Choquet Order and Simplices

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1145 Gerhard Winkler

Choquet Order and Simplices with Applications in Probabilistic Models

Springer-Verlag Berlin Heidelberg New York Tokyo

Author Gerhard Winkler Mathematisches Institut, Universit&t MLJnchen Theresienstr. 39, 8000 M6nchen 2, Federal Republic of Germany

Mathematics Subject Classification (1980): primary: 4 6 A 5 5 secondary: 1 8 B 9 9 , 2 8 C 9 9 , 4 6 E 2 7 , 5 2 A 0 7 , 6 0 G 0 5 ISBN 3-540-15683-6 Springer-Verlag Berlin Heidelberg N e w York Tokyo ISBN 0-387-15683-6 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wart", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214613140-543210

PREFACE

These

lecture

notes

o r d e r on l o c a l l y simplices were

with

The r e a d e r mathematics proofs

convex

by the author recent

I had

attending

in p r o b a b i l i t y

thorough

measure

- like R.D.

Bourgin's

down

the

in the last

and l o c a l l y

to have

There

some

monograph

will

convex

familiarity

systems

is little

proofs

to be f a m i l i a r

of C h o q u e t

The c e n t r a l

with

or no

(1983).

study.

probably

with

of

So the

skip a lot

the b a s i c

vector

of

results

is a student

t h i r d of his

and the s p e c i a l i s t

theory

but not n e c e s s a r y

theory.

1983.

is a s s u m e d

properties

of inverse

in 1982 and

a seminar

The r e a d e r

of t o p o l o g i c a l useful

surveys

of c e r t a i n

an e x a m i n a t i o n

in m i n d when w r i t i n g

are r a t h e r

of details.

a summary

spaces,

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

obtained

overlap

present

spaces.

ideas It is

the C h o q u e t

theory.

I am i n d e b t e d Hele,

to J.P.R.

H. Kellerer,

but not least H.v.

Christensen,

E. Kolb,

S. Dierolf,

R. Koteck~,

Weizs~cker

G. Godefroy,

Z. Lipecky,

D. P r e i s s

for their help and u s e f u l

Chro and last

comments.

CONTENTS

INTRODUCTION

................................................

CHAPTER

0°

NOTATIONS,

CHAPTER

I.

BASIC

1.1.

CONCEPTS

NONCOMPACT

representing

barycenter

CONVENTIONS

map

..........

CHOQUET

measures

THEORY

...

sets

1.3.

Choquet

order

...................................

1.4.

Boundary

I75.

Simplices

FOUR

Measures

2.4.

3.

in

Extension

of to

Nets

measures

Inverse

3.3.

The

the

order;

46

order

convex

measures

OF

live

of of

tight

from

Borel

the

o-algebra

in Choquet

SIMPLICES simplices

simplices

dimensional

on 54

case:

from

measures

...

weak

Borel

o-

........... order

57

......

62 68

75

...................

76

.....................

79

a historical

LIMITS

OF

SPACES

OF

MEASURES

Inverse

limits

of

spaces

of

measures

kernels

bounded

.....................

INVERSE

substochastic

54

...................

of measures

monotone

intersection limits

sets

strong

LIMITS

finite

ORDER

Choquet

of

tight

algebra

3.2.

38

....................................

in Choquet

The

4.1.

smaller

tightness

INVERSE

28

...............................

CHOQUET

Uniform

3.1.

4.

OF

sets

of

21

.............................

.......................................

smaller

above 2.3.

measures

ASPECTS

13

..............................

convex

2.

6

and

Measure

2.2.

CHAPTER

AND

1.2.

2.1.

CHAPTER

FROM

Barycenters, the

CHAPTER

DEFINITIONS

I

note

..

............

88

94

and

...........................

95

VI

4.2.

A criterion

4.3.

Examples mechanics

and

limits

statistics

........................

Specifications

4.3.3.

Projective

4.3.4.

Further

examples,

an

problem

OF

inverse

theory,

4.3.2.

open

of

probability

Entrance

boundaries and

of M a r k o v their

statistical

Gibbs

fields

complements

..

105

statistical

processes states

108

.......

108

.........

110

.................

117

and

...............................

118

....................................................

123

..................................................

130

REFERENCES

SUBJECT

from

nonemptiness

4.3.1.

APPENDIX

INDEX

for

SYMBOLS

INDEX

............................................

140

...............................................

141

INTRODUCTION

C h o q u e t o r d e r is one of the b a s i c presentation.

tools in the t h e o r y of i n t e g r a l re-

C o n s i d e r a c o n v e x set M in some l o c a l l y c o n v e x

space E. An e l e m e n t x of M is the b a r y c e n t e r

linear

of a p r o b a b i l i t y m e a s u r e

p

on M - or it is r e p r e s e n t e d by p - iff

l(x)

= IM l(y)

for e v e r y c o n t i n u o u s

dp(y)

linear functional

1 on E.

One asks

for those sets M w h e r e e v e r y e l e m e n t has

such a r e p r e s e n t a t i o n

w i t h a m e a s u r e p l i v i n g on the e x t r e m e

p o i n t s of M.

equivalent

s e r v e d G. Choquet,

formulations,

and m a n y o t h e r s

C h o q u e t order

to a n s w e r this q u e s t i o n on v a r i o u s

The t h e o r y for c o m p a c t c o n v e x c l a s s i c a l books

of R.R.

(1971). L. A s i m o v

and A.J.

S i n c e the m o n o g r a p h s Lusky

Phelps

sets

(1980)

of R.D. B o u r g i n

describe

(1983)

m a i n s n o t too m u c h to say a b o u t this topic.

chapter

It o n l y

appears

and E. A l f s e n

later d e v e l o p m e n t .

and B. F u c h s s t e i n e r and W. presently

there re-

So we do n o t p u r s u e this

in the r e m a r k s

r o u n d i n g off

I.

T h e e m p h a s i s of the p r e s e n t n o t e s o r d e r in its own right, finally

Edgar

levels of g e n e r a l i t y .

(1969)

(1981) w h i c h c o v e r the n o n c o m p a c t s i t u a t i o n ,

a s p e c t of C h o q u e t order.

G.A.

is s u r v e y e d in the n o w a d a y s a l m o s t

(1966), G. C h o q u e t

Ellis

In its d i f f e r e n t

to b r i d g e

our p e r s o n a l

is d i f f e r e n t .

then i n v e s t i g a t e s y s t e m s of s i m p l i c e s and try

the gap to a p p l i c a t i o n s ,

taste f r o m p r o b a b i l i t y

applicability,

we have to w o r k

of i n t e r e s t in the t h e o r y To be m o r e specific,

We s t a r t s t u d y i n g C h o q u e t

theory.

w h i c h we c h o o s e a c c o r d i n g To a v o i d r e s t r i c t i o n s

in a f a i r l y g e n e r a l

setting which

itself.

I will c o m m e n t on the single c h a p t e r s below.

to

in is also

2

As we will

speak

space which them

are

inner

"measures"

"outside" other

only

functions.

q.

probability

regular

in t h i s

the mass

measure

about

w.r.t,

compact

introduction.

of s o m e m e a s u r e

So b o t h m e a s u r e s

measures

sets,

the

first

iff

I ~ dp

integrals

how

p is d i s t r i b u t e d

compared

with

tested

~ I ~ dq

Many

of b a s i c

are

of t h e

important

First,

we

centers

exist

and depend

class

of

Choquet

by means

for e v e r y

to be

sets

theory

property

that

for

as o n e w o u l d

collected: already

are

should

some

of c o n v e x

are

instance,

supposed

maximal

as t h e

do not

the Dirac

in C h o q u e t

of c o n e s

the d e s c r i p t i o n

to d e c i d e

those

we have

o u t of t h e called

by m e a n s

in On

out

a bary-

This convexity.

as t h i s

point

sets)

the dilation

of c o n d i t i o n a l

It

is

problem

(on a r b i t r a r y

of f u n c t i o n s ,

the

in m i n d .

of an e x t r e m e

order

on which

It t u r n s

set.

measure

As s o o n

space

considered

on such a set have

measure

of C h o q u e t

bary-

convex

be e x c l u d e d .

order.

sometimes

theorem:

locally

applications

fall

it

literature.

On the one hand,

to c o v e r

should

but

I is

published.

we have

to live.

of c o m p l e t e n e s s ,

by m e a n s

used by Choquet,

Next,

order.

Chapter

papers,

Haydon's

that each measure

formulations

available

not even

enough

itself.

a l o t of s p a c e

of the u n d e r l y i n g

behaviour

that

order

in v a r i o u s

R.

continuous

convex

t h a n q in C h o q u e t

requires

and prove

be l a r g e

version

definitions

This

some

this barycenter

equivalent

to C h o q u e t

on the measures.

to r e q u i r e

expect

p smaller

scattered

u p to n o w as w e l l

is a w e a k

guarantees

settled,

and

hand pathological

and

are

barycenters

the m e a s u r e s

sufficient

center

far

~,

in the p r e s e n t l y

in the c o m p l e t i o n

such

the o t h e r

a gap

facts

continuously

of s e t s

call

devoted

in the p r o o f s ,

introduce

and

properties.

at least partially

only hidden

kind

exist

two c h a p t e r s

a collection fills

call

measures

function

The

simply

order

are

Choquet

l e t us

convex

Write

p < q

provided

on a l o c a l l y

is are

order

3

expectations and

a new

verse

to

which as

the

link

between

Choquet

characterization

due

to H.v.

Weizs~cker.

the

are

near

in t h e

a simplex

iff

boundary

Several

results

one

would

sets.

measure

they

not). the

second

of m e a s u r e s

which

convex

Subsequently, Borel

A close

a-algebra

study

examined

result

measures). converge

to

in c h a p t e r

of

sets

see

are

bounded

nets

that

above

uniformly

in

convex

this

has

when

order

revealed.

in C h o q u e t

order

in c o n n e c t i o n convex

sets

This

result

with they will

be

We

of

a greatest the

only

basis

smaller (but

fails

a strong sets

in

consequence

that

concentrated compact

also In

on

sets.

a measure

to t h e

(increasing

have

have

has

spaces.

existence

on

convex

the

extension

convex

is

2. N a i v e l y ,

compact

convex

an

locally

live

sets

by

topology is

space

Choquet

chapter

order

convex

is a t t a c k e d :

most

convex).

measure

w.r.t,

one

and

with

(by a m e a s u r e

tight

problem

metrizable

are

on m e a s u r e

we

Choquet

it w e a k l y .

the

In c o n -

the

one

close

new.

in C h o q u e t

section,

with

On measure

if

as

convex

of

relation

presented

exhausted

are

exhaustively

are

prefer

locally

really

be

the weak

decreasing

order

is

- those

mass

is m e a s u r e

the

can

of

for

nothing

Measures

set

and

it

is a c o n -

simplices.

barycenter

- if

smaller

true

extension

connection

classical

By way

set)

the

far,

is

the

derived

measures

this

is

latter

measures

we

S of a

martingales

their

introduce

a subset

and

The

"have

definition,

course

Choquet

that

are

are

order

boundary

which

points"-and

- of

So

concerning

and

elements

properties

study

theoretic

(and

In t h e

a measure

ions

its

then

order

context:

of

property:

measure.

we

each

expect

as

tightness

the

lattice

Fortunately,

soon

We

extreme

is e x a m i n e d .

New

sets

the

present

useful

simplices

order.

in C h o q u e t

to C h o q u e t ' s

natural

one

is

dilation

maximal

possible

trast

as

which

strong

reprove section

nets

on

have

one? the

2.4,

been

representing lower of t h e

bound

and

investigat-

3.

introduction

to c h a p t e r

3,

let me

pose

an

elementary

problem.

Consider

a decreasing

Euclidean

n-space

triangle,...). answer

dimensional

Choquet

happens

The a n s w e r

Choquet

3.3,

as E.B.

in C h o q u e t

topologies

is r e m e d i e d a simplex properties improved,

The

fields

the

that

in all r e a s o n a b l e of C h o q u e t

inverse

order d e r i v e d

nets

to c o n s t r u c t

(1968)

c o u l d prove. as often

for application: artificially

is based

2. The

This

results

show.

of mathematics.

In chapter

4, they are by way of example

at first glance

(and it proves

simplices) : inverse spaces

and

models

from p r o b a b i l i t y

exist

purely

theory,

boundary.

Further

structure

of sets of Gibbs

are

limits

methods.

applied

of standard

in passing;

The

of

with n o n c o m p a c t

and s t a t i s t i c a l

that a p u r e l y m e a s u r e - t h e o r e t i c

measure-theoretic

inverse

that we w o r k e d

is t r e a t e d

from v a r i o u s

measure-theoretic.

the c o n s t r u c t i o n

sketched

states

in models

This may be i n t e r p r e t e d

statistics

it means

examples

about

in the c a t e g o r y

kernels.

for M a r k o v p r o c e s s e s

by purely

looks

to be c o n v e n i e n t

substochastic

One may argue

inherent

of the results

limits

is

cannot be

3 are

simplices

nuisance

on the

in chapter

is a c o n s e q u e n c e

reason-

simplices

studied

following

limit

a compact

limit of n o n c o m p a c t

in chapter

(1952)

the inverse

are compact.

The proof

the

of i n f i n i t e

However

is an o b s t a c l e

situations.

as c o u n t e r e x a m p l e s

development.

in q u e s t i o n

line-segment,

solution

is again

Vincent-Smith

impossible

the

elementary

simplices

in

surprisingly,

M o r e generally,

and G.F.

sets

Not

compact

for d e c r e a s i n g

Choquet

compactness

or even

in w h i c h

to a p r o b l e m w h i c h

e.g.

"yes"

simplices.

Davies

theory,

by showing

structures

a simplex?

we r e v i e w the h i s t o r i c a l

it may be c u m b e r s o m e able

is also

of simplices

a point,

of B o r o v i k o v ' s

s y s t e m of c o m p a c t

simplex

In section

I now mean

intersection

(a t r a n s l a t i o n

compact

of an inverse

S I m S 2 D S 5 m ...

(by a simplex

Is their

is "yes"

is included).

sequence

Borel

in c o n c r e t e mechanics;

of D y n k i n ' s

entrance

the g e o m e t r i c a l

in more

problem

detail.

should be solved

But the view taken here allows

a

v e r y n a t u r a l and t r a n s p a r e n t proof a n d a b e t t e r u n d e r s t a n d i n g of the problem.

D. Preiss'

of G i b b s s t a t e s and 4.3.2). useful

derivation

of a w e l l - k n o w n c r i t e r i o n

is a n o t h e r e x a m p l e in the same s p i r i t

In summary,

I wanted

to look at p r o b a b i l i s t i c

for e x i s t e n c e

(cf.

s e c t i o n 4.2

to s h o w that it can be p l e a s a n t and objects

through geometrical

spectacles.

CHAPTER NOTATIONS,

Notations, collected chapter

definitions in this

I should

DEFINITIONS

and

chapter. notice

0 AND

conventions The

the

reader

CONVENTIONS

used who

conventions

throughout

skips

the p a p e r

it a n d

in 0.3,

0.6

starts

and

are

with

0.7 t y p e d

in

italics.

0.1

We d e n o t e

integers

by

the

~

, the

set of r e a l

~+

and E u c l i d e a n

real

numbers

0.2

By a m e a s u r e

gative bra

on a m e a s u r a b l e

a-additive

F. A m e a s u r e

signed P2

by

finite

set of p o s i t i v e

measure

set

by

n-space

space

~ , the

set of

set of n o n n e g a t i v e

~n.

(X,F)

we a l w a y s

(with real

of the

by

~ , the

by

a probability

function

numbers

values)

measure

mean

on the

iff p(X)

f o r m q = Pl

a nonnea-alge-

= I. A

- PZ w h e r e

Pl

and

are m e a s u r e s .

Denote

by

B(X,F)

the

space

P(X,F)

the

set of p r o b a b i l i t y

M + (X,F)

the p o s i t i v e

cone

M(X,F)

the

signed

The

numbers

function

p is c a l l e d

is a set

natural

integral

noted

space

of real

of

of a m e a s u r a b l e

b y IX f(x) dp(x),

abbreviate

it by p(f).

p,(A)

the o u t e r

measure

:= sup

bounded

measurable

measures

of m e a s u r e s measures function

fX f dp or For A c X

{p(B) : B 6

on

F - including

F , BOA},

the

zero-measure,

F.

f f dp.

inner

on X,

F,

f on X w . r . t ,

simply the

on

on

functions

p £ M(X,F)

is de-

If f is b o u n d e d ,

measure

is

we

p*(A)

:= p(X)

the p - c o m p l e t i o n

Fp

The u n i q u e

of

- p.(X~A)

;

F is

:= { A c X :

extension

p,(A)

= p*(A)}.

of p to

F

is a g a i n

denoted

by p.

P If X o c x tension

and p 6 M + ( X o , X ° n F), of p to

If x 6 X,

F with

the p o i n t

then

p without

measure

we o f t e n

further

~x is d e f i n e d

identify

the c a n o n i c a l

ex-

remark. by

~x(B)

:= IB(X)

for e v e r y

B6F. Let

(X,F)

and

(Y,G)

be m e a s u r a b l e

P

: X ~ M(Y,G),

is c a l l e d

a

(sub-)stochastic

(i)

P(x,.) 6 M + ( Y , G )

(ii)

P(x,Y)

(iii)

the m a p p i n g s

the c o m p o s i t i o n

P induces

kernel

f r o m X to Y w h e n e v e r

for e v e r y

x 6 X,

=< I) for e v e r y

P(.,B) kernel

:= Sy Q(y,B)

a l~near

mapping

x6X,

are m e a s u r a b l e

for e a c h

f r o m Y to some

PQ is a s u b s t o c h a s t i c

PQ(x,B)

A mapping

x ~ P(x,')

= I (P(x,Y)

If Q is a s u b s t o c h a s t i c

spaces.

kernel

B 6 G.

measurable

space

f r o m X to Z d e f i n e d

dP(x,y)

for e v e r y

B 6 H.

f r o m M(X,F)

to M(Y,G)

via

M(x,F) 9 ~ ~ uP 6 M ( Y , G ) , uP(B)

and

a linear

:= SX P(x,B)

mapping

from

d~(x)

B(Y,G)

for e v e r y

to B(X,F)

B6 G

via

B(Y,G) 9 f ~ Pf 6 B(X,F) , pf(x)

:= P(x,f)

:= ~y f(y)

dP(x,y)

for e v e r y

x6X.

(Z,H), by

T o p o l o g i c a l s p a c e s are assumed to be s e p a r a t e d t h r o u g h o u t

0.3

the paper. T h e u s e of t e r m s

"Suslin"

and

"Lusin"

spaces

is n o t u n i q u e

in l i t e r a -

ture.

Definition. a.

X is s a i d

metric b.

to b e a P o l i s h

on X compatible

X is c a l l e d

tinuous c.

Let X be a topological

bijective

X is s a i d

continuous

Lusin

are

also

called

Now,

we

and

the

Remark. the

~

space

iff t h e r e

called

spaces,

an i m p o r t a n t

space

isomorphic

o-algebra

We have

Borel

given notion

is a

space

Y and a con-

is a P o l i s h

space

Y and a

e.g.

the definition of a Polish

theory

spaces;

cf.

spaces

in H o f f m a n n - J ~ r g e n s e n

measure

(X,F)

and Suslin

theoretic

is c a l l e d space as ~

(1970).

notion.

a standard

(Y,t) , i.e.

Borel

there

I are measurable

space

is a w.r.t.

F

b y t.

can be given

If X is t o p o l o g i z e d

there

X is c o m p l e t e .

"standard"

to a P o l i s h

generated

from measure

in w h i c h

is a P o l i s h

: Y ~ X s u c h t h a t ~ as w e l l

characterization

standard

iff t h e r e

are often

A measurable

topological

notions

topology

and

f r o m Y o n t o X.

"analytic"

it is B o r e l

iff it is s e p a r a b l e

f r o m Y o n t o X.

spaces

can define

bijection

space

mapping

surjection

Definition.

the

to b e a S u s l i n

Remark.

iff

Lusin

with

space

space.

most

space.

as w e l l .

to d e f i n e chapter

the

space of real

Cb(X,t)

the

space

"B-spaces"

Bo(X,t)

the B a i r e - o - a l g e b r a ,

measure

Dynkin which

4.3.4.

continuous

of b o u n d e d

in l i t e r a t u r e ,

A purely

In fact,

b y t, t h e n w e d e n o t e

¢(X,t)

common

by

functions

elements generated

from

o n X,

C(X,t),

by Cb(X,t),

using

theoretic

(1978)

uses

coincide

only

with

B(X,t)

the B o r e l

o-algebra,

B(X,t)

the

space

B(X,B(X,t)).

If

and

(Y,t)

are

(X,s)

is d e n o t e d product

0.4

by

s × t.

topology

If t h e r e

on H

X.

i61

~

T-smooth

iff

the

topological are

o-algebra

generated

spaces,

the p r o d u c t

topological

spaces

is d e n o t e d

by

H

by

t,

topology

(Xi,ti) , i 6

on X x y I, the

t..

i61

L e t X be a t o p o l o g i c a l

is c a l l e d

i.e.

space

for e v e r y

with

topology

decreasing

net

t. A m e a s u r e

(Ai)i61 of

p on B(X,t)

closed

sets

we h a v e

p(

N

Ai)

i£I

= inf{p(Ai):

It is s t r a i g h t f o r w a r d an

increasing

w.r.t,

net

to s h o w

of

p and d e n o t e

lower by

for a T - s m o o t h

semi-continuous

f its

supremum.

f f dp = sup{ S f i d P : (the For -

integral

may

a Y-smooth

called

A measure

measure

support

tight

functions

on X,

integrable

Then:

p there

+~).

is a s m a l l e s t

closed

set of

full m e a s u r e

tight

iff

C compact}

for e v e r y

B 6 B(X,t).

is T - s m o o t h .

by

the

M+(X,t)

the p o s i t i v e

set of t i g h t

measure the

the v a l u e

is c a l l e d

P(X,t)

M(X,t)

(fi)i61 be

p: L e t

i6 I}

= sup{p(C) : C o B ,

measure

we a b b r e v i a t e

measure

of p.

p on B(X,t)

p(B)

Every

attain

i 6 I}.

cone

probability of t i g h t

measures, measures

,

linear

space

M+(X,t)

- M+(X,t) .

- including

the

zero-

10

A

subset

A of

p-completion

0.5

X is c a l l e d of B(X,t)

If t h e r e

0-algebra

A vector are

write

vector x

for

every

is n o d a n g e r

and

Only

0.6

universally

is

over

convex

a 7 , . . . , a n 6 ~+

with

space

elements.

An

iff extreme space

is c o n v e x element

C ~

{x}

boundary

H then

the

of H c o n t a i n i n g cone is

iff

said

it

to b e

convex

strict H*

inequality

(E,T)

topological Let

M be

smallest

the

closed

is t h e

S(M)

denotes

the by

hull

forth.

field

will

n Z

all

set

C

ex C.

a.x-. A

C of H convex.

of

is t h e

A real

subset

or

C of

combinations

a

extreme

points of

smallest iff

~+-C

function

of

its

or a n e x t r e m e

is a s u b s e t

is a c o n e

topology

~ ~

is e x t r e m a l

If M

the

in t h e

xl,...,x n iff t h e r e

convex

collection

is

be considered.

of v e c t o r s

c o M of M

~ af(x)

0 < a < I;

a

and

of E.

~-algebra cone

it

a linear

convex cC

f on

is c a l l e d

and

subset a convex

a convex

set

is

strict~ Z convex iff

space

of

separated!)

Subsets The

+ (1-a)f(y)

affine

dual

(real,

convex

the

so

it

iff

space.

a subset

real

a convex

subset

algebraic

dual

suppress

i=I

denoted

holds

denotes

Z(M)

x of

(1-a)y)

x,y £ C and

denotes

0.7

+

we

and

it c o n t a i n s

in a d d i t i o n

f(ax

whenever

A

p o n X.

~

iff

convex M.

is

the

is c o n v e x ; and

measure

a. = 1 such that x =

i=I vector

M(X)

combination

n Z

iff

confusion,

C(X) , P(X),

spaces

the

of

tight

measurable

always

closed

subset

of

always

locally have

of

all

sets"

functionals

convex

the

hull

E containing

"cylinder

equality

always

the

holds.

H.

convex

of

iff

relative

E'

topology.

c o M of M

is t h e

M°

o n M, ~

space,

generated

: M ~ ~

of

the

by

E',

form

its

C

11

= m a x ( I / ( - ) + c i) for

some

Two properties

Lemma. a.

Let M be a subset space

in the p o i n t w i s e b.

Suppose

w.r.t°

that

p,q6

maximum

S(M)-

hence

is a v e c t o r

a. a n d c o n t a i n s

their

In fact, hence of

I < i < n,

space

b y S(M)

E. Then:

is a v e c t o r

lattice

= f~ dq

contains

lattice b.

The

is Z(M).

two

proves L

-

of a n y

(~ + ~); of

the

this

shows

its e l e m e n t s ,

is a v e c t o r

So by the uniqueness

Choose

~ also

p = q.

a.

:= S(M) - S(M)

p is e q u a l

implies

~ and

= 2 ( ~ v ~)

value

is i n t e g r a b l e

~ 6 S(M)

functions

I ~ - ~I

the absolute

space

~ 6 S(M)

for every

with

which

theorem,

every

assertion

to q o n t h e o - a l g e b r a

now compact

lattice

s e t s Cn, n ~ ~ ,

by

of t h e

generated such

that

=

I

=

q(C).

is p o s s i b l e ,

O

(c)

every

=

since

p and q are

tight.

We

show now

that

z(c).

f 6 Cb(C)

Z(M)-measurable:

If(x)

is the p o i n t w i s e

for e v e r y

n 6 ~

theorem

some ~n

the S t o n e - W e i e r s t r a 8

sup x£C

limit

there

of a s e q u e n c e

is b y

the

lattice

f r o m L, version

6 L with

I - ~n(X) I < -. n

n By uniqueness on

B(C).

for

C

choice

B

convex

spanned

such that

the constants.

that

p(c)

This

S~ dp

contains

extension

union

,

~t

be needed.

locally

~ v ~. M o r e o v e r ,

S(M)

L e t us n o w p r o v e

o n M b y L,

will

of the

P(M)

S(M)

that

Daniell

c .6 ~

order.

Obviously

pointwise

S(M)

S(M) - S(M)

p a n d q. T h e n

Proof.

and

1 . 6 E'

n 6 ~ .

of t h e c o n e

The vector

where

This

of t h e e x t e n s i o n completes

of t i g h t

the proof

of b.

Baire

measures,

p and q agree

12

If H is a linear

space and G a s u b s p a c e

logy on H g e n e r a t e d

If

is

(X,t)

by the m a p p i n g s

a completely

by the f u n c t i o n a l s

cally c o n v e x the symbol

0.8

space.

~i ~

As s t a n d a r d

spaces,

measure

theory

is the topo-

g 6 G.

space,

t h e n M(X) i s always

o(M(X),Cb(X)) ~ ~ ~(f),

Convergence

in this

f 6 Cb(X), to p o l o g y

which makes

it a lo-

will be i n d i c a t e d

by

~"

references,

Hoffmann-J6rgensen(1970) Lusin

x ~ g(x),

regular topological

endowed w i t h t h e weak t o p o l o g y generated

of H* then o(H,G)

we use Dugundji(1968)

and Schwartz(1973)

Tops@e(1973)

topology,

for the theory of Suslin

and also Schwartz(1973)

and Schaefer(1973)

for general

for t o p o l o g i c a l

for t o p o l o g i c a l vector

spaces.

and

CHAPTER

BASIC

There

are

theory. called

several

Some

We will

the

work

chapter cations.

sections. 3;

If o n e

a standard

to p r o v i d e

ions.

So I had

Nothing

which

are

in t h i s The

are not

a systematic

chapter reason

published

is s c a t t e r e d

indicate

the c o n n e c t i o n

Section

L e t us may

of the c e n t r e

Definition.

barycenter

of

notion of

investigations

further

in

search

the

the

for

of t h e p r e s e n t of t h e

at l e a s t

foundat-

those

subsequent

basic

studies.

the proofs

are

f r o m v. W e i z s ~ c k e r ( 1 9 7 7 ) journals;

also

Several

sketch

a lot of

remarks

the h i s t o r i c a l

reading.

measures

some material

convex

p £ P(E)

in

setting,

and

of a b a r y c e n t e r .

measure

spaces.

for a p p l i -

papers.

material,

for

of B a n a c h

treatment

mathematical in v a r i o u s

Bourgin(1983)

be established

scope

results

be

it is u s e f u l

together

x in a l o c a l l y

the p r o b a b i l i t y

will

Nevertheless,

representing

of m a s s

A point

new.

related

suggestions

the basic

the

to u n d e r s t a n d

and hidden

Barycenters,

start with

think

1.1.1

and contain

1.1.

to p u t

in c u r r e n t

sets may meanwhile

for t h e

exhaustive

that we need

with

which

show that

It is b e y o n d

is r e a l l y

in C h o q u e t

subsets

in s u c h a g e n e r a l

myself

is,

fields

convex

convex

framework,

indispensable

arguments

background

closed

and

THEORY

and Alfsen(1971).

4 we will

is idle.

CHOQUET

special

It is a p p r o p r i a t e

to c o n f i n e

which

elaborated.

general

is i n t e r e s t e d

paper

concepts

Phelps(1966)

in c h a p t e r

reference

for

of c o m p a c t

for b o u n d e d

in a m o r e

2 and

surveys

the c a s e

- like

theory

following

FROM NONCOMPACT

excellent

covering

classical

describes

the

CONCEPTS

I

iff

the barycenter

Intuitively,

distributed

space

map

one

in space.

E is c a l l e d

the

14

(i)

each

1 6 E'

(ii)

l(x)

= f 1 dp

We also

be called

barycenter

map

(provided

Remark.

We note

three

they are unique

Example

I.

simple

the b a r y c e n t e r

then

special

If the m a s s

p,

1 6 E'.

formula

x b y r(p).

The expression

a n d the m a p p ~ r(p)

in t e r m s

since

E'

of t h e P e t t i s

combination

separates

condition

(i)

is f u l f i l l e d .

cases.

is c o n c e n t r a t e d

is x i t s e l f . of p o i n t

2.

Similarly,

More

masses,

let p =

--

~ i=I

t h e n x is the b a r y c e n t e r

(i)

a i I l(x/)

and

points.

in o n e p o i n t

generally

x - i.e.

if p =

then

r(p)

=

Z i< n

Example

the

integral

Z

a

i 0,

xi

~

a

= I;

i=I

of p iff

I < ~

for e v e r y

1 6 E',

for e v e r y

1 6E'.

/=1 (ii)

l(x)

In t h i s

case,

Example

~.

E

:= M ( X ) .

If ~ 6 M(X) nition

=

~ a/l(xi) i=I

x =

a.

~ a i x i in t h e w e a k i=I

Consider

Recall

that

some completely

M(X)

is t h e b a r y c e n t e r

1.1.1

reduces

~(f)

topology

is e n d o w e d

o(E,E').

regular

with

o f p 6 P(M(X))

the

space X and let topology

then condition

to

= f ~(f) dp(v)

for every

f 6 Cb(X) .

o(M(X), (ii)

Cb(X)).

in d e f i -

15

In p r o p o s i t i o n more b.

general

1.1.2 we will

functions

It is e a s y

to see t h a t e x

to X i t s e l f

(1970),

11.1).

the e x t r e m e

points

~(f)

such

that

= rex

barycenter

that map

now

P(X).

formula

functions, are written

For example,

with

= {~

: x C X}

x

P(X)) from

has

set

I : x ~ ~x

P(ex

formula

not only

for m e a s u r a b l e

: g 6 /I (~)}.

Then

onto

hold

for b o u n d e d

functions.

As

on p r o b a b i l i t y

Suppose

on e x

with measures,

L e t X be a c o m p l e t e l y

p 6 P(H)

setting

these

theory,

in t h i s

regular

further

the e v a l u a t i o n

with

yields

Hence

the barycentricontinuous

lecture

notes

we confine

special

space

latter

dp(~)

our-

context.

and g

t h a t H is a s u b s e t

: X ~ of

~ in H the b a r y c e n t r i c a l

holds

gets

for e a c h then

B 6 B(X) .

~ is t h e b a r y c e n t e r

the

in t h e

map

barycenter

g = IB o n e

= fH ~(B)

if the

on

P(X) . F u r t h e r ,

formula

Conversely,

-I

~ , ~ ~ f g d~

for every

~(B)

~oi

may not be adequate

Proposition.

function.

-I

P(X).

if o n e d e a l s

3 should

~oi

P(X))

1.1.2

In p a r t i c u l a r ,

is

(Tops~e

formula

pol 6 P(X).

s u c h a g a p c a n be f i l l e d

satisfies

this

barycenter

to s h o w h o w

: H ~

that

transformation

selves

{~ 6 M+(X)

for

~ to t h e m e a s u r e

of the m e a s u r e

a side-glance

a Borel measurable

also

and

injection

integral

the barycentrical

also

holds

P(X) vlf) d~ol -I (~)

in e x a m p l e but

formula

~ 6 P(X) ; l i f t

The

is a b i j e c t i o n

1.1.1(ii).

this

the natural

each p 6 P(ex

In a p p l i c a t i o n s ,

cal

P(X)

~ is the b a r y c e n t e r

it is c l e a r

form

via

Choose of

that

f.

homeomorphic thm.

see,

of p.

18

Proof.

From

the

~(f)

Let

U be

~(U)

= fH~(f)

an o p e n

~ and

~(U)

standard

of

: f 6 C}

p are

of

dp(v)

subset

= sup{~(f)

0.4).

By

definition

class

for

every

set

each for

= fH ~(U) dp(~)

monotone

topology

X and

for

T-smooth

the

C

{f 6 Cb(X)

since

each

the

same

reason,

for

every

open

arguments

we

have

f 6 Cb(X) -

:=

~ 6H,

on M(X)

the

: 0 s f s IU}.

~ is

Y-smooth

Then (cf.

hence

subset

U in X.

barycentrical

formula

holds

+

for g-

any

h 6 B(X)

= - (g ^ 0),

The

partial

the

belief

fies

the

Example and

success that

of

Borel

measurable The

ex.

formula.

example

~.

¢

measure

I = r(p)

for

O(v)

:= m

bounded

(for

the

proof

measures

~

I one

some

x

¢(I)

= 0.

p concentrated

on

function

. Set

affine

H

support ~ satis-

:= P ( [ 0 , 1 ] )

where

~

is t h e

- moreover

is r e f e r r e d the

the

sin-

points

hand,

3,we

: x 6 [0,1]},

it

is

to A l f s e n ( 1 9 7 1 )

extreme

other

In e x a m p l e {~

might

S

with

1.On

[]

true.

([0,1])

and

reader

coincide

%1 ex H =

has

affine

to be

s

= g v 0 and

proposition

measurable fails

g

standard.

to C h o q u e t ( 1 9 6 2 )

on H b y

the

is

preceding

This

is d u e

consequently

Lebesgue

the

for

rest

¢ is o b v i o u s l y

point

3),

in

The

(universally)

a function

part

convergence

g itself.

achieved

every

This

define

(cf.

for

barycentrical

4.

21).

by monotone

hence

gular

p.

and

for

have

of H

the

seen

that

hence

X

O(r(p))

Especially The

proof

1.1.3

second

is d u e

to

Proposition.

embedded Then:

the

= ©(I)

in

its

= 0 % I = p(~D) .

part

R.

of

Haydon

Suppose

completion

the

following

proposition

will

be

needed.

(1976).

that

E and

the that

locally M

convex

is a b o u n d e d

space subset

E is of

E.

,

17

a.

for every

b.

the b a r y c e n t e r

r

is a f f i n e

nal

exists

in E,

~ E ,

p ~ r(p)

Note

that the barycenter on E - which

map

is n o t o n l y

is t r i v i a l l y

true

continuous

- but even

in t h e

in t h e o r i g i -

topology.

I.

of pr0~osition

L e t us

Suppose

first

that

isomorphic

and denote

holds

since we may

of the

x = (xi)i61

p ~ r(p)

E[

and

is t h e b a r y c e n t e r topology

w.r.t,

for e v e r y

p ~ p o p r i-1

is

i6

limit.

I. B u t

and the barycenter

Since

(Ei,

Assume

by x i

now

is n o r e -

pri)i61

to c h e c k

topology

that

o n E is t h e

p ~ pri(r(p))

is the c o m p o s i t i o n ~ E.

from

to a d i -

, the barycenter

r ~ : P(Pri[M])

of

III.4.4

isomorphic

the

if the m a p s

such a map map

limit

Denote

Proposition

it is s t r a i g h t f o r w a r d

if a n d o n l y

spaces

inverse

in E i. T h i s

(algebraically)

family

it is

of pri[ ~ ] in E i i n s t e a d

o f p in E.

the

is t h i s

by assumption.

inverse

spaces.

of Banach

i n t o E i b y Pri.

the closure

u s t h a t E'

spaces

exists

(Ei)i£ I

itself

of E

of B a n a c h

S i n c e E is c o m p l e t e

pri[ ~ ] is d e n s e

a reduced

is c o n t i n u o u s

continuous

tinuous

tells

take

spaces.

that E

projection

space

to the c a s e

of a f a m i l y

-I of p o p r i , w h i c h that each

Schaefer(1971)

projective

limit

II.5.4) . A s s u m e

. E is t h e n c a l l e d

limit

for B a n a c h

the c a n o n i c a l

additionally,

rect

the p r o b l e m

to a n i n v e r s e

the b a r y c e n t e r

striction,

1.1.3.

reduce

1.1.3

(Schaefer(1971),

2.

r(p)

map

: P(M)

topology

Proof

E.

the barycenter

and continuous.

Remark. weak

p 6 P(M)

map

are of

which

is c o n -

by assumption.

Consider

now a Banach

B o f E.

Denote

by

into E"

. For

each

1 ~ p(1),

1 6 E'.

space

E. We m a y

choose

N" II t h e n o r m

of E " a n d

suppose

We

p 6 P(B) show

the e l e m e n t

first

that

r(p) 6 E "

M to be t h e u n i t

ball

t h a t E is e m b e d d e d is d e f i n e d

by

18

r

is

: P(B)

continuous

a covering

~ E"

w.r.t.

, p ~ r(p)

~(M(B)

(Ua)a6 A of

B by

is

paracompact

by

Stone's

is

a partition

of

unity

VIII.4).

Recall

(i)


0 and

consider

A metrizable IX.5.3)

(Ua)a6 A

, so

set there

(Dugundji(1968),

means is

of

continuous,

the


a. T h e n

of the H a h n - B a n a c h

theorem

and

convex.

Choose

(x,a) ~ G(~) a. B y t h e g e o m e t r i there

is 1 6 E'

and c £ ~ such

that

l(x) +

Necessarily unbounded

ca

c > 0,

above

s o m e x' 6 M w i t h

< inf{l(x')

since

+

ca'

f o r x' 6 M

whenever ~(x') < ~.

~(x') < ~;

:

(x',a') 6 G ( ~ ) a } .

the

set

{a' 6 ~ : (x',a') 6 G(~) a}

but we may assume

that

there

is

is

30

Set

k

: = -c

-I

k (x)

and

for e v e r y

1 and

+ b

b := c-ll(x)

=

a

x' 6 M

k(x')

+ b

=

-c


0 b e c a u s e

f to C,

measurable

fl a n d

f2 a r e u n i -

(C\ex

x 6 C,

x 6 C ~ ex C.

on C a n d a s s u m e C)

such

of L u s i n ' s

that

theorem.

t h a t p ( e x C) < I. T h e n

f; a n d

f2 a r e

continuous

Define

1

Tx

C ~ G

and

Let now p be a boundary there

there

if x 6 ex C

f r o m C to C x C.

fl (x)

C~ex

the universally

function

(fl (x)

I ~ ( x + y)

(x,y) ~

:= Ex for x { K, T x

:= 2 ( ~ f ; (x) + ~f2(x))

for x 6 K.

41

Then

,dp Hence

p % pT,

-

but by

= theorem

1,3.6.d.

assumption

t h a t p is m a x i m a l

c.

now that

Assume

pO.

in C h o q u e t

C is S u s l i n

to t h e o r e m

we have

r contradicting

the

order.

and measure

1.3.6.d.

p ~(r[A

]) = ~(A

Let n o w p be arbitrary.

~. such that p(~i) > 0 and : ~T by 1 3 . G ;

qi (B)

) = I.

C h o o s e p a i r w i s e d i s j o i n t in o c o m p a c t sets p( U ~i) i6~4

= 1. T h e r e is a p - d i l a t i o n T w i t h

set

::

I

IC Tx(B)

dp(x)

for every B 6 B(F,o).

57

Since

q

=

~

qi ' each

qi

dilation

(to be

to p o i n t

I, p(- N C/) is c o n c e n t r a t e d

The

announced

2.1.3

subset

a.

M

b.

whenever

Proof. ceding

q6

P' (E)

that

a.

to g e t

a.

~r(q)'

converse,

we

2.2.

have

bability

pleasant measures

to n o r m a l i z e ! ) ,

on M,

hence

also

locally

following

2.2.4).

The

salient

measure

convex

that

Set

assume

that

1.1.3

p 0 there

tight

by M the

condition

sup{p(C)

is a c o n v e x

in M'

this may

set of i r r a t i o n a l

[0,1]

compact

in M'

subset

where C of M'

p 6 Q.

is t r i v i a l l y

: C c

convex-tight

fail

numbers

fulfilled

convex

in M!

in M'

for e a c h

and compact}

:= [0,1].

p 6 P(M)

in M' :

= I .

But obviously

sup{p(C)

The

proof

2.2.2 locally

of the

: C c M

following

Proposition. convex

convex-tight.

space Then

Proof.

Let

result

Suppose

the

compact

contains

further

} = 0 .

the m a i n

that

arguments.

subset

a fixed

of the c o m p l e t e

measure

p £ P(M)

is

set

convex-tight

F denote

and

t h a t M is a b o u n d e d

E. A s s u m e

Q := {p 6 P(M)

is u n i f o r m l y

convex

the

: p < q}

in E.

family

of all

continuous

semi-norms

y on E

satisfying

sup{y(x-y)

F generates I.

the

: x, y 6 M}

topology

By assumption,

of M s u c h

that

(I)

q(M~Ci)

of E.

for e v e r y

~ 2-2ic.

~ I .

Choose

i 6~

now

there

~ > 0.

is a c o m p a c t

convex

subset

Ci

59

Fix

now

some y 6 r. The

Mi¥

: E

is c o n t i n u o u s

Let

B

¥

~ ~ + , x ~min{y(x-y)

and

Mi¥ 1 M

(2)

~

denote

- C, b e i n g

the c l o s e d

C Y. := { x 6 E
2-n

The

(Pn)n~1

net

unit

ball

B of

on the u n i t 1

;

(~)

ball

of E d e c r e a s e s

is m e a s u r e

convex;

in

in C h o q u e t

P(B),

order.

The

w

we h a v e

Pn ~ ~x'

where

x =

Z

2-~e

.

i>I Hence there

~x

is the g r e a t e s t

is no g r e a t e s t

Increasing

nets

to

remarks.

several

Remark.

Lemma.

Let

Combining

(P,~)

a least

a least

upper

this

Proposition. creasing

are n o t

L e t us q u o t e

in P has has

lower

chain

lower

bound

bound

in

treated

sequel;

from Edgar(1978),

upper

bound.

Then

2.4.2.

Since

x ~ E,

P(B N E) .

in the

be a p a r t i a l l y

lemma

we c o n f i n e

ourselves

2.3:

ordered

set.

every

subset

Suppose

that

of P w h i c h

every

chain

is d i r e c t e d

bound.

lemma

with

theorem

Let M be a m e a s u r e in

by t h e o r e m

(P(M),i

Pkj w-~ qj

and

for

k > j,

hence -] Pki w~ qj o~ji which

shows

for

k ~ j,

that -I

qi = qj°~ji" 3.

Because

of the that

= P°~ZI

and

theorem

4.

N o w we

(I,~)

(qi)i6i

system qi

center

the n e t

show,

exists,

for e v e r y

21 on p.

is c o u n t a b l y i.e.

i 6 I

generated,

there

the p r o j e c t i v e

is a u n i q u e

(Schwartz(1973),

p 6 P(S)

corollary

limit such

on p.

81

75).

that

p is the u n i q u e

boundary

measure

on S w i t h

bary-

x.

Choose

q 6 P(S)

with

r(q)

= x

and

fix

i 6 I. T h e n

r

= xi;

furthermore q o ~ i-1 < p o ~ 1 This

relation

will

see

Every

being

true

= qi . for e a c h

i 6 I

is e q u i v a l e n t

functional

f on

E =

to

p < q, as we

immediately:

continuous

linear

H

i61

E.

is of

the

form

83

jCJ ljoprj

f =

where

J

is a finite

the c a n o n i c a l upper b o u n d

subset

projections

Since every

lj

are

(Schaefer(1971),

in

thm.

g 6 S(E)

ljo~kj o~ k(x)

for every

is of the form

max(fi +

see from the last formula

Hence

I, the

Ej

prj

and the

IV.4.3).

are

If k is an

for J then

jcJ

every

of

that

q 0

all

are in

:= {x 6 ~ d : d i s t ( x , S ) < ~}

being

convex

the c o n v e r s e

L e t a be an e x t r e m e if it is a m e m b e r

- the same is true for the s i m p l i c e s

S i. T h i s

inclusion. k af

p o i n t of S. The v e r t e x

of some s e q u e n c e

is said to b e l o n g

(ak)z6~4which

converges

C a~ 1= co {ak:

0-_ P ( - , R ( . , I F ) )

= I F.

same

R(-,1F)
0})

F ~ R(-,IF) : F ~ G

= I{y:R(y,F)

there

and

> 0} £ B(Y,G).

corresponds

similarly

to

therefore

G ~ P(-,1 G)

canonically

a map

~

a map

: G ~ F. F r o m

(**)

follows

~o~

hence F;,

= id G

~ = ~-I.

inverse

manner

implies

= idF,

of R i m p l i e s ~(FI) c ~ ( F 2 ) .

and p r e s e r v e s

countable

monotonicity

of ~,

Hence

F onto

~ maps

set o p e r a t i o n s

and

i.e. G in a also

its

~.

2. A s s u m e

now

directly.

that

all

singletons

F i x x £ X. F r o m

I = £x({X})

Hence

#o~

Monotonicity

F Z £ F, F I c F Z

one-to-one

and

the m e a s u r a b l e

Gx

has m e a s u r e

first

= f R(y,{x})

equality

in

We c o n s t r u c t (*),

the m a p

it f o l l o w s

that

dP(x,y).

set

:= {~ 6 Y:

one w.r.t.

for y 6 G x r e s u l t s

the

are m e a s u r a b l e .

R(y,{x})

P(x,-).

= I} = {y 6 Y: R(y,-)

Writing

out

the

= ex }

second

equality

in

(*)

in

I = ~y({y})

= f P(x',{y})

d R(y,x')

= f P(x',{y})

dex(X')

=

= P(x,{y}),

from which ton

we c o n c l u d e

that

P(x,.)

= ~y;

in p a r t i c u l a r

G x is a s i n g l e -

{~(x)}.

The mapping

~

same way,

construct

we

: X ~ Y

is w e l l - d e f i n e d a measurable

and clearly

mapping

~

measurable.

: Y ~ X

which

In the fulfills

98

~o~(x)

= x

by

second

the

verse

We

Borel

4.1.3

i 6 I

that

the

from

X~ j

X.

and

every

y 6 y

if

turn

we

of o w n

interest.

4.1.4

Lemma.

G = B(Y,t) ,

(iii)

f is c o n t i n u o u s

is a P o l i s h

By definition,

topology

s × r. H e n c e

is m e a s u r a b l e

with

topology

with

that

theoretic

(*) a n d in-

of

a standard t e r m s (4.3.4).

net and

spaces

4.1.3,

Pji

of m e a s u r e s . kernels

to be a s t a n d a r d such that

further

kernels

substochastic

limit

let for

Assume

substochastic

are Borel

of t h e o r e m

Pi,

Borel

{y} 6 G

for

isomorphic.

we

formulate

some

results

Borel

space with

o-algebra

r. F o r e v e r y

measurable

mapping

t on Y such

f

that

space,

w.r.t,

there

that

coincides

G(f)

(Y,G)

before

be given.

system

inverse

is a t o p o l o g y

(ii)

G-algebras

in

t h u s ~ is t h e

generated

with

t h a t Y is a s t a n d a r d

space with

Y such

Recall

c a n be c h o s e n

and

(Y,t)

space

together

(X,F)

(i)

Borel

y 6 Y;

(Xi,Fi)

inverse

is a n o t h e r

Suppose

from Y to Z there

equality

in m e a s u r e

together

is an

to t h e p r o o f

Z a Polish

Proof.

space

(X,F)

(X,F)

chapter.

be a countably

Borel

Moreover,

then

for e v e r y

exclusively

(l,&)

limit

Before

G and

of t h i s

, i ~ j,

(Y,G)

first

D

((Xi,Fi))i61

an inverse

space

result

Let

family

I, e x i s t s .

= y

of t h e

is c l e a r .

a standard

to

because

~o~(y)

c a n be d e f i n e d

Theorem.

every

i6

rest

the main

space

x 6 X

equality

of ~. T h e

state

Then

for every

t a n d r.

is a P o l i s h

G = B(Y,s). the Borel

by lemma

On

topology Y x Z

o-algebra

s on the

the product generated

12 in S c h w a r t z ( 1 9 7 3 ) , p.

standard of the Borel

by the product 106,

the graph

:= {(Y,f(Y)) 6 Y × Z: y 6 Y}

w.r.t.

B ( Y × Z,s x r).

As a Borel

set

in t h e P o l i s h

space

99

(y x Z,s x r), cit.,

thm.

stronger cit.,

the g r a p h

2 o n p.

than

cot.

95).

So t h e r e

G(f) N (s x r)

2 o n p.

the projections. topology

is a L u s i n

101).

Since

space

in the r e l a t i v e

is a P o l i s h

and generating

Denote pry

by

pry

topology the

: G(f)

is a b i j e c t i o n ,

same ~ Y

topology t'

o n G(f)

o-algebra and

(loc.

(loc.

pr z : G(f)

we can define

~ Z

a Polish

t on Y by

t

Obviously,

:= { p r y [ G ] :

G 6 t'}.

we have

B (Y,t)

=

= {pry[B]:

BEB(G(f),t')}

= {pry[B]:

B6 B(G(f),G(f)

=

~ (Y,s)

= N (s × r))}

=

=

=G.

Writing

f in the

of the

t-t'-

is c o n t i n u o u s

Remark.

The

countable theorem

Next,

form

f = przopr~1,_

continuous

map

w.r.t,

topologies

the

assertion

base

and

40 o n p a g e

we note

see t h a t

and

the

f as t h e

composition

t'-r-continuous

map

pr z

t a n d r.

can be derived

a countable

family

o

for every of maps

topological

space

Z with

f from Dellacherie(1980),

210.

three

By ~ w e d e n o t e

p r y-I

we

simple

statements

the e v a l u a t i o n

about

o,algebra

on

measures M+(x,F)

on P o l i s h generated

spaces. by the

maps

M+(X,F) 9 ~ ~ ~(f),

4.1.5

Lemma.

Let

a.

M(X,B(X,t))

b.

M+(X,t)

c.

Z coincides

(X,F)

f 6 B(X,F).

be a Polish

space.

Then:

= M(X,t) ;

in its w e a k with

the

topology Borel

is a P o l i s h

o-algebra

space;

for the w e a k

topology

on M ÷ ( X , t ) .

100

Proof. a.

Schwartz(1973),

theorem

b.

Bourbaki(1969),

proposition

c.

The

tely

fact

that

9 on p a g e 5.10.

Z is c o n t a i n e d

in the B o r e l

from Hoffmann-J~rgensen(1970),

since

the w e a k

topology

has

122.

IV.4.A.7

a countable

o-algebra a n d the

follows

converse

immedia-

holds

base,(Parthasarathy(1967),

thm.

II.6.6).

o

L e t V be a p o i n t e d subset

convex

H of V is c a l l e d

is convex.

4.1.6

V is c a l l e d

Lemma.

If

cone

a cap

in some iff

conves

it is c o m p a c t

well-capped

(X,t)

locally

iff

space.

and c o n v e x

it is the u n i o n

is c o m p l e t e l y

regular

then

of

A

nonempty

and

V~ H

its caps.

M+(X,t)

is w e l l -

capped.

Proof. 1.

For

locally

compact

in P h e l p s ( 1 9 6 6 ) , 2.

The

general

proposition result

let

CoX

be the u n i o n

such

that

~(X~C)

topologies same

Borel

o-compact

follows

straightforward.

of p a i r w i s e

= 0. C t o g e t h e r

disjoint with

C i N t, i 6 ~ , s a t i s f i e s sets

and

tight

measures

M+(C,t')

is w e l l - c a p p e d

cone

containing

~ I B ( C , C N t)

u since

{~ 6 M+(X,t) : ~ ( X ~ C )

the

the as

the p r o o f

Choose

compact

c a n be f o u n d

t)

(Bourbaki(1969),

Ci,

convex

5.8).

Obviously

i6

~ ,

I a n d has Hence,

Any

set

homeomorphic

and

t' of the

2.1.1.

as well.

a compact

is a f f i n e l y

sets

in p a r t

C N t by l e m m a

defines

~ 6 M+(X,t)

sum t o p o l o g y

conditions

M+(C,C N

and

M + ( C , C N t) = 0}

(X,t)

11.5.

cone

containing

spaces

in

cap

the

in this

M+(x,t)

to its c o m p l e m e n t

is c o n v e x ,

Remark.

the

a

Lemma

4.1.6

(the d e f i n i t i o n

holds

of the w e a k

also

if

topology

v. W e i z s ~ c k e r / W i n k l e r ( 1 9 7 9 )

a class

is c o n s i d e r e d

turn

whose

members

(X,t)

out

has of

is an a r b i t r a r y then

separated

to be m o d i f i e d ) .

subcones

of

spaces

to be w e l l - c a p p e d .

space

In

of m e a s u r e s

101 The A

following p in

a convex

x 6V~

{0}.

It

imply

that

y

V

ray

lemma

is d e n o t e d

4.1.7 and

V

space

1.

by

exr

If V

in

the

Fix

relative

question

The

extreme

ray

a well-capped

cone

in

the

V~exr

elements

page

V

is

((1982),

the

form

z EV

union

all

of

unpublished).

{ax:

x E p, y,

in

a ~ 0}

and

x

extreme

a locally

topology

then

{(x,y)

the

of

y 6 V

where = y

+ z

rays

of

convex exr

V

space is

a Lusin

x

L x L

in

+ y

and

v.

with

exr

V

Then

L

:=

x + y E L

there

the

then

is

a Lusin

91).

assumption

role

6VxV:

page

set;

H containing

x,

no

a Borel

2 on

a cap

88;

plays

subset

relative

, theorem

all

a closed

iff The

and

:=

of

proportional.

choose

R

Weizs~cker

a subset

and

closed

v.

V is

v 6 V

is

H.

topology.

that

(Phelps(1966),

is

is

space

show

to

V.

(Schwartz(1973)

o-compact L

z are

a Lusin

space

cone

and

is

We

due

is c a l l e d

Lemma.

Proof.

is

simple

that

are the

proof).

U nH n6~ contained

cone

V

is in

in

Hence

= v}

thus

o-compact.

set

Prop =

:=

{(x,y)

=

6 V x V:

U

{(x,y)

x and 6 V

y are

x V:

x =

proportional} ay

or

y =

=

ax}

aE[0,1] is

a closed

subset

of

V x V

(consider

convergent

nets!).

Hence

Rnp

is

a subset

R.

In

a Lusin

a Lusin space

of

space

and

each

:=

{ (x,y) 6 v x V:

the

o-compact

space,

each

compact

set

Borel

(Schwartz(1973) Suslin

x + y

= v,

R and

x

as

and

y are

R

= R~

np

- in p a r t i c u l a r

, theorem space

2 on

page

is m e t r i z a b l e

each 95)

not

proportional}

Prop

it

compact hence

(loc.

is

open

- subset

in is

a Suslin

cit.,

cor.

2 on

102

page

106);

hence

Furthermore,

compact

in a m e t r i z a b l e

Putting

things

2.

set

The

every

M

together,

subset

space

we get

of a Lusin

every

that

:= {(x,y,v) E V x V × V:

R

x +y

open

subset

is

np

space

is m e t r i z a b l e .

is o f t y p e

F O-

o-compact.

= v, x a n d y a r e n o t

proportional}

is a B o r e l

set

in

M is o - c o m p a c t . tells

us that

Lusin

space.

We are here. for

gave

4.1.3.

and will

Polish

lemma

t2

4.1.5,

on

X2

that

need

is a B o r e l

net

topology;

the

and

there tI

on

R

of

np

cor.11)

then

set a n d t h u s

remark

a

is in o r d e r

from chapter

topology.

~

the

XI

Hence

topologies

also

3 only

theorem

- for w h i c h

is the n a t u r a l

on

such

space

By lemma

P2,1"

4.1.4,

such

this procedure,

we

Fi = B ( X i , t i ) , i E ~ ,

(ii)

M(Xi,F i) = M(Xi,t i) , i E ~ ,

(iii)

Pji

since

sequence.

that

that

being

F I = B(Xl,tl).

measurable

Reis

w.r.t.

is a l s o m e a s u r a b l e there

P2,1

is a P o l i s h

: Xj ~ M+(Xi,ti) , i ~ j, is c o n t i n u o u s .

top-

: X2 ~ M+(X1'tl)

we get a sequence

that

(i)

order

M+(xI,F I) = M + ( X l , t ; )

M+(XI,FI),

F 2 = B(X2,t2)

i E ~ , such

results

is a g e n e r a t i n g

the kernel

o-algebra

with

A technical

for w e a k

I = ~

we observe:

Repeating

ti,

4.1.3.

the w e a k

B(M+(Xl,tl) ) .

and

is c o n t i n u o u s . topologies

section

in 2.4.4.

topology

the evaluation

F2

w.r.t.

with

proof

generated

in t h e w e a k

and

logy

assume

a Polish

theorem

in the v e r s i o n

of t h e o r e m

Choose

v 6 V the

(Saint-Raymond(1976),

to the v - a x i s

we will

spaces

simple

calling

F2

4.1.3,

and

in a c o u n t a b l y 1.

to p r o v e

convex

a short

We may

fixed

D

is o n l y n e e d e d

Proof

For every

of Arsenin

its p r o j e c t i o n

theorem

locally

2.4.2

A result

now ready For

(V × V) x V.

of P o l i s h

103

2.

The

sets

Si are Polish fine

:= { v 6 M + ( X i , F i ) :

simplices

affine

in the

v(Xi) ~ I}, i 6 locally

convex

~ ,

spaces

Ei

M(xi,ti).

:=

De-

mappings

~ji : Sj ~ S i, ~j ~ ~jPji" i ~ j. It f o l l o w s Si

together

verse 3.

from

system

Denote

(iii)

with of

that

the

affine

continuous

space

topologies

~(M(Xi,ti),Cb(Xi,ti))

the

limit

even

maps

forward capped

4.1.6,

S

~

E.

on E.

:= l i m S. c E

each

argument

simplices

are o b v i o u s l y

and by Because

is a S u s l i n

set

exr

shows

~/S

ex S ~ {0}

serve

that

M+(Xi,t i)

of the c o n e s

(Phelps(1966),

that

that

the c l o s e d

an in-

T the p r o d u c t of

corollary

simplex

of the

3.2.5,

- in fact

proposition

is a L u s i n is a L u s i n

11.4).

space

space.

by

To

is w e l l - c a p p e d .

subcone Since

lemma

this

~/S ~/S

4.1.7.

end,

/

(I)

4.

~ji

The

it is

Polish.

By l e m m a

the

are c o n t i n u o u s .

simplices.

by E the p r o d u c t

inverse

~ji

the m a p s

A straight-

of E is a l s o w e l l is a P o l i s h From

this,

it is s u f f i c i e n t

space,

we

infer

to o b -

%

ex S = ~ e x r

Now a candidate

~+-S

N {(~i)

for the

X

:= ex S ~ {0},

t

:= X N T,

F := B(X,t)

inverse

~i(Xi) + I}) U

6 S:

limit

ex S ~ {0}

space.

The m a p p i n g s

4.1.5

measurable

can be d e f i n e d .

Set

,

Pi : X ~ M+(Xz,t i) = M+(Xi,Fi), Since

{0}.

is a L u s i n Pi

w.r.t.

are

space

by p a r t

continuous

F and

x = (~j) ~ ~i' i 6 ~ . 3,

(X,F)

is a s t a n d a r d

as p r o j e c t i o n s

the e v a l u a t i o n

o-algebras

- hence on the

Borel

by l e m m a spaces

104

M+(Xi,Fi).

This

shows

to the m e a s u r a b l e 5.

spaces

are

substochastic

kernels

(X,F)

from

(Xi,Fz).

:= { p 6 M + ( e x

barycenter

m a p on

? : M 0

(~j) in

and

k 6 ~

Mi

such that

i 6 ~

Pi

a n o r m de-

is g i v e n such that

is nonvoid, into

with kernels

O(M(Xi),A(Xi))

whenever

~(X k) ~ a

relatively

i ~ j;

for each

~ 6 M k-

T h e n the m e a s u r a b l e space X is nonempty. Proof. limit.

(I)

We may pass over to a s u b s e q u e n c e w i t h o u t c h a n g i n g the inverse So, assume that

eM,,sj(Pji)~~ 2 -i

for all

i ~ j.

107 oo

There

(~(J))

is a s e q u e n c e

in

M

:=

N

M.

such

that

i=I

vi(J) = v(J) j Pji Choose

some

whenever

clusterpoint

(~i)

i < j.

of

(~(j) )

in

M a, w h e r e

Ma

is the

co

closure

of M

in

~(M(Xi),A(Xi)).

U

From

(I)

follows

that

i=I I~i - ~jPji I 0

v 6 M.

o(M(X),B(X)) .

[]

from probability

theory,

statistical

mechanics

statistics

A few concrete

examples

sidered,

inverse

where

b > 0

space,

from different systems

of

fields

spaces

of

stochastics

of m e a s u r e s

arise

are

con-

in a n a t u r a l

way.

The

first

part

m a y be v i e w e d There,

the

fields further

4.3.1

Let

detail.

and

I and

kernels

Kolmogorov"

general

fields

the

section

processes.

second

4.1.3

Then

is c o n c l u d e d

specification

theorems

and proposition

It

example.

for a g i v e n

and well-known

like de Finetti's.

of Markov

of t h e r e a l

of s t a n d a r d

stochastic

important

of t h e o r e m

boundaries

I be a subset

set

to the m o r e

of M a r k o v

turn out

4.2.1.

projective

by remarks

This

statistical

concerning

a n d an o p e n p r o b l e m .

Entrance

parameter

boundaries

set of G i b b s Two

theorems

sketched

models

a family

of the

consequences

includes

are

entrance

as an i n t r o d u c t i o n

in some

simple

examples

with

structure

is s t u d i e d to be

deals

Borel state

line with

spaces. spaces

Pij" i ~ j

processes

from

Then X,

usual

order

every Markov

process

c a n be d e s c r i b e d X •

to

X•J

((Xi,Fi))i6 I

and

with

time

as a f a m i l y

satisfying

the

"Chapman-

equations

(CK)

Pik = PijPjk '

(see e.g.

Kuznecov

(1981)).

The kernels

P, , rJ

are

called

of

transition

109

probabilities at t i m e

and

j having

can be viewed so,

one

has

If

(X,F)

Pij(x,F) started

in x at t i m e

as an i n v e r s e

to take

the

together

the entrance

preted

as t h e b e g i n n i n g through

system

reversed

with

called

entrance

is i n t e r p r e t e d

which

i. B y

(CK) , e a c h M a r k o v

of s p a c e s

natural

the kernels

boundary

as the p r o b a b i l i t y

of measures

order

P.

is the

of t h e p r o c e s s .

of some a path

trajectory

gets

on

into

process

(if o n e d o e s

I in d e f i n i t i o n inverse

An element

limit,

4.1.1).

it is

x in X is i n t e r -

of the process ~

to h i t F

o r as t h e

X i. A m e a s u r e

~ 6 P(X)

is

iEI the

"initial

distribution"

the entrance X

are

the usual

Theorem spaces From

boundary

4.1.3

that

the entrance

the e x t r e m e

coherent

families

and

e

for M a r k o v exists

remark

I has

a minimum

the probability

6

processes and

we

a then

measures

on

~

measures

Borel

state

a standard

Borel

space.

that

measures,

it c o n s i s t s

of

i.e.

~iPij, i < j}.

P(X/) : ~j =

of p r o b a b i l i t y

standard

learn

of p r o b a b i l i t y

i6I

with

is a g a i n

thereafter,

families

(~i)i6i

{

X

If

distributions.

boundary

and the

X = ex Coherent

is s i m p l y

initial

shows

the p r o o f

of the process.

hence

are

called

entrance

laws.

This

entrance

hension

of

additional versed

the

assumptions

constructed

connected

c a n be

the d u a l

- can be used

of t h e

found

- i.e.

the

boundary

does

(1971).

in D y n k i n ( 1 9 7 8 ) .

chain

to c o n s t r u c t

Dynkin

of t h e

A compreUnder

the process

"exit boundary"

is a l s o D y n k i n ( 1 9 7 8 ) ; S.E.

is a l s o

of transition

b y E.B.

Markov

from the entrance

for that

structure

was

its a s p e c t s

this procedure

Closely (Pi])

of

process

A reference where

some

in t i m e

original

boundary

reversed

Kuznecov

re-

of t h e

process.

(1974)

shows

not work.

the

following

probabilities s e t M(P)

problem.

satisfying

of all M a r k o v

Fix

(CK).

a family

One

processes

P =

is i n t e r e s t e d with

transition

in

110

funct i o n

P, i.e.

all those p r o b a b i l i t y

(A;~i) where

~i

are

of the

past,

(strict)

the

= ~i,~i(A) projections

generated

on

(i.e.

F>i

every the

by all

with

iE

~

I,

F>i

is

on

F~i the

~j, j > i, and

finite

dimensional

X =

U

X.

with

F>i, ~ - a . s .

A6

process),

~j, j ~ i ,

by a l l

future, G_e n e r a t e d

bility measure

for

measures

is

the

o-algebra

o-algebra • ~,x

of the

is the proba-

distributuions

given

by

Piil (x,dXl)Pili2 (x;,dxf)'''~in_lin(Xn_;,dXn), i I