Topology and Borel structure

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TOPOLOGY AND BOWL STRUCTURE

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N0RTH-HOLLAND MATHEMATICS STUDIES

10

Notas de Matematica (51) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Topology and Bore1 Structure Descriptive topology and set theory with applications to functional analysis and measure theory

J. P. R. CHRISTENSEN University of Copenhagen

1974

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

0 NORTH-HOLLAND PUBLISHING COMPANY

- AMSTERDAM - 1974

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.

Library of Congress Catalog Card Number: 73-93099 ISBN North-Holland: Series: 0 7204 2700 2 Volume: 0 7204 2710 x ISBN American Elsevier: 0 444 10608 1

PUBLISHERS: NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD.-LONDON

SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:

AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

PRINTED IN THE NETHERLANDS

MOTTO: All this have I proved by wisdom: I said, I will be wise; but it was far from me. That which is far ofA and exceeding deep, who can find it out? Ecclesiastes, ch. 7, verses 23-24

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TABLE OF CONTENTS Foreword Chapter 0 Introductory remarks, with basic definitions and theorems Chapter 1 Souslin schemes and the Souslin operation. Properties of Souslin sets.

14

Chapter 2 Theorems of separation, Isomorphism and measurable graph theorem. Uniformization theory, standard and universal measurable spaces.

30

Chapter 3 Properties of topologies and Borel structures on function spaces and on spaces of compact and closed subsets of a Hausdorff topological space.

50

Chapter 4 Measurable section and selection theorems with applications to the Effros Borel structure.

78

Chapter 5 Continuity of measurable ‘homomorphisms’. Baire category methods.

85

Chapter 6 Measurability properties of liftings. Some negative and positive results.

105

Chapter 7 Continuity of measurable homomophisms. Measure theoretic methods. A measure theoretic zero set concept in abelian Polish groups.

112

Chapter 8 Miscellaneous exercises, open problems and research programs.

125

References

131

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FOREWORD

We shall discuss in this book selected topics from descriptive topology and set theory,in particular the theory

of analytic spaces and analytic measurable spaces.We shall also examine a number of recent applications of this theory. The main weight w i l l be on these applications and we do not intend to give a rounded and complete coverage of descriptive topology and set theory (a formidable task).A reasonable survey of this area of mathematics can be found in 1163

and [23] ,and in the references given there.The results contained in the present book are increasingly useful to workers in potential theory and probability theory and may also have substantial applications to functional analysis. There has been a considerable revival in the theory of Souslin or analytic sets.Thi8 revival are above all due

to the development in probability theory,more precisely,

to the theory of Markov processes.??utherrnore,the theory of integral representation in convex compact sets led Effros

to the introduction of a particular kind of Bore1 structure (named after Effros) which generalizes some old work of Hausdorff on the topology of compact s e t s to arbitrary closed sets (in a sufficiently ,,nice,,space). The content of the book is a revised version of lecture notes from a course in the subject given by the author in the fall 1972 .The book is designed with the double purpose both to be useful for students as a comparatively easy readable introduction to the field and also helpful for

FOREWORD

4

research workers in this rapidly expanding area of mathematics.Furthermore the book has been an opportunity for the author to publish for the first time several new research results in the field.The exposition should be mainly selfcontained assuming only rudiments of general topology

and set theory (naive set theory).Some of the chapters assume also a rudimentary knowledge of measure theory. It is a pleasure for the author to thank his students for many helpful remarks improving the exposition.With everlasting patience they pointed out many serious errors.In particular this work could not have been done without the encouraging interest and many helpful remarks the author received from stud,scient. Bjsrn Felsager.During the lectures he pointed out several errors and suggested some improvements. I am a l s o thankful to Edward G.Effros and

Gustave

Choquet for encouraging parts of the research results presented in this book.Furthermore I am thankful to my scientific advisor Esben Kehlet whose deep knowledge of the literature was very helpful for me. This book would not have appeared without the encouraging support and interest the author received from Prof. Heinz Bauer and without his recommendation the book would probably not have been accepted by the publlsher.For this I owe him many thanks. Discussions with my Danish colleagues in particular Bent Fuglede,Fleming Topsrae and Hoffmann-Jsrgensen was very stimulating for the research carried out in the book.

CHAPTER 0 INTRODUCTORY REMARKS, WITH BASIC DEFINITIONS AND THEOREMS

We give some basic definitions.The concept of analyticity is defined for topological spaces and measurable spaces, A few fundamental theorems is proved and some problems are discussed.A topological analogue of the Cantor diagonal procedure is developed and applied to an easy proof of the fact that the space of continuous functions on the irrationals is not analytic with the topology of compact convergence.

We shall often concern ourselves with the properties of so-called Borel structures on a set X

.

There is a strong analogy between this concept and the concept topology on a set.Many concepts involving Borel structures have an evident topological analogue.We call the pair

(X,s )

a measurable space or a Borel space .This

should of course not be confused with the concept measure space,which means that a measure on the

6-field is given.

INTRODUCTORY REMARKS

6

One will often have several Borel structures on the same set,the re1rti.m between which are important in an investigation.Natural1y ,all Borel structures lie between

,

a coarsest,the diffuse structxre defined by

8=[X,03

and a finest,the discrete structure with

consisting

3

of the set of all subsets of X.

A measurable space ( X , a ) for all x&A

but

is called separated if

x,ybX (xfy) there exists

A63

such that

y+A ,It is called separable if there is a

sequence

which generates

a

; and it is called

countably separated if there exist a separable subfield which is separated

.

If a topology on X is given in advance,Borel measu.rable without further specification w i l l always mean with respect to the

6-field generated by the open sets.

Concepts like the Borel structure of a subset and the product of Borel structures are defined similarly to the analogous topological concepts.For example,

tr

i6 I (Xi,ai) is defined as the set product equipped with the coarsest

Borel structure that makes all projections measurable,i.e. the Borel structure generated by the cylinders

INTRODUCTORY REMARKS

I '

where

is a finite subset of I and A i d a i

WARNING! One often sees in the litterature on the subject the mistake of without further ceremony setting the

product of Borel spaces defined by a topology equal to the Borel structure generated by the topology of the product space.This latter is,in general,finer,even with finite products.However,this error does not as a rule cause major disasters as the two structures are equal for countable products of

.

,,small,, spaces e.g.

separable metrizable

spaces

Proof:It is clear that a subset A

of I=[O,l]

is separated and separable.Assume conversely that we have a generating sequence is separated.We define

Ane$

for

f:X -9 I

3

,and that

(X,a)

by

The function f, a s a pointwise limit of measurable functions is measurable (it is left to the reader to verify that sums and pointwise limits of sequences of measurable functions are measurable ) AS

.

i ~ ~separates j points in

x ,f

is injective.

INTRODUCTORY REMARKS

8

To show that

f-'

is measurable (with respect to

the Borel structure on the subspace) it suffices to show that f(An)

is measurable with respect to that structure.

But

is precisely the set of'

f(An)

a decimal representation with the n'th

tion equal to f(X)

1.But this shows that

tef(X)

which have

figure of the fracf(k)

is equal to

intersected with a finite union of half-open. inter-

vals.This concludes the proof.

We shall see later that smothness is preserved by surjective measurable mappings with countably separated images. This is a fairly deep theorem.We shall d l s o be able to conclude from some results in the seque1,that whenever a sub-

set of the unit interval is smooth in the subspace Borel structure,it is a projection of a Borel set in IL.Smooth Borel spaces are in many ways analogous to compact Hausdorff topological spaces,for example,a measurable surjective and injective mapping from a smooth Borel space to a countably separated space is automatically an isomorphism (a result which lies considerably deeper than its topological analo-

gue ).After this book was completed the attention of the author was dram to a recent paper (M.Orkin,A Blackwell space which is not analytic,Bull.Acad.Polon. Sci. (20)

9

INTRODUCTORY REMARKS

p.437-438 (1972)) from which it follows that this property is not equivalent to smoothness.

We shall in what follows concern ourselves in particular,among Hausdorff spaces,with analytic Hausdorff topological spaces.

We shall later show that the Borel structure of an analytic topological space is analytic.The converse is false, the real line with the Sorgenfrey topology is

a31

example

of a Hausdorff topological space whose Borel structure is analytic without the topology being analytic (we leave to the reader the verification of this non trivial fact,note that

fIx,q

of the point

I

a

x

> x]

forms a basis for the neighbourhoods

in the Sorgenfrey topology).However it

is possible to prove the deep theorem that the converse is true for metrizable spaces which a r e separable (indeed for all spaces which are homeomorphic with a subset of an analytic space).After this book was completed the attention

INTRODUCTORY REMARKS

10

of the author was drawn to a recent paper

(Z.Frolik,A mea-

surable map with analytic: domain and metrizable range is quotient,Eull.Amer.Math.Soc.

(76),1112-1117,(1970) part C.)

which shows that in the metrizable case separability is implied by the analyticity of the Bore1 structure. There are two reasonable definitions for a Hausdorff topologicai space being preanalytic :

Evidently

i) implies ii) ,but the reverse implica-

tion does not seem to have been proved.

It is left for the reader to show that closed (open) subsets of analytic (Po1ish)spaces are analytic (Polish). We just indicates the proof in the case of an open subset OCQ

of a Polish space

(X,@).Let

d

be a complete

metric on X generating the topology of X ;then the metric D on the set 0 defined by: D(x,y)=d(x,y)+l (dist(x,X\O))-’

-

(dist(y,X\O))-’ I

generates the subspace topology and is complete (we may assume X\O@

).

Most of the spaces one meets in functional analysis are analytic or even standard if they are defined in a ,,reasonable,, way on the basis of suitable ,,nice,,spaces. There are,however,some apparently well-behaved function

INTRODUCTORY REMARKS

11

spaces which rather surprisingly are exceptions to this rule.We give an example of this type at the end of the chapter.

kw

is the space of

> i l l

sequences of positive integers

(natural numbers) bearing the product topology (of the discrete topology on crete topolcgy

k

).Since

is Polish with the dis-

( a suitable metric is d(x,y)=l

if x+y

and d(x,y)=O else) and a countable product of Polish spaces is Polish

(if dm

are complete metrics on the faxtor

a complete metric on the product space) the space

km

is Polish.This is a particular important Polish space,among other reasons because of the following theorem.

Proof:It is sufficient to assume that X is a complete separable metric space with the metric induction on

k

, we

d

. By

can find for each finite ordered

set of positive whole numbers A ( (nl,. ,nk)) such that:

.

(nl,..,nk)

1) X=,UA( (n)) i A( (nl,. ,nk,l

closed sets

I)=&( (nl, ,nk-l,n))

It is almost immediate from 1) that and surjective.This concludes the proof,

0

is continuous

INTRODUCTORY REMARKS

It can be shown (using expansions in continued fractions) that

h@ is homeomorphic with

the space of irrational

numbers in the unit interval (with subspace topology).

Proof:Suppose that such a mapping Define the realvalued function

f(x)=

e (x)(x)+l

f

on

X

8

existed.

by

It follows almost immediately from the assumptions that is a continuous function on such that

f

X .Hence we may choose x o € X

o ( x o ) = f .Then a contradiction is obtained by

inserting in the equality

ax,) (x)=f(x)= e(x) (x)+l

the value x=x0

.This concludes the proof.

13

INTRODUCTORY REMARKS

Proof of corol1ary:We note that if

(X,@) is

a Hausdorff topological space satifying the first axiom of

countability then a function tinuous on

f

defined on

X

is con-

X, if the function has continuous restriction

to every compact subset.This follows from the observation

that the members of a convergent; sequence together with the limit forms a compact set. Now the corollary is an immediate consequence of theorem 0.2

and theorem 0 . 3

in theorem 0.3

.The paving

is the paving of compact sets.

is the continuous

Later on we shall prove that C(N") image of a subset

SCi" .But as we see, S

cannot be choo-

sen analytic. We shall prove later on that if X

is metrizable and

is analytic,then X is a countable union of compact

C(X)

sets.This is a rather deep theorem and does not seem to have a proof as simple as that of the corollary,which can be considered as a special case since

kw

is not a countable

union of compact sets (it is a good exercise to show this using the Baire category theorem ) .

Notes and remarks on chapter 0: The chapter contains some of the basic principles from

[ f b ]and

[dpresented in a manner determined by the preferen-

ces and intentions 09 the author.There is no completely accepted standard terminology.What we have called a smooth

(or analytic) measurable space is the same as a countably separated Blackwell space.Theorem 0.3

is an unpublished

result due to the author.The corollary can be found in [?6] with a proof that is considerably more involved.

CHAPTER 1 SOUSLIN SCHEMES AND THE SOUSLIN OPERATION PROPERTIES OF SOUSLIN SETS

We show how Souslin sets may be charaterized both by means of semicompact pavings and by means of Souslin schemes.A few of the most important properties of the Souslin operation and Souslin sets are stated and proved. One of the most important technical aids f o r the theory we are about to develop has already appeared in chapter 0. This is a Souslin scheme. A multiindex

p = ( ~ ~ ~ . . ~ pis ~ )a finite ordered set

of positive whole numbers (i.e. a mapping of a finite section of r)J

into

h

).In the following P represents the

set of ell multiindexes. P

is naturally countable.

As the union is uncountable we cannot be sure that S(A)

will belong to the 6-field generated by the values

of the Souslin scheme.That this is indeed not always the case will follow from later results.

I5

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

f

Let

be a paving on the set X .By S(

g)

we un-

derstand the set of all Souslin sets that can be defined with the help of a Souslin sheme with values in

$

.

We shall in what follows use from time to time the concept o f a semicompact paving { on a set

x

.

Semicompactness is'evidently preserved if the closure of

f

with respect to countable intersections is taken

.

Less obvious is Theorem ------- l.l:Ef

k

n B v #0

v.1

such that

Bk6 :A

Proof:Let

%=A k

for all

k.Let

h Bv€a

Y:.l

#

is a semicompact paving-:;

VAnk k with Ai&[

and

J

be an ultrafilter on

X

for all k .Then we have in particular

ik , 1 5 ik <, nk .Then k' 0 (since it is a filter set ) and thus by the semiA k e T for some

and so

v compactness of u=9

$

f

00

we conclude

set is contained in

w

nA? # 0 ,but the latter

v=i lV

r\ Bv .This concludes the proof.

Vs1

An important property of semicompact pavings is given by the following theorem.

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

16

Proof:Note t h a t

C

i s t r i v i a l l y correct.We choose n y G f l f ( A n ( ? A . ) . I t i s clear that f - ' ( y ) n ?Aj = n:q J:f J ;Pi fi(f-'(y)nAj) i s non empty,as t h e r e e x i s t s an x e A 0 jzr fiA 'j j. 1 such t h a t f ( x ) = y ,so x e f-' ( y ) r) A . ) .From t h e semicomI:? J (Y, p a c t n e s s of we deduce t h a t t h e r e i s a n x e n(f-' ( y ) f) A j ) n

QD

A(

f

Then

ti)

zI

If

(Xi,

AI*Pi-1 ( A i ) ic is the i ' t h

ti

and t h i s

h-9

concludes t h e -proof.

a r e paved s e t s we d e f i n e t h e product

fi

paving

J.1

0.

y=f (x)c f (0 An)

as a l l s u b s e t s o f

fcI

,where

is finite

icI TT Xi

, AiG

fi

The sum paving

zIfi

i s d e f i n e d as a l l s u b s e t s

+

isI X

t e n as a f i n i t e

(we will use

Xi

f o r t h e , , d i s j o i n t , , union t o

d i s t i n g w i s h from t h e union of t h e s e t s )

i

pi

a r e 6 - f i e l d s and we mean t h e product

, , d i s j o i n t , , union of t h e s e t s

the designation

where

ti ,and

projection.To avoid confusion we w i l l w r i t e

@ i f the iCI 6-field. o f the

o f t h e form

,, d i s j o i n t , ,

which can be w r i t -

union of s e t s

Ai€

ei ,

t r a v e r s e s a f i n i t e s u b s e t of t h e i n d e x s e t

D i r e c t sums

I

.

of Borel s p a c e s can, of c o u r s e , b e d e f i n e d

analogously,and t o avoid confusion we will u s e t h e d e s i g n a t i o n @ when we a r e r e f e r r i n g t o a d i r e c t sum of Borel spaces

.

H

SOUSLIN SCHEMES AND THE SOUSLlN OPERATION

It easily follows from the earlier results that sums and products of semicompact pavings are again semicompact.

This is extremely important in what follows. If

f

is a paving then

f6,

, , f G 6 , fbS

f6

,etc. will

designate the sets of countable unions,countable intersections,countable unions of countable intersections,countable intersections of countable unions, etc. ,all of

f

sets.

A particular important semicompact paving is the pa-

? defined as the set [ne YI nl=PlY..,nk=pkf

ving d o n ,

N(P)=

is a multiindex ,together with

of sets of the form where p=(p,,

iip",0(

.

.,pk)

The following results shows that the Souslin operation can be expressed also using semicompact pavings.Since some proofs is much easier using Souslin schemes,it is important

to have both equivalent definitions. a set and f a paving on --------- X be --------------------X which is closed with respect to finite intersections ........................................................and contains both the empty set and X.Then the following con................................ ....................... ditions are equivalent for A G X : .......................... Theorem 1.3:Let

1) A C S ( t )

2) A

set in ------

~xi".

.

is the projection on X of a -------------------

(fx&&

3) There, exists a semicompact paving T o n a set ................................. --------

Y such that A is the projection on X of ----a ( f x r ) , g --------.................... set in X S Y ------

.

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

18

P r o o f : 1) ==$ 2 ) .Let A=S(B) ,where B

Souslin scheme.We

define D((nl,..,nk))

B((nl,..,nk)).It

is an

as equal to

.

is clear that S(D)=S(B)=A og

Now set C= utwT\D((n,,.. ,nk))$N( (nl,.. ,nk))CXrBm naI K.1 At this point we need a small lemma.

00

P r o o f of the 1emma:If

xEA((n1,..,nk))SuA(p)

9

Conversely,if x

Q

n=(nl,.. ,nk,..)c

~&

.

x Q fl A( (nl,. ,nk)) kd

.

then

.

for all k .This proves

(UA(p)),then we can clearly find

*l?&

such that xgA( (nl,.. ,nk)) f o r all k ,

as the assumptions imply that if

xpA(p)

and

xgA(q)

,

then one of the multiindexes is a segment of the other. This concludes the proof of the lemma. A s the Souslin scheme D((nl,..,nk))$N((nl,..,nk))

evidently fulfills the conditions in the lemma,we have

n ( u D( (nl,..,nk))%N( (nl,..,nk))> x:f ntNR Hence C e (frdks and the projection 00

C=

on X 2)

p(C) of C

is obviously precisely A .

==I$ 3 )

3 ) === 1)

. Clear.

. The assumptions imply that there exist

A(n,k)

OOOOQ

and B(n,k)

such that

B= 0 uA(n,k))(B(n,k) n.1 kci

We may assume that the paving

and

A=p(B).

is also closed under fini-

te intersections and contain f Y , 0 3 ,

19

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

We now define Souslin schemes 2 and B by n x A ( (nl, ,nk))=flA( j ,n. ) and similarly vf J B((n,,..,n,))=&B(j,n.) .It is easy to see that we have 4

o ,

A

J

S.1

B= clawby.. ,nk)) x @ A

nq

w

Indeed ,let x e B that

ice (nl,.

,nk))

n

be the smallest n such j h x € nP((n,,.,nk))ltB((nl,.,nk)).

and

xeA(j,n)ZB(j,n),then

The other inclusion is even easier to prove.We have now expressed the

6s-operation on the product sets by a Sous-

lin scheme on the product set. Select a fixed n=(nl ,..,nk,..)6 N

‘W

r

h

Ck=A((nl,..,nk))SB((nl,..,nk)).Let

4=[$x]xB

defined by

semicompact paving.As

We obtain:

be the paving on X x Y

, Be$-3.

xQ X

p-’(xMCkt

$

4

is evidently a

,we can apply theorem 1.2.

00

00

60

1

%

.We set

9nk) )

P(l?,C,)=~P(ck)=n(~nl, where A”(p) is empty if B(p)

is empty,and is otherwise

h

equal to

A(p) .It is now clear that A=p(B)=S(A*)6

4)==+3).If

AhS(f)t%),then

there exist a set Z with

f such that A is the ce ( ( f r $ t ) x # L~ .NOW

a semicompact paving onto XIY

of a

semicompact on Y X Z onto

x

of a

and

(f~”dp),~

S ( f ).

p(A)

projection q(C) %=Fx$’is

is thus the projection

set in X x ( Y A Z )

.

3)==+ 4). In the proof of 3)==$ 1) ,we have already expressed a 65 -operation on product sets by means of a Souslin scheme.This completes the proof of theorem 103

.

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

20

Proof: 1) .Let and

B are

CC-

8

and

t h e S o u s l i n scheme

D( (n1, for

D

N=S(A)

by

D((n))=XXB((n))

k 1 1 .It i s almost t r i v i a l t h a t

,

(S(f

,"g)6s,hence

We may assume t h a t and c o n t a i n s

such t h a t

m d so

due t o

4

P

A=p(B)

4)

Using

3)

3) Let

.

com2act pavings g o o

Cn

=

EL€

with a semi-

Y

, where

1Y903 ,Hence t h e paving f x $

An€ S ( $ )

.

lies in

R

3))66 ,C S ( t r g6A )-

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

of theorem

t o prove

.

S(&)c:S(S(b))

1) Be ( S ( f ) X S (

t h e s e c o n d i t i o n s and t h e r e f o r e

and

S(D)=N%M

A € S ( S ( f )).Then t h e r e e x i s t s a s e t

%

and

9n2k+l) )=A( ( n 2 , , n z k ) ) X B( ( n l , 9n2k-l))

,n2k) )=D( ( n l ,

compact paving

A

S o u s l i n schemes respectively.We d e f i n e

2 ) It i s c l e a r t h a t f c _ S ( f )

Let

M=S(B) ,where

and

1.3

3)

a l s o fulfills

imply

B 6 S( f X

we s e e t h a t i t now only remains

be a sequence.There e x i s t hr

#,

2).

such t h a t

%=p(Cn)

Yn

with semi-

,where

fl V E ~ X F ~

Ka4 -7

f , I?&,€ n

.We c o n s i d e r

Y=vYn

n

equipped

21

SOUSLIN SCHEMES A N D THE SOUSLlN OPERATION

with the product paving

flh

thus belongs to the OcloDOc

D= O n u E n nrl

Wrnq

Thus nAn=p(D) n-4

X

D&

(5

.

r=+c

The union is a little easier.Let the situation be as above but define Y=+Yn the

,,disjoint

,,

n

and

n

union of the Cm

.D is defined as

.Evidently

, we

have

we was able to shift the order of the union and intersection because the union is ,,disjoint,, .This finishes the proof. It is now clear that if X \ E of

f

sets for every

is a countable union

E€f,then 6 ( f ) C S ( % ) ( 6 ( f ) of

a paving is the generated Bore1 structure).That this inclusion can be proper cannot be proved until later on. As not a l l

6 -fields are closed with respect to the

Souslin operation,it is useful to have a

sufficient con-

dition for this to be true.The following theorem is sufficient for most purposes. If

3

is a 6-field on a set X ,we call a set A

hereditary if all subsets of A therefore depends upon

3

are

8

sets (the concept

).It is very easy

t o see that

any countable union of hereditary sets is hereditary and any subset of an hereditary set is of course hereditary (the hereditary sets forms a 6-ideal ).If

M C X is

any subset then a 3-hull ( o r simply hull if no confusion is likely to arise) is a set for any

Aea with M G A

B e 3 such that M g B

we have

B\A

and

is hereditary.

SOUSLIN SCHEMIIS AND THE SOUSLIN OPERATION

22

Since a 3-hull is unique modulo hereditary sets we also speak about the 8-hull of a set

M S X .La.teron we

shall develop tools by means of which it may easily be shown that in the special case of the unit interval with the Bore1 structure generated by the usual topology,the hereditary sets are just the countable subsets and only measurable condition in the following theo-

subsets has a hull.The

rem is therefore very strong.

Proof:Let A and M=S(A).Let

be a

R

lin scheme A: A

be a monotone

$-hull of M.We modify the Sous-

rn

V,.OA( “CN

A ( (nl,.,nk)) =

3 Souslin scheme

3’9

(nl,.,nk,ml,.,mj))

We select a 3-hull of 2 ( p ) 4

by obtain a Souslin scheme B

.

for each multiindex and there-

.We can evidently obtain

without difficulty that: A

1) B

is monotone

2) Bh(p)C A ( p )

.

for each mdtiindex p 6 P

.

Now define the Souslin scheme D and the set D_ by:

We now have :

23

SOUSLIN SCH1.MES AND THE SOUSLIN OPERATION

Indeed, 2) implies S ( $ ) S S(A)=M .If X Q B

and

it can therefore not be true that there exists an m that xg%((m))

and if

A

xbB((nl,.,nk))

A

such that x e B( (nl,.,nk ,m))

an m

.

x E S(hB)GM

x#M such

there exists

,as that would imply

But that one of these countably many conditions is not satisfied means precisely that x belongs to the union. It is an immediate consequence of the definition of

2

that we have:

From this we may conclude that every Do

is hereditary hence

thus M=B\ ( B \ M)e Let

8

3)

implies that

D(p) set and B\M€

3 ,and

.This concludes the proof.

( X , a ) be a measurable space and

u:$-$[O,@c

a countably additive set function (i.e. a finite measure). $(u)

is defir-ed as the set of all subsets

for which there exists

B1CA
with

A

of X

B1,B2Ca

'and

u(B2\ B1)=O. is precisely those subsets which can be repre-

a(u)

sented as the union o f a set B 6 (262with u(C)=O

.

The universal completion

2

&

and a subset of a set of

2

is defined as:

u

where the intersection is taken over all finite measures u

If M C X rable hull of

is an arbitrary set we define the

u-measu-

M as a set

and

with minimal u-measure. B null sets

.

BE

3

containing M

is of course unique modulo

u

.

SOUSLIN SCHEMES AND THE SOllSLlN OPEKATION

24

It f o l l o w s e a s i l y from theorem 1 . 5

au

therefore

a(u)

that

and

a r e closed w i t h r e s p e c t t o t h e S o u s l i n

operation. Let

(X,

0)

be a Hausdorff t o p o l o g i c a l space .Ag X (in X ) if

of t h e first category

A

i s contained i n a

countable union o f nowhere dense c l o s e d sets.The

is the 6-field of subsets A Oe@

such t h a t

of

X

sets).Of cc.urse t h e

RP-field

f o r which t h e r e e x i s t

i s of t h e first

AaO=(A\ 0) V ( O \ A )

category (observe t h a t

is

i s unique modulo first c a t e g o r y

0

BP-field c o n t a i n s t h e Bore1

structure

generated by t h e topology. A reasonable t o p o l o g i c a l analogue o f u n i v e r s a l measu-

r a b i l i t y i s t h e f o l l o w i n g concept.Let r a b l e space. The s e t

S C_ X

r a b l e if f o r any mapping

be a measu-

(X,a)

i s c a l l ed u n i v e r s s , l l y f:Y

BP-measu-

-9 X from a Hausdorff topolo-

( Y , @ ) which i s measurable from 6 ( 6 )

g i c a l space we have t h a t

is

f-'(S)

BP-measurable

in

BP-measurable

8

.It w i l l

Y

follow from t h e n e x t theorem t h a t t h e 6 - f i e l d

to

of u n i v e r s a l l y

s e t s i s a l s o c l o s e d under t h e S o u s l i n opera-

tion. Let

AGX

be a n a r b i t r a r y s u b s e t of t h e Hausdorff

t o p o l o g i c a l space x€X

(X,

8 ) and

define

A*

as t h e s e t of

w i t h t h e p r o p e r t y that every neighbourhood

intersects

A

U

of

x

i n a s e t which i s n o t of t h e first c a t e g o r y

( i t i s t h e n s a i d t o be of t h e second c a t e g o r y ) . A* d e n t l y closed m.d hence

BP-measurable.

i s evi-

2s

SOUSLlN SCHEMES AND THE SOUSLIN OPERATION

Proof: Let

be a s e t such t h a t every p o i n t

McX

has a neighbourhood

xeX

category ( i , e . M* = 0 ).Let

Ui

of t h e f i r s t

MQU

with

U

be a maximal f a m i l y

(iCI)

( u s e Zorns lerma) o f pairwise d i s j o i r - t open s e t s w i t h

ieI

of t h e f i r s t category f o r a l l Let

O=.OUi

lrr

t e r i o r such that

be c l o s e d s e t s w i t h empty in-

and

FYGcl(Ui)

o f t h e maximality o f t h e empty i n t e r i o r . F n = c l (

u

n , indeed, Fn\ 0

and

UinMC

(ieI)

Ui

~ C I

for all

.

)

, X\O

Qo

UFY . I n w i e w

11.1

must have an

a l s o has an empty i n t e r i o r

i s a closed s e t w i t h empty i n t e -

r i o r , s o i f i n t ( F n ) # O was true,we should have f o r some

i I .But

,M

PD

MSuFnU(X\O) n73

X\ A* ,which con-

A \ A* .The above reasoning shows t h a t

t h e f i r s t category r e l a t i v e t o

X\A*

.

A \ A*

i s of

But a s e t which i s o f t h e f i r s t category r e l a t i v e t o

an open s e t which c o n t a i n s i t i s obviously o f t h e f i r s t category r e l a t i v e t o t h e whole space. We remark t h a t a s e t only i f

.

i s o f t h e first category.

We now apply t h i s t o t h e open s e t tains

int(p)Ol,$43

i n t (Fn) /1 Ui C FFO Uid, FY ,which e v i d e n t l y

leads us t o a conflict.It is c l e a r t h a t Thus

MOUi

C*\C

C g X

i s BP-measurable i f and

i s o f t h e first category.Indeed i f t h i s

26

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

is the case

then

C*o C= (C \ C*)U(C*\ C)

category,hence the set int( C*)4 C

C_

( c* \

is of the first

c) u ( c* \

is of the first category.Conversely assume an open set 0 such that C A O Then C* = 0"

and hence

shows that C * \ C

C*

that there exist

is of the first category.

C C_ (O*\O)

C=O*

\

int( C*) )

.

is of the first category

U( C A

0 ) which

Next we remark that a set of first category obviously is hereditary relative to the

BP-field ;the converse is not

true as a discrete topological space shows.

Let now C G X with M L C

C BP-measurable.We

and

then have (A*IJ (A\ A*) ) \ C

C_

(A* \ C) I.) (A\ A*) C

(C*

\

C) d (A\ A * ) ,

and the last, set is of first category according to the above

is a hull of A uld

remarks.This shows that A*U(A\A*) thereby finishes the proof.

A s an important application of theorem

we find that the

1.5

and

1.6

BP-field is closed with respect to the

the Souslin operation.In a measurable space the 6-field of universally BP-measurable sets is also closed under the Souslin operation. Let

(X,9, u) be a measure space with a finite positive

measure.For any subset A c X

P(A) = inf iu(B)I

BE3

WE

,

define

ACB]

.

It is almost immediate to see that this is a set function with the properties W

for any increasing sequence of subsets of y(Bn)=

p (pp%, n**

for any decreasing sequence of

3

sets.

X

and

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

21

Since this set function (outer measure) is also increasing the following theorem may be applied and yield an alternative proof of the fact that the universally measurable sets are closed under the Souslin operation.

Proof:Let A be equal to Souslin scheme with values in

f

S(E) ,where E is a

.We can assume without

w has its values in

l o s s of generality that

A l s o we may assume that E

I+,l].

is monotone,Now let e > 0

be given,then by induction we choose a sequence of positive

whole numbers n P v(Ak) 2

in a manner ensuring that fox any k:

v(A)-e 9

cc,

where A k = u [

r\E((ml,.,%))lmc

V:f

OI

,1 S mi <, n. 1 f o r i=l,.,k]

.

SOUSLIN SCHEMFS AND THE SOUSLIN OPERATION

28

That t h i s can be achieved i s an easy consequence o f condition 2 ) .For the s e t ca

B= G u $ E ( (ml,.

B Bk=

and

c, U

3)

P(B) 2

that

V(A1-e

,

i s the i n t e r s e c t i o n o f t h e s e t s ‘V

, m v ) ) I m e N ,1 S mi <_ ni, i=l,. ,v]

u [ E ( (ml,

Bk contains Ak

(we a l s o make use of condition 1) ).

B :

We a l s o have another r e p r e s e n t a t i o n o f OD

4

1

‘V

, m v ) ) Ima N ,1 5 mi ini ,i=l,, v

i t follows by the use o f since

defined by

B

‘lo

B = u f / vI= Ei ( ( m l , . , ~ ) ) l m t N

,1 5 mi L n i , i = 1 , 2

,...3 .

The decisive point o f the p r o o f i s the v e r i f i c a t i o n of

h

6

B IB .Now assume t h a t

B = B .Evidently

m P e Nvp

m?

ni

1

,.,m:

)). P ,we may assume (by considering a s u i t a b l e sub-

can choose a sequence As

xeB .Then we

such t h a t

xsE((m:

sequence i n s t e a d i f necessary) t h a t t h e r e e x i s t s , f o r each i

a

ti

m:

with

x€E((tl,,,tk))

-+

ti .But i t then f o l l o w s c l e a r l y t h a t

for all

k.

From the f i r s t r e p r e s e n t a t i o n of

B

we can see t h a t

B 6 fg ,while the second r e p r e s e n t a t i o n shows t h a t As

e

>

0

,

B4A

.

was a r b i t r a r y t h i s concludes t h e p r o o f .

Motes and comments on chapter 1: Most of the p r o o f s i n t h i s chapter a r e taken f r o m

[f61.

we hope t h a t we succeeded i n simplifying them t o some extent. The r e s u l t s a r e the harvest o f about arch a r i s i n g out o f

Lebesgue‘s

t h a t the p r o j e c t i o n of a B o r e l

50

years

rese-

, , p r o o f , , f o r the ,,theorem,, set

in

R2

onto one o f the

axes i s a Borel set.There proved t o be an e r r o r i n Lebegue’s

29

SOUSLIN SCHEMES AND THE SOUSLIN OPERATION

argument,and attempts to repair the proof ended with the discovery that Lebesgue’s

, ,theorem,,was

incorrect

(as we shall see in the next chapter).The Souslin scheme and the Souslin operation was invented in the course o f these investigations (see[Mand

L%])

.The decisive breakthrough

occured in 1917. was discovered by

The fundamental theorem 1.5 rajc-Marczewski ( see[&$

in 1929

.

Szpil-

Theorem 1.7 (there are a number of variants) is due

to

Gustave Choquet.When it was discovered in 1952 (see[+]) it represented a decisive step forward in potential. theory

pi).It also has important applications

(see also [ 5 ] , [ 6 ] ,

in the field of stochastic processes to show measurability

of certain ,,stopping times,,. Compactness arguments appear throughout the theory. The axiom of choice therefore assumes a natural importance in everything that we have done and plan to do.In general, the presentation (like most of today’s mathematics) is based on naive set theory.We shall however come across examples

of important theorems equivalent with set-theoretical axioms of a similar status as the continuum hypothesis.We shall point out any such case specifically, The

,,disjoint,,union

+

Xi is constructed by ear-

marking the elements of each Xi before forming the union. This can be expressed precisely by defining

CHAPTER 2 THEOREMS OF SEPARATION, ISOMORPHISM A N D MEASURABLE GRAPH THEOREM. UNIFORMIZATION THEORY, STANDARD AND UNIVERSAL MEASURABLE SPACES

We prove the fundamental theorems giving conditions for measurability of mappings between smooth Borel spaces. The two extremes of isomorphism types of analytic measurable spaces are defined and studied.

We start by defining some measurable spaces for which we can develop a reasonable theory.

We assumed in the definition that eG8.It does indeed seem unlikely that this condition could be omitted or that it is automatically fulfilled.But if we assume that ( X , a )

is countably separated (i.e. there exist Bnt

5 Bn{

8

such that

separates points in X) ,then we shall l a t e r prove

that f G $

is automatically satisfied if

is semicompact.If

9s S( f

)

and

f

X is uncountable,we shall prove that

( X , a ) ,given the above assumptions,is automatically Borel-

isomorphic with the unit interval I=lO,l] ,when the latter

THEOREMS OF SEPARATION, ETC.

31

is endowed with the natural Bore1 structure (generated by the usual topology).This,however,requires more theory than we can cover in this chapter.

It turns out that the assumption of countable separation plays a role analogous to Hausdorff’s separation axiom in topology.The following theorem is a major reason for the assumption’s importance. Theorem ------- 2.1:The following conditions are equivalent for a measurable seace ....................... ---

(X,3 ) :

----------- --- ------

1) ( X , a ) is countablg seearated. 2) The ------diagonal ---- D= i ( x , x )

I

x 6 X)

-is B @ a mea---

n

Proof: 1) ==+ 3 ) .Let

Brie$

be a separating

sequence.Then we have the following representation of the

..

.

and this shows that the graph is

3 ) ==$2)

a@ameasurable.

.It suffices to consider as f

mapping of X

onto X

.

the identical

2) ==? 1).We select sequences An,BnE ,$ such that

D e 6([AnX

Bn$ ).It is then clear that fa,Bn] is a

32

I H I OKLMS 01 SLPARATION. FTC

countable separating subset of

3 .This concludes the p r o o f .

The next theorem is vital for the isomorphism theorems.

Proof:It is easy to see that we can content ourselves with

showing that two disjoint sets

can be separated by disjoint sets

B1,B2Ea

( X , a ) is semicompact.Let

1) Assume

paving with

3 S S( -& )

fG3 be

,We may assume that

f

a semicompact is closed under

finite intersections and contains iX,0) .Choose monotone Souslin schemes n

A;=S( A; )

A

A1

and

A

A2 with values in f

such that

i=1,2 .Then we have

Suppose that A1

and A2

cannot be separated by

sets.Then it is easy to see that there exist v

and

u

such that

..

-

cannot be separated by find

m,m’6

(B

im such that

sets either.By induction we can , f o r all

p

,

THI OK1 M S 01 SLPARATION. LTC

3

cannot be separated by n

sets,in particular

and

A1( (ml,. ,mp))

.

A

33

A2( (mi,. ,mb))

cannot be separated by

$

sets.But these two sets are

3

sets,so we can deduce that

A1( ( m y for any

m

4;) )+

,ap)) A2((mi,

p .The semicompactness of

f

now implies

U

(~,A1((m1,.*,mp)))n (p2((m;,**,mp)#0 in conflict with A1/)A2 #0

(X,$) is a countably separated Black-

Assume now that well space.Let f:Y -+X

and

.

(Y,4) be a semicompact measurable space surjective and measurable

(f
semicompact

paving closed under finite intersections and containing w , 0 3 such that

A sS(f)

).As in the proof for theorem 0.1

,

we can construct with the help of a separating sequence Bn€P an injective mapping

g: X -$I =cO,lI (we shall

only later on be able to prove that this will automatically

be a Borel isomorphism onto the image).We see now that h=g o f

is measurable from Y

with

S2 = 0 .Theorem 2.1

S1fl

Gr(h)E (where

9

I .Let S1,S26

S(3) ,

states that

8@9

is the Borel field on I ) .In particular we have

Gr(h) 45 since

to

s(&Ab,

4 @ $Q

S(&E

f)

because

countable unions and intersections

S(,$xg)

is closed under

and contains both pro-

duct sets and their complements (which are unions of at most

3

product sets ).Now

f-’(Si)

i=1,2

belongs to

S(c&),so:

THEOREMS OF SEPARATION, ETC.

34

Theorem 1.3 i=1,2 .As

can now be used to show that g(Si)6S(g)

(1,g)evidently is a

sets C1,C2

space,we can find disjoint i=1,2. .But then we have well as

g-l(C,)n

semicompact measurable

g-1 (Ci)zSi and

g-’(C2)=E

with g(Si)C_Ci g-’(Ci)83 ,as

.This concludes the proof.

We can now prove the measurable graph theorem.This result is one of the most important in fhis chapter.Together with the projection theorem and the separation theorem it forms the most often used tool to show measurability.

P r o o f :Assume first that

and that

(Y,d)is a

(X,3 ) is semicompact

countably separated Blackwell space.

1) =+ 2 ) = 3 3 ) has already been

shown by theorem 2.1

.

THEOREMS OF SEPARATION, ETC.

3)

+ 4) .Let

and

f.

35

be a semicompact measurable space

(Z,$)

a semicompact paving with

gc

furthermore that the paving

f

sections and contains )Z,0)

.Let g:Z -+Y

measurable mapping.We define g((x,z))=(x,g(z)).It

S(

f

) ,suppose

is closed under finite inter-

2:

be a surjective

X F Z --+ X J Y

is easy to see that

is

by

$@%-+$@A

measurable. Then we have z-’(Gr(f))O

(xrg-’(A))€ s ( a @ ? ) = s ( $ A $ ) = s ( a X f

Note that the equality S ( a @ f ) , = S ( $ x g )

1-

is obtained as

in the last part of the proof of theorem 2.2 .As the projection of the above set onto X theorem 1.3

f-’(A)€S(@,by

.

is

f-l(A),we find

4) =+l) f o l l o w s immediately from the separation theorem as

f-’(A)

and

if

A E ~ .

The second Cam

f-’(Y\A)

are disjoint sets in S ( a )

and the last part of the theorem is pro-

ved in a very similar way.

THEOKEMS 01: SICPAHATION, ETC

36

Proof:

f(X)

is a countably separated Blackwell

space when it has the subspace Borel structure.The theorems now follows from the preceding ones, It follows from this theorem that a point-separating sequence BnE$

in a countably separated Blackwell space

( k , a ) generates &) ( in particular the B o r e l structure is separable ! ).Recently attention to a paper

J.E.Jayne has drawn the authors (M.Orkin: A Blackwell space which is

not analytic,Bull.Acad.Polon.Sci. (20) 437-438 (1972)) which shows that this property does not characterize countably separated Blackwell spaces between countably separated measurable spaces,but it may be that it characterizes them between coanalytic measurable spaces ( a coanalytic measurable space is a space which is isomorphic with the complement

OT

a Souslin set in

Let

1=p,1]) .

3) be a countably separated Blackwell space,

(X,

We know that there exist an injective measurable mapping

f: X -9 I and from the isomorphism theorems it follows that

f is automatically a Borel isomorphism of

onto f(X)zI ; a l s o it follows that

f(X)

(X,a )

is a Souslin

set in I (with respect to the paving of compact sets or equivalent with respect to the paving of measurable sets). It follows from theorem 1.3 in I = L O , l ] sets in Ix

that the Souslin sets

is precisely the projections on I of Borel

k“

.Since the continued fraction representation

defines a homeomorphism of

f”

onto the irrationals of the

unit interval and since the irrationals f o r m s a Ga in I

,

we see that the projections onto I of Borel sets in I? Nm

TH€
37

are just projections onto I of Borel sets in I2 and conversely.Since it follows from later results in this chapter that every Souslin subset of I is Blackwell in the subspace Borel structure (see the remarks following theorem 2. G ) the countably separated Blackwell spaces are precisely the smooth Borel spaces. Let (X,@) be an analytic topological space.Then X2 with the product topology is also analytic and hence a Lindelaf space ( A Lindelaf space is a Hausdorff space with

the property that for every collection of open sets there exists a countable subcollection with the same union.This property,like compactness,is preserved by surjective continuous mappings.It is easy to show that a topological space with a countable base is Lindelaf.In particular separable metrizable spaces are Lindelaf).It is a simple consequence of the Lindelaf property that the product Borel structure on X2 is precisely the Borel structure generated by the product topology.In particular the diagonal in X2

is mea-

surable with respect to the. product Borel structure.It now follows from earlier statements that the Borel structure of X is analytic.

We shall now investigate more closely Borel isomorphism types of analytic measurable spaces.We need among other things the following theorem for this purpose.

THEOREMS OF SEPARATION, ETC.

38

P r o o f : We define h(x)=f(x)

M=z((g o f)k(X)\ ((g

0

R.0

and h(x)=g-'(x)

f)ko g)(Y))

on X\M=Mc

mapping).It is easy to see that morphism of X

on the set

( (g o f)'

h

is the identity

is indeed a Borel iso-

onto Y .This concludes the proof.

An analytic Borel space (X,3 ) is called universal if there exists for any other analytic Borel space set

Be3

which is Borel isomorphic to

immediate consequence of theorem 2.5

(Y,&)

(Y,t$).It

a is an

that there exist at

most one universal analytic Borel space (modulo isomorphism).

But the existence of such a space is ,of course,not evident. We postpone the proof of its existence to later on.

We shall now prove some surprising results which roughly speaking show that most of

the important topological spaces

occuring in functional analysis have the same Borel structure.We say that an analytic measurable space is standard if it is either countable (and hence discrete) or Borel isomorphic with

I=[O,lJ

endowed with the natural Borel

structure (generated by the usual topology).A Hausdorff topological space is called standard if it is the continuous injective image of a Polish space.From the fol-lowingtheorem

we may in particular conclude that the Borel structure of a standard topological space is standard and that the converse holds for separable metrizable spaces.As a very special case all uncountable Polish spaces are Borel isomorphic to the unit interval !

THEOREMS OF SEPARATION, ETC.

39

P r o o f : First we consider the standard case of the

theorem.We prove first Lemma ----- 1 :If -- f:X

-+ Y

is an injective continuous

rable subset of Y. ---------------

A1,A2 be disjoint subsets

P r o o f o f lemma 1:Let

of Y which are analytic in the subspace topology.Then there exist B1,B2t a ( Y ) by the topology Indeed,let

(the Borel structure generated

Blfl B2 =0 and

) such that

g:h@-+A1

and

'OD

h:N

- jA2

AiC Bi

i=1,2.

be continuous and

THEOKEMS 01: StPARATION, I
40

surjective.If the suggested separation was not possible, we would be able to find (by induction)

.

g ( N ( (nl,. ,nk)))

that

.

n,m€kO such

h(N( (ml,. ,mk)) )

and

could not

be separated by Borel sets for any k (note that the argument is very similar to the proof of theorem

2.2).Since

Y is Hausdorff we may find disjoint open sets in Y 03

with

g

and

g(n)bU h

we find

h(N((ml,.,mk))) c V

and

Now let

and

oneB

in X .Then f(On)

U and V

h(m)BV .Using the continuity k

such that

g(N((nl,.,nk)))sU

This is obviously a contradiction.

be a countable base for the topology and

f(X\On)

are disjoint analytic

sets in Y,so we can choose Bn€ & Y )

such that f(0,)

is contained in Bn and f(X\On) 0 B,=0 coarsest topology on Y finer then

.Now let $) be the such that Bn is 3

4

d

open for all n.The open sets in the

?

topology are pre-

cisely the subsets of Y which can be written as a countable union of

finite intersections of

It is now clear that all

? open sets

Bn’s

f

3 sets

.

are measurable in

.It is equally evi-

the Borel structure generated by dent that

and

is a homeomorphism of

X

onto a subset of

h

(Y,?),The proof of lemma 1 is therefore finished as soon as we have proved :

Proof of lemma 2:Let

d

be a complete metric on

THEORIMS OF SEPARATION. ETC.

X

41

which i s compatible with t h e (subspace) topology.For

each

xbX,choose an open s e t

On(x)

in

such t h a t

Y

XQ

On(X)

0

diam(On(x)QX ) < l / n

and

and assume t h a t

On(x)

yEM.Then t h e r e e x i s t

y e O n ( x n ) , f o r each as

.Define

n .We see that

xnc X

and

M a On, n.9

with

d(xn,xm)sl/n

+

l/m

,

y € On(xn) 4 Om(xm)#Oand (because of X ' s d e n s i t y ) t h u s

On(xn)40m(xm)/)X# 0 .Now l e t property,we have t h a t

y=x,

xn

-+ x w

.But t h e n

.From t h e Hausdorff M=X

and t h i s con-

cludes t h e proof of both lemma 1 and lemma 2 It i s now c l e a r t h a t

2)

+3)

.

.To prove t h a t

3)

= j 2 ) ,

we w i l l start by proving :

Proof o f lemma 3 :Let

& be

the smallest collec-

t i o n o f t h e type described i n t h e 1emma.Let l e c t i o n o f those A,B€$

;

since

A & d f o r which

that

49

AY B €

(X\A)=AC&&.Let

now

A C , B C e d we conclude t h a t

AUB=(ACI) B ) U ( A /)BC)V(A/) B ) € d

implies

be t h e col-

,hence

(AOB)c=Acfl BC

.By i n d u c t i o n i t i s now e a s i l y s e e n

i s s t a b l e under f i n i t e unions.Hence from t h e e q u a t i o n

i t i s e a s i l y seen t h a t

9

i s c l o s e d with r e s p e c t t o countable

unions.The lemma i s an immediate consequence.

THEOREMS OF SEPARATION, LX"T

42

It remains to be shown that the paving consisting of sets which are standard in the subspace topology is closed with respect to countable intersections and countable dis-

h C X be standard in the subspace topolo-

joint unions.Let

fn:Pn -+X

gy and let

be injective continuous mappings 09

from the Polish space Pn onto An .Set P=7[Pn 1:)

1.

D= (p,)

d

space P

and

,D is closed in the Polish

PI Vv:fl(pl)=fv(pv)j

and is therefore itself Polish.The mapping

M

f:D -->nAn h'r

defined by

f((pn))=fl(pl)

An's

jective and continuous.If the

is surjective ,in-

are pairwise disjoint,

we can obtain an obvious injective continuous mapping from the disjoint topological sum of

the spaces Pn onto the

union of the ph ' s .This concludes the proof of f:X -+Y

l)==)3).Let

3)=+ 2).

an injective measurable mapping

be

from the Polish space into the countably separated measurable space

%&&

and

defined by every ph

be-

(Y,d).We select a separating sequence on Y

consider the topology

ing both open and closed.This topology is evidently separable and metrizable,let d with

(y,

d)

extension of the metric Now

Y compatible

T A (Y1)- yh(y2 1/lOn n be the completion of Y with the natural

(we may define d(y1,y2

and let

)=eI

be a metric on

Gr(f)G X % ' Y

d

VCf

which is also denoted by

d

.

is a Bore1 measurable subset (since f

is measurable ),thus standard in the subspace topology. As

f(X)cY

is the injective continuous image of Gr(f)

by the projection onto Y,we deduce

f(X)ca(Y),and hence

f(X) € 6(fAn$) .This shows in particular that 1)=j3 ) The proof of the ,,standard,,part of theorem 2.6

.

THEOREMS OF SEPARATION, ETC.

43

will now be completed i f we can show that any uncountable Polish space is Borel isomorphic to the unit interval.Let

(X,@ )

be an uncountable Polish space and \

base for the topology o f

X .Let K=[O,l\N

On a countable ; this particu-

lar space will play a fundamental role in a good deal of the following material.With the usual product topology and product group structure,K is a compact metrizable abelian group ; it may be convenient occassionally to identify in elements of K

the natural way

with subsets of the natu-

ral numbers.

We define a mapping f : X -$K

by

f(x)(n)=x

It follows from the above results that f

X onto a Borel subset o f

isomorphism o f

T o every finite ordered set

1 ’s

(x) On is a Borel

K

e=(ely..,ek)

.

.

of 0’s or

(let us call this a Cantor index),we can find by in-

duction closed uncountable sets F(e) in X such that i) F((ely..,ek,l))2F((ely..

.

yek)) for all k

ii) F( (el,. ,ek) ) 0 F( (fly.., f k ) )=0 iii) diam(F( (el,.. ,ek)))S

We now define

g:K -3 X by

It is clear that g theorem 2.5

l/k

is

.

g(k)=

if

e#f

n F( (?((I),.., W ) ) . CQ

V.9

injective and continuous.By using

we now can complete the proof of the standard

part o f theorem 2.6 , It is obvious that we by a slight modification of the above argument may obtain that every uncountable analytic topological space contains a subset homeomorphic to the Cantor group K

.

The proof of the

, ,analytic,, part

of theorem 2.6

is

rHI OH1 MS 01 SI PAKATION. FTC

44

very similar.Severa1 times in the sequel we shall make use of the fact (which is a little stronger then 2 ) = 3 3 ) ) that if f:he-j X

is a continuous mapping from

Hausdorff topological space

G@

into the

(X,0 ) then we have

(10

f ( N )=nqaflcl(f(N((nl,.. ,nk)))).In X"

particular every subset

of a Hausdorff space which is analytic in the subspace topology is necessarily BP-measurable (remember theorem 1.6).

From the above results it follows that any smooth Borel space is Blackwell .If a smooth Borel space is uncountable it contains an uncountable standard (hence measurable) subset,

Proof:Every analytic measurable space is B o r e l isomorphic with a

S(d) subset of h*

(use theorem 0.1,

theorem 2.4 and theorem 2.6).

Proof of 1emma:Let A

be a monotone Souslin

sheme whose values are sets which are analytic with respect to their subspace topologies.We define

45

THEOREMS OF SEPARATION, ETC.

We see without difficulty that the set of analytic sets

is closed with respect to countable unions and intersections (by a very similar argument to that in the proof of the last theorem where it is shown that the standard sets are closed under countable intersections and countable disjoint unions). Hence B

is analytic.This in turn implies that S ( A ) is

analytic,as it is equal to the projection of B

onto X.

This concludes the p r o o f of the lemma. S( 4) set in

Every

im is thus

the projection onto

km

of a closed set (the graph of a continuous function)

in

?Kd"

.Let On

ha

in

M

be a countable base for the topology

.We define

is closed ,so its projection Q

onto the first

and third coordinates is analytic in haafi" .Every analytic subset of

im is a

section of this projection Q,hence

isomorphic with a measurable subset of Q Sy=

i(x,y)

I

(a section

(x,y)eQT is even closed in the subspace topo-

l o g y which generates the Borel structure of

Q ).It is now

clear from the above remarks that Q with the subspace structure is a universal analytic measurable space.Now assume that Q was standard. Q would then be a Borel measurable subset of

iv"* N-

.Then D=[nB?@((n,n)&Qi

would be Borel measurable,as the injective cor,tinuous image

of a standard set.In Particular D

should be analytic but

this contradicts the fact that D

cannot be aSection of Q

(Cantors diagonal principle).This finishes the p r o o f .

THI
46

We now know two isomorphism types of analytic uncountable measurable spaces.This is indeed almost all that is known at the time of this writing (1973).

If one assumes

; I

set-theoretical axiom of a similar

standing as the continuum hypothesis,one can prove the existence of at least one more isomorphism type of uncountable analytic measurable spaces

(see l.IY], P F ]

, [?Oj

) .This j.s

not ,however,a very satisfying result,and it seems likely that there are continuously many isomorphism types and that this can be proved with the help of the usual axioms (to which we count the axiom of choice).The outline

following is an

of a proniising way of attack.

We call an analytic measurable space

3) a type

(X,

space if there exist an isomorphic imbedding of X a (automatically analytic) subset A

I

val

,such that the set

IlA

I

onto

of the unit inter-

has a coarser Borel

structure under which it is standard.If

I\A admit a coar-

ser Rorel structure under which it is an analytic measurable space we c ~ l 1 (X,2 )

a type

I1 space.A type

space is of course a space which is not of type

I11

I or I1

.

We do not know whether or not every type I1 space is type I.

It is an important fact that any Borel isomorphism between subsets of standard measurable spaces may be extended to a Borel isomorphism between measurable subsets (see[/63 theorem 26,

p.194) .Using this one easily shows that if I

A

(11) subset of a standard measurable space

X\ A

is a type

(X,a)then

has a coarser Borel structure under which X \ A

standard ( analytic)

.

is

THEOREMS OF SFPARATION. ETC.

41

In the next chapter we shall construct examples of type

I and type I1 spaces which are not standard.0ur guess is that the universal analytic measurable space is of type

111 and that there exist type I1 spaces which are not of type 1,but this is entirely open. In complete analogy with the calculations on cardinal numbers,we can define cowltable sums and countable products

of isomorphism types of analytic measurable spaces.Let and u

s

be the uncountable standard and the universal iso-

morphism type respectively.For any isomorphism type a

we have

u+a=ua=u and if a

is uncountable also

(the latter result follows from

S+S=S

,

s+a=a

and the fact that

every uncountable analytic space can be shown to contain an uncountable standard subset).It is an open question whether

sa=a for every uncountable a (i.e. whether the pro-

duct of an uncountable analytic space with

I

is Bore1

isomorphic with the space itself) .It is similarly open whether

a2 =a and whether a+a=a for an uncountable iso-

morphism type a.

.

Notes and comments on chapter 2: Parts of the content are inspired by[f6]but

we have

tried to select material which will be needed later and to simplify the proofs to some extent.The proof for lemma 1 under theorem 2.6 is (as far as known to the author) new, but the result is well known.The basic results on isomorphism and uniformization theory seems to be due to Lusin, Hausdorff , H a h n and Kuratowski

(see [?8), L V ] , [ 2 7 ] , CZZ] ) .

THEOREMS OF SEPARATION, ETC.

48

In the early theory,the work was done on metriaable spaces.But

Cartier discovered in the middle of the s'ixties

that the assumptions of metriaability could usually he omitted.The applicability of the theory was thereby enormously Increased,as many of the most important topological spaces in functional analysis turns out to be analytic o r even standard

.

New phenomena can however appear.It is,for example, easy to show that the limit function of a pointwise convergent sequence of measurable functions from the measurable space

(X,

3)

into the topological space

(Y,@) is measu-

rable if Y is metrizable,but it is quite open whether the limit function need he measurable even if

(Y,6 )is ana-

1ytic.In a recent paper the author has shown together with Niels Johan Mmch Andersen (see[2])

that if

(Y,0 )is ana-

lytic the limit function is measurable with respect to

S ( 3 ) I) CS(8)

.If a l s o

3) is smooth this result implies

(X,

that the limit function is measurable. The existence of a universal analytic measurable space and theorem 2.5

seem to he mathematical ,,folklore,,,There

are several references to them in the litterature (see [ZF]) but we have been unable to determine who should have the honour for them. The type theory is due to the author.Since we have not been able to prove our hypotheses concerning the types the main motivation for the definition lies in the fact that we can construct explicitely type I and type I1 spaces which are not standard.The type arithmetic is of course

THEOREMS OF SEPARATION, ETC.

49

not much interesting as long as we knows two isomorphism types only. An important reason for the study of analytic spaces is the isomorphism theorems,as well as the possibility of

,,measurable selections,, o f

several types (see chapter 4).

By limiting oneself to working with andytic spaces,one can also obtain the measurability of several of the ,,stopping times,, that appear in the theory of stochastic processes. Generally speaking,many measurability problems which cannot be solved at all for completely general spaces can be solved relatively simply when one is working with ana-

lytic spaces.This is not an empty advantage if one is working in the usual naive set theory (to which we count the axiom o f choice).But it is possible to choose a system of axioms f o r set theory such that every subset of (at least

, ,small,,

topological spaces) a space is measurable with

respect to any measure (see Igf]).Then

of course the axiom

of choice is no longer valid but a weaker version still may be assumed.

We think however that one cannot find a better motivation for the study of analytic spaces then this,that it gives a nice and deep mathematical theory.

CHAPTER 3 PROPERTIES OF TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES AND ON SPACES OF COMPACT AND CLOSED SUBSETS OF A HAUSDORFF TOPOLOGICAL SPACE

The a b s t r a c t t h e o r y of t h e preceding c h a p t e r s i s used f o r a d e t a i l e d s t u d y of ( d e s c r i p t i v e ) p r o p e r t i e s of Borel s t r u c t u r e s o n spaces of closed s u b s e t s and on f u n c t i o n spaces.Furthermore t h e sane study i s done f o r propertiies of topologies.The E f f r o s Bore1 s t r u c t u r e i s d e f i n e d and some of i t s fundamental p r o p e r t i e s a r c s t a t e d and proved.

(X,@)

Let

be a Hausdorff t o p o l o g i c a l space.Let

p0

be t h e , , s p a c e , , of compact s u b s e t s ( i n c l u d i n g t h e empty

*

set. X

means t h e compact s e t s without t h e empty set).We

a X

s h a l l d e f i n e a topology on

which we s h a l l c a l l Hausdorff’s

topology.This concept i s probably due t o Hausdorff.If we o c c a s i o n a l l y f i n d i t convenient t o a d j o i n t h e empty s e t t o

a X

, i t w i l l be considered as a n i s o l a t e d p o i n t ( o r measu-

r a b l e s e t i f we a r e d e a l i n g w i t h Borel s t r u c t u r e s ) . I n t h i s manner we t a c i t l y c o n s i d e r i n t h e s e q u e l 6

X0

as being defined a l s o on

.

Hausdorff‘s topology c o a r s e s t topology on t h e s e t sets

iKG?

each s e t

I

Kn0#0]

cad

on 4

X

iKE?

structures on

A

i s d e f i n e d as t h e

such t h a t b o t h of the

I

KS0)

are open f o r

0 €0a

I n t h e s e q u e l we always r e f e r t o t h i s topology and t h e Borel s t r u c t u r e generated by t h i s topology i f we speak

TOPOLOGltS AND B0RI:L STRUCTURES ON FUNCTION SPACES

51

?

about continuity,measurability and similar concepts in without further specific-t' a Ion.

A similar topology could naturally be defined on the fi

space X

of closed subsets.But we shall see later on that

this topology would even for Polish spaces not possess reasonable properties (it would be too fine).

and Kl#K2 .We may assume

Proof: Let K1,K2€X that there exists an

x€K1

such that x+K2 .From the

Hausdorff property of @ and the compactness of K2 follows that there exist open sets I K e X I KSO,!

and

are disjoint open sets containing K1

and

K2 respectively 4

Let F c X

K2 O2 .Then

and

sets in

0

$ K a?

A

I K00,#0

.

6 be # compact and set M%YF

(i€I) be a covering of M with 4

with

1

X € 01, O l n O2 =0 A

0,,02e@

it

defined by

CKE

x\

x I

&' KE

K .Let Oi

sets.The system of

oij

[I*I is finitej

6 compactness forms an 4% 0 open covering of F .From the 0 of F we can now easily obtain a finite subcovering of' M, hence M

is compact in X

Conversely let

6

FgX

,. be

closed and assume that

the union of the sets in F is compact in X .Let Ka (aeD)

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

52

be a u n i v e r s a l n e t i n

F .Let

OCX

sets

GnKa=O

eventually.Now

such t h a t

Ge&'

i s compact ( M defined as above)

t h e compactness o f G

be open and

M

be t h e union of a l l

and i t i s easy t o see t h a t

i m p l i e s tha$

G n K # 0 .Then

K

we have

i s non empty.Let G O Ka&

Ka

eventually,

G g O (at t h i s

because we could otherwise conclude t h a t point i t i s used t h a t

X=M\O

i s a u n i v e r s a l n e t , s i n c e this

Ka e i t h e r belongs t o o r not belongs t o an a r b i t r a r y Liven s e t e v e n t u a l l y ) .Let H € 0 be open i s e q u i v a l e n t with t h a t

K 5 H. Suppose t h a t

and

Ka$H

eventually.We choose

H G M \H .Let

x=lim x a choose an open neighbourhood U

xa6 Ka\

eventually since

x

;then of

belongs t o

x

x€M\H

.We may

such t h a t

0 .But t h e n

UQKa=O

xa4U

even-

t u a l l y . This g i v e s a c o n t r a d i c t i o n which concludes t h e proof

3.1

o f theorem

.

If t h e space

we d e f i n e

(X,@)

i s m e t r i z a b l e with t h e m e t r i c

Hausdorff's m e t r i c

d*

on

6

X

d*(A,B)=sup E d ( a , B ) , d ( A , b ) I aeA,beBj

d,

by

.

I t i s very easy t o show t h a t t h e topology generated by

Hausdorff's m e t r i c Let

Fa

X

in

be a n e t o f closed s e t s i n t h e Haus-

C ---9 F

if and only i f f o r every

e x i s t s a neighbourhood of' intersect x

c

by

Fa

of

topology.

( X,@ ) .WE d e f i n e a convergence concept

d o r f f space n

( a e D)

i s p r e c i s e l y Hausdorff's

x

F ,and moreover f o r

a intersects

Fa

x E X\F

there

which e v e n t u a l l y does n o t

x e F every neighbourhood

eventually.

The collvergence topology i s defined by c o n s i d e r i n g

53

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

Fs? and

as closed if and only if : Fa (aeD) is a net in Fa

C ---+

F

always implies Fe $- .It is easily

seen that the convergence topology is finer then Fell's topology,defined by requiring every set of the form EFe? 1 F A O # 0 $ and

and

IF€? I FnK:0]to

be open,where 0 6 0

K is compact in X .Flachsmeyer has shown that Fell's

topology in general is strictly coarser even in 6-compact metrizable spaces (seeLff])

.Fleming Topscae has shown that

this can also be the case in Polish spaces.Tops0e has conjectured that the convergence topology equals Fell's

topo-

logy if and only if the space is locally compact (in any case if it is also separable and metrizable).

Later on,we shall see that the Effros structure would not have nice properties if A

was allowed to be open.It

is very easy to show that Fell's Effros Borel structure if X

topolcgy generates the

is metrizable and separable

(it is not clear whether or not the convergence topology generates the Effros Borel structure), The above definition should be considered as no more than provisional in the case where the topology of X

does

not have a countable base.It can easily be shown that the structure is countably separated if and only if the topology of X

has a countable base.0n the other hand there are

TOPOLOGltS AND BORI-L STRLICTURES ON FUNCI'ION SPACES

54

cases

(i.e. a separable Hilbert space with weak topology)

of analytic Hausdorff spaces with no countable base but with a ,,natural,,smooth Borel structure on the space of closed subsets.We do not know whether or cot every analytic Hausdorff space has a

, ,natural,,smooth Borel

structure

on its space of closed subsets.Although a later argument may suggest a possible definition (i.e. as inductive limit), we have not been able to show that this definition gives 6

smooth Borel structure in the general case of an analy-

tic Hausdorff space.Therefore and since our arguments in the present chapter only works if the space in question has a countable base we shall d o in this book with the above definition of the E f f r o s Borel structure.

TOPOLOGIES AND BOREL STRUCTURES ON FUNmION SPACES

55

Proof:Let 2 be the compact d-completion of X for the natural extension of the metric

(we a l s o write d d

to a metric on

-X

) .We may obviously identify

isometrically with the set M

(? ,d*)

for which

(?,a*)

of those points F

F n X is dense in F .Let On

in be a

*

countable base for the topology of X .It is easy to see that

Now the set

I xeF3

{(x,F)

C=

-

A

is obviously closed in X S X ,hence measurable in the product Borel structure since both spaces are separable and metrizable (in fact even compact metrizable).It follows that is analytic in

%%a and e

is analytic in X

3

ceding setinto X .c\

consequently that the set

since it is the projection of the pre-

. It is now clear that

CI

since fF B X I Fc ?\On$

in X

M is analytic

is closed

.

Since the identification mentioned above was isometric (in particular a homeomorphism) we see that

A

(X ,d*) is an

analytic metric space.It is easily seen that Fell's is coarser then the d*

topology

topology.Hence the Effros Borel

structure is coarser then the Borel structure generated by the d*

topology.Let

Gn

be a. countable base for the

topology on X .Then

& d tI F G X \ G ~ ] I

ni

;J3

is a separating sequence of Effros measurable sets.The iso-

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

56

morphism theorem (theorem 2.4) now i m p l i e s t h a t t h e two s t r u c t u r e s i n q u e s t i o n coincide. If

because

X

i s 6-compact t h e n

On

i s an open s e t i n a

X f i On

compact metric space.#

t h e n a countable i n t e r s e c t i o n o f hence

M

i s a l s o 6-compact is fi

X

6-compact s u b s e t s i n

,

i s standard.

Suppose now that

Hn

d e c r e a s i n g sequence ( s e e lemma

2

of

X

-X

i s Polish.Then t h e r e e x i s t s a of open s e t s i n

t h e proof of theorem

gory theorem i m p l i e s t h a t

FeM

i s dense i n F f o r every n

m

X=nHn

with

-1

2.6).Baire8s

Hn/r F

i f and o n l y i f

.

cate-

Obviously we have A

IF6X

(the

0,

's

I

FOH,

dense i n

F]=

a r e a b a s i s f o r t h e topology o f

v

Since t h i s f o r every and a c l o s e d s e t i n

is the

?,M

union

f

of an open A

is a

as b e f o r e ) .

G6 s e t i n X

.The proof

i s now complete as soon as we have shown

Proof of 1emma:Let in

X

f o r all n .Set

D1=

w

C ~ ) e T O n I l(a*

&KOn n*1

where

b/n,m: %=% 3

Because D1 i s c l o s e d i n t h e P o l i s h space Polish.Clearly t h e p r o j e c t i o n p of D1

On

* ;?;on

i s open

, D ~i s

d e f i n e d by

57

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTlON SPACES

P ( (xn))=xl

is a homeomorphism of D1

onto D.This con-

cludes the proof.

We denote the coordinatewise ordering in $la by

Let

5

.

be an arbitrary set equipped with the partial order

S

.

5

S

is (by our definition) an analytically ordered set

p:iw-*S

if there exist a mapping

such that the follo-

wing conditions are fulfilled i) n i m

==+

(4

p(n)<

ii) For all s 6 S The order relation

there exist

<

.

neb- such that

s<.q(n).

is called analytic.To be analy-

tic is a very weak condition on an order relation.If there exist a majorant the ordering of course is analytic.Recent-

ly the author has shown that an ultrafilter on N

is ana-

lytically ordered by the relation 2 if and only if the ultrafilter is trivial.1-t is a good exercise to show this using the theory in the

following chapters together with

the non trivial (but easily obtained)result, that the union of a system of closed sets in a separable metric space is a Souslin set with respect to the paving of closed sets if the system in question is analytically ordered by inclusion

. It is e l s o an musing exercise to show that any well

ordered set of cardinality less then the reals is analytically ordered. The main motivation for o u r definition is the following result.This result is the main theorem of turns out

this chapter and

to be very useful for the proofs of non measura-

bility in a number of cases.

TOPOI.OGIL<SAND ROREI. SlKUCTURES ON FUNCTION SPACES

58

Proof : 1) ===)2) on X .We consider Let Kn

4

,

Let

d

be a complete metric

equipped with Hausdorff's

be a d*-Cauchy sequence.We define K,

easily seen that K,

Let Q C X

of

X

Q

in

metric (of course the completeness of the metric

is essential for the compactness of

4

.It is

--*K@

is compact and that

d*,

as the

set of accumulation points for sequences xneKn

the d*

metric

K,

),

be a countable dense set.The finite subsets

then together forms a countable d*

dense subset of

A

.This shows that X equipped with the metric

d*

is

\* A f:N --> X be

a complete separable metric space.Now let

a continuous surjective mapping (use theorem 0.2).We define \o

A

'p: N - - j X

by

q(n)=u5€(m) Im€k= The set of m 6 h Y

and

with

m 5 nl m 5 n

.

is compact in 'N@

59

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

and the image of this set under in

. It is now clear that @

4&

X

f

is therefore compact

has the properties requi-

red in the definition of an analytic ordering. 4

x

cp:N’-+

2 ) ==+ 1) Let

be a mapping which is increasing

and swallows compact sets.First we show that the space X

d

is separable.Suppose this is not the case.Let

be a

metric on X.Then there exists an uncountable family xi“X (itI) and We define

an e h:k*-+

h(n)={ i e I The mapping

1

>

0

with

%(I)

by

xi

Q

d(x.,x.)le 1

3

.

f o r all ifj

.

fJ7(n)3

kw

h goes from

into the set a(1)

finite subsets of I ,and it is easily seen that

of

h

is

increasing and swallows finite subsets of I .Thus,the properties of

are similar to those of

h

may assume empty values.

We now define the relation

+

9

,except that

between pairs of mul-

tiindices and between multiindices and elements of

n dm

writing

if and only

There exists an p s n

the set

h

if n

N- by

is a segment of

m.

n 6 h W such that for all multiindices

1

U{h(rn)

p-(

mi

is infinite.If this

was not the case ,we could easily show using the properties of

h

and the Lindelaf property of

at most countable.We may choose i €h(n ) ,where the

P

P

implies that

mum

i ‘s

P

n

P

P

that

n

in

---)

hw

was and

(the coordinatwise supre-

,which is finite because

n - 9 n ).This contradicts the fact that P mes finite vaLues.Henc2

I

are all different.But this

lid E h( sup cP )

of the sequence

n

i*

h

only assu-

X is separable and therefore has

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

60

the Lindelnf property.We shall make use of this result in the remaining part of the proof. The next step of the proof consists of constructing a Souslin scheme

of non empty open subsets of X and

A

a strictly increasing (with respect to the relation 4 )

-+

q: P

mapping

of the set of hdtiindices into it-

P

self,such that the following conditions are fulfilled:

K

ii) For every compact set n € N O with q(p)4n

and

A(p)

Kgy(n)

.

there exists an

The essential idea in the proof is to make use of the elementary property of metric spaces that for any sequence Kn

of compact sets fulfilling

the set

S K n U 1x3

Let p,

pact

XQ

X

is compact.

be given.We wish to find a multiindex

and a real number

set K

exists an

-+ 0 ,

supfd(x,y)Iy.K,g

rx

> 0 such that for every com-

contained in the open ball S(x,rx\

ng

k”

with

px
and

Suppose this was not possible.Let

.

KcfYn)

pvtP

there

be a sequ-

ence of multiindices such that every multiindex is assumed at least once.We choose for any

ne?

with

Kvc, S(x,l/v) such that pv+

n

.The set

K=

Kv$Q(n)

uK 08

Vd

is then compact and therefore exists an n 6 N”

U fx3 v with

K C ?(n) .This obviously gives a contradiction. We now make use of the Lindelraf property of find xn 6 X covering of

such that the open balls S(x ,r )

X .Next we set A((n))=S(xn,r

X

to form a

x*

) ,q((n))=p,

xn n This concludes the first stage of an induction proof of

.

61

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

the existence of the Souslin scheme A.The next stage consists of countably many similar steps,one for each A( (n)) q:P -3 P

Note that the mapping

with respect to the relation Let;

‘x

be the

Let

is strictly increasing

.

d-completion of

-X \ (cl.in z(X\A(p))).We that every A ( p )

-=(

.

X

and

then have B(p)nX=A(p)

B(p)= (note

is open). be the set of those

MES?

y€y

for which

we have (P)

There exists an

if yeB((nl,.,,nk))

.

nai

such that yeB((n))

then there exists an n

and

such that

YE B( (nl, ,nk,n)1 Clearly that Xk€

XG M .For y e M we may choope n 6 ka such for every k .Next we choose

y&B((n,,..,n,))

A ( (nl,. ,nk))

q((nl,..,nk))4

mk

with and

d(y,xk)Sl/k ,and Xk 8 Cp(mk) .Since

increasing we have that m=sup mk $xktC_Q(m) . A s

Hence

we deduce that

y&X

.

q(m)

(P)

q

with is strictly

is an element of

hw.

is a compact subset of X

Now we easily sec that the set of those do not possess the property

mke ?i

y ~ ?which

is equal to a countable

union of intersections between open and closed sets,hence a countable union of closed sets.Since

X is a

Gs

set

c

in the Polish space X it is itself Polish.This concludes the proof of 2 ) ==$ 1).

3)-9

2) This is an immediate consequence of the preceding

results. 2)

==+ 3 )

Immediate

TOPOLOGIES A N D BOKEL STRUCTURES ON FUNCTION SPACES

62

1) =+4)

This is an easy consequence of the preceding re-

sults.

4) ==+ 2) We choose a precompact metric

d

on X ,As

is a measurable subset of the analytic metric space 6

the Bore1 structure of X

is analytic and

6

X

(?,d*),

A

X. is analytic

in the subspace topology.Therefore there exists a surjective d*-continuous mapping from

k*

onto

A

2

.In the same manner

as i.n the proof of 1) ==$ 2) we now see that 1) is fulfilled.This concludes the proof.

We call a subset S C , X space in X

(X,@)

,,almost compact,, if A S S

implies that A

of

S in X

SSX

and

A

closed

is compact.We leave for the rea-

der to verify that if X compact set

of a Hausdorff topological

is metrizable then every almost

is conditionally compact (the closure

is compact in X ) ,

Proof:We define

4

X by

q:N*--+

According to the preceding remarks this is the closure of an

,,almost compact,, set, hence

$P

has compact values.

TOPOLOGIES AND BOREL STRUCTURES O N FUNCTION SPACES

Obviously

q7

63

fulfills the conditions in the definition of

an analytic ordering.Now the theorem follows from theorem 3 . 3 . This concludes the proof. Let

(X,f )

be a paved set.We let CS(

f

)

denote the

set of all complements of sets in S ( f ).If we assume that

9

is closed with respect to finite intersections and

contains [X,0? it is clear that

S ( f ) 4 CS(

f

)

is a 6-field,

since it is closed with respect to complements and countable intersections and unions (and is non empty because it

contains ~X,D] ).

Proof:We may under the assumptions of the theorem choose a sequence

En6 f

such that

S(fE#n CS(fEn<)

is a countably separated Borel structure

.We may assume

that finite intersections of sets in the sequence again belong to the sequence and moreover that fE,.{:'fX,03 let

@ denote the coarsest topology on X

.We

for which the

sets En

are both open and closed.Equipped with this t o -

pology, X

is a separable metrizable space.A metric d

on

TOPOLOGIES A N D BOREL STRUCTURES ON FUNCTION SPACES

64

X which generates the topology is given by go

XE

d(x,y)=x I Y E (XI(Y)(/lon n.9 n n We remark that >En$ forms a base for the topology fl. To prove this it is obviously sufficient to show that for every n the set X \ En of sets Em.Let

0

is an (of course countable) union

y€X\En .Then

Hence there exist a subsequence

Eyj6S(IEnf)r)CS(jEnj) Em

such that {yf

P

.

is

precisely equal to the intersection of the sets in the sequence.The semicompactness now implies the existence of Em with y € E m and En 4 E m s

.

We now choose a system A(.,.) quence En

such that for every n

a covering of

of elements in the se-

X with sets whose diameters are less then

l/n .Let F S X

be a closed set which for every n

be covered by finitely many

F

ultrafilter on there exists mn

,

setg A(n,m) .If

such that A(n,mn)

n A(n,mn)#P 00

that

n

belongs to the filter.

f

we conclude that

.Indeed from the assumptions it even follows

n A(n,mn) .o

n.r

can

is an

it can be seen that for every

From the semicompactness of the paving Wl

form

the sets A(n,.)

consists of exactly one point

x .It is

f-

now clear that ,$ converges to x .An application of theorem 3.4

now shows that

the Bore1 structure of

@

is Polish and consequently that

is standardofhe remaining part of

the proof follows immediately from the isomorphism theorem. The next theorem characterizes the 6-compact metric spaces by the behavior of their coverings.Remember that \

K=tO,l)N

is the Cantor group;with its usual product topology

and product group structure it is a compact metrizable group.

65

TOPOLOCILS AND BORE1 STRUCTURES ON FUNCTION SPACES

P r o o f : 1) ==$ 3 ) Let

w

,where the

X=,I;'Kv

Kv 's

00

S=nfxeK I KvC

a r e compact sets.Obviously,we have

v-l

But every s e t i n the i n t e r s e c t i o n i s open i n

(D

0 x(n)On{

nr*

K (use t h a t

any open covering o f a compact s e t has a. f i n i t e subcovering).

3 ) ===22)

2 ) ===)1) Let

be a precompact metric on

d

-X

w i t h the topology.Let

X

(we use the n o t a t i o n

of

open i n have

3 ).We

to

d

.

Entirely t r i v i a l

-X

S=IxcK

I X

define

Gn=?\cl.in

.Now l e t

j e c t i v e continuous mapping. W e define

F={ (A,n)e.$sk*l in

X

o( v:fu

i s closed.Let

F

that

A

n € k m such that of

F

\o -jS

f:N

Y=z \ X

On

be a sur-

and

f ( n ) ( v ) G v ) = O j .It i s e a s i l y seen C C _ Y be compact.Since every point

i s a s t r i c t l y p o s i t i v e d i s t a n c e away from

(using the f a c t t h a t

is

'X(X\On).Gn

.From t h i s i t follows t h a t we

CUx(n)Gn\ W

d-completion of

a l s o f o r the canonical extension

d

GnnX=O,,

and

be t h e compact

compatible

X

C

we see

i s a base) t h a t t h e r e e x i s t s an

C fl( U f ( n ) ( v ) G v ) = O.Hence the projection

i n t o the f i r s t coorCinate i s p r e c i s e l y

6

Y ,which

therefore i s a n a 1 y t i c . h a p p l i c a t i o n o f theorem 3 . 3 now shows t h a t

Y

i s P o l i s h ,hence a

G8

set i n

.But since

.

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

66

X=x \ Y ,we can conclude that X is a countable union of

-

closed sets in X .Since f

is compact,this completes the

proof. The next result is very closely related to the preceding one ,both in content and method of proof.

Proof:Let logy

0

On

be a countable base for the topo-

is closed with respect to On rnck be a dense sequence in k . F o r

and assume that

finite unions.Let f ,g 6 C(X) Clearly D

we define is a pseudometric on C(X).If

find an xo 6 X some a 6 k and

such that

f(xo)(, a-e

<

ffg ,we can a+e

5 g(xo) for

e > 0 ( o r the inequality may be obtained

67

TOPOLOGIES AND BOR1:L STRUCTURES ON FUNCTION SPACES

in the reverse direction).By choosing a suitable small On with xo& On

(use the fact that both f and

tinuous and that

lo,\

g are con-

is a basis) and a suitable k we

see that the sum defining D(f,g) has a strictly positive term.D(f,g) is therefore strictly positive.It follows that

D is in fact a metric. in D metric.Let C S X P VOn such be compact .We can then find a covering C 4 w V < e for all v=l,..,p that supilf(x)-f(y) I Ix,ye on Suppose now that fn

(e

--jf

V

3

> 0 is an arbitrary fixed number) ,It is now easily seen

that we have

supt Ifn(x)-f(x)l IxcC{

<

2e

for all n 2 N

for some sufficient large N .The topology generated by the

D metric is therefore finer then the topology of compact convergence. Each f aC(X)

induces a function

by the definition

8 ( f1 (n,k)=sup[I f( x)-rk I Clearly

, @defines

xeonjA1

@(f)(n,k)

on

i2

.

a homeomorphism o f

(C(X),D) onto

a subset of the space of realvalued functions on

k2

(equip-

ped with the topology of pointwise convergence ).It makes

no difference to the definition of the metric D supremum is only taken over

XQ

if the

Onfl S ,where S C X is some

fixed countable and dense set.It follows f r o m this that is measurable if C(X)

0

is equipped with the topology of

pointwise convergence and the Borel structure generated by this topology.In particular ,it follows that the topology of pointwise convergence,the topology of compact convergence and the D-topology have the same Borel structure.This

68

‘TOPOLOGII:S AND UOKLL STRUCTURI S ON FUNCTION SPACES

structure evidently is separable and separated. CL)

1) ==$ 2) Let X= v:+ V Cv ,where the C, are compact in X. w Assume S= u Sv ,where for each v, S v t C, is a countable y: 1

dense subset of Cv .Let the elements in S be ordered in .We define q:C(X) - - 3 R 4 by q(f)(n)=f(sn). a sequence s

P

From the preceding remarks,it f o l l o w s that

onto a subset of

isomorphism of C(X)

is a Borel

h’ ,if the first

space is equipped with the Borel structure generated by the topology of compact convergence and the

second space

with its natural product Borel structure.If tric on X

m

d

is a me-

compatible with the topology,then

q (c(x))=f)jr,.R

b

N

Yr9

Ih$%UP(SUP

bt-ruI I S u ~ C v , d ( S t ’ S u ) ~ l / P ~ ) = O ~

It follows from this that q(C(X))

is measurable in

4 R ,

0

in particular that the subspace Borel structure is smooth. (C(X),D) is consequently an analytic metric space,since it is separable and its Borel structure is smooth.Since the D-topology is finer then the topology of compact convergence, we draw the desired conclusion 2) Clearly

,

3)

and

.

4) are equivalent,as there exists

a strictly increasing homeomorphism of 3-ao,m[

onto 30,00[

(e.g. the function exp). 2) = = j3 ) Let

T:

bw-+ C+(X)

be surjective and continuous,

with C+ (X) equipped with the topology of compact conver-

q:p-+C+(X)

gence.We define

by

(ip(n)(x)=inffT(m) (x)lmcim,m ,< ni

.

It is easily seen that the conditions in 3 ) 3 ) ==+1) Let

d

are fulfilled.

be a precompact metric on X and

be the compact d-completion of X .We define

B 4

W:hm-+XB by

69

TOPOLOGIES AND BOKIL STRUCTURES ON FUNCTION SPACES

W(n)=cx (z\s(x,q(n) (x)1 ). shows that 2 \ X

An elementary application of theorem 3.3

is Polish ,hence Gb

in

2

. X=z \ (i\X)

is therefore

6-COmpaCt.This completes the proof, We now are able to answer the natural question about which conditions are necessary and sufficient to ensure that the set of closed sets contained in a given set is measurable.

Proof: 1) ==j 3) Let

be a precompact metric

d

on X which generates the topology.Since the d*-topology generates the Effros Borel structure ,there exists a surjective d*-continuous mapping define

q:r--+ [F6?(

,$7(n)=Uif(m)I

FsS{

.We

by the equality

n e N m and

It is easily seen that

f : --j ~ fFe31 F C S )

40

m

L

nj

.

is increasing and that it

swallows every closed set contained in S

( that the values

of $b‘ are closed in X is not trivial but f o l l o w s easily

'L'OPOLOGIES AND UOREL STRUCTURES ON FUNCTION SPACES

70

from the fact that im 6 ? Im define Y=(cl.in X ) ( S ) \ mapping

4

-+ X,

\ *

P:N

S

< ni

is compact in

?I.We )

Z=(cl.in %)(Y) \ Y .The

and

defined by

(V(n)=(cl.in f)(Y)/1 (cl.in % ) ( $%'(n))

is obviously in-

creasing,and its values are compact sets contained in Z. Conversely,let C E Z be compact.Making use of the compactness of

C

and the density of

we can find a sequence points in C

S

in

which has exactly the

sne S

as accumulation points.Now

xsnf U (C A S )

(cl.in ): ( S )1 Z,

is closed in X

(cl.in X) (!fsnf) =

and obviously contained \cp

in S.We deduce that there exists ne N

such that (y(n)2 C.

An application of theorem 3 . 3 now shows that

hence a G~

set in

Z

is Polish,

( c ~ i F)(Y). n Y is consequently

6 -compact. 3 ) ==+ 2) and

2) ===>1) follows almost immediately from

the preceding results.

Proof:If

39 C X B A

is an analytic subset,then

the subspace Bore1 structure on

4

Xp

equals the Effros

71

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

Xp .Assume this is indeed the.case.

A

Borel structure on Then for A

in

, 1H ~

I

$ 0

Xp I

6

= [He

H g F3

is therefore analytic

X p .Theorem 3 . 8 now shows

X6 and hence also in

F can be made

is measurable.

HSFf

n

A

that from

I H 'F$

e^x,

Xp"5-H

F 6%@

closed by adding a set disjoint

F which i s 6 -compact with respect to

topology.

A

If ,conversely this is true for every F d X o ,then n

HO X p

I HE Ff

29

is measurable in

.This implies that

h

4

the subspace Borel structure on Xp induced from ,,X

is

precisely the Effros Borel structure .Since this implies that ?p

is smooth in its subspace Borel structure,it h

must be an analytic subset of XCO.The proof is now obviA

h

is always ously complete if we can show that X6 \ X? A analytic in Xg ,since we then only need to apply theorem 2.2. A

4

h

X6 \ X p is analytic in X6

To see that precompact metrics

generating the @

dl,d2

,we choose and

topo-

logies respectively.We may assume that dl 2 d2 .The mapping

4

a:XQ

--+

A

Xb

which is defined as the closure in A

topology of a given F e X6 L(F,a(F),x)6?&

is :d

x Zpr

X

continuous.

- j d;

Ix&F,xg a(F){

is therefore measurable in the product space and the proh

jection on X,

o f this set i s precisely

therefore is analytic in

A

Xg

Xp ,which

h

A

Xa\

.This concludes the proof.

The above theorem makes it possible to construct explicitely analytic measurable spaces of type I 4

are not standard.E'oL, if both Xd, and and

?*

CI

is not measurable in X@

$J, then

which

are standard

?,

\

?p

ped with the subspace Borel structure induced from

,equipA

X@,

12

TOPOLOGII<S AND BOKLL STKUCTURES ON FUNCTION SPACES

is an analytic measurable space of type I and not standard.

An analytic non-standard measurable space of type I1 may be similarly constructed.1f we consider a continuous surjec-

tion

f:X

-+ Y from a Polish space (X,,9) onto an ana-

p)

lytic non-standard metrizable space (Y, in a similar way as before that h

analytic type I1 subset of X D

it can be shown

%\If-’(A)

IAs?’~

is an

.It is not known whether

or not every type I1 space is also type

I

although

we feel strongly that this is not the case in the examples that we have constructed. It also follows from the preceding results that the isomorphism theorem is not valid for coanalytic measurable spaces. Clearly,$

with intersection as composition is an

abelian semigroup.The intersection operation seems to be rather discontinuous in any reasonable topology on

.

It is therefore natural to ask for conditions ensuring that the intersection operation is at least Effros measurable.

13

TOPOLOGIES AND BOKEL STRUCTURES ON FUNCTION SPACES

be a countable base for @ .We

Proof:Let On

define

On/) Ad3

v:?D -+

K=\O,&N

by

w(A)(n)=l

for

and zero otherwise.

is a Cantor index (a finite sequence

If e=(el,.",ek)

of zeros o r ones),we define N(e)=lxeK

I

x(i)=ei,i=1,.,k{

.

The paving consisting of sets of this type forms a basis for the topology of K. (/J-'(N( . ) ) is easily seen to be

1 is injective, is a Bore1 isomorponto y(z0),which is analytic in K.We now

measurable.Since A

hism of Xe,

T(X)=(~ x(n)On)' aD

IX0

T : K ---)

define

by

n-*

(C

denotes

complement)

,f'

To show that

is measurable (which is not always

the case) it suffices to show that Now it c a n be seen that the set of

(

y

x e K such that

($ o

y

is measurable.

)-'(N( (1)) )

o

is precisely

(P

O1c

(Ix(n)On .From the proof

nit

of theorem 3.6 ,it follows easily that this set is measurable if O1

is 6-compact.If,therefore ,X (

and metrizable ,we can conclude that

is

W o $9)"

6-compact ( N ( (1)) )

is

measurable arid nnalogously that

( y o f)-' e

(N((e)) )

is measurable for every Cantor index

.It follows from this that

measurable.From the equation

(y

op

and therefore q9 is

Al)B=f( p(A)+p(B)-p(A)W(B)),

it follows that the intersection operation is measurable. he set

5x6 K I

olc

(D

Ux(n)onj

n*

is coanalytic in K

since it is the complement of the projection onto I( of

the intersection

Of

an open and a closed set in X X K .In

a similar Imnner to before it can now be shorn that the

74

TOPOLOGIES AND BOREI. STRUCTURES ON FUNCTION SPACES

intersection operation always has the weaker measurability property mentioned in the theorem. Suppose now that the intersection operation is measurable and

d

-X

is a precompact metric on X compatible with

the topology.Let The set D in

it2

be the compact d-completion of X.

of pairs of disjoint sets is measurable

in particular analytic.The mapping (A,B) -3 (cl.in %)(A) n (cl.in z)(B) maps D measurable into the set of closed (compact) subsets of x \ X

.An argument very similar to the proof of theorem 3.8

coricludes the proof. It might be that a separablemetric space with the property that every closed set with empty interior is b7compact is automatically 6-compact. If this could be proven theorem 7.8 would give a simpler proof of a slightly stronger r e s u l t (sectionwise measurability would be enough). A

The mapping h:A -3 cl.(X\A) measurable if

(X,&)

of X 0

into itself is

is a metrizable analytic space since

h-l(fAQi'XO I ACF) )=be?,

I

,

F'4.A)

the latter set being convergence-closed.The equation dA=Ahh(A) now shows that the boundary mapping is measurable if X is 6-compact and in the general case measurable from the 6-field spanned by analytic sets to the Effros Bore1 structure.It seems probable that the boundary operation

is measurable if and only if the space is 6-compact,but we have no complete proof of this hypothesis. Notes and remarks to chapter 3 : There exist in the literature several investigations of topologies on the space of closed subsets of a Hausdorff topological space ( X , 6 ) (see f.ex. [14]

).The known

7s

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

topologies with reasonable properties are all nor.-IIausdorff when X is not locally compact.Even f o r Polish spaces,it seems to be entirely open whether o r not the intersection o f all d*

A

topologies on X is Hausdorff (d

through all precompact metrics on X

is running

compatible with the

topology).It might be that this topology is equal to the convergence topology.It also seems to be open whether o r not the supremum of all d*

topologies is analytic.

In most of our results about the Effros Borel structure obtained in. this chapter,the metrizability is essential for the proofs.For an arbitrary analytic Hausdorff space

(X,@) we have a surjective continuous mapping f:?-.j The mapping

f*

$- -+?

defined by

X

.

f*(A)=cl.(f(A))

is obviously measurable with the definition of the Effros structure adopted in the present book.Since

f*

also is

surjective the Effros Borel structure is Blackwell (seela]). It may be more reasonable to define the Effros structure as the finest structure making

f* measurable.In a few

cases where the space in question does not have a countable base we have been able to show that the Borel structure so defined is countably separated.In general however we do not know,in the case where the analytic Hausdorff space has not a countable base, what is the ,,reasonable,, definition of the Effros structure and how

(some of) our results should

be obtained.

It is a remarkable fact that the proofs of the fact that the Effros structure is standard in the Polish and in the

6 -compact case are entirely different.No necessary

76

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

and sufficient condition for the Effros structure to be standard has yet been found.It might be that the condition is (at least in the metrizable case) that the space is a Gga

in its completion.We feel strongly that there exist

standard spaces for which the Effros Borel structure is not standard but even this is open at the time of writing. The definition of the Effros Borel structure and the content of theorem 3.2

are due to Edward G o Effros in

the Polish case and to the author in the general case.Effr0s.s paper,surprisingly,is from as recent as 1965 (see 1931). The general case is treated in Theorem 3 . 3

L 8 1 and

[ ]

is due to the author and the essential

parts of it are published in

Lp].

The author has recently

received some information from Jean Saint Raymond which proves that he has obtained the essential content of the theorem by an entirely different method (of the same degree of difficulty).His work is not published at the time of writing but it seems to have been carried out almost simultaneously with o m work.The result gives a new and powerful method to show non-measurability in a number of cases where the variants of the diagonal method cannot be used o r only gives results for special cases, Theorem 3.4

is an unpublished result due to the author.

The theorem partly overlaps a result of Hoffmann-J~rgensen (see k91)

0

Theorem 3.5

is also an unpublished result due to the

author.It seems probable that if B Q

3

is countably sepa-

rated in the subspace Borel structure induced from

8 ,then

11

TOPOLOGIES AND BOREL STRUCTURES ON FUNCTION SPACES

B

must be standard and

f n B GBfi

8

.But we have not

for the time being been able to modify our proofs to yield this result. The theorems 3.6

, 3.7

work due to the author.In

and

[s]

3.8

are unpublished

,it is shom that the set of

closed sets contained in the open unit ball of an infinite dimensional separable Hilbert space is not measurable with respect to the Effros Bore1 structure.0f course,this is also implied by theorem 3.8 Theorem 3.9 and theorem 3.10

c 45

.

is unpublished work due to the author is a collection o f results from

183 and

0

It is a promising project to try to carry over the results of chapter 3 (possibly substantially modified) to arbitrary analytic Hausdorff spaces.We know positively that some of them does not carry over (at least not without substantial modification).It seems however that for a completely regular analytic Hausdorff space (X,@) it is true that C ( X )

with the topology of compact convergence is

analytic if and only if the space X is 6-compact.We have at yet not been able to find a complete proof of this conjecture in any direction

.

CHAPTER 4 MEASURABLE SECTION AND SELECTION THEOREMS WITH APPLICATIONS TO THE EFFROS BOREL STRUCTURE

W e s t a t e and prove a general a b s t r a c t s e c t i o n theorem

due t o

Hoffmam-Jci?rgensen ( s e e [46],p8*) from which i t seems

t h a t all known measurable s e c t i o n and s e l e c t i o n theorems may be deduced.This theorem i s applied t o t h e E f f r o s s t r u c t u r e and t o some s e l e c t i o n the0rems.A negative r e s u l t concerning existence of measurable i n v e r s e s i s given.

t e r s e c t i o n of these s e t s contains a t most one point ..................................................

MEASURABLE SECTION A N D SELECTION THEOREMS

Proof: We define the Souslin scheme 0

79

by

for all multiindices p a I? .Next the Souslin scheme H

is

defined by induction as follows : n-1

and

H( (n))=A( (n)) \ $G( (k))

-.

1cC I

By induction it is easily seen that

3

cef

H(p)

$' =

for all mdtiindices p e p

By induction we easily show that the Souslin scheme H is monotone decreasi-ng and

P*q

H ( p ) n H(q)=0

if neither

nor q < p . This implies

fi u

S ( H ) = u.1 ( p 8 4 tH(P))

hence

.

S(H)=BE.~'

Let now

m=(ml,..)6h00

x 15 X by

mk=min in I

For each

be arbitrary.We define inductively

CXIM

k > 1

(ml,. ,mk-l ,n)

we then have

[xln

5

A( (m,,.,mk))#O

,hence

using the transitivity of cU we obtain :

3c.

G(

(m17

Since for p 4 P

0

9mk))

0

either Lx] g G ( p )

or [x]

0

G(p)=0

we con-

clude (by induction on k):

1x1 A A( (ml, and

[x]OH(p)=0

,mk))=[XI 0 H( (ml, ,mk) if

and from the assumption

p 4 m .From this it follows : ii) it even follows that Cx] fl B

MEASURABLE SECTION AND SELECTION THEOREMS

80

c o n t a i n s just or,e point.Hence

B

is a section f o r

cy

and

t h e proof i s f i n i s h e d . I n a p p l i . c a t i o n s of t h e p r e c e d i n g theorem t h e S o u s l i n scheme

w i l l o f t e n be a S o u s l i n scheme whose values a r e

A

non empty open s e t s i n a complete s e p a r a b l e m e t r i c space (X,d) such t h a t t h e f o l l o w i n g c o n d i t i o n s are s a t i f i e d :

(nl,snk))

2)

l/k

It f o l l o w s e a s i l y from t h e L i n d e b f p r o p e r t y t h a t a S o u s l i n scheme w i t h t h e s e p r o p e r t i e s always e x i s t i n a complete s e p a r a b l e m e t r i c space.

E f f r o s measurable.

I -

P r o o f : F i r s t we c o n s i d e r t h e case where

i s a P o l i s h space.Let

d*

topology.Let

S=t(x,F)bXS? I x & F i Clearly duct space

X

be a precompact m e t r i c on

d

compatible with t h e topology.We c o n s i d e r now Polish

( X , @)

h

S C_ X% X

.

f

with

the

be t h e s e t

S is c l o s e d and t h e r e f o r e P o l i s h i n t h e proX K P .Let

D

be a complete metric on

S

81

MEASURABLE SECTION AND SELECTION THEOREMS

generating the topology.By induction we now choose a Souslin scheme A

of non empty open sets in S

with the

properties:

ii) D-diam(A( (nl,.,nk)

)5 l/k

We define the equivalence relation (x,A)-(y,B) If

GgS

G=9

A=B

.

on S

Cro

by

is an open set we easily see that

is open in S ; note that this of course depend of the special definition of S .We remark that the equivalence classes are closed in S .It is now clear that all conditions for the applicability of theorem 4.1 are fulfilled If we take the Borel field

3

to be the Borel structure genera-

ted by the topo1ogy.A Borel measurable section for the equi-

valence relation cu is the graph of a choice function,which is measurable since its graph is measurable.This concludes the proof of the

,,Polish,,part of the theorem.

In the general case let

be surjective and

h:??-*X

continuous.We consider the injective mapping h,:? defined by

h,(A)=h-’(A)

.The mapping h,

-+

6

is in general

not Effros measurable but it is measurable from the Borel structure to the Effros Borel structure.To see this we only need to remark that

hF1 ( I F E N - I F/)H#0] )=CAE? I h

and the last set is analytic in X

lytic in

x

AAh(H)#0)

since

( HGN* is an open set ).If

f

h(H)

9

is ana-

is a measu-

MEASURABLE SECTION A N D SELECTION THEOKEMS

82

rable choice function on

2 we put

g(A)=h o f o h,(A)

and

n

it is clear that g is a choice function on X with the desired measurability properties.This concludes the proof.

It can be shown that in the Polish case there exist a sequence fn

that can

fn(A)

of Effros measurable choice functions such

is dense in A

for each closed set A .This

also be shown for a 6-compact metric space.We do not

know whether or not there exist always an Effros measurable choice function in the case of an analytic metrizable space; our conjecture is that this is not the case.

Proof:In the first case we consider the mapping h

h:Y -+ X

defined by

h(y)=f-’(y) .This is indeed a closed

set since the graph of f is closed.The assumption implies that this is an Effros measurable mapping and we obtain a measurable inverse by composing with an Effros measurable choice function.In the last case we obtain by a similar argument an inverse function which is measurable from the

MEASURABLE SECTION A N D SELECTION THEOREMS

83

coarsest Borel structure containing the topology and closed under the Souslin operation to the Borel structure generated by the topology.In particular this function is univer-

sially measurable. Suppose that every continuous surjection from a Polish space onto a compact metrizable space admit a Borel measuC1,C2 C X

rable inverse.Let

be two disjoint coanalytic

.

sets in a standard uncountable measurable space

(X,$))

Let @ be a compact metrizable topology on X

generating

the Borel structure

2

.Let f,g

from Polish spaces (Y,%) and (X\C2)

(Z,e) onto

if S L Y

h(s)=f(s)

now

k:X

sets

S=

Y

+

and h(s)=g(s)

and

Z onto (X,&) if S S Z .Let

-+ S be a Borel measurable inverse of h .The

B,=k-’(Y),

B2=k-’(Z) are Borel measurable in X;

they form a partition of X and we have C2S

( X\ C, )

respectively.We define the continuous surjection h

f r o m the disjoint topological sum

by

be continuous surjections

ClsB2

and

B1 .Hence every pair of disjoint coanalytic sets may

be separated by Borel sets.Now we look over the proof of n

theorem 3.9 .In the argument which shows that X,,\?= analytic we could let a(F)

is

be in an Effros measurable sub-

h

set of Xp .Then we would obtain that the natural imbedding from

%

into

A

X9 maps Effros measurable sets onto

coanalytic sets.The above separation would now show that this imbedding is measurable in contradiction with theorem

3.9

if both

z0

and

4

Xp

are standard and the further

conditions in theorem 3.9 are not fulfilled.This contradiction concludes the proof.

MEASURABLE SECTION AND SELECTION THEOREMS

84

Notes and remarks to chapter 4: The section theorem is as stated due to Hoffmann-Jmgensen and in [f6] he shows how t h e most commonly used section and selection theorems may be deduced from it. The existence of Effros measurable choice functions is due to the author.It seems however that these results was known for compact spaces even before the Effros Borel structure was invented. The negative result concerning non existence in general of a Borel measurable inverse is due to the author and

J.E.Jayne in collaboration.

It is most probable that there does not need to exist Effros measurable choice functions in the case of an analytic metrizable space;we have not been able to decide this question,it may be related to the problem under what conditions the Effros structure is standard. A very important application of the section theorem

is to find a Borel set in a Polish group which intersect

-

every sideclass of a closed subgroup in exactly one point. \

In K=[O,f

the Cantor group we have the seemingly

nice equivalence relation k co h <==+

1-N

1

defined by

k(n)#h(n)]

is finite .It is an

amusing exercise to show that this equivalence relation does not admit a universally measurable section nor does there exist a universially BP-measurable section (indeed no BP-measurable section).This may be,shown using the results of the following chapters.

CHAPTER 5 CONTINUITY OF MEASURABLE ‘HOMOMORPHISMS’ BAIRE CATEGORY METHOnS

We state and prove several results of the type that measurability and a certain algebraic property for a mapping together may imply continuity.The methods are Baire category argwnents.Most o f the results for groups carry over to the case where we assume universially measurable instead of BP-measurable although new (measure theoretic) methods must be used.Surprisingly it can be shown that this does not hold for the last results of the present chapter. Thus they belong to a special category although they are analogous to the preceding results. The first result which states continuity of a homomorphism fulfilling a measurability condition seems to be due

to Banach (see[3

1).Although the method

of Pettis (seepy])

is a considerable improvement (because it can be applied

to more general classes of groups and can a l s o be used in the proofs of some open mapping theorems) we shall consider a slightly modified version of Banach‘s proof,because of its brevity and elegance. By a topological group dorff topological space (G,&) sition

o

( G , o , & ) ,we understand a Haus-

equipped with a group compo-

such that the mapping

(g,f) -+ g

0

f-’

is a continuous mapping from G L

into G .We denote by

e

86

CONTINUITY OF MEASURAHLE ‘HOMOMORPHISMS’

the neutral element in the group G (eG ‘.ifany confusion is likely to arise),It is well known that the set of neighbourhoods of

e

(in the sequel ,,neighbourhood,,without

further specification always mean neighbourhood of e ) is a filter

with the properties

i.1

Q!V =b{

ii) For each U C 29 there exists a V

GV

that V o V-’C U iii) For all g& G and V € v Conversely,if a filter

, gVg-h

such

.

fulfills these conditions

then there exists exactly one topology on G

with which

G is a topological group with neighbourhood base

.

The assumption that all of the topologies are Hausdorff makes some proofs a little easier ,but is not of great importance. A left invariant metric

on G is a metric with

d

(LI) d(gh,gf)=d(h,f) f o r all shown that the topology on G

g,h,fEG .It can be

is always generated by a

family of left invariant semimetrics (a semimetric is a metric without the property x=y <==9d(x,y)=O ) in particular that the topology of a topological group is always completely regular (see [2U],

p 68

).The topology is gene-

rated by a left invariant metric if and only if the filter of neighbourhoods has a countable base.The topology is gerated by a twosidedly invariant metric if and only if the filter of neighbourhoods has a countable base of sets invariant under inner automorphisms (an inner automorphism is a mapping of the form

Q

-j

h o g o h-’ ,heG ) .

87

CONTINUITY OF MEASURABLE ‘HOMOMORPHISMS’

A topological group is called Polish if it is Polish when considered as a topological space.A Hausdorff topological space is called a Baire space if every open set is of f: (X,&) --? ( Y ,

the second category.If from the Baire space

(X,@)

p)

is a mapping

(Y,p )

to the Hausdorff space

with a countable base,then

f

BP-measurable if and

is

only if there exists a set A c X

containing a dense

set such that the restriction of

f

to

A

G6

is conti-

nuous on A. To see this let

Pn€p

On€v be such that

be a countable base and let

OnA f-’(Pn)

is of first category.

OD

A=X\ ( n-. I, (on& f-’(Pn)) .)

We set

Now let

f:G -+H

be a Bore1 measurable homomorphism

from the Polish group G

to the separable metrizable group

H .Suppose we may choose a sequence and

f(gn) k P ,hliere P

Now let A S G

Let

set such that

gng -9 g

f

is con00

A - n<’AT)A. 1-n=t and gng,g&A .Hence

f is also continuous on

g € A 1 .We have

gn -9 eG

with

is a suitable neighbourhood in H.

be a dense Gs

tinuous on A.Then

%€G

f(gng) -9 f ( y ) ,which obviously gives us a contradiction ( f(gn) -3 f(eG)=eH ).This concludes the proof of the fact

that

f

is necessarily continuous.The following result

(Pettis ’s lemma) is a considerable advance.

CONTINUITY01: MEASURABLE ‘HOMOMORPHISMS’

88

Proof:Let

060 be such that

first category.Choose such that

W-’s

g-’O

gc0

But

(04 Oh) d ( A /?Ah)

,

is of the

0

and an open neighbourhood

.Then for every

O n Oh ZgV ; in particular

A4

0 L! Oh

h GV

V

we have

is a non empty open set.

is of the first category,hence A n Ah

is non empty (note that every open set is of the second catagory since otherwise G would be locally of first category and therefore of first category in contradiction with the asswnption that A heA-’ o A , hence V C A-’

is of second category).This shows

o A

or

71”s

A o A-’

.This con-

cludes the proof. A topological group is called analytic if G

lytic when considered as a topological space. G 6 -bounded if G

is ana-

is called

can be covered with countably many left

translations of every neighbourhood.

Proof:Let U be a neighbourhood in H. We shall

89

CONTINUITY OF MEASURABLE ‘HOMOMORPHISMS’

investigate f”(U)

and show that this set is a neighbour-

hood in G .Let V be an open neighbornhood in H that V-lo V C U .We choose a sequence hn6 H OD

H= ,1! hnV .Now we have an no

such that

*

such that

1/

f-’ (h V) ,hence there exists n f-’(h V) is of the second category. G=

ht9

Applying Pettis‘s lemma,(note that f”(h surable) we obtain

such

(f”(h

V))”o(f”(h

V) is BP-meaV ) ) S f”(V-’O

v)

f-’(U) ,where the first set is a neighbourhood.This obviously shows that f-’ ( V )

is a neighbourhood.Every analytic group

is 6-bounded,being a Lindelwf space when comidered as a topological space.The remaining part of the proof now proceeds in the same manner ,making use of the fact that the image of an analytic set by a Bore1 measurable mapping is analytic and therefore has the BP-property (of course we also make use of the fact that both groups are analytic). A semigroup of continuous mappings of a topological

space into itself acts transitively on the space if every point may be transferred to any other point by a mapping in the semigroup. From the following theorem it

c a n be seen that the

preceding result to a certain extent carry over to the case where a Hausdorff topdogical space is given with a transitive group of homeomorphisms of the space onto itself. In the proof of the following theorem the regularity assumptions on the topology is very essential.It may however be that another proof could be made without this assumption.

It may also be that this condition is automatically fulfilled.

CONTINUITY OF MEASURABLE ‘IIOMOMOKPIIISMS’

90

$)s 0

t h e n we a l s o have

f tT

every,

i s both

, tide ,--~-,,-,--,if,,------

a n ~?-+a ---------continuous .

8 -+p

A b i j e c t i v e mapELbg

eveq ----

fE T

+X

h:X

which commutes w i t h

&’ -+@

i s automatically

(X,&) ....................... t-----------------a b l e base o r a n a l z---------t i c a n d r e g ---ular.

h

i s Borel measurable and

Proof:We choose a dense respect t o

&)

ybU

of

@-+?

is

A

P be a ?-open

We choose an

Gg

set

AGX (with

GOU

continuous.Let

b-neighbourhood of

@-open s e t

be a r b i t r a r y and y.Then

i s r e g u l a r w i t h a coun-

topology) such t h a t t h e r e s t r i c t i o n o f t h e

i d e n t i t y map t o and l e t

continuous i f

U G

such t h a t

x&A

in

x

PflA=U/)A

X.

.Let

be an open d-neighbourhood

i s of t h e second c a t e g o r y i n

X (with

respect t o

@-topology) i n p a r t i c u l a r non empty.To s e e t h i s

we choose

f€T

that

such t h a t

f (U) I) f ( G ) 4 A

f ( y ) € A .It i s e a s i l y s e e n

i s a @-neighbourhocd of

f (y)

in

t h e subspace topology hence i t i s of t h e second category. The c o n t i n u i t y of GAU

f

and

f-’

therefore implies t h a t

i s of t h e second category.Hence

GO P

i s non empty.

91

CONTINUITY OF MEASURABLE 'HOMOMORPHISMS'

We have shown that y

belongs to the ?-closure

of

P .From the regularity of the ?-topology it f0llCJk.sthat

every point

xbA

tity mapping from

is a point of continuity for the iden(X,&')

(X,$)).Because

to

such continuity points is

T

the set of

invariant,it o n l y remains

to be shown that a set A with the properties above may be choosen in the case where both

To see this let Let Gn let

Q

h:h*-+X

be

0

and

are analytic.

surjective and ?-continuous.

be a countable base for the topology of

be the coarsest topology on X

sets Fn=h(Gn)

are open and closed.Clearly

and

such that a l l

Q

is separable

and metrizable.Moreover Q is finer then the ?-topology on X and every Q open set is contained in the coarsest 6-field containing the

d

topology and closed under the

Souslin operation (since this is the case for Fn).Now we easily obtain the desired set. To prove the last part of the theorem we remark that every f € T h-'(&')=P-'(O)

is continuous with respect to the topology

I

Od$.Hence

( l ' h

@ ) c @ according to the

result above.This concludes the proof of theorem

5.3

.

CON1 INUlTY OF MLASURABLE ‘HOMOMORPHISMS’

Y!

Proof:Let

h:he-+ G

be a continuous Surjection.

There exists an n6hm such that Lategory in

G

h(U)

for every neighbourhood

is of the second

U

of

n

.

Indeed,i.fthis was not the case,the Lind-ellafproperty in I? was of the first category in itself.

would imply that G

We may assume h(n)=eG .Let now ence of neighbourhoods of at this point.Let

V

n

that

h(U

P

forming a neighbourhoodbase

be a neighbourhood in G

mother neighbourhood ,with Since h

Up be a decreasing sequ-

and

W

WW-lC v

is continuous,there exists a

p

such

)sW ,hence

h(U )h(U )-’gV .Hence the sets P P ,which according to Pettis’s lemma are neigh-

h(UP)h(Up)”

bourhoods (note that h(U ) is amlytic and therefore has P the Baire property),form a countable neighbourhood base in G

. An application of the remarks before theorem 5.1 now

shows that G If G

is metrizable.

admits a metric

d

whicb is twosidedly in-

variant and generates the topology,we can consider the completion G2

-G

of

d-

(g,h)-+ g o h from

(G,d).The mapping

to G is uniformly continuous (this makes use of the

twosided invariance of

d ).From this it easily follows

that the group structure on G manner to a group structure on 5

can be lifted in a natural such that

becomes

-G (with respect to the subspace topology),G

a topological group equipped with the d-topology.Since is analytic in

G has the Baire property and we can apply Pettis’s lemma. If G

I

is of the second category in G ,then we see from

Pettis‘s lemma that G is open and therefore also closed

CONTINUITY OF MEASURABLE 'HOMOMORPHISMS'

93

(as the complement to an open union of sideclasses).Since

-

G is dense in G first category in Fn

of closed subsets of G

that G s

(I)

OFn

h '

G =

,this shows

.If G

with empty interior such

.Again using the density of G

the fact that G

is of the

-G, there exists an increasing sequence in

5

(and

is of the second category in itself) we

easily obtain a contradiction.This concludes the proof of theorem 4.4

.

The group of homeomorphisms of the unit interval onto itself with uniform convergence is an example of a topological group which is not complete in m y left invariant metric, although there exist a complete metric on the group compatible with the topology and making the group a complete separable metric space.This example seems to be due to Dieudonn6.We do not know whether a group fulfilling the first set of conditions in theorem 5.4 is Polish.

------ j e J .Then ---- H and --- J are, closed and the sum is topologically direct. .......................................... A n analytic subspace S C V of a complete metrizable .................... ....................... topological vector space V is automaticdlz----------closed and ........................ --------------sentation --------- g=hj with h 6 H and

CONTINUITY OF MEASURABLE ‘HOMOMORPHISMS’

94

Proof:Let

p

be the projection on H along J,

According to the assumptions Gr(p)= l(g,h) d G x G This shows that that

p

is a homomorphism a n d

p

.

I h e H , gh-’e J

p has an analytic graph and therefore

H and J

is continuous.Hence

We may assume that V

are closed.

is separable.If

S

fulfills

the conditions we may choose an algebraic complement 9’ which is 6-compact in particular analytic (as a countable union of finite dimensional subspaces).From the preceding results we conclude that

S and

S’

are closed and an

elementary application of the Baire category theorem now concludes the proof of theorem 5.5 Most of the preceding results carry over

(at least

partially) to the case where universal measurability is assumed,although new (measure theoretic ) methods must be used.Surprisingly enough,this turns out not to be the case with the following results (see the next chapter) although they are somewhat analogous to the preceding ones. Let

(Mn,Qn) be a sequence of Polish spaces.We con-

sider the space M = T Mn

equipped with the (Polish) pro-

duct topology.We consider on W

the equivalence relation

defined by :

X-Y

==+ {nei

I x(n)+y(n)j

The equivalence class con’hining

A mapping xcoy

f

defined on M

===) f(x)=f(y)

x

is finite

.

is denoted

by

[xJ

is said to respect co if

.

The following result,which is in essence contained in a theorem of Oxtoby

(see L20l) ,is entirely analogous

.

95

CONTINUITY OF MEASURABLE 'HOMOMORPHISMS'

to the so-called zero-one law in probability theory.

-M=

Proof:We define

Mi, L3

-

,where Mi, j=Mi .M is

equipped with the product topology,which of course is Polish.Let e

M

T be the countable group of homeomorphisms of

which are defined by a permutation of finitely many

indices some

(i,j) such that each

(i,k) for some

k .Define the projection

(Px)(n)=x(n,l) .Since P

by

open,P-'(A)

On

If

P:fJj: -? M

is continuous,surjective and

set in 'M .Then B=A t(P-'(A)) trT set in 3 .Let c=fx&M I T(X) is densej.

is a dense

is also a dense

(i,j) is transferred to

G$

Gs

z,

is a countable base for the topology of 00

C = 0 T(On) .This shows that

then

. COB

nv

set in

M

choose

y e C O B .For

14flA

C

is therefore non-empty,and we

x

=

Py

Gs

can

it is easily seen that

is dense.

We may choose a dense Gs set A striction of f

is a dense

respects

to

f &

,

A

such that the re-

is continuous on

A .Because

the preceding results shows that

f

is

constant on A .This concludes the proof. A

mapping

T:X=fO,l\N

-+

(G,+,@) ,where

G

is

CONTINUITY OF MEASURABLE 'HOMOMORPHISMS'

96

an abelian topological group,is called finitely additive if

qP(x+y)=q(x)+@y)

for all dlejoint

x,yeK (we

shall in the sequel often tacitly identify elements of K

h

with subsets of

).

is called countably additive if

where the limit holds uniformly in to the uniform structure on all invariant

0

G

xCK

with respect

generated by the set of

continucus semimetrics ( the uniform struc-

ture induced by the group structure).The element is defined by

en E K

en=(O,o.,l,O,o.o.) ( 1 is in the n'th

position). Theorem ------- 5.7:Let --ping from gmzg

K='$I,lIh

p

be a finitely additive m a p

into the separable metrizable abelian

(G,+,@) .Then ----

cp is countablx-----------------additive if the re_----------

-----------striction of Cp to any velg --- BP-measurable. ---------

compact subgroup of

Proof:We may assume that d

G

K

Ls-re&atz-

is complete and

an invariant (complete) metric on G generating the

topology.We first show that P

W (XI=6-im Xq(en)x(n) a:1

is well defined since the limit exists uniformly in x e K with respect to

(y =

d (it does not; follow from this that

! ).Suppose that this is not true.We easily see

that there exists an

e

> 0

and a sequence

ykbK

of elements in Xo=fxdK 1 f n a h I x(n)=ljis finitei

such

CONTINUITY OF MEASURABLE 'HOMOMORPHISMS'

that

ykyh=O

if

We choose a dense

h#k

E

tion of @ to A

and such that

set

AgK

by

x,(n)=x(n)

otherwise,and xk -9 x ,and

if

xi(n)=x(n)

otherwise.Then both

xL-+ x

xk

if

OBut

yk(n)=O yk(n)=O

and

,hence

p(xL) -3 p(x)

is countable,

is invariant with respect to trans-

A

lations with elements in KO .We choose xk,xi6K

d(p(yk),O)> e.

such that the restric-

is con1;inuous.A~ KO

we may assume that

91

tinuous on K,Now the mapping

and define

and xk(n)=l and

xi(n)=O

xi belong to A Q(xk)

p(xk)-

This contradiction shows that

x6A

(/

-+

?(X)

,

and

p(yk)

-++

is well defined and con-

~ ' ( x ) = p ( x ) - ~ ( x ) has

the same measurability properties as 4p and is finitely additive.Furthermore

4.6

q'(en)=O

for every

now shows that there exists a dense Gs

such that

cp'(x)=g

n .Theorem set A g K

for all x B A .At this point we need

a small lemma.

Proof of 1emma:We define g( (a,b))=a-ab

and

h( (a,b) )=ab

g,h:K2 - 3 K

.These mappings are

continuouslopen and surjective.Hence g-' ( A ) are dense

e-l(A)n

Ga

h"(A)

by

and

h-' (A)

sets in K2 .We choose (a,b) in the set 0 (AXK)

which is a dense Gh set in parti-

cular non-empty.Now we put x=a, y=g((a,b))

and

z=h((a,b)).

CONTINUITY OF MEASURABLE 'HOMOMORPtlISMS'

98

It is easily seen that

x,y,z

together fulfills the

requirements of the lemma and this concludes the proof of the lemma. Since we have 7' (x)=g shows that

g+g=g ,hence

choose

x 6 (u+A)4 A

X,YG A

and

on

K) .Hence

q'(u)=O+O=O Kv= i x t K I

x+y=u

for all x% A,the above lemma g=O

and set

.Let

~ = ( l , l , . ~ ~.We kK

y=x-u .We then have

(with respect to

the group structure

xy=O and x+y=u ,and consequently .Let

vb K

be an arbitrary element.

vn:x(n)< v(n)i

is then a compact subgroup of

K . A repeated application of the argument above shows that p'(v)=O

.This proves that

=y/

,which gives us the

de;;i.red r e s u l t and concludes the proof of' theorem

5.7

.

It is easily to show that the assumption that the group is separable and metrizable is not very essential,

It is enough to assume that the image of group valued measure 4p

K

is 6-bounded in

by the

G .This is

even automatically fulfilled if we assume that Borel measurable.For details of proof

see 6 2

is .In $his

paper it is shown that any finitely additive Borel measurable group valued measure is countably additive. Many promising possibilities exist for applications of theorem 5.7 .We give a few examples which far from ex-

haust them.In what follows,we assume a little measure theory.

CONTINUITY OF Mk:ASURABLE 'HOMOMORPHISMS'

99

Proof:We define a finitely additive measure on

3 by

An€$

v(A)=L( xA).Clearly

if u(A)=O .Let

be a sequence of disjoint sets.We define h:K -9 I,"

It follows from Lebesgue's that

v(A)=O

v

h

theorem on dominated convergence

is continuous from the usual topology on K

the weak topology

6(L@,L,).Hence

4p'x)=L(h(x))

to

is a fi-

nitely additive universially BP-measurable function on K. Theorem 4.7

shows that

v is countable additive.The Ra-

don-Nikodym theorem now implies that v is induced by an L1

function.As it follows from theorem

5.2

that L

is continuous in norm and since every function in Lb" c a n be approximated in norm by stepfunctions,we caa deduce that L

is induced by a L1

function.

CONrlNUlIY 01 MI ASUKABLI 'HOMOMOKPHISMS'

I00

d (@ )

Proof:Let @(the

be the field generated by

0

smallest paving containing

and closed with respect

to complements and finite unions).We intend to show that

A( @)

DQ

for every

FcD

closed set

and each

such that

e

> 0

there exists a

u(D \F) <, e

.The paving

D 6 (R( 0 ) with this property

consisting of all sets

contains the closed sets and is closed with respect to finite intersections.It is therefore sufficient to show

G

that every open set has this property.Let set such that

be an open

is non-empty and let

A=X\ G

precompact metric on X

d

be a

compatible with the topology.

For e > 0 ,we define

Se=L x e X

I

d(x,A)=ej

.Let

en

be a strictly decreasing sequence of positive numbers tending to

zero and assume that

We set

S=US ~, 00

( x ) = u ((

x ---3 ( the

d*

and define

nao

c)x(n)Se

(pn=f

n

u(Se )=0 for all n n. 47:" -3 R by

.

) U A)-u(A)) .As the mapping

x(n)Se )UA is d* continuous and n topology generates the Effros Borel structure,

it follows that 4 p is Borel measurable on K

.

4p is ob-

viously finitely additive.Theorem 5.7 now implies u(S)=O

.

Now set R(n)='jxe X

1

en,,

R(l)={xoX

I

dCx,A) 2 el{

2 d(x,A) 2 en]

for

n 2 2 and

bD

As before,we define q , ( x ) = u ( ( n x(n)R(n))UA)-u(A)) ~ and use theorem

5.7

to show that

tive,But this implies that G

v1

is countably addi-

can be approximated in

measure from inside with finite unions of the R(n)'s,

u

CONTINUITY OF MEASURABLE ‘HOMOMORPHISMS’

101

thus with closed sets a strictly positive distance away from A,

To obtain the above conclusion it would have been sufu

ficient to assume that

was measurable with respect

to the 6-field generated by the sets analytic with respect to the Effros Borel structure. Now let The measure setting

2

denote the compact d-completion of X.

u

induces a Borel measure

u(B)=u(Brl X) ,where B T:A

The mapping

3

stem of sets in

X

-+

is analytic in

AllX

Ti

on

-X

by

d

is any Borel set in X

.

n

h

to Xc .The sy-

goes from X

intersecting a given analytic set in 6

X

,as it is obtained by projecting

TxX

an analytic subset in

onto the first coordinate

axis. From this it follows that

h

is measurable if X

T

has the Effros Borel structure and

4

X

the Borel structure

generated by a11 sets which are analytic with respect to the Effros Borel structure. The preceding result now implies that C on (i.e.

d(2 )

with resepct t o the paving of compact sets De

for every

pact set C C D this that

is ,,tight,,

5

&x)

and

such that

e

> 0 there exists a com-

u(D\C) 5 e ).It follows from

has a unique extension

v

from

J(?)

to a countably additive probability measure on a ( z ) . A s

v

is tight on the whole Borel field,it follows that v=E (note that v and

‘u

coincide on compact sets and that

both are probability measures ).This shows that u is countably additive.If conversely u

is a countably additive

CONTINUITY OF MEASURABLE 'HOMOMORPHISMS'

102

probability measure on the Borel field of X easily seen that u

is

then it is h

d*

upper semicontinuous on X

,

hence in particular measurab1e.Thi.s concludes the proof.

(X,6)

Let

h

a subset

IgX

be a Hausdorff topological space.We call

an ideal if the following conditions are

satified : i) X e I

and

1#0

ii) A G I

and

BcA

iii) A,B 6 1

===$

==+ A U B BI

.

B6I

An ideal I is maximal if it is not contained in a strictly bigger ideal.This condition is seen to be equivalent to the property that A imply

c~.(A')EI

.

e 2 ,A91

AfX

and

together

Proof:Using the Hahn-Banach theorem.,weconstruct a finitely additive probability measure field in X mality of

satisfying

u

on the Borel

A € I ==+ u(A)=O .From the maxin

I it follows easily that ,for A e X ,u(A)=O

implies that A

belongs to A

I .Therefore u is Effros

measurable as a function on X .An application of theorem

CONTINUITY 01' MEASURABLE 'HOMOMORPHISMS'

y.9

103

concludes the proof,The remaining part of the theorem

is proved in a similar m y

.

It would have been sufficient to assume in the above results universally BP-measurability with respect to the Borel structures in question.

At the time of writing ,the author has under preparation some applications of an improved version of theorem

5.7 which roughly speaking state that in an abelian topological group,subseries convergence is independent of the topology and only depends on the Borel structure. Notes and remarks to chapter 5: The fundamental result contained in theorem 5.1 obtained by Pettis in 1950 (seeC2y]).Theorem

was

5.2 is ta-

ken from [5f6] and is a well known application of PettiS'S 1emma.Theorem 5.3

and

5.4 are unpublished results due

to the author.It can be shown that most locally convex t o pological vector spaces occurring in functional analysis are analytic o r even standard (see196J) .Hence theorem 5.4 explains why many of them are of the first category in themselves. Theorem 5.5

is a (probably known) easy application

of theorem 5.2 .It is not known whether or not every coanalytic hyperplane 3

in a separable Banach space V

is necessarily closed but it follows from theorem 5.5 that any analytic H is closed,and so in particular that

H is closed if H is Borel measurable.Closely related to this is the interesting open problem of whether or not

CONTINUITY OF MI:ASUKABLI

104

‘IIOMOMOKPI11SMS’

every first category hyperplane in V

is closed

(V being

a separable Banach space)(note that every coanalytic hyperplane is BP-measurable and therefore (by Pettis lemma) of the first category),It is easily seen that the problem is equivalent to the problem of whether or not every hyperplane is of the second category in itself. The theorems 5.6-5.10 are due to the author and were published inbO].In

this paper it is also shown that they

do not carry oveL‘ to the case where wiiversal measurability is assumed.

To the authors mind one of the most interesting open problems in abelian topological groups is the following hypothesis:

,, Let

(G,+,@)

be an abelian topological group which

is non-trivial (i-e. contains more then just O).Let an ultrafilter on G.Then there F E P such that

dense in G

F-P={g eG

I

be

exists a filter set

g=fl-f2,

fl,f2C F]

is

,,

If this hypothesis could be proved it would yield a significant contribution to the ambitious program to find some structure theory af abelian Polish groups.It would yield the existence of a non-trivial continuous positive definite function on the group. The above problem is not related to the work in the present book.

CHAPTER 6 MEASURABILITY PROPERTlES OF LIFTINGS

SOME NEGATIVE AND POSITIVE RESULTS

Using the continuum hypothesis we prove that a linear lifting having nice measurability properties with respect

to universal measurability exist .Such a lifting can however not be multiplicative and we show that a lifting has

bad measurability properties wTth respect to universal BP-measurability,

The next theorem is the key to all our positive results.

It was published in [@]with

only an indication of the idea

of proof.Since the result is useful in other connections it would be interesting to have a proof independent of the continuum-hypothesis.

Gn= cl.conv. (

I fn,fn+,,.. 3

)

MEASURABILITY PROPERTIES OF LIFTINGS

106

Proof:Since the set of all probability measures 2alef we may defined on has cardinality at most

3

choose for each countable ordinal number w

uu,

a probability

such that each probability is choosen at least once.

By transfinite induction we choose,for each countable ordinal w of

for

fn's

such that

%

of finite convex combinations

ex€ G,

conv. (

and such that

exist

for each

we have

2 c:(x)

x6 X

~ycz(x)

and

almost every xe X

ordinal W <

for all

c(yn

a sequence

Y ,en+, (x),

,

nb N

and

5

..

) 5 conv. ( en( x) ,en+, (x), W

W

)

.Suppose this has been done

for all countable ordinals less then the countable ordinal

y

.Then there are two cases.If

we choose a sequence

up

increasing and with limit

4)'

is a limit ordinal

of countable ordinals strictly .We consider

g =cup P P Since this is a uniformly bounded sequence of universially

measurable functions there exist a weak accumulation point as element in g in L ~ ( x , $ , ~ ? ) if we consider gP 2 L (X,a,u ).But then we may choose a sequence of finite g; 7 convex combinations of gp's (with index number > p ) converging

uy almost everywhere to

g .If we put

! c = g; clearly we have d0ne.A similar but even easier argument concludes the proof if is not a limit ordinal.

Let K={O,liN

be the Cantor group;with its usual pro-

duct topology and product group structure K

is a compact

metrizable abelian group.We consider K with the Bore1 structure generated by the topology.

107

MEASURABILITY PROPERTIES OF LIFTINGS

We define a sequence of‘ continuous functions on K by n

fn(x)=(l/n)

v:1

Using the preceding theorem o n

x(v)

this sequence we obtain a finitely additive probability \

measure defined on the subsets of N which equals the arithmetic density of a set whenever this density exists. This measure is universially measurable but of course not countably additive.We may a l s o conclude using a result from chapter 5

that theorem 6.1

is felse with universal

BP-measurability instead of universal measurability.

(X,d)

Let space and consider

be a separable and separated measurable

u a probability measure defined on Lm(X,a,u)

8

.We

with the weak topology b(L”,L1

)

and the Borel structure generated by this topol.c;gy.This

Borel structure is of course smooth.

Proof:Let finite subfields of

an

be an increasing sequence of

3 with t,2n (10

generating

3

.Let

MEASURABILITY PROPERTIES OF LIFTINGS

108

be a linear version of the conditional expectation

(Pn

with respect to

an

I If1 I-

1(4pnf)(x)l<

for all

.We may assume

x B X .Clearly the function

is Borel measurable on L-3 X

by integrating

f

(the values is calculated

over the sets in

now follows froni theorem 6.1

(f,x) -- ( pnf)(x)

2, ).The

theorem

and the martingale convergence

theorem. We note that a linear lifting with the properties in cannot be mutiplicative if the measure u

theorem 2

is continuous.This follows from

G

Proof:Let

be the class of all equivalence

classes of measurable subsets of X (modulo u-nulsets). The topology on G

defined by the metric

d(A,B)=u(AA B) is Polish

(this metric is even complete ; we use the no-

tation

D=(A\B)U (B\A)

from

AA

G

into L”

).The natural imbedding map

defined by

A

-9

xA

is continuous

(but not a homeomorphism onto the image).Hence this imbedding is a Borel isomorphism of G Lm .We note that

G

is an abelian Polish group with

d-topology and the composition law (A,B) - - j A d B

onto a Borel subset of

MEASURABILITY PROPERTIES OF LIFTINCS

q : A -+

The mapping

(-l)L(XA)

109

is a character

on G which is universially measurable according to the preceding remarks ( G is identified with its image under the natural imbedding ).This implies that

is d-conti-

nuous (see next chapter ).From the continuity of 4p it follows easily that

on

L

in the theorem,Since L

G

has the form stated

is norm continuous (since L

multiplicative) and each

f c L”

is

can be uniformly approxi-

mated with stepfunctions this concludes the proof of theorem 6 . 3 The next theorem is in a striking contrast with theorem 6.3.

measurable on -------------

Loo

Proof: It follows from the preceding chapter (theorem 5.8) that if the linear functional

L on L”

is measurable with respect to the Bore1 structure mentioned in the theorem Let BnGp

,then L

is induced by a function in L1

be a sequence the charateristic functions of

which are weakly dense in we have that

L” .For almost every

xeX

.

MEASURABILITY PROPERTIES OF LIFTINGS

I10

If

the set S

of all x e X

is universially BP-measurable on L" for

there exist

u

such that

f

-+

(pf)(x)

is not a zero set

such that

xoB S

(cP(X

))(xo,=3: (xo) for a l l n Bn Bn with (yf)(xo)=
for all

~ C L -,in particular

But this implies that

[ xo3

tion of

equals the charateristic func-

g

u-almost everywhere This contradiction

with the assumption that

u

is continuous shows that

S

is a z e r o set.

Some problems with relation to the preceding results remain open,Of course it f o l l o w s from theorem 6.4 that f o r no --

lifting the mapping

(f,x) -+ (yf)(x) is measu-

with respect to %he product Bore1 struc-

rable on

Lgx X

ture ( u

is assumed to be conti.nuous ).But it may be that

one is able to find liftings for all

f 6 La.

9 with

pf

3-measurable

This is in fact known if the continuum-

hypothesis is assumed

(seeL993) but is open without the

continuum-hypothesis. Notes and remarks to chapter 6: The results in this chapter are unpublished results due to tke author. \

Let K=t0,ljN be our familiar Cantor group.The function

fn

on K

defined by

fn(x)=x(n)

is conti-

nuous in particular measurable .Using our results in

MEASURABILITY PROPERTIES OF LIFTINGS

chapter 5

111

and the next chapter it is easily to show

that no accumulation point for the sequence of functions with respect to the topology of pointwise convergence fn is universially measurable or un.iversiallyBP-measurable. In fact no accumulation point can be measurable with respect to the Haar measure o r can be BP-measurable with respect

to the topology.This observation is probably due t o Bourbaki .This shows that it is essential in theorem 1 to take convex hull before making the closure.It is an interesting problem whether or not theorem 1 may be proved without the continuum-hypothesis All

.

,,liftings,, considered in the present chapter

are linear liftings in the sense of Ionescu-Tulcea (seebd). Since

,,good,,applications of liftings need the liftings

to be not only linear but a l s o multiplicative it is of course annoing that no multiplicative lifting can have even the weakest possible measurability property. All known methods to show existence of liftings are

very non-constructive.0ur negative results this since a

,,explains,,

,,reasonably constructive,, method ought to

yield a lifting with reasonable measurability properties.

CHAPTER 7 CONTINUITY OF MEASURABLE HOMOMORPHISMS. MEASURE THEORETIC METHODS. A MEASURE THEORETIC ZERO SET CONCEPT IN ABELIAN POLISH GROUPS

We give a measure theoretic analogue of Pettis's

lemma and apply this to prove measure theoretic analogues of some of the results in chapter 5 .Since measure theore-

tic methods are applied the relevant measurability concept is universal measurabili.ty.The concept of zero set for the Haar measure is generalized to arbitrary abelian Polish groups

.

Let

(G,+,P)

be an abelian locally compact topolo-

gical group.We reproduce some elementary results in Harmonic analysis from LqO3which we shall use without proof. There exist a (modulo multiplication with a positive real number ) unique positive Radon measure

u

on

G

,

invariant with respect to translations.We shall use the following elementary but important property of this measure (the Haar measure ) :

CONTINUITY OF MEASURABLE HOMOMORPHISMS

This statement is

113

,of course, a direct analogue o f

Pettis's lemma and we shall consider analogous applications

to show continuity of universially measurable homomorphisms. It can be shown that there exist universially measurable sets which are not BP-measurable,and conversely.The following results therefore only partially overlap with the previous ones.

= t s C S

I (s+A. ) O A . # 03 =0

lo

Proof:We choose a sequence

.

in€N

such that

each positive whole number is repeated infinitely many times.Let us assume that the conclusion of' the theorem does not hold.By induction,we can choose a sequence

- Ain n for each o f the

with the properties : sn#Ai d(r,r+ sn) i 2-n

different group

ST

,

sn € S

and moreover

p-1

sums

r

of

( v= l,.o,n-l ) .We consider the Cantor \

K=IO,lIN

.We shall make considerable use of the

fact that K is a compact metrizable abelian group equipped with the usual product topology and product group struc-

CONTINUITY OF MEASURABLE HOMOMORPHISMS

114

80

ture.We define the mapping

W

W:K - + S

is clearly continuous.The sets

by

v-’ (Ai)

l(x)=xx(n)sn. n: q therefore form

a countable covering of K

with universially measurable

sets.Hence there exists an

io such that

V-’(Ai ) 0

has a strictly positive Haar measure in X .Then

y l ( A i )-p(Ai 1 0

is a neighbourhood in a

peh

such that

0

K .This implies that there exists

encF’(Ai

) - v-’(Ai 0

n 2 p

(eneK is the element defined by

where

1

)

for all

0

en=(O,.,O,l,O,..),

is in the n’th position ).Hence we have

into consideration how the group operation in K sn€ Ai 0- Aio eludes the p r o o f .

that

taking

is defined )

for n 2 p .This contradiction con-

Theorem 7.1 is sufficient to show continuity of universially measurable homomorphisms in a manner similar to the proofs in chapter 5 .However to obtain a complete ar,alogue of

Pettis’s lemma in the non-locally compact case,

it is necessary to improve the result, We shall define a generalization of the zero sets for

the Haar measure to the case of an arbitrary abelian Polish group. It turns out that this can be done in a way such that many theorems from Harmonic analysis (suitably modified ) carry over to the case of a non-locally compact abelian Polish group.We already have the topological zero sets (first category sets ) in this case.By contrast with the topological zero sets,our zero sets can be used in the

CONTINUITY OF MEASURABLE HOMOMORPHISMS

I15

differentiability theory for Lipschitz mappings between Banach spaces.This may turn out to be the best justification for them.It can be shown that there exist dense Gr sets which are measure theoretic zero sets.Although there are some strong analogies between the two zero set concepts, there is no direct connection.An essential difference is that our zero sets (which we shall call Haar

zero sets)

depend upon both the group structure and the topology, while the topological zero sets only depend on the topology. The Haar measure itself cannot be generalized in a reasonable manner.There do exist invariant measures in the non-locally compact case,but they are neither unique nor 6-finite

.

In the sequel and

d

,

(G,+,

6)

is an abelian Polish group

is an invarian'b metric on G

compatible with

the topology (therefore automatically complete ).We call a universially measurable set

AGG

a Haar zero set

if there exists a probability measure

u

on G (not

unique) such that

xA

x A * U = O

,

denotes the characteristic function of the set A ;

xA

the convolution of the function is the function

(XA

*

U)(X)=

xA* u

on

G

and the measure u

defined by

f G )!A(x+y)u(dy)

.

Therefore our definition is the same as requiring that every translate of the set A sure

u .The measure

u

is a zero set for the mea-

is called a

,,testing measure,,

CONTINUITY OF MEASURABLE HOMOMORPHISMS

116

f o r the set A .We shall show that this zero set concept

in fact gives a generalization of the iero sets f o r the Haar measure in the locally compact case.Assume that G locally compact and let As

h

denote the Haar measure.

is 6-finite,we can use the Fubini theorem to show

h

that for any universially measurable set probability measure

s

is

(

u

AGG

and any

on G :

5 lA(x+y)h(dx) )u(dy)= s(

XA(x+y)u(dy) )h(dx)

If A is a zero set in our definition

and

u

.

a tes-

ting measure for A ,then the right hand side of this equation is zero.Then the left hand side is also zero and this can be seen to imply that A

is a zero set for the Haar

measure (note that the Haar measure is translation invariant). Conversely,if

A

is a zero set for the Haar measure we

obtain a testing measure

u

for

A by choosing u with

density with respect to the Haar measure. Motivated by the preceding result, we call a zero set in our sense a Haar zero set (zero set if no confusion is

likely to arise).It is quite easy to show that a finite union of Haar zero sets is a Haar zero set (by convoluting the testing measures corresponding to each of the sets). The next result is somewhat more delicate.

CONTINUITY OF MEASURABLE HOMOMORPHISMS

Proof:Let G

sures on

?

that

117

denote the set of probability mea-

Without proof we shall make use of the fact

is a Polish space equipped with the so-called weak

topology induced by all functionals of the form

5 f(x)u(dx)

u -3 where Let

is a bounded continuous function on

f

D

be a complete metric on

topology.We choose a sequence on

un

G ) (see[?8]).

generating the weak of probability measures

such that % is a testing measure for %.Then

G

we also have that

vn

is a testing measure for

An

if

vn is any probability having density with respect to a translation of % is obviously an abelian semigroup

.

with convolution as the composition.We denote the neutral element

e

. It is easily seen that

lity measure with mass 1 lation

u;

in

e

is the probabi-

0 .We first choose a trans-

of % such that any neighbourhood of

has a strictly positive

<

measure.For

r

0

> 0 we then

have that is a probability measure which converges weakly to for

r 4 0

. For every weak neighbourhood of

e

e

,we

can then find in this neighbourhood a testing measure with

Xh*

vn=o

By induction,we now choose a sequence

xAn*vn=O

and

vn

0

D(r,r*vn) S 2-n ,where

convolution of different

...

vn6p r

with

is any

v ‘s (p=l, n-I ).Now v=vl P is well defined and we have v=xn * vn * yn for any n

*

.

..

CONTINUITY OF MEASURABLE HOMOMORPHISMS

118

It follows from this (using Fubini's theorem) that a testing measure for each particular An

v

is

and therefore

for A .This concludes the proof of theorem 7.2. A,BCG be two universiE&&x ------measurable --_---------------sets.Let F ( A , B ) = i g e G I ( g + A ) n B is not H a m ---- F ( A , B ) is ----oeen ----in G (possib&X-"EQ). Then

Theorem 7.3:;~:

Proof:Assume viously have A=B

g GF(A,B).For

Zero].

C=( g + A ) n B ,we ob-

g+F(C,C)c F ( A , B ) .Hence we may assume that

and that

a Haar z e r o set.It is moreover

is not

A

sufficient to prove that

is in this case a neigh-

F(A,A)

bourhood.Suppose that this is not the case.By induction, we choose gne G

with

d(r,r+gn) 5 2-n

where

,in a manner such that

%cfF(A,A)

is any sum of different

r

00

gp's

( p= l,o.,n-l ).We define

A'=

A\(U(g,+A)) n:r

from theorem 7.2

that

A'

is not a Haar zero set.We

define the mapping

CQ:K

viously @

-+

is continuous.As geG

there exists a

by

G A'

such that

; it follows cp

@x)=xx(n)g,

.Ob-

n: i

is not a Haar zero set, q - ' ( g +A')

has a strictly

positive Haar measure.In the same manner as in the proof of theorem 7.1 for anv

,

it can now be seen that

gn& (A'-

A') \

n 2 p ,for some sufficiently large number p 6 N .

But this implies

(g,+A')nA'#

0 ,and we get a contradiction

which completes the p r o o f of theorem 7.3 If A G G

.

is a universially measurable set which is

not a Haar zero set,then

O€F(A,A)CA-A

and

A-A

is

CONTINUITY OF MEASURABLE HOMOMORPHISMS

119

consequently a neighbourhood. This is a direct measure theoretic analogue to Pettis's lemma and can be used in a similar way as in chapter 5

to

show continuity of universially measurable group homomorphisms.lf

G is not locally compact,the preceding results

immediately show that every compact set is a Haar zero set. In the non-locally compact case ,there does therefore not exist a (countably additive) probability measure

u(A)=O

u

with

for every Haar zero set A ,nor one such that every

zero set for

u

is a Haar zero set.This is an easy con-

sequence of the ,,tightness,,of every countably additive pro.bability measure with respect to the paving of compact sets

.

It does not seem to be known whether o r not every universially measurable hyperplane in a separable Frechet space is closed.It follows of course from the preceding results

that a universially measurable hyperplane is a Haar zero set.0ur problem is thus the measure theoretic analogue of the similarly open problem of whether o r not every first category hyperplane is closed.These problems seems to be

of the same degree of difficulty.The proof of the following theorem is an adaptation of a similar reasoning shown to the author in the category case by

W.Roelcke (oral com-

munication).It is of course not trivial at all that there exist non-universially measurable hyperplanes,but this follows immediately from the theorem.

CONTINUITY OF MliASURABLE HOMOMORPHISMS

120

Proof:Suppose that

ineI

is a sequence of

bT1 (0) is universially ln L n = n bT1 (0) .Each Ln is a univer-

different indices such that each Q)

measurable.#e set

v:n

lV

sially measurable (proper) linear subspace and therefore a Haar zero set.Clearly the ,mion of the Ln's

is the whole

of E .As E is not a zero set we have arrived at a contradiction. Let A f:A -3

B

and B be two real Banach spaces.The mapping is said to be a Lipschitz mapping if there exists

a C > 0 such that for all x,yCA :

I

f(x)-f(y)

I I < cl lx-Yl

.

B is by one possible defi-

nition a Radon Nikodym space if every Lipschitz function f

from the reals into B

is differentiable almost every-

where with respect to Lebesgue measure. It c a n easily be shown that € o r example all reflexive Banach spaces are Radon Nikodym spaces,but there exists separable Banach spaces

(e.g. L1 ) which are not Radon

Nikodym spaces. It c a n be shown that there are Lipschitz functions which are not differentiable on a dense

set .Hence b the topological zero sets are not well suited f o r differenG

tiability theory.The following recent result of the author

CONTINUITY OF MEASURABLE HOMOMORPHISMS

and

121

Soren Frisch Kier may therefore turn out to be the

best justification for introducing the measure theoretic zero sets.

Proof:We shall just sketch the proof of this result very brief1y.A clzssical theorem of Rademacher is that the theorem is true if A

is finite dimensional.

In the proof of Rademacher‘s theorem it is easy to show using the Fubini theorem that the directional derivative exist in every direction for almost all x (with respect to Lebesgue measure),The main difficulty is to show that the directional derivative for almost all x depends linearly on the direction a

. We have a very short proof

of this using convolutions with smooth functions with compact support

e

In the general case,we choose E 1 G E E 2 ~

e O . O

,an in-

122

CONTINUITY OF MEASURABLE HOMOMORPHISMS

creasing sequende of finite dimensional subspaces whose union is dense.The set Dn is the set of all x G A

for

which the directional derivative rf(x,a) exists f o r all aoEn

and depends linearly on

acEn .By using the finite

dimensional Rademacher theorem on every sideclass of En is a Haar zero set (the testing meawe prove that A\Dn sure is a probability equivalent with the Lebesgue measure). The set D is the intersection of the Dn's.This

concludes

the proof. The preceding result may be proved also for Frechet spaces and the Lipschitz condition need only to hold ,,locally,,

Notes an remarks to chapter 7: The first result known to the author about continuity

of universial3.y measurable homomorphisms is due to Douady (see L38) .Douady's method can only be applied to linear operators between locally convex topological vector spaces. Theorem 7.1

is due to the author and can be used to show

that a universially measurable homomorphism between abelian Polish groups is continuous are due to the author

(seeLqq]). Theorems 7.2

(see[?2]).It

and 7.3

seems rather probable

that the whole of this theory can also be established in the case of a non-abelian Polish group,but some difficulties arise which are certainly not trivial and we have not at yet been able to overcome them.But a l s o there are interesting open problems in the abelian case.

123

CONTINUITY OF MEASUKABLE HOMOMORPHISMS

We do not know whether o r not every family of pairwise disjoint universially measurable non-zero sets is at most countable.Nor do we know whether o r not F(A,B) is non empty if A

and

B are both universially measurable

non-zero sets (this is of course the case if A=B ,because then O€F(A,B)).To our mind the most interesting and important open problem about the Haar zero sets is whether o r not the zero sets are preserved under bijective mappings which in both directions fulfill a Lipschitz condition.

A measurable group a Borel structure

b

is measurable from G2

(G,o,

3)

is a group equipped with

such that the mapping

(g,f)-+ g o f-1

with the product structure to

G

.

In such a group we may define a measurable set to be a zero set if there exist a probability measure defined on such that any

8

(twosided) translate of the set is a zero

set for this probabi1ity.A finite union of zero sets is a zero set .Even if we assume that the group is abelian and the Borel structure is smooth it is not true that a countable union of zero sets is necessarily a zero set. Theorem 7.4 is as stated due to the author and Sfaren Frisch K i m .This result is very closely related to some recent investigations of Mankiewicz

(seep6]),which are

simultaneous and independent of our work.The main difference is that the zero sets were not known to Mankiewicz .Detailed proofs and some applications will probably appear somewhere.

We state the following hypothesis which we are only

CONTINUITY OF MEASUKABLI: HOMOMORPHISMS

I24

able to prove in very special cases: ,,Let A be a real separable and reflexive Banach space and

FcA

an arbitrary norm closed subset.?lhen

for almost every point x a A

,,

in F

there exist a nearest point

.

If this is assumed and the unit ball of A

is strictly

convex then we are able to prove that there exist a unique nearest point for almost every point in A .We have tried

to attack this problem by differentiating the function f(x)=inffIIx-ylI If A

I

Y E F ~

is not reflexive there always exist a counter-

example F which may be choosen to be a hyperplane.0ur attention has been drawn to this consequence of a well known result of James by

Arne Brrandsted

Let H be the Hilbert space of real quadratically integrable functions on the interval from 0 to define a Lipschitz function B f(x)=J sin(x(t) )dt

.

f

on H

7 .We

by

It can be shown that f is not strongly differentiable in any point result sense

,f

(Frechet differentiable).According to our is differentiable almost everywhere in a weaker

(so-called compact differentiable).This example is

taken from [ 4 1and is due to

Sova

.

The above example show that theorem 7.5

cannot be

improved (to Frechet differentiability) without imposing some further conditions on f .Possibly convexity would do

.

CHAPTER 8 MISCELLANEOUS EXERCISES, OPEN PROBLEMS AND RESEARCH PROGRAMS

We discuss some exercises and open problems some of them already mentioned in the book.The word exercise means that the author claims to have a complete proof of the stated theorem,while the word problem means that the solution

is not known to the author. 1) (Problem) Let

( X , a )

be a coanalytic measurable space

with the property that every bijective measurable mapping from

(X,a)

(Y,&)

onto a countably separated measurable space

is a Borel isomorphism.Is then ( X , a )

necessarily

smooth ?

k

2) (Exercise) Let

(the sets [Cx,a[

I

be equipped with the Sorgenfrey topology

aei

and

x

<

a{

forms a basis for

the filter of neighbourhoods at the point

x ).Show that

the Borel structure is the usual one hence standard but that the topology is not even analytic ! 3 ) (Problem) Suppose that the Hausdorff topological space

(X,@) is a surjective continuous image of a separable metrizable space.Is then

(X,@) homeomorphic to a subset

of an analytic topological space ?

4) (Exercise) Let and

v

ving

t~ $

(.,a)

be a measurable space.Let

u

be finite positive measures which agree on a paclosed under finite intersections and contai-

EXERCISES, OPEN PROBLEMS AND RESEARCH PROGRAMS

I26

ning { X , 0 )

.Show that

u

and

v

s(&)

agree on

!

5) (Problem) Is the universal analytic measurable space of type I

or type I1 ?

6) (Problem) Is the non-standard type I and type I1 analytic measurable spaces constructed in chapter 3 not universal ?

7) (Problem) Does there exists on C(h*)

a coarser Bore1

structure (then this generated by the topology of compact convergence) such that

C(k")

equipped with this is stan-

dard or analytic ? 8) (Problem) Is the equalities

sa=a

, a2=a

and

a+a=a

t r u e for every uncountable ison;orphism type of analytic measurable spaces ?' 9 ) (Problem) Let

(X,9) be a measurable space and

fn:X -$ Y a pointwise converging sequence of measurable

3)

mappings from (X,

(Y,@).Is

into the analytic topological space

the limit function necessarily measurable ?

10) (Problem) Let

(X,@)

be a separable metrizable space, A

The infimum topology on X is (by our definition) the in-

#.

tersection of all d* -topologies on X where

d

is a

precompact metric compatible with the topology.1~ the infimum topology Hausdorff ? Under what conditions is the infimum topology equal to the convergence topology ? 11) (Problem) What is the condition on the analytic metri-

zable space

(X,&')

which is necessary and sufficient to

ensure that the Effros structure is standard ? 12)

(Exercise) Show that the union of a family of closed

127

EXERCISES, OPEN PROBLEMS A N D RESEARCH PROGRAMS

sets in a separable metric space is a Souslin set with respect to the paving of closed sets if the family in question is analytically ordered by

C

!

13) (Exercise) Show that a free ultrafilter on

3

is not

analytically ordered by 2 !

14) (Exercise) Show that any well ordered set of cardinality less then the reals is analytically ordered !

15) (Problem) It is a general and seemingly difficult problem to obtain the results of chapter 3 (with modifications) f o r non-metrizable analytic spaces ?

16) (Exercise) Let

(G,o,

0)be

the group of all homeornorp-

hisms of the unit interval I=[O,q

onto itself.The topology

on G is that of uniform convergence.Show that ( G , o , @ ) is a Polish group which is not complete in any leftinvariant metric

(This example is probably due to Dieudonne ) !

17) (Problem) Let H E A be a hyperplane in a real separable Banach space A .Is then H necessarily closed if H is of the first category in A ? 18) (Problem) In chapter

6 we show that there exist a

universially measurable finitely additive probability measure

01!

h

which equals the arithmetic density whenever

this exist (hence is not countable additive ).It should be interesting to know whether or not this can be obtained without the continuum hypothesis and to investigate further the properties of such measures

19) (Problem) Let Polish group and let

?

( G , + , b ) be an abelian non trivial

4I-

be an ultrafilter on G .Does there

128

EXERCISES. OPEN PROBLEMS A N D RESEARCH PROGRAMS

almys exist a filter set dense in the group

Fe$

such that

F-F

is not

G ?

(Problem) Is every universially measurable hyperplane

20)

in a separable real Banach space necessarily closed (note that it is automatically a Haar zero set ) ? f: A e-3 B be a bijective mapping

21) (Problem ) Let

between two separable Banach spaces.Suppose that fills z Lipschitz condition in both directions

f

ful-

.Does then

preserve the Haar zero sets ?

f

(Problem) Let A

22)

be a real separable and reflexive

Banach space and F S A an arbitrary norm closed subset.

Is it true that there exist a nearest point in F for alin the space A (with respect to the

most every point x

Haar zero sets defined in chapter 5 ) ? f:X -3Y

2 3 ) (Exercise) Let

be a surjective open and

continuous mapping from the analytic space

2).Assume that

(Y,

(X,@ )

set in

onto

is a Baire space .Show that

the image of any dense GZ set in X Gf

(X,@)

(Y,p ) and that

contains a dense

(Y,p) is a Baire space !

24) (Problem) The following may be a promising program for the application of Baire category arguments in the theory of countable discrete groups.Similar results may be obtained and similar questions may be posed f o r countable modules

over countable rings

.

Let F be the free group on countable many generators gl,g2,

..... . F

is of course a countable group.Let

M

be the set of all normal subgroups of F .By identifying

129

EXERCISES. OPEN PROBLEMS AND RESEARCH PROGRAMS

a normal subgroup H 6 M with its characteristic function and considering the topology of pointwise convergence we obtain a ,,natural,,compact metrizable topology on M

.

Let now P be any grouptheoretical property (for example to be a simple group,a solvable group or a finite group). We

define

M(P)=

{ HEM

F/H has the property P (

.

In a l l cases occuring in applications it will be easy is Bore1 measurable in M .By an argu-

to show that M(P

ment very similar to the proof of the topological zero one law of chapter 5 we have been able to prove the Theorem -------

:

zf

M(P)

is --

BP-measurable then either

M( PI

G&

A group

Se4

has residually the property P if for

G

any finitely many elements

giEG i=l,.,n

exist a normal subgroup H g G has the property

P .If P

with

gi$H

gi# eG and

there G/H

is hereditary (i.e. is preser-

ved by taking subgroups ) it can easily be shown that density of M(P)

in M

is equivalent with the statement

that any finitely presented group has residually the property P .Since every finitely generated residually finite group is Hopfian and there exist finitely presented nonHopfian groups this shows that M(P)

is not dense if

P is the property of being a finite group.

We have, in mind f o r applications of this in particular

the Pro2ertY PH not t o contain any subgroup isomorp-

130

EXERCISES, OPliN PR0BLI;MS AND RESEARCH PROGRAMS

hic to H where H easy to show that

is a finitely generated group.It is M(PH)

is a G8

set in M .It would

be very interesting to have a proof of the conjecture

that M(PH) is dense if H is infinite.We have only been able to prove this in the case where H is not recursively presented (equivalently : is not imbeddable in a finitely presented group ) . It is also not very difficult to show that M(P) is a Ga

if P is the property of being an amenable group.

We have not been able

to decide whether o r not M(P) is

dense in this case. Generally speaking it seems that one is running into hard problems,when one is trying to decide what of the two possible alternatives of the theorem does actually hold for a concrete property.However we still hope that this ,,method,,may give some new results in group theory. 2 5 ) (Problem)

It has been shown that one may assume as a

settheoretical axiom the hypothesis that every subset of a (reasonable small) space is universially measurable.Then the axiom of choice is no longer valid but a weaker version apply.It should be very interesting to know whether or not the results of C3fido hold with BP-measurability instead of measurability with respect to a measure.If this was the case one could show using o u r results that any finitely additive measure defined on a 6-field is automatically countably additive.0f course this also would exclude the axiom of choice in its strong form.

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