Topology and Borel structure

TOPOLOGY AND BOWL STRUCTURE

This Page Intentionally Left Blank

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

This Page Intentionally Left Blank

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

This Page Intentionally Left Blank

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

that we have

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