Analytic spaces and dynamic programming : a measure theoretic approach

theoretic approach

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Thiemann, J. G. F. (1982). Analytic spaces and dynamic programming : a measure theoretic approach. (Memorandum COSOR; Vol. 8208). Technische Hogeschool Eindhoven.

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-Memorandum COSOR 82 - 08

Analytic spaces and dynamic programming; a measure theoretic approach

by

J.G.F. Thiemann

Eindhoven, the Netherlands November 1982

A MEASURE THEORETIC APPROACH

by

J.G.F. Thiemann

The measure theoretic part of the theory of analytic topological spaces is develloped within a purely measure theoretic framework and applied to dyna-mic programming.

AMS(MOS) subject classifications (1980): 28AOS, 28A33, 49C20, 60B99, 90C39. Key words: analytic sets, Souslin sets, probability measures, dynamic pro-gramming.

§ 1. Introduction

Analytic topological spaces are employed for the solution of certain measur-ability and selection problems arising in dynamic programming. Since these problems are of a measure theoretic nature it is only the measure theoretic properties of these spaces which is relevant, while their topologies serve merely as an auxiliary structure from which these properties are derived

(and in terms of which the spaces are defined).

In this paper the measure theoretic part of the theory of analytic topolo-gical spaces is develloped (in fact generalized), within a purely measure theorectic framework without any appeal to a priori given topologies.

To this end one of the measure theoretic properties shared by analytic topologi-cal spaces is taken as the defining property for a class of measurable spaces, called analytic measurable spaces. Since most applications of analytic topo-logical spaces to dynamic programming are based on the property mentioned, the applicability of the theory is not seriously reduced by this general-ization. The absence of topological arguments however results in a simpler and more uniform theory.

The paper consists of three parts. The first part, comprising the Sections 2 to 7, provides the prerequisites needed for the remaining parts of the paper. It deals with measurable spaces in general and, besides some new results, it contains many well-known facts. The latter are included in order to make the exposition self-contained so that it can serve as an introduction to the subject. In fact, besides the Radon-Nikodym theorem and the martingale con-vergence theorem, only elementary measure theory has to be known in advance.

In the second part, the Sections 8 to 11, analytic measurable spaces are introduced and several of their properties are discussed, notably those which are material to the application of the theory to dynamic programming, which is, by the way, the subject of the third part.

As mentioned above, many known facts have been incorporated. As new

how-ever may be considred a result on Souslin sets comprised in Proposition 12,

as well as the concept of an analytic measurable space as such and some properties of this kind of spaces. Nevertheless on many occasions our line of reasoning was inspired by existing ones, the books by Christensen and Hoffmann-J~rgensen being our main sources. As to the application to dynamic

programming we followed Bertsekas & Shreve and Hinderer.

§ 2. Preliminaries

By

m

we denote the set {1,2,3, ••• } of natural numbers equiped with the

a-alg.ebra of all its subsets. lR denotes the set of real numbers endowed with

the a-algebra generated by the collection of all intervals. lR is the set

lRu {-oo,oo} equiped with the usual ordering and with the a-algebra generated

by the collection of intervals [a,b] (a,b € lR) • Addition and

multipli-cation of lR are extended to lR in an obvious way subject to the convention that ~ + (~)

=

-~ and 0 • (-00)

=

0 • 00

=

O.

Bya function on a set E we mean a mapping of E into lR. A function will

be called positive when its values belong to [0,00]. When~ is a mapping of a

-1

set E into a set F, then ~ denotes the mapping of the class of all

-1

I

~ B:= {x e E ~(x) e B}. For any set E the identity mapping on E is

de-noted by idE' Sometimes a mapping will be dede-noted by its argument-value

2

pairs separated by the symbol f"1-. Example: x »' x + I (x e JR.) denotes the

mapping defined on JR. which assigns to x the value x2 + 1.

Let E be a set. Then for every subset A of E the complement (with respect

to E) of A is denoted by AC• When A is a class of subsets of E, then

Ac := {Ac

I

A E A}. Moreover Ad (resp. As' A

_o'

Aa) denotes the class of

those subsets of E which can be written as a finite intersection (resp. finite union, countable intersection, countable union) of members of A,

while a(A) stands for the a-algebra generated by A, i.e. the smallest

a-algebra of subsets of E which contains A. The class A is called separating if for each pair (x,y) of distinct points of E there exists

A E A such that either x e A and y

I

A or x

l

A and yeA.

Let E be a set and let E be a a~algebra of subsets of E. Then the measurable

space (E,E) is called countably generated if E is generated by a countable subclass. (E,E) is called separated, when E is separating and countably separated when some countable subclass of E is separating.

To simplify the notation we shall as a rule not mention the a-algebra of a measurable space, provided that no confusion can arise. The product of a

family of measurable spaces will always be supposed to be endowed with the product a-algebra. A subset of a product space will be called a cylinder when it equals the intersection of finitely many subset each of which

de-pends on only one coordinate. A cylinder subset of a product of two spaces will also be called a rectangle. The product a-algebra of the product of

the measurable spaces (E,E) and (F,F) is denoted by E 8 F. On the other hand,

for arbitrary classes

A

resp. 8 of subsets of E resp. F,

A

x 8 stands for

the collection {A x B A E

A,

B E

8}

of subsets of E x F. When ~ is a

pro-bability on the product E x F of two measurable spaces E and F, then the

-1

marginal of ~ on E (resp. F) is the probability ~ 0 ~ on E (resp. F), where

~ is the projection of E x F onto E (resp. F).

We conclude this section mentioning Dynkin's theorem, a proof of which can

be found in [Ash, p. 169]. A collection V of sets is called a Dynkin class

when:

i) V A,B E V EA ~B. A \ B € VJ

ii) when (A) IN is an increasing sequence in

V

then U A €

V.

n nE n

n

Proposition 1. Let C be a collection of subsets of a set E such that

Cd

=

C and E E C. Then a(C) is the smallest Dynkin class which contains C.

§ 3. The space of prohabi1ities

In this section the set of all probabilities on a measurable space is equiped with a certain structure, turning it into a measurable space with some desirable properties. These spaces of probabilities play a

pre-dominant role in the sequel; not only do many notions and results find their most natural formulation in terms of this structure, but also results

bea-ring on individual probabilities often can be derived more easily when these probabilities are consid.ered members of the measurable space of all proba-bilities.

Definition. Let E be a measurable space with a-algebra

E.

The measurable space

E

is defined to be the set of all probabilities on

E,

endowed with the a-algebra! which is defined as follows:

! is the smallest a-algebra on

E

such that for every A E E the function

P» p(A) on

E

is measurable with respect to

E.

'"

Our notation is ambiguous, because the space E depends on the set E as well as on the a-algebra

E.

When clarity is needed, notably when more then one a-algebra has been defined on E, we therefore shall write (E,E)'" instead of

'" _E.

The following proposition is a frequently used tool to check measurability of mappings into spaces of probabilities. Considering probabilities to be functions on a a-algebra, one could loosely phrase this proposition as follows:

a mapping into a space of probabilities is measurable iff it is measurable pointwise on a sufficiently large class of measurable sets.

Proposition 2. Let (E,E) be a measurable space and let A be a subclass of E which generates E and which is closed under formation of finite

inter-sections. Then

i)

E

is the smallest a-algebra on E such that for every A E

A

the function

P ~ p(A) on

E

is measurable

ii) a mapping ~ : F +

E

from an arbitrary measurable space F into

E

is

Proof

i) Let B be the smallest a-algebra on E such that for each A E A the

function ~ ~ ~(A) is measurable on (E,B). It follows from the definition

of

E

and from AcE that

BeE.

Now the sets A E E for which the function ~ ~ ~(A) is measurable on

(E,B) constitute a Dynkin class V which contains Au {E}. As the

collec-tion A u {E} is closed under the formacollec-tion of finite interseccollec-tions it

follows from Proposition 2 that V ~ a(Au{E})

=

E. SO for every A E E

the function ~ ~ ~(A) is measurable on (E,B), which implies that

B :>

E.

ii) The "only if" part is a trivial consequence of the definition of

i.

Let therefore for every A E A the mapping

W

A :

i

+ E be defined by

~A(~) := ~(A) and suppose that the mapping

W

A 0 ~ is measurable.

Denoting by F (resp. R) the a-algebra of F (resp. E) we have by i) :

'::! -1 -1'"

t: = a u ~A R and consequently ~ E = a

AEA -1

u ~ -1 (~~lR)

AEA

because (~Ao~)

ReF

for every A E A. SO ~ is measurable.

Some prominent examples of applications of the foregoing proposition are

the following. Let (E,E) be a measurable space and let 0 : E +

E

be defined

o

by o(x)(A) := l

A(x) (ME, xEE). Then 0 is measurable (In the sequel we

shall write 0 instead of o(x) in order to reduce the number of parentheses). x

Also any product probability ~ x v depends measurably on the pair (~,v) as

can be seen by application of Proposition 2, taking for A the collection of measurable rectangles.

Finally let ~ : E + F be a measurable mapping of a measurable space E into a

probabi--I . ,...

lity on F and ~ ~ ~ 0 ~ 1S a measurable mapping of E into F. In particular

we may consider the case where ~ is the projection of a product space onto

one of its components. The foregoing then implies that the marginals of a probability depend measurably on that probability.

Certain properties of a measurable space E are inherited by the space E, as is examplified by the following proposition. More examples will be met with in the sequel.

Proposition 3. Let (E,E) be a measurable space. Then

E

separates the points

of

i.

When (E,c) is countably generated, then

(E,E)

is countably generated

and countably separated.

Proof. Let ~1'~2 E E and ~1

F

~2' Then ~l(A) < ~2(A) for some A E

E.

Hence {~ E

i

I

~(A) ~ ~2(A)} is a member of

E

which separates ~1 and ~2'

Now let (E,E) be countably generated and let C be a countable generating

subclass of

E.

Then by Proposition 2 the a-algebra

E

is generated by the

functions II ~ ~(A) (A € Cd) and hence by the countable collection

{{ll €

i

I

~(A) ~ r}

I

r € ~ , A € Cd} • As

(i,E)

is separated, this

count-able generating collection must be separating as well.

o

Identification of points of a measurable space which are not separated by measurable sets yields a measurable space whose a-algebra is isomorfic to

the a-algebra of the original space. Such an identification therefore does not affect the measurable space of probabilities either and for that reason it is inessential in many situations. We don't however restrict ourself

to separated spaces, because separatedness is not conserved under the for-mation of measurable images,which turns out to be inconvenient.

Finally we remark that the theory which was dealt with in this section can easily be generalized to include bounded measures, not necessarily being probabilities.

§ 4. Universal measurability

There is no essential difference between a probability ~ on a a-algebra E

and its extension on the completion E~ of E with respect to ~, because this

extension is unique. Also measurability with respect to the o-algebra E and measurability with respect to E amounts to the same thing when ].I-null sets

~

are neglected. Now the completion E of E depends on the probability ~ and

].I

one may ask wether there exists a a-algebra which can be looked at as a completion of E for all probabilities on E simultaneously.

The subject of this section is to prove that such a completion does exist and that the measurability notion associated with it has some nice stability properties.

The usefulness of this generalized measurability concept will become evident in the subsequent sections where certain not necessarily measurable sets and mapping emerge which turn out to be measurable in generalized sense.

Definition. Let (E,E) be a measurable space and for each probability ~ on E

A subset of E is called universally measurable i f it belongs to Ell for every

pro-bability J..l on E. The collection of all universally measurable subsets of (EtE) is

denoted by

U(E).

A mapping ~ : (E,E) + (F,f) of E into a measurable space F

is called universally measurable if ~-lB is a universally measurable subset.

of (E,E) for every B E f.

Proposition 4. Let (E,E) be a measurable space. Then U(E) is a a-algebra.

When

A

is a a-algebra of subsets of E such that E

cAe

U(E) , then every

probability on

E

can be uniquely extended to a probability on

A.

Proof. We have U(E)

=

n

E , where the intersection is taken over all

proba-11 J.l

bilities J..l on E. Consequently U(E) is the intersection of a collection of

a-algebras and therefore it is a a-algebra itself.

Now let

A

be a a-algebra such that

E

~

A

c

U(E)

and let p be a probability

on

E.

Then J..l can be extended on

E

and hence on the sub-a-algebra

A

of

E •

11 U

Let 11' be an extension of p on

A

and let A E

A.

Then A E

E ,

so there exist

u

B

1,B2 E

E

such that B} cAe B2 and p(Bt)

=

p(B2). This however implies

U(B₁)

=

11'(B

t) ~ p' (A) ~ pI (E2)

=

P(E2, so J.l'(A) is uniquely determined by p.

From the arbitrariness of A it follows that

u'

is the unique extension of

p on

A.

o

As has been argued earlier identification of points which are not separated by measurable sets is inessential in most cases. The following proposition implies that this identification commutes with the completion of the

a-algebra of measurable sets to the a-algebra of universally measurable sets. So again identification is inessential.

Proposition 5. A pair of points of a measurable space is separated by the measurable sets if it is separated by the universally measurable sets.

Proof. Let x and y be points in a measurable space (E,E) which are separated

by the collection U(E) of universally measurable sets. Then the

probabili-ties Ox and 0y' which are defined on the a-algebra of all subsets of E, do

not coincide on U(E). By the preceding proposition this implies that these

probabilities don't coincide on E either and that therefore the points x

and yare separated by

E.

o

Proposition 6. Let E and F be measurable spaces, , : E ~ F universally

mea-surable and B a universally meamea-surable subset of F. Then

~-lB

is a

univer-sally measurable subset of E.

Proof. Let

E

be the a-algebra of E. We have to show that ~ -] B E E~ for

every probability~ on E.

-]

Let therefore ~ be a probability on

E.

Then ~

0,

is easily seen to be a

probability on (the a-algebra of) F.

Now B is a universally measurable subset of F, so there exist measurable sub-sets B 1,B2 of F -} such that rp BI such that BI C -} -) C rp B c ~ B2 B c and -I -I

B2 and (uorp )B_I =(~orp )B

2, and hence

- I · -} -I

~(~ B

I)

=

~(cp B2). The sets ~ BI and

-I

cp B2 are universally measurable and therefore belong to E~. From this we

-1

deduce that also rp BEE.

Jl

o

Corollary 7. The composition of two universally measurable mappings is un1-versally measurable.

Proposition 8. Let (E,E) be a measurable space. Then U(U(E»

=

U(E), i.e. in the measurable space (E,U(E» every universally measurable subset is measurable.

Proof. Apply Proposition 4.1 to the spaces (E,E) and (E,U(E», taking for

~ the identity on E.

Proposition 9. Let E be a measurable space and A a universally measurable subset of E. Then the function ~ ~ ~(A) is universally measurable on E.

Proof. Let v be a probability on E and let A be defined on the a-algebra E of E by A(B) := ~f ~(B)v(d~). Then A is easily seen to be a probability.

E

As A is univ.ersally measurable and therefore belongs to E_A, there exist B} ,B2 E E such that B} cAe B2 and A(B2 \ B})

=

O.

Consequently we have, by the definition of A:

E E

We also have for all ~ E E the unequalities ~(B}) ~ Jl(A) ~ _{Jl(B 2). So} ~(A)

=

~(B2) for v-almost all ~ E E.

Now B2 E

E,

so Jl(B

2) is a measurable function of Jl. Therefore the function Jl ~ Jl(A) is measurable with respect to (E) and hence, by the arbitrariness

v

of v, even universally measurable.

o

§ 5. Souslin sets and Souslin functions

Let

A

be a class of subsets of a set E. In general there is no simple

con-struction principle by which the members of a(A) can be obtained from those

of

A.

The only thing we know is that for every A E a(A) there exists a

countable subclass

AO

of

A

such that A E a(A

O)'

In this section a construction principle, the Souslin operation, is

con-sidered which, applied to a class

A,

meeting certain conditions, yields a

class of sets which contains a(A). Moreover the class in question is not

too large, since it is itself contained in the a-algebra of universally

mea-surable sets derived from a(A).

Definition. Let

A

be a non-empty collection of sets and let I be the set

of all finit.e sequences of natural numbers.

By

S(A)

we denote the collection of sets S such that

S==U

lN n nElN kElN

for some mapping A I +

A

(depending on S).

S is called the Souslin operation and for every measurable space (E,E) the

members of S(E) are called Souslin sets of (E,E).

Proposition 10. Let A be a non-empty collection of subsets of some set.

Then

ii) when

A

c

SeA)

(in particular when

A

is closed under complemen-c

tation) then

cr(A)

c

SeA).

Proof.. Let (B.). 'T be a sequence in A and let A :

=

1. 1.E.1Ll nl'.'.'~

Then

A

=

U

nl,···,n. .

1<. 1.EE

So

Acr

c

SeA)

and

Ao

c

SeA).

Next let A E

S(S(A)).

Then

A = U

n

B. 1. (resp.

n

iEJN B.) • 1. B (resp. B_k). n 1 U U

n

A g(l), ••• ,g(k);h(k,l), .•. ,h(k,~)

=

g:E+E h:E xE+E (k,R.)EE XE

U

n

~k(n)'

n:E+E kEE

where for each k the index Fk(n) depends on only finitely many components of n.

Now let v : E u {a} + E be strictly increasing and such that for each k E E

Fk(n) depends on nl, ••• ,nv(k) only. If then follows from the countability of

the sets E v(k)-v(k-I) (kEE) that

A =

for suitable indices G k•

To prove ii) we merely observe that by i) the collection

{S E SeA)

I

SC E

SeA)}

is a sub-cr-algebra of SeA) which contains A.

o

As an immediate consequence of Proposition 6 i) we have that for every class

A the classes (S(A»cr and (S(A»o are equal to

SeA).

Proposition 11. Let (E,E) and (F,F) be measurable spaces, let A be a

sub-class of E such that cr(A)

=

E and A c

SeA),

and let B be a subclass of F

c

such that cr(B)

=

F and B c

S(B).

c

Then S(AxB)

=

S(E®F).

Proof. For each A x B E A x B we have

[S(S(A)

x B)J c

[SeA

x B)J

= SeA

x B) •

s s

So (AxB) c S(AxB) and this implies, by Proposition 10, that

c

cr(AxB) c S(AxB) and therefore S(E®F)

=

S(cr(AxB» c

S(AxB).

The reversed inclusion is obvious.

As a particular case of the foregoing proposition we have S(ExF)

=

S(E®F).

Let (E,E) be a measurable space and (F,F) a subspace of (E,E). Then

F

=

{AnF

I

AEE} and consequently S(F)

=

{SnF

I

SES(E)}. So every ~ouslin

subset of the space (F,F) is the trace on F of a Souslin subset of (E,E).

When in addition F is a Souslin subset of (E,E), then we consequently have

S(n c

S(E).

Again let (E,E) be a measurable space and let E be generated by a subclass A.

Defining Al := A u Ac we have by the preceding proposition (taking a

single-ton for F)

SeE)

=

SCAt)'

Now every member of SeAl) is derived from some

sequence in AI' in a fixed way, so the cardinality of S(AI) does not exceed

:N

IAI

I·

Consequently, when E is countably generated, then

SeE)

has at most

continuous cardinality.

Besides the set theoretic properties mentioned above, Sous1in sets do have some useful measure theoretic features.

Proposition 12. Let (E,E) be a measurable space. Then

i) Each member of

SeE)

is universally measurable.

ii) {ll €

E

I

ll(A) ~ a} €

s(i6

for every A €

S(E),

a E ]R..

iii) When C is a class of subsets of E such that C c E c S(C), then

peA)

=

sup {u(B)

I

B E

C

so and B c A} for every II €

E

and every

universally measurable AcE.

Proof. For each n E ~ and k € IN let n/k denote the restriction of n

to {l, ..• ,k}, i.e. the finite sequence nl""'~' Moreover let n/k ~ m/k

standforVQ,€{I, .•• ,k}nR,~mQ,.LetA= U

n

:N

belonging to E. nE:N klE]N

Let us define the auxiliary sets

Bm/'J. := U

{k~1

An/k 1 n IE :N1 with n/'J. :;;; mh}

:N

for all m IE]N and fl, IE :N.

The sets Bm/i are measurable and for each m Bm/t is decreasing in t. More-over for each m we have ~ Bm/t c

A.

For let x € ~ Bm/t " Then for each t E: E there exist n E: Et with n/ t :::;; m/ t such that x € n A /k •

k:::;;i n

As

a consequence

{n € EE In:::;; m and

is non-empty and decreasing in to Moreover the space {n E: EE In:::;; m} is a compact subspace of EE. These facts imply the existence of n € EE such

that x E: n A / for all t and, hence, such that x € n A /k" So x E: A.

k:::;;t n k kE:E n

Now let U € E and a € lR. Suppose that

(1) 1l (A) ;;::

*

a •

For every i € E there exists an m E: EE such that for every !L E: E

and hence

(2) Vi€E

The existence of such m for every i follows, using induction on i, from the

*

continuity of 1l on increasing sequences of sets and the relations

U

{k~E

An/k I n € EE with n_{l :::;;} m1 , ... ,nt - 1 :::;; mt - 1}

=

= lim t U

{k~""'T

An/k I n € EE with n] :::;; m1,·"· ,nt

~

m_{i } .} m

The statement (2) is equivalent to

(2') llE

n u n

{VEE!V(Bm/t»a-I/i}.

iElN mElNlN tElN

As Bm/t is decreasing in R. we have ll(~ _Bm/t)

=

!!:

~ ll(Bm/R.) for each

m E lNlN. Together with (2) this implies ViE IN 3 mE

JJN

II

(~

Bml .1'.)

~

a - Iii

and hence sup u(n B IR.) ~ a •

IN R, m mElN

As A :::> n Bml R, for every m €: lNlN, we can conclude

R,

(3) _{sup ll(n Bin)}

IN R. mk

mElN

Now (3) is easily seen to imply (1), so for every II €: E and a € lR the

statements (1), (2), (2') and (3) are equivalent.

i) Taking for a the value ll*(A) the equivalence of (1) and (3) gives

*

ll*(A) ~ II (A), which implies that A is ll-measurable. As 1.1 is arbitrary,

A is universally measurable.

ii) The equivalence of (1) and (2') (together with the universal

measurabi-lity of A) implies that the set {v € E

I

v(A) ~ a} equals the

right-....,

hand member of (2'), which is a Souslin subset of E.

iii) Taking once more for a the value 1.1 * (A) we have by equivalence of (1)

and (3) that ll(A)

=

sup ll(~ B_m/t) •

mElN1N k

When there exists a subset C of

E

such that

E

c

S(C),

then

to C, which in turn implies that the sets

n

Bm/t belong to

t

The statement under iii) follows from this for A €

S(E),

hence for

A €

E

and consequently for universally measurable A.

Proposition 13. Let (E,E) be a measurable space. Then the a-algebra of

universally measurable subsets of E is closed under the Souslin operation,

Le. S U(E) = U(E).

Proof. It follows from the prec.eding proposition applied to the space (E,U(E» that every member of S U(E) is universally measurable with respect to U(E) and therefore belongs to U(E) as a consequence of Proposition 8.

The class of Souslin sets of a measurable space is not in general a

a-algebra and measurability of mappings and functions with respect to this class therefore is not defined. Nevertheless many facts ,concerning Souslin sets can be most easily expressed with the help of a certain kind of

functions, called Souslin functions, which are in a sense comparable to measurable functions.

Definition. Let (E,E) be a measurable space and f : E + lR. Then f is

called a Souslin function i f {x €

EI

f (x) > a} e: S(E) for each a € lR.

o

o

The class of Souslin functions on a measurable space is closed under certain

operations as stated in the following proposition. However, when f is a

Souslin function, the function -f need not to be so, because the complement of a Souslin set not necessarily is a Souslin set again.

Note that in the following proposition we use the conventions ~ - ~

=

-~

and ±~ • 0

=

0 mentioned under the preliminaries.

Proposition 14.

i) Let (f) _{n n} ~T be a sequence of Souslin functions on some measurable space.

.Il.~

Then sup f , inf f , limsup f and liminf fare Souslin functions as

n n n n

11 n n n

well.

ii) Let f and g be Souslin functions on some measurable space. Then f + g

also is a Souslin function. ~Vhen in addition f and g are positive, then

f • g is a Souslin function as well.

Proof. Let (E,E) be the measurable space mentioned above.

i) For every a e:

m.

we have

{x e: E

I

sup fn(x) > a} = _{U {x e:} E

I

f (x) _n > a} e: S(E) _0'

=

S(E)

n nEE

and

{x e: E

I

inf fn(x) > a}

=

U

n

{x EEl fn(x) 2 a +

11m}

e:

n mEE nEE

€ S(E)oO'

=

S(E) •

So sup f and inf fare Souslin functions. From this and the equalities

n n

n n

limsup f

=

inf sup f and liminf f

n m n

n nm2=n n

follows.

=

sup inf f the remainder of i)

m

ii) For every a € lR we have {f + g> a} =

U {f > r}

n

{g > a - r} to S (E) dO' = S (E) ,

r€Q

so f + g is a Souslin function.

For f and g positive and a ;:;: 0 we have

{fg > a}

=

U r€Q r>O

{f > r}

n

{g > ~} € S(E)

r

which implies that fg is a Souslin function.

Besides the a-algebra of universally measurable sets there is another a-algebra of subsets of a measurable space which displays equally nice features. The remainder of this section is devoted to the latter a-algebra. The results will not be used in the sequel.

For any measurable space (E,E) let L(E) be the smalles a-algebra of

sub-o

sets of E which contains E and which is closed under the Souslin operation

S.

Following Bertsekas and. Shreve we call L(E) the limit-a-algebra of (E,E)

and we call its members limit-measurable subsets of (E,E) [Bertsekas

&

Shreve, p. 292J.

All properties of the a-algebra of universally measurable sets derived up to now are shared also by the a-algebra of limit measurable sets, as will be proved presently.

Since the role of universally measurable sets and universal~y measurable

mappings .in the remainder of this paper is based entirely on these proper-ties, the adjective "universally measurable" can be replaced everywhere by

"limit measurable" without affecting the validity of the results. One pro-perty of the limit measurable sets, which is not shared by the universally measurable ones and which is of use sometimes (cf. the proof of Proposition 43), is expressed by the following equality:

L(E)

=

U

{L(E

_o)

I

EO countable subclass of E} •

To prove this equality we merely note that the righthand side is a a-algebra

which is closed under

S.

To prove the analogy claimed above we firstly observe that, due to

Propo-sition 13, for every measurable space (E,E) we have

L(E)

c U(E). From this

inclusion the analogues of the Propositions 4 and 5 for the a-algebra of limit measurable sets easily follow. As to Proposition 6 we remark that for

a limit measurable mapping ~ : (E,E) + (F,F) the collection

{B c F

I

~-]B

E

L(E)}

is a a-algebra which is closed under

S

and which

con-tains F. Proposition 7 is a simple consequence of 6 again, while 8 follows

directly from the definition of

L.

To prove the analogue of Proposition 9

is slightly more laborious:

Let (E,E) be a measurable space and let B be the collection consisting of

those members A of U(E) for which the function ~ ~ ~(A) on

E

is limit

mea-surable, i.e. measurable with respect to

L(E).

B is easily seen to be a

Dynkin class. Zorns lemma implies that among those subclasses of B which

contain E and which are closed under the formation of finite intersections

there is a maximal one, say

A.

By Dynkins theorem we have a(A) c

B

and the

We now consider the space (E.A). Since E cAe U(E), it follows from

Propo-sition 4 that the probabilities on E and those on A can be identified in an

obvious way, so the spaces (E,E)~ and (E,A)~ are composed of the same set

E

of probabilities. Now AcB, so for every A € A the function ~ ~ ~(A) on

E

is measurable with respect to L(E) and consequently

A

c L(E). It follows

from Proposition J2 ii) applied to the space (E,A) that for every A E

SeA)

we have Va E JR {~ €

E

I

~(A) ;::: a} € SeA) and, as SeA) c S L(E)

=

L(E),

also that ~ ~ peA) is limit measurable on E.

Due to the maximality of

A

again we conclude from this that

SeA)

=

A.

SO

A

is a a-algebra which contains

E

and which is closed under

S

and it

there-fore contains

L(E).

We thus have proved the analogue of Proposition 9, that for every limit

measurable subset A of E the function p ~ peA) on

E

is limit measurable.

§ 6. A generalized integral

A measurable function f defined on a measurable space E can be written in

+ +

the form f - f ,where f and f are positive functions defined by

f+(x)

=

max {f(x),O} and f-(x)

=

max {-f(x),O}.

When p is a probability on E then f is called quasi-integrable with respect to ~ when at least one of the integrals

J

f+dp and

f

f-dp is finite. In this case one defines

As is easily seen, a measurable function defined on some measurable space is quasi-integrable with respect to every probability on that space only if it is bounded from above or from below. In order not to be forced to con-sider bounded functions only or to demand quasi-integrability in advance every time, we generalize the integral concept so as to be applicable even to functions which are not quasi-integrable.

Definition. For every universally measurable function f and every

proba-bility ~ on some measurable space we define

I

fd~ to be equal to

f

fd~ if

f is quasi-integrable with respect to ~ and equal to - 0 0 in the other case.

Using the convention = - =

=

-00 and ±= • 0

=

0 we conclude that

I

fd~

depends additively and positively homogeneously on f. In particular we have

In general however

f

-fd~ differs from -

f

fd~.

Some results on measurability of an integral as a function of parameters are valid also for the generalized integral defined above.

Lemma 15. Let Ebe a measurable space and f a function on E. When f is

universally measurable (resp. Souslin, measurable), then ~ ~

I

fd~ is a

universally measurable (resp. Souslin, measurable) function on

i.

Proof. We prove the statement on Souslin functions. We first consider the

case that f is positive. Then f can be written in the form

~

f

=

lim t 2-n

I

so

J

fd~

=

lim

+

2-n

~

n m=l -n ~{f>m·2 } for each ~ € E • -n

For each m,n E: :IN the set {f>m·2 } is Souslin, which implies by Proposition 7

that ~{f>m·2-n} isa,Souslin function of ~. Applying Proposition 14 we

con-clude that rfd~ also is a Souslin function of ~.

As any bounded function differs only a constant from a positive one this result also holds for bounded Souslin functions f.

Now let f be an arbitrary Souslin function and let

fn := inf {nt, sup {-ml,f}}

m (m, n E: :IN) •

Then for each m,n € :IN fn is a bounded Souslin function and

f

fd~ =

m

lim lim

J

fnd~. _m It now follows from Proposition 14 that

f

fd~ is a

n m

Souslin function of ~.

The statement on (universally) measurable functions f can be proved in a

similar way.

0

Proposition 16. Let E and F be measurable spaces and f a function on E x F.

When f is universally measurable (resp. Souslin, measurable), then

(x,~) ~

J

f(x,y)~(dy)

....

Proof. We prove the statement on Souslin functions; the other cases can be treated in a similar way.

For every x E E, U E F we have

and the same for f •

Consequently

J

f(x,y)u(dy) equals

J

fd(o xu), which is a Souslin function

x

of Ox x u by Proposition 15.

The result now follows from the fact that

a

X].I depends measurably on

x

(x,u) (cf. examples following Proposition 2) and that a measurable mapping followed by a Souslin function gives a Soualin function again.

§ 7. Semi-compact classes

o

In this section we introduce the concept of a semi-compact class. ItCountably compact" might have been a more suitable adjective for these classes, be-cause their defining property is precisely the set-theoretic feature

exhibited by the class of closed subsets of a countably compact topological space.

The usefulness of semi-compact classes lies in the fact that a-additivity of functions defined on algebras of sets can be deduced from certain approxi-mation properties of semi-compact classes.

Definition. A collection

A

of sets is said to possess the finite

inter-section property if every finite subcollection of A has a non-empty

inter-section. A collection

A

of sets is called semicompact if every countable

subcollection of A which possesses the finite intersection property has a

non-empty intersection.

Lemma

17.

When

C

is a semicompact collection of sets, then

C

_so is

semi-compact as well.

Proof. Let

B

be a countable subclass of

C

_s which has the finite intersection

property, let C :=

uC

and let U be an ultrafilter on C containing

B.

Then

every B E

B

is a finite union of members of

e

which belongs to

U

and, hence,

at least one of those members of

C,

C

B say, belongs to

U

as well. The

col-lection {C

B 1 B E B} is contained in U and therefore has the finite

inter-section property. As it is also a countable subclass of

C,

it has a non-empty

intersection. Now for each B E B we have C

B C B, so oB is non-empty as well.

The arbitrariness of

B

implies that

C

s is semicompact.

Let B be a countable subclass of

e

so such that oB

=

~. Now

B

= { 0 mEE C _run n 2~ ~,~EE finite subset

nEE} for a suitable choice of Com E

Cs'

Hence

=

oB

= ~ and, as

C

_s is semicompact, n Com

=

~ for some

(n,m)EI

I of E2. This implies that B has a finite subclass the

inter-section of which is empty_ From the arbitrariness of B it follows that C_so

is semicompact.

o

Proposition 18. Let

A

be a semicompact algebra of subsets of some set.

Then every positive additive function on A is a-additive.

Proof. Let {A

n n E 'N} be a countable collection of mutually disjoint

members of

A

such that u

nE'N

A E _n

A.

Then the collection

has empty intersection and, being a subcollection of

A,

co

{n~

An

I

m E 'N }

is semicompact.

Consequently u A _n =

0

for some m E 'N, which in turn implies that only

n=m

finitely many members of {A

I

n E 'N} are non-empty.

n

The foregoing implies that additivity and a-additivity are equivalent for

positive functions defined on

A.

Next we introduce the auxiliary space D, the socalled Cantor space. The

role played by D strongly resembles the one which the space ~ of real

numbers plays in maJ;ly measure theoretic arguments. In fact D and ~ can be

o

shown to be isomorfic measurable spaces [~ertsekas & Shreve, proposition 7.16].

The space D however is better suited to our needs.

Definition. The measurable space (D,V) is by definition the productspace

IT (D ,V), where for each nEE

nE'N n n D n := {O,t} and Vn is the a-algebra

con-siating of all (four) subsets of D •

n

Proposition 19. The a-algebra of D is generated by a countable semicompact algebra.

~. Let C:= {{x ED

I

Xi = j} l i E 'N, j = O,]}. Then C

countable and semicompact. By Lemma 17 the same holds for

C

_{sd •}

under complementation, the collection

C

_Sd is an al~ebra.

is generating,

As

C

is closed

§ 8. Analytic spaces

The importance of analytic topological spaces (for a definition see [Hoffmann-J~rgensen, Ch. III §lJ) for dynamic programming is due to certain properties

of their Borel-cr- alg~bras. In our measure theoretic approach we have taken

these properties as a starting point and in fact we have chosen them as the defining properties of the class of measurable spaces to be studied in the sequel. These spaces therefore are a generalization of the analytic topolo-gical spaces as far as their measure theoretic structure is concerned and we therefore call them analytic measurable spaces or, shortly, analytic spaces.

Definition. A measurable space F is called analytic if for every measurable

space E and every Souslin subset S of E x F:

i) the projection SE of S on E is a Souslin subset of E,

ii) S contains the graph of a universally measurable mapping from SE to F.

The utility of analytic spaces for dynamic programming is mainly due to the property expressed in the following proposition.

Proposition 20 (Exact selection theorem). Let E be a measurable space, F

an analytic space, S a Souslin subspace of E x F for which every section

S _x := {y E F

I

(x,y) E S} is non-empty and f a Souslin function on S.

Let g E + JR be defined by g(x)

=

sup f(x,y) and let

YES _x

Then g is a Souslin function and T is universally measurable.

Moreover for every universally measurable function h on E such that

{

< g(x) if -<XI < g(x)

h(x)

=

g(x) else

there exists a universally measurable mapping ~ E + F the graph of which

is contained in S and such that for each x E E:

f(x,~(x»

{

~

g(x) if x E T

h(x) else •

Proof. Let E (resp. F) be the a-algebra of E (resp. F).

For every a E ]R the set {x EEl g(x) > a} is the projection on E of the

Souslin subset {(x,y) E S

I

f(x,y) > a} of E x F and therefore is a Souslin

subset of E, because F is analytic. This implies that g is a Souslin function.

Next let A := {(x,y) E S

I

f(x,y) = g(x)} and B := {(x,y) E S

I

f(x,y) > h(x)}.

Then and A

=

S n [n ({r<f} u ({g~r} x F»J rEQ B

=

S n [u ({r<f} n ({h~r} x F»J • rEQ

From the analyticity of F we conclude that the projection ~ (BE) of A(B)

on E is universally measurable with respect to U(E) and that there exists

a mapping a(resp. 8) from ~ (resp. BE) to F which is universally measurable

with respect to U(E) and whose graph is contained in A (resp_ B). Moreover

it follows from the definitions of A and B that ~ U BE

=

E.

Now T

=

~, so T is universally measurable with respect to U(E) and

there-fore with respect to

E,

by Proposition 8.

Finally the mapping ~ E + F defined by

_

_ { a(x) if x E

~

~(x)

Sex) else

is easily seen to have the desired properties.

Taking for f a Souslin function defined on E x F and attaining the values

o

and 1 only, the exact selection theorem reduces to the definition of

ana-lyticity of F. Consequently the validity of the exact selection theorem characterizes analytic spaces.

D

We shall show that the class of analytic spaces is stable under the construc-tion methods for measurable spaces most commonly used and that it therefore contains many of the measurable spaces encountered in practice. We start with a number of lemmas.

Lennna 21. The space]) is analytic.

Proof. Let E be a measurable space and S a Souslin subset of E x D. Let for each k € ]N' the partition P k of ]) be defined by

=

n.

L (i=l, ••• ,k)}ln€ {O,I}k}

and let

P:=

u P

k• Then by Proposition 11 S can be written as k€]N'

where the A's are measurable subsets of E and the B's belong to

P.

For each nand k B is contained in a member of P_k or equals a finite

n

1, ••• ,~ union of members of P

k; in any case B _{n 1,·· "~} can be written as a countable union, say u Bt , of members of P each of which is contained in

R..€]N' nl'···'~ some member of P k • Hence we have S

=

BR. )

=

n₁ , •• "~

=

u

n

u

=

u

u

n

=

u

n

where n J+ (l(n) ,Sen»~ is some surjection of IN onto IN x IN. Hence

S

=

u

n

(A' x Bf )

IN nl""'~ nl""'~

nElN b:lN

where for each nand k

A'

is a measurable subset of E and

n l ,··· ,~

B' is a member

n1, .. ·,nk of

P

which is contained in some member of

P

k•

This implies that for each n E lNlN

point of D.

We now define for each nand k:

Then

s

== n k€lN B' if k

n

2=1 n1 ""'~

contains at most one

Now P is semicompact (being a subclass of the semicompact algebra

construc-d · h f ) IN h ' 1 ' ,

te l.n t e proo of 19 , so for each n E IN we have t e l.Inp l.cat1.ons:

k

n

B' r/:

0

t=1 _nI,···,nt An r/:

0 ..

n) , ••• ,~

.. n

kElN

Let x ESE' Then there is some y E ID such that (x,y) E 5 and hence

X E n A" • _On the other hand, when x belongs to the last

IN nl""'~

nEJN kEJN

set, then there exists n E JNJN such that x E n A" , which implies

n 1 ,. "'~

u

n

Bt .J:.

0

and hence the existence of y E ID with YEn BI •

k€ IN n 1 ' ••• ,~ kE IN n 1 ' ... ,~

Then (x,y) E

n

(A" x B' ) c S, so the point x belongs to

kEJN nl""'~ nl""'~

The foregoing implies that SE equals U n An and, consequently, is

n k nl'·"'~.

a Souslin subset of E.

Next we show that S contains the graph of a universally measurable mapping

For each n E JNJN and k E IN we def ine

C U A*

Then SE C U A* and for each n € JNJN and k E IN A *

mE IN m Ii n 1 ' ••• , ~ mEJN n_{1 , · · · ,~m}

Now let x E SE and let p E IN be defined recursively by the statement:

"for each k E ]N Pk is the smallest natural number such that x E A*

PI,···,Pk Then

n

A" ::>

n

k€]N PI'''',Pk kE]N

and hence n Bt .J:.

0.

This implies that the last intersection

con-kE]N Pl'''',Pk

tains precisely one point, say ~(x), of F.

Moreover we have (x, cp (x» €

n

A" k€ ]N PI"'" Pk x B' c S • PI' • •• ,Pk "

The graph of the mapping ~ SE + ID defined above is therefore contained

in S.

To prove the universal measurability of cp let k EO 'IN and B EO: P K" Let N be

h f k . h '

t e set 0 those n EO 'IN for wh~c B c B.

n_{1,· ..} ,~

Then the following four statements are equivalent cp e B, Bf c B,

x PI, ••• ,Pk X e k U

n

neN £=1

*

-J

As each A is a Souslin subset of E and N is countable, cp B belongs to the

a-algebra generated by the Souslin subsets of SE and, hence, is universally

measurable. 0

Lemma 22. Let E and F be measurable spaces, let F be countably separated

and let cp : E + F be measurable. Then the graph of cp is a measurable

sub-set of E x F.

Proof. Let A be a countable collection of measurable subsets of F which

separates the points of F. Then for every (x,y) EO: ExF we have

(x,y)

i

graph cp

*

y ~ cp(x)

*

3A e A[yeA A cp(x)

t

A v yiA A cp(x)eAJ

*

(x,y) e U «cp-1A)c x A U cp-1A x AC) •

AeA

So the graph of cp is the complement of a countable union of measurable

Definition. Let (E,E) and (F,F) be measurable spaces and ~ : E + F.

A right inverse of ~ is mapping

4 :

~E + E such that ~ 0

4

is the identiy

on

~E.

The mapping

~

'is called strictly measurable if E ..

~-lF.

A mapping

is called an isomorfic embedding when it is strictly measurable and inject-ive.

Lemma 23.

i) Let (E,E) and (F,F) be measurable space and let ~ : E + F be strictly

-I

measurable. Then S(E) .. ~ S(F) and every right inverse of ~ is

measur-able.

ii) Let E be countably generated and ~ : E + ID measurable. Then there

exist a strictly measurable mapping

4 :

E + ID and a measurable mapping

X : 4E + ID such that ~ .. X 0 4.

Proof.

i) S(E) .. S(~ -I F)

=

~ -1 S(F) because ~ -1 commutes with the operations U and

O. Let

4

be a right inverse of ~. For every measurable subset A of E

there exists a measurable subset B of F such that A ..

~-lB

and hence

such that

Consequently

4

is measurable.

ii) Let

C

be a countable collection generating the a-algebra E of E and let

n 1+ C be a mapping of IN onto C. Moreover let V be the a-algebra of ID

n

and for each n € IN let D := {y € ID

I

Y =1}. Define 4 : E + ID by

n n

C = l/J-ID and consequently a{C

I

n E IN} ,. 4 -} a{D

I

n E IN} or

equi-n n n n

valently E = CP-1V. SO l/J is strictly measurable.

Now let

x,x'

E E be such that

4x

=

4x'.

It then follows from the

defi-nition of

4

that x and x' are not separated bye, hence not by a(e)

and a fortiori not by cp-1V, which is contained in a(e). So cpx and cpx'

are not separated by

V

and therefore coincide, because (ID

,V)

is a

separated space. The foregoing implies that for any right inverse I; of

4

we have cp

=

cp 0 I; 0

4.

By i) we know that I; is measurable and the

map-ping cp 0 I; is therefore measurable as well.

o

Let (E,E) be a countably generated space. It follows from point ii) of the

foregoing lemma, taking for ql a constant mapping, that E can be mapped into

ID by a strictly measurable mapping. Consequently E is isomorfic to the a-algebra of measurable subsets of some subspace of ID.

Proposition 24. A measurable space F is analytic iff for every measurable

mapping cP : F + ID the range of cp is a Souslin subset of ID and cp has a

universally measurable inverse.

Proof. Let F be analytic and cp : F + ID measurable. As ID is countably

separated, the graph of cp is a measurable subset of F x IDand hence a Souslin

subset of F x ID. The range of cp is the projection on ID of the graph of cp

and therefore is a Souslin subset of ID. Moreover the graph of cp contains

the graph of a universally meas.urable mapping from cpF to F, which obviously

Next let (F,F) be a measurable space having the property stated in the proposition, let (E,E) a measurable space and let S be a Souslin subset of

E x F. Then, by Proposition 11, 'W'e have S €

SeE

x F) and therefore

S e:

seE

x e) for some countable subclass of F.

No'W' let ~ : F + ID be strictly measurable 'W'.r.t.

aCe)

(and therefore

mea-surable 'W'.r.t. F) and let ~ : E x F + E x ID be defined by

~(x,y) := (x,~(y». Then ~ is easily seen to be strictly measurable 'W'.r.t.

-1

E 0

aCe).

It now follows from Lemma 23 that S

=

~ S' for some Souslin

sub-set S' of E x ID, or equivalent ly ~S

=

S' n (EXqlF). The definition of F

implies that ~F is a Souslin subset of ID. So E x (jlF is a Souslin subset of

E x ID and this in turn implies that ~S is the intersection of t'W'o Souslin

subsets of E x ID and consequently is such a set itself.

From the definition of ~ it follows that the projection SE of S on E equals

the projection of S' on E. The last set however is a Souslin subset of E, due to the analyticity of ID •

Moreover there exists a universally measurable mapping a : SE + ID the graph

of which is contained in

st.

Also, by the definition of F, there exists a

universally measurable inverse

a

of ~. The mapping

a

0 a SE + F therefore

is universally measurable. Also for every x € SE 'W'e have

~(x,aoax)

=

(x,cp .. a III ax)

=

(x,ax) e: S' ,

-1

hence graph (Soa) c ~ S' = S.

Let F be a countable generated analytic space. By the remark following Proposi-tion 23 F can be mapped into ][) by a strictly measurable mapping. As F is analytic, the range of this mapping is a Souslin subset of ID. These facts imply that the a-algebra of F is isomorphic to the a-algebra of some Souslin subspace of ][). When in addition F is separated, then the strictly measu~able mapping mentioned above is injective and, hence, an isomorfism. So every countably generated and separated analytic space is isomorfic to a Souslin subspace of ID.

Now we are able to prove the stability properties of the class of analytic spaces announced at the beginning of this section.

Proposition 25. The class of analytic spaces is closed under the formation of measurable images, Souslin subspaces and arbitrary products.

Proof.

i) Let a measurable space F be the image of an analytic space under a

measurable mapping ~. Moreover let ~ be a measurable mapping of F into ID. Then the range of ~ equals the range of ~ 0 ~, which is a Souslin

sub-set of ID by the preceding proposition.

Also ~ 0 ~ has a universally measurable right inverse

x.

Hence, ~ 0 X

is a universally measurable right inverse of ~.

From the preceding proposition it now follows that F is analytic. ii) Let F be a Souslin subspace of an analytic space F', let E be a

measur-able space and let S be a Souslin'subset of E x F. Then S is the inter-section of E x F and some Souslin Subset of E x F'. Now E x F is a Souslin subset of E x F' itself, so S is a Souslin subset of E x F'

From the analyticity of F' we deduce that the projection SE of S on E is

a Souslin subset of E and that S contains the graph of a universally

measurable mapping from SE to F'.

This facts imply that F is analytic.

iii) Let (F.,F.). I be a family of analytic spaces and let F := IT F ••

~ 1 1€ 1

iEI

Also let (E,E) be a measureable space and S a Souslin subset of E x F.

The set S may be supposed to be composed of a countable number of

measurable rectangles in E x F and every measurable subset of F belongs

to some cr-algebra which is generated by a countable number of measurable

cylinders in IT

iEI

F .• So there exists a countable subset J of I and for

].

each i € I a countable subclass Ci of Fi such that Vi€I\J C. = {cp,F.}

]. 1

and such that S € S(Ex 0 cr(C.». Now for each j € J let!p. : F. -+ ID

. I 1 J J

].€

be strictly measurable w.r.t. cr(C_j) and, hence, measurable w.r.t. F_{j •}

Moreover let q> : F -+ IDJ be defined by (<p(y». := <po (y.) and

J J J

~ : E x F -+ E x ID by ~(x,y) := (x,<p(y». Then

4

is easily seen to be

strictly measurable w.r.t. E 0 0 cr(C.).

. I 1

].€

In order to show that F is analytic we now can proceed as in the second

part of the proof of Proposition 24 (almost verbatim, with ID replaced

by IDJ) provided we know that qJF is a Souslin subset of IDJ , that IDJ

is analytic, and that !p has a universally measurable right inverse

B.

N<?w QlF

=

II

jeJ

q>.F. and for each j E J the set q>.F. is a Souslin subset

J J J J

of ID, because F j is analytic. The countability of J implies that <pF is

a Souslin subset of IDJ •

Due to the countability of J again, the space IDJ is isomorfic to the

Finally for every j € J let 8_j be a universally measurable inverse of ~j and

let for each i € I \ J Y i be a point of F i' Then the mapping B : IDJ

~

F

defined by B(z). := ~ { B. (z.) ~ ~ y. 1 if i E J if i E I \ J

is a universally measurable inverse of ~.

o

The class of analytic spaces is not closed under the formation of projective limits, even not of countable projective limits as the following counterexample shows.

Let E be a subset of (ID ,V) which does not belong to $(V) (such a set exists,

because S(V) has continuous cardinality).

Let (V) _{n n€} :IN be an increasing sequence of finite sub-a-algebra' s of V such

that V

=

a( u Vn) and let En := On n E for each n E :IN.

n=l Then OnE

=

a( u n=l E ) • n

So OnE is the projective limit of the sequence (E,En)'

Now for each n E :IN En is finite, so (E,En) is analytic.

E however is not analytic, not being a Souslin subset of ID.

The stability properties of the class of analytic spaces imply that many of

the spaces encountered in practice are analytic. To begin with the space ]R

is analytic.

co

Proof. Let ~ : ID ~ [O,IJ be defined by ~(x)

=

I

x • 2-n• Then ~ is a

sur-n

n=l

jection. Moreover ~ is measurable, because for each n x depends measurably _n

The same holds for the measurable subspace (0,1) of [0,1J and consequently for lR, which is the image of (0,1) under the measurable mapping

1 ]

x 1+

x-I -

x'

o

Also the space :IN of natural numbers endowed with the a-algebra of all its subsets is analytic, being a measurable subspace of lR. Consequently for

I I

every index set I the spaces lR and:IN as well as their measureable

sub-spaces are analytic.

The remaining part of this section is devoted to probabilities on analytic spaces.

Proposition 26. Let (F,F) be an analytic space and let G be a countably

generated sub-a-algebra of F. Then there exists a universally measureable

mapping ~ : (F,G)~ + (F,F)- such that for every probability ~ on G the

pro-bability ~(~) is an extension of ~ on F.

Proof. The space (F,G) is a measurable image of the space (F,F), and

there-fore is analytic. By Proposition 23 there exists a mapping W : F + D which

is strictly measurable w.r.t. G. As W also is measurable w.r.t. F, there

exists a right inverse X of

W

which is universally measurable w.r.t. F.

Now let q> (F,G) ~ + (F,F) be defined by '" ~(~) := ~ 0

W

-1 0 X • -1

For every A E F the set X-1A is a universally measurable subset of the space

~F, consequently the set ~-l(x-lA) is a universally measurable subset of (F,G),

. -1 -1

which in turn implies by Proposition 9 that p(~ (X A» depends universally

measurably on p. So from the definitions of (F,F)- and universal