now, r e p e a t t h i s

T+(C,)

<

and so f o r t h . I f t h i s procedure stops a t some

p u t B = Cm, i f i t does n o t s t o p we have found a d i s j o i n t sequence

m

and

Cm t h e n C,c

A

398

Appendix

T ( C ) t n. Then p u t

with

m

+

-

= l i m i n f T(

N

-, o3

u C and g e t t h e c o n t r a d i c t i o n

C =

EN

N

u C ) = T(C)

m=l

m

E

16

.

0

Signed measures and p o s i t i v e measures a r e c l o s e l y r e l a t e d :

Theorem:

5.2 Let

_ .

T

measure

be _ an R_ T

t

valued s i g n e d measure. Then t h e r e a r e a p o s i t i v e f i n i t e ___

and a p o s i t i v e measure

T-

such t h a t --

T

=

T

t

-

T-

.

Proof: D e f i n e .+(A) T

t

= s u p { ~ ( B )I B c A, B E E l

i s p o s i t i v e and a d d i t i v e on z

.

Then i t i s q u i t e obvious t h a t

. Actually,

a- monotony i s q u i t e obvious

too. B u t c e r t a i n l y many readers a r e aware o f t h e f a c t t h a t i n qeneral o r d e r complete l a t t i c e s one has t o be e x t r e m e l y c a r e f u l i n u s i n q t h e d i s t r i b u t i v e law f o r sup and i n f i n an u n r e s t r i c t e d manner, so t h e y m i g h t w o r r y about u- monotony. Therefore we p r e s e n t t h i s d e t a i l .

T

t

i s monotone ( i . e .

A c B i m p l i e s T'(A) 9 T'(B)), so i t remains t o prove t h a t f o r an a r b i t r a r y i.ncreasing sequence An i n I and f o r B c u An we have nEN T(B) 5 l i m i n f n +m We d e f i n e

Bn = B f l An

for all

T

t

(A,)

n. The

Bn

a r e i n c r e a s i n g and

B =

u B.,

n€N

Therefore we o b t a i n t

T ( B ) = l i m i n f r ( B n ) 5 l i m i n f T (B,) 5 n + ~ n -,= 5 l i m i n f T t (A,) , s i n c e Bn c An f o r a l l n +W "

F i n a l l y , t h e n e g a t i v e p a r t o f t h e measure i s o b t a i n e d b y d e f i n i n g

-

T-(A)

= T(A)

-

.+(A),

A E I

0

n.

Measures and the Riesz Representation Theorem

399

I t should be remarked that the assumption on boundedness for signed measures plays an essential role in this proof, since for T+ ( A ) = + the above definition of T - would make no sense a t a l l . By the usual standard procedure (as described in A2 ) one can define an integral f d r with respect t o a positive measure X

T

w

.

Of course, i f the measure i s not f i n i t e one has t o r e s t r i c t the construction t o the positive cone

If

Bd z+(X) = {f E Bd Z(X)

of the bounded z- measurable functions Bd we then define

If X

where

I X

d~

t T ,T-

d T

If

=

X

d r

t

-

are the measures given by

2 01

For a signed measure

E(X).

If

d.r-

, f

Theorem

5.2.

X

R- valued linear functional on

defines an

E Bd Z+(X),

The integral

Bd I t ( X

with the

fol 1owing additional properties : (i)

j

(ii)

I

X

X

-

d T

d

T

i s bounded , i . e . sup1 is

U-

If X

d r 10 I f 5 l X } < +

monotone, t h a t means, i f

T

fn

-

i s a bounded

increasing sequence in Bd z+(X) then

I (sup f n ) d r X

nEN

=

I

lim inf fn d.r n-rX

.

Changing the point of view, one sees that every u- monotone bounded R- valued functional p on Bd z+(X) defines a signed measure T by T(A) =

p(lA)

,

A

E

E

.

And, no wonder, the integral given by T turns o u t t o be the functional So, we have, canonically given, a one-to-one correspondence between the

p

.

400

Appendix

bounded

R-

u- monotone

R-

Bd I + ( X )

valued f u n c t i o n a l s on

valued s i g n e d measures on

. This

X

and t h e cone o f

correspondence i s l i n e a r and monoBd E+(X)

tone ( w i t h r e s p e c t t o t h e a p p r o p r i a t e p o i n t w i s e o r d e r s on

and I).

Now, one m i g h t ask i f t h e fundamental Riesz R e p r e s e n t a t i o n Theorem c a r r i e s o v e r t o R- valued measures. I n f a c t t h e answer i s a f f i r m a t i v e [340] i n t h e sense t h a t i f

X

i s compact t h e n f o r each monotone l i n e a r

t h e r e i s an R- valued B a i r e measure on

X

p : C(X)

which r e p r e s e n t s

theorem can be o b t a i n e d as an a p p l i c a t i o n o f t h e Loomis

-

.

p

-,R

This

S i k o r s k i Theorem

i n s e c t i o n 1.2.3, b u t s i n c e we a r e g i v i n g i n c h a p t e r I 1 4 a f a r reaching g e n e r a l i z a t i o n o f t h i s r e s u l t we do n o t p r e s e n t t h e d e t a i l s here. B u t a l ready i n t h e c o n t e x t o f t h e Riesz Representation Theorem a word o f warning seems a p p r o p r i a t e because i n general t h e r e p r e s e n t i n g measures do n o t have t h e r e g u l a r i t y p r o p e r t y o f Remark A 3 . 5 . I n fact, regularity f o r a l l these measures i s e q u i v a l e n t ([3391) t o R b e i n g weakly U- d i s t r i b u t i v e .

R

i s s a i d t o be weakly

{bn,k

1n

~ Ek N I w i t h sup i n f nEN k m

u- d i s t r i b u t i v e i f f o r e v e r y bounded subset

bn,k+l b

nyk

- b n,k < =

inf &NN

where NN i s t h e f a m i l y o f f u n c t i o n s

we have sup b nEN n,cp(n)

cp : N

-,N

.

I n t h e r e a l - v a l u e d case a p a r t i c u l a r consequence of t h e Riesz

-

Represen-

t a t i o n Theorem i s t h a t e v e r y f i n i t e B a i r e measure on a compact space can be extended t o a r e g u l a r Bore1 measure. T h i s r e s u l t even f a i l s f o r weakly u- d i s t r i b u t i v e Dedekind complete v e c t o r l a t t i c e s [3401. Another fundamental

r e s u l t which does n o t c a r r y over t o t h e general s i t u a t i o n i s t h e HopfKolmogorof Extension Theorem ( o r e q u i v a l e n t l y t h e Daniel1

-

Stone Theorem).

Matthes proved t h a t under t h e a d d i t i o n a l assumption o f weak U- d i s t r i b u t i v i t y t h e Hopf-Kolmogorof Theorem remains t r u e and M a i t l a n d W r i g h t [339] showed t h a t t h e e x t e n s i o n p r o p e r t y s t a t e d i n t h i s theorem i s i n f a c t e q u i valent t o

R

b e i n g weakly

U-

distributive.

Measures and the Riesz Represntation Theorem

401

A t the end of this section we would like t o mention that in the general lattice-valued case the boundedness condition in the definition of sinned measures was n o t superfluous. Example (private communication of A. Schep):

-.

Let R be equal t o the space co consisting of a l l sequences converging t o 0 for n

-,

(a,)nEN

co i s an order complete vector l a t t i c e

with respect t o the obvious order. Let m be the Lebesgue measure on [0,11 and take a n orthonormal sequence ( r n ) in L 2 ( m ) such t h a t , f o r a l l n , lrnl

= 1

m- almost everywhere. The Rademacher functions for example

have t h i s property. NOW, take for z the Bore1 subsets of [0,1] T : E + c 0 by

This i s always an element of

and define

co , because i t i s the sequence given by the

Fouriercoefficients (with respect t o

( r n ) ) of

lA

.

B u t s u p { ~ ( AI) A E 11 , i f i t existed, would be given by the constant sequence (

1

1 lrnl

dm)n

E N = ( l , l ,...)

,

0

which i s certainly n o t a n element of

co

,

On the other hand a standard calculation shows t h a t tone.

T

i s in f a c t u- mono-

This Page Intentionally Left Blank

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C. P o r t e n i e r : C a r a c t k r i s a t i o n de c e r t a i n s kspaces de Riesz, In: Skminaire Choquet ( I n i t i a t i o n a l ' a n a l y s e ) 10e anne no. 6 (1971), Paris

254.

E.T. Poulsen: A s i m p l e x w i t h dense extreme boundary, Ann. I n s t . F o u r i e r 11 (1961), 83-87

255.

A. de La P r a d e l l e : A Dropos du mkmoire de G.F. Vincent-Smith s u r 1 l a p r o x i m a t i o n des f o n c t i o n s harmoniques, Ann. I n s t . F o u r i e r 19 (196!), 355-370

256.

W. Prenowitz, J. J a n t o s c i a k : J o i n Geometries, Undergraduate Texts i n Mathematics, S p r i n g e r Verlag, Berlin-Heidelberg-New York, 1979

257.

J.B. P r o l l a : Approximation o f vector-valued f u n c t i o n s , Mathematics Amsterdam-New York 1977 S t u d i e s 25, N o r t h H o l l a n d Pub1

258.

J.D. Pryce: Weak compactness i n l o c a l l y convex spaces, Proc. AMS 17 (1966) , 148-155

259.

J. Radon: T h e o r i e und Anwendungen d e r a b s o l u t a d d i t i v e n Men enf u n k t i o n e n , S i t z . B e r . Acad. Wiss. Wien 128 (1919) , 1083-112

260.

J. Rainwater: Weak convergence o f bounded sequences, Proc. AMS 14 (1963), 999-1004

261.

F. Riesz: Sur l e s syst&nes orthogonaux de f o n c t i o n s , C.R. P a r i s 144 (1907), 615-619

262.

--

263.

--

264.

-- Sur c e r t a i n s systemes s i n g u l i e r s d ' k q u a t i o n s i n t k g r a l e s , Ann. 1 ' E c o l e Norm. Sup. 28 (1911), 33-62

265.

--

266.

--

267.

.,

4

S u r l e s o p e r a t i o n s f o n c t i o n e l l e s l i n k a i r e s , C.R. P a r i s 149 (1909), 974-977

Acad. S c i .

Acad. S c i .

Untersuchungen uber Systeme i n t e g r i e r b a r e r Funktionen, Math. Ann. 69 (1910), 449-497

Demonstration nouvel l e d ' u n theoreme concernant l e s o p e r a t i o n s f o n c t i o n e l l e s l i n e a i r e s , Ann. 1 ' E c o l e Norm. Sup. 3 1 (1914), 9-14

Sur quelques n o t i o n s fondamentales dans l a t h ' e o r i e g 6 n 6 r a l e des o p e r a t i o n s l i h a i r e s , Ann. o f Math. 4 1 (1940), 174-206 G. Rod'e: E i n e a b s t r a k t e V e r s i o n des Satzes von Hahn-Banach, Arch. Math. 3 1 (1978), 474-481

418

268.

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279.

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283.

H.H. Schafer: Banach l a t t i c e s and p o s i t i v e o p e r a t o r s , S p r i n g e r V e r l a g (Grundlehren Bd. 215), Berlin-Heidelberg-New York 1974

284.

H. Schubert: Topolog e, Teubner Verlag, S t u t t g a r t 1969

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287.

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M a t r i c e s o f zeros and ones, B u l l . AMS 66 (1960), 442-464

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290.

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291.

S. Shermann: On t h e theorem of Hardy, Littlewood, Polya and Blackwell,

292.

T. S h i r o t a : A c l a s s of topological spaces, Osaka Math. J . 4 (1952), 23-40

293.

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294.

R . S i k o r s k i : On an analogy between measures and homomorphisms, Ann. SOC. Pol. Math. 23 (1950), 1-20

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297.

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Ch. S t e g a l l : Optimization o f Functions on c e r t a i n s u b s e t s of Banach spaces, Math. Ann. 236 (1978), 171-176

300.

M . H . Stone: A general theory of s p e c t r a 11, Proc. Nat. Acad. S c i . USA 27 (1941), 83-87

301.

--

302.

--

303.

--

304.

V . Strassen: The e x i s t e n c e of p r o b a b i l i t y measures with given marginals, Ann.Math.Statist. 36 (1965), 423-439

305.

W . H . Summers: A r e p r e s e n t a t i o n theorem f o r biequicontinuous completed t e n s o r products of weighted spaces, Trans. AMS 146 (1969), 121-131

306.

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Boundedness p r o p e r t i e s i n f u n c t i o n l a t t i c e s , Can.J.Math. 1 (1949), 176-186

50-58

Notes on I n t e g r a t i o n IV, Proc.Nat .Acad.Sci. USA 35 (1949),

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420

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308.

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309.

--

R e p r k e n t a t i o n i n t ' e g r a l e dans l e s canes convexes, C.R. P a r i s 286 (1978), A 515 - A 518

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310.

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312.

F. Topsoe: F u r t h e r r e s u l t s on i n t e g r a l r e p r e s e n t a t i o n s , S t u d i a Math. 55 (1976), 239-245

313.

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--

The n a i v e approach t o t h e Hahn-Banach theorem, Comment .math .spec. Val. I , dedic. L. O r l i c z (1978), 315-330

314.

M. V a l a d i e r : Sur l e thior'eme de Strassen, C.R. (1974) , 1021-1024

315.

F.A. Valentine: Konvexe Mengen, B I - Verlag, Mannheim-Zurich

316.

V.S. Varadarajan: On a theorem o f F. Riesz concerning t h e form o f l i n e a r f u n c t i o n s , Fund. Math. 46 (1959), 209-220

317.

-- Measures on t o p o l o g i c a l spaces, AMS T r a n s l a t i o n s 48 (1965), 161-228

318.

G.F. Vincent-Smith: The Hahn-Banach theorem f o r modules, Proc. London Math.Soc. 17 (1967), 72-90

319.

Acad.Sc. P a r i s 278 1968

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A Choquet boundary t h e o r y f o r measures t a k i n g values i n a Stone algebra, J . London Math.Soc. 44 (1969) , 553-558

320.

R.C. Walker: The Stone-Czech c o m p a c t i f i c a t i o n , Ergebnisse Math. und Grenzgebiete Vol 83, Springer-Verlag, B e r l in-Heidel berg-New York, 1974

321.

H. Watanabe: Simplexes on a l o c a l l y compact space, Nat.Sci.Rep. Ochanomi zu Univ. 23 (1972), 35-42

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3 2 2 . ~ L - b l e i 1 : L ' i n t e g r a t i o n dans l e s groupes topologiques e t ses a p p l i cations, Hermann, P a r i s 1953 323.

E.C. Weinberg: Free l a t t i c e ordered a b e l i a n groups, Math.Ann. 159 (1965) , 217-222

324.

M.D. Weir: Hewitt-Nachbin Spaces, Mathematics S t u d i e s Vol. 17, North H o l l a n d Publishers, Amsterdam-Oxford-New York 1975

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326.

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D.V. Widder: Necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e represent a t i o n o f a f u n c t i o n as a Laplace i n t e g r a l , Trans. AMS 33 (1931), 851-892

330.

--

--

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The Laplace Transform, P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n

331.

G. Winkler: On t h e i n t e g r a l r e p r e s e n t a t i o n i n convex noncompact s e t s o f t i g h t measures, Math.Z. 158 (1978), 71-77

332.

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336.

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337.

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On t h e t h e o r y o f a p p r o x i m a t i o n by p o s i t i v e l i n e a r o p e r a t o r s i n v e c t o r l a t t i c e s , i n : F u n c t i o n a l A n a l y s i s : Surveys and r e c e n t r e s u l t s ( E d i t o r s : K.D. B i e r s t e d t , B . F u c h s s t e i n e r ) N o r t h - H o l l a n d Mathematics S t u d i e s 27 (1977), 73-87

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J.D. M a i t l a n d W r i g h t : Stone-Algebra v a l u e d measures and i n t e g r a l s , Proc. London Math. SOC. 19 (1969) 107-122

339.

-- The measure e x t e n s i o n problem f o r v e c t o r l a t t i c e s , Ann.Inst. F o u r i e r 21 (1971), 65-85

340.

--

module homomorphisms,

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E i n e Bemerkung zum D e s i n t e g r a t i o n s s a t z von Strassen, Arch .Math. 28 (1976), 98-101

Vector L a t t i c e Measures on l o c a l l y compact Spaces, Math.Z. (1971), 193-203

120

422

References

341.

J.D. Maitland Wright: Measures w i t h values i n a p a r t i a l l y ordered v e c t o r space, Proc. London Math. SOC. 25 (1972), 675-688

342.

K. Yosida: On v e c t o r l a t t i c e s w i t h a u n i t , Proc.Imp.Acad. Tokyo 17

343.

M. Zippin: The s e p a r a b l e e x t e n s i o n problem, I s r a e l , J.Math. 26 (1977), 372-287

(1941), 121-124

AUTHOR INDEX

Alexandrov Andenaes

248 80

Ando

111

Anger

78

Edgar

250 111, 171, 304, 335

Edwards

111

Ellis Fell

172

Asimov

111

Fischer

72

Aumann

75

Frolik

338

Baker

173

Banach

Gale

72,73,248

Bauer

76, 258, 286, 303, 304

Bernau

173

54, 84

Glicksberg Goodner

G o u l l e t de Rugy

Berns t e i n 343

Hahn

Bishop

Halperin

21, 269, 303, 304

Blackwell Bleier

173

173

Boboc

304

173

Hasumi

75

Helley

72

368

Herglotz

303

Hervg

74,78

Buck

248

Cartier

239, 248, 337, 338, 339

H i 1b e r t

Hustad 172

Chatterji

199,249

368

303

Hewitt

250

75

Hardy

Bonsall Bou

72 75

Kadison

170

Kakutani

89, 157, 162, 174, 248

Choquet

76, 173, 249, 269, 303, 304, 335

Katetov

338

Cignoli

82

Kaufmann

75

Kantorovit c h

Conrad

173

Kelley

Cornea

304

Kendall

Cotlar

75

Klee

Daniel1

246

Davies

304

de Leeuw Diestel

269, 304 249

73

75 345

172

Kolmogoroff Konig Kranz

379

28, 29, 37, 44, 75, 76, 79, 84

Korovkin

167

75

Dinges

75

Lembcke

Donner

78

Lindenstrauss 423

198

73

Bochner

Bourgin

248, 313

73

78 77

424

Author Index

L i ttlewood Loomis

173

Shermann

88, 100

Manki ewi cz Markov

Shilov

250

Shirota

248

Matthes Meyer

Sibony Siciak

400 172, 304

Mokobozki Mrbwka

173

303 337 237, 239 304

Sikorski

237, 239, 304

338, 339

Simons

88, 100 78, 300

Stegall

199

Nachbin 73, 74, 244, 249, 337

Stone 97

v. Neumann 393, 394

Strassen 83, 173

Neumann 44, 247, 287

Thomas

250

Polya

Topsoe

75

173

Portenier

198

Uhl

249

Radon 248

Valadier

Riesz 172, 248 Rod5 79

Vincent

-

Rodriguez Ryser Saks

Salinas

85

Weil 74, 78

44

-

Smith

Weinberg

173

v. Weizsacker

248

Wenjen

Schep 397, 401 E. Schmidt

72

S.eevers

22,75

Semadeni

173

312

368 250

338

Winkler Wittstock

250 82

W o l f f 44, 176 Maitland Wright

171, 311, 400

SUBJECT INDEX A(K) 267 absorbing functional 149 a b s t r a c t integral 386 adapted cone 237 additive 3 a f f i n e 35, 90, 267, 283 a f f i n e interposition 34 AL-cone 154 Alexandrov compactification 338, 341 Alexandrov-G1 icksberg theorem 313, 330 algorithm 80, 82 AM- cone 154 AM- space 89, 157, 162 archimedean 170 arithmetic mean 24 Ascol i ' s theorem 342 B(X), Bo(X) 381 Bor(X) 382 Baire ( X ) 382 Bdt(X) 382 Bdt+(X) 399

boundary 40, 252, 255, 288 Cone(u) 221 Conv(K) 253, 267 &(K) 259 co(X) 256 ConvB(X) 297 Convc(X) 300 C*- algebra 170, 358 cancellation law 8, 107 Cartier-Fell-Meyer Theorem 88, 121 central l i m i t theorem 371 character 88, 139, 158, 270, 276, 285, 315, 350, 363 Choquet boundary ( C h ( K ) ) 259, 287, 298, 300 Choquet-Meyer Theorem 281 Choquet-order 275 Choquet-simplex 88, 173, 286 Choquet's Theorem 268, 312 commodity s e t 58, 60 common sense 65 compatible 62, 66 complementary set 318 Baire's category theorem 100, 171, completely monotonic function 341 310 complex 1 inear 34, 355 Bai re-measurabl e 99 , 301, 392 concave 35 Baire measure 228, 239, 282, 296 Baire space 171, 335 cone 6 consumer s e t 56 Banach l a t t i c e 150 consumption 54, 57, 60 Banach Stei nhaus theorem 72 consumption measure 56 Bauer-simplex 286, 378 continuous order r e l a t i o n 243 bimeasure 49 convex combination 256 binary intersection property 72, 77 convex cone 6 Birkhoff-Ulam theorem 171 convex function 34, 253 Bochner's Theorem 375 convex hull 125, 257, 297 Bochner-Weil Theorem 363, 378 425

426

Subject Index

convex s e t

8, 33, 90, 102, 125, 253,F-

311, 335 c o n v o l u t i o n 364 countable covering

202, 224

107, 179

Fi

113

FT

117

VF(X) 191 c o u n t a b l e decomposition 177 c o u n t a b l e i n t e r s e c t i o n p r o p e r t y 230 Face(p) 221 Daniel 1-Stone Theorem 203 , 209 , 220 ,face 220, 226 246, 400 F a r k a s ' Lemma 29 F- compact, F- c o m p a c t i f i c a t i o n 315

decomposable 179 decomposition p r o p e r t y 203 Decomposition Theorem 184

F- Hewitt-Nachbin-space 315 f i n i t e c o v e r i n g 27 F i n i t e Decomposition Theorem

demand measure 61, 66 D i n i boundary 287, 291 Dini c h a r a c t e r Dini-condition

F i n i t e Sum P r o p e r t y

315 189, 268

222 f i r s t countable

(FSP)

302

D i n i cone 189, 203, 210, 268, 287, 291, 308, 328

fixpoint 254 f i x p o i n t boundary

D i n i continuous

Flow Theorem 49, 51 Ford-Ful kerson Theorem

Dini lattice D i n i ' s Lemma

210, 232, 314

198 164, 189, 240

D i r i c h l e t s t a t e 220 D i s i n t e g r a t i o n Theorem 42, 44, 83 d i s t r i b u t i o n p l a n 62, 66 d i s t r i b u t i v e law 101 Dominating Extension Theorem downwards d i r e c t e d

11

186

ex(K) 256, 267 Edward's s e p a r a t i o n theorem

343 54, 84

F o u r i e r - c o t r a n s f o r m 365 F o u r i e r - t r a n s f o r m 365 F- pseudocompact

315

F- r e a l compact , F- r e a l compacti f i c a t i o n 315

f r e e a b e l i a n semigroup

25

f r e e l a t t i c e cone 88, 122, 125, 285 functional 9 171

envelope 103, 277 exposed element 80

F ( x ) - exposed 253 Gale-Ryser Theorem G- a n t i t o n e

exposed selfmap 252 E x t e n s i o n Theorem 12 e x t r e m a l l y disconnected

17

88, 107,

54

106

Gelfand-Naimark Theorem 359, 378 Gelfand Representation 360, 378 go, 274, 351 75, 97, 157 Gelfandtransform

extreme p o i n t 80, 142, 170, 256, 267, 340, 378

character 139 g e n e r a l i z e d Hewi tt-Nachbin space

extreme r a y

generated

(F,<)*

142

(F,3#

,

(F,

0,

102

107, 182

geometric s i t u a t i o n - i n v a r i a n t 252

r

253, 267

313

Subject Index

G- montone G- o r d e r

106, 179

l a t t i c e cone homomorphism,

106, 179

isomorphism

g r e a t e s t l o w e r bound

G- space

427

17

120

l a t t i c e cone subhomomorphism

174

l a t t i c e - g e n e r a t e d cone

Haar measure

363

l a t t i c e homomorphism

Hahn-Banach p r o p e r t y 74 Hahn-Banach S e p a r a t i o n Theorem

33

196 l a t t i c e state

74, 94, 120,

87

Hahn-Banach Theorem

2, 34, 72

Lebesgue decomposition theorem

Hewitt-Nachbin space

313, 337

Lebesgue Dominated Convergence

H e w i t t l s Theorem Hilfsfunktional

201

Theorem

307

123

136, 145

395

385

L6vy-Khintchine f o r m u l a

345, 376

homogeneous 9 L i n d e l o f space 230, 298, 334, 339 Hopf-Kolmogorov E x t e n s i o n Theorem 400 l i n e a r 9, 43, 241 hypo1 inear 78 l o c a l i z e d o r d e r s t r u c t u r e 43, 5 1 i m p o r t c a p a c i t y 56 l o c a l l y compact a b e l i a n group 363 Improved R e p r e s e n t a t i o n Theorem infinitely divisible i n j e c t i v e element

74

in t e g e r - v a l ued f l o w

85

inverse preorder

102

287 22, 23, 303

Jordan decomposition

(K)

259

262

170

Theorem

87, 95, 157, 164, 170 Katetov-Nachbin Theorem

244

167, 176 89, 167, 175

Krein-Milman Theorem 393 40

93, 101

83 84 199,

249

28, 215

Kakutani-Krein-Stone-Yosida

Krein-Milman P r o p e r t y

100, 400

m a r t i n g a l e convergence p r o p e r t y

Kadison-Kakutani-Stone Theorem

l a t t i c e cone

a,B

M a r r i a g e Theorem

Jensen measure

Laplace e q u a t i o n

259

marginal measures

James' Theorem

LP(m)

Max(K)

Mx

i n v o l u t i o n 355, 364

Korovkin f a m i l y

255, 258

Max

89

K o r o v k i n ' s theorem

Loomis-Si k o r s k i Theorem max(K)

M ~ x ( K ) 259

111

I n t e r p o s i t i o n Theorem invertible

292

345, 371

199 80, 257, 267

mass z e r o

242

max-boundary max f l o w

40

- m i n c u t theorem 54, 84

F(X)- exposed f a m i l y maximal measure 275 maximal

maximal l i n e a r f u n c t i o n a l

254

140, 211,

225, 281 maximum p o i n t

259

Maximum P r i n c i p l e max-stable

20, 40, 256, 267

189, 191, 274, 291

Mazur-Or1 i c z Theorem meagre

99, 307

34

428

Subject Index

measure 383 minimal f i x p o i n t boundary 81, 254 minimal t o t a l demand 64, 66 monotone, 3, 43, 349, 357 monotone convergence property 385 m u l t i p l i c a t i v e 22, 350, 355 m u l t i p l i c a t i v e cone 349 Nachbin family 247 natural preorder 106, 155 negative cone 107 negatively generated 107, 151 networks 49, 54, 82 neutral element 2, 8, 22 normed l i n e a r functional 213, 242 Norm Theorem 13 order, a- pointwise, a- decompos i t i o n 118 bounded 2, 96 complete 2, 96, 305, 396 dual 113, 153 u n i t 90, 290, 356 u n i t cone 90 u n i t cone homomorphism, isomorphism 91 order u n i t functional 90, 150 order u n i t homomorphism 91, 95 Prob(n) 274 P a r t i a l Decomposition Theorem 183 p a r t i a l l y decomposable 180 Phragrnen-Li ndel of Pri nci pl e 20 PA- space 74 pointwise infimum, supremum 97 polish 231, 297 polyhedric cone 31 Pontryagin d u a l i t y 363, 370 p o s i t i v e 362 p o s i t i v e cone 107 order order order order order order

p o s i t i v e dual cone 113, 154 p o s i t i v e flow 55, 57 p o s i t i v e l y homogeneous 43 p o s i t i v e l y independent 26, 30 p o s i t i v e type 367 possible flow 55, 57 possible plan 62, 66 Poulsen simplex 340 preorder 2 production capacity 54, 60, 61, 62, 66 pseudocompact 252, 296 Radon-Ni kodym-Property ( R N P ) 198, 249, 250 Radon-Ni kodym theorem 47 , 395 Rainwater's theorem 304 raw material bound 62, 66 real compactification 315 real -1 i near 355 regular measure 228, 391 regular open s e t 171 representation 184 Representation Theorem 210 representing measure 179, 203, 258, 281, 291, 305, 328, 390 Riemann-Lebesgue Lemma 374 Riesz Decomposition property 172 Riesz i n t e r p o l a t i o n property 172 Riesz-Konig Theorem 34, 227 Riesz property 88, 107 Riesz Representation theorem 37, 204, 208, 227, 392 R- valued measure 100, 305, 396 R- valued representing measure 308 u- completion 233, 382 supp(m) 230, 258 Sandwich Theorem 2 , 4 , 10

Subject Index

s a t u r a t i o n bound 58, 61, 66 s e l f a d j o i n t element 355 semicontinuous 8, 38, 192, 212, 244, 253, 258, 288, 298, 334, 349, 382, 386 semigroup 2 s e m i i n t e r p o l a t i o n property ( S I P ) 16, 19, 122, 207 semi 1a t t i c e - c o n e 102 seminorm 33 semisimple 366 Separation Theorem 33 set of domination 318 set of s t r o n g domination 318 Shilov boundary 261 signed representing measure 213 Simons' Convergence Lemma 287 simplex 286, 378 s i m p l i c i a l cone 88, 131, 281, 378 s i m p l i c i a l map 131 s p e c t r a l radius norm 352 S- s t a t e 306 s t a t e 90, 204, 362 s t a t e space 90 Stone-Czech compacti f i ca t i o n 98 , 171, 229, 248, 315, 338 Stone space 87, 171 Stone-Weierstrass Theorem 96, 274, 344, 360, 367 s t r i c t l y decomposable 180 s t r i c t l y representing measure 203, 213 s t r o n g l y exposed p o i n t 199 s t r o n g maximum point 258 subaddi t i ve 3,43 subcone 8 s u b - l a t t i c e cone 125

429

s u b l i n e a r 9 , 43, 74 s u b m u l t i p l i c a t i v e 351 subsemigroup 8 Sum Theorem 14 sup-boundary 331 s u p e r a d d i t i v e 3, 43 super1 i n e a r 9 , 43 s u p p o r t 228 symmetric 355, 362, 365 Three C i r c l e s Theorem 21 t i g h t measure 49, 231, 240, 249, 297 Tong-Katetov Theorem 244, 387 t o t a l maximal production 64, 66 truncated 233 USC(n) 38, 227 u c ~ ( x ) 192, 212, 235 Universal Property 115 u n r e s t r i c t e d d i s t r i b u t i v e law upper integral 389

101

vector l a t t i c e 2, 95, 96, 207, 209, 305, 396 Vector Valued Representation Theorem 308 weak Choquet boundary 259 weak-Dini-condition 197 weak Dini cone 196, 213 weakly monotone 349 weakly U- d i s t r i b u t i v e 247, 312, 400 weight f u n c t i o n 218 X- monotone 43, 52 Y- monotone 220 zero-one matrix 85 zero-one measure 316 z e r o - s e t f i l t e r 338 Zorn's lemma 4 , 46, 211, 254, 276, 306

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