Analytic spaces and dynamic programming : a measure-theoretic approachCitation for published version (APA):Thiemann, J. G. F. (1984). Analytic spaces and dynamic programming : a measure-theoretic approach. Centrumvoor Wiskunde en Informatica. https://doi.org/10.6100/IR5458

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ANAL YTIC SPACES AND DYNAMIC PROGRAMMING

J.G.F. THIEMANN

ANAL YTIC SPACES

AND

DYNAMIC PROGRAMMING

A MEASURE-THEORETIC APPROACH

ANALYTIC SPACES

AND

DYNAMIC PROGRAMMING

A MEASURE-THEORETIC APPROACH

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR

MAGNIFICUS, PROF.DR. S.T.M. ACKERMANS, VOOR EEN

COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN

DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 8 JUNI 1984 TE 16.00 UUR

DOOR

JOHANNES GEORGIUS FRANCISCUS THIEMANN

GEBOREN TE HOORN

1984

CENTRUM VOOR WISKUNDE EN INFORMATICA, AMSTERDAM

Dit proefschrift is goedgekeurd

door de promotoren

Prof.dr. F.H. Simons

en

Prof .dr. J. Wessels

CONTENTS

INTRODUCTION'

I. MEASURE-THEORETIC PREREQUISITES

1. Preliminaries

2. Spaces of probabilities

3. Universal measurability

4. Souslin sets and Souslin frinctions § 5. Semicompact classes

§ 6. Measurability of integrals

II. ANALYTIC SPACES

§ 7. Analytic s~aces § 8. Separating classes

§ 9. Probabilities on analytic spaces

III. DYNAMIC PROGRAMMING

§ JO. Decision roodels

IJ. The expected utility and optimal strategies

REFERENCES

INDEX

SAMENVATTING

CURRICULUM VITAE

6

JO

14

25

31

34

49

55

75

81

90

91

93

95

INTRODUCTION

Analytic topological spaces are used in dynamic programming in order

to avoid certain measurability problems. In this monograph we give a measure­

theoretic alternative for these spaces serving the same purpose. Befare

giving an overview of the contents, we briefly describe the measurability

problems encountered in dynamic programming and indicate how they have been

solved.

Consider a system that passes through a sequence of statea in the

course of time and suppose that a controller can influence each of the

transitions of the system to a new state by taking certain actions. At each

transition the new state of the system depends on the old state and on the

action chosen by the controller in a stochastic way; it is the probability

distribution of the new state that is determined by the old state and the

action, rather than the new state itself. Also, for each realization of

this process, i.e. for each sequence of statès and actions, a utility is

defined, that is, a number repreaenting the desirability of the realization.

Now the controller tries to choose his actions such as to maximize the

expected utility. In general, this implies that the action to be chosen at

each instant of time depends on the state the system is in at that time.

The sequence of these choice functions, one for each transition of the

system, is called a strategy.

When the state space, i.e. the set of states the system can be in,'

and the action space, i.e. the set of actions available to the controller,

are finite or countably infinite, and when only finitely many transitions

of the system are considered, no difficulties arise in defining the expecta­

tion of the utility. However, when these spaces are uncountable, this is no

longer the case. For the expected utility to be definable it seems necessar~

that the state space and the action space are measurable spaces and that the

choice functions in a strategy as well as the utility are measurable func­

tions. In the construction of strategies yielding maximal expected utility

one fueets measurability problems that can be described in their simplest

form as follows: If f is a measurable function of two variables, then, in

genera!, supy f(x,y) is not a measurable function of x. Also, when the

supremum is attained for each x, there does not necessarily exist a measur­

able function ~ such that sup f(x,y) = f(x,~(x)). y

Blackwell was among the first who paid attention to these problems.

He took Borel spaces as state and action spaces, a measurable utility and

measurable strategies. Later on, in a paper by Blackwell, Freedman, and

Orkin, this formalism was generalized: analytic state and action spaces, a

semianalytic utility and analytically measurable strategies. Another

generalization is due to Shreve, who took Borel spaces again, a semianalytic

utility and universally measurable strategies (for the papers in question

see the references). In the last two formalisms the measurability problems

mentioned earlier do no longer appear.

The use of Borel spaces and analytic spaces is unsatisfactory in so

far as topological conditions are imposed on the system in order to avoid

difficulties that are measure-theoretic by nature. Moreover, the theory of

analytic spaces is far from trivial, and quite remote from the things one

expects when turning to dynamic programming. In this monograph we shall

develop, within a purely measure-theoretic framework, those parts of the

theory of analytic spaces that are material for dynamic programming, and we

shall show how they can be applied. The formalism for dynamic programming

treated in the following is quite general: the utility need not be the sum

of single-step utilities but may be an arbitrary function of the realiza­

tion, while the restrictions imposed on the choice of the actions do not

concern the actions themselves but rather the probabilities on the action

space on which the choice is based.

The monograph is divided into three chapters. In Chapter I the

measure-theoretic prerequisites are introduced, the main topics being

univeraal measurability and the Souslin operation. This chapter is self­

contained: apart from the Radon-Nikodym theorem and the martingale conver­

genee theorem only elementary measure theory is required. In the second

chapter analytic measurable spaces are introduced. The defining property of

these spa~es is common to all analytic topological spaces, and our measure­

theoretic approach is therefore a generalization of the topological one.

Also for this chapter no a priori knowledge is needed, and, consequently,

it may serve as an introduetion to the subject, However, only those topics

are treated that are needed for the applications in Chapter III, or that

serve a good conception. The final chapter is devoted to dynamic program­

ming. Although familiarity with this subject is not needed for the under­

standing of this chapter, it will certainly add to its appreciation. So,

this part of the monograph should not be taken as introductory. The topics

treated have been chosen so as to give the reader a good impression of the

use of analytic measurable spaces. Consequently, results that are based on,

say, a particular structure of the utility or on the choice of particular

strategies are not considered.

Although there are some new results in this monograph and the purely

measure-theoretic approach as such may be considered as new, on many occa­

sions our line of reasoning was inspired by arguments found in the litera­

ture; our main sourees were the books of Bertsekas & Shreve, Christensen,

Hinderer, and Hoffmann-J~rgensen (see references).

CHAPTER I MEASURE-THEORETIC PREREQUISITES

The measure-theoretic prerequisites needed for the understanding of

the theory of analytic spaces and their applications'are collected in this

chapter. In its first section only rather well-known facts are recalled

(often with an indication of a proof) and some notations are introduced.

In the secoud section sets of probabilities are equipped with a structure

such as to make the theory of measurable spaces apply to them. The, perhaps

less familiar, subjects of universa! measurability and Souslin sets are

treated from scratch in the sections 3 and 4, respectively. Also, in

sectien 4, the first new result appears, viz. Proposition 4.8. In section 5

a nontopological compactness notion is introduced and applied to probabili­

ties. Finally, in section 6, we derive a result on measurability of inte­

grats that play a central role in Chapter III.

§ I. Preliminaries

I) The terms: set, collection, and class all stand for the same thing.

Which of them is used depends merely on the role played by them in the

argument. A set is called aountable when it is finite or countably

infinite.

Let E be a set. Then for every pair A,B of subsets of E the complement

(with respect to E) of A is denoted by Ac and the difference A n Be of A

and B by A \ B. When A is a colleetien of subsets of E then

Ac :=A u {Ac I A € A}. Moreover, Ad (As• A0, A0

, respectively) denotes

the colleetien of those subsets of E that are finite intersections

(finite unions, countable intersections, countable unions, respectively)

of members of A, while crA stands for the cr-algebra generated by A, i.e.,

the smallest cr-algebra of subsets of E that contains A. Colleetiens like

(As)ê will be simply denoted by Asó etc. For each colteetion A of sub­

sets of some set we have A8d = Ad

8: the inclusion A

8d ~ Ads is obvious,

whereas the reverse inclusion fellows from this by complementation and

de Morgan's rule.

2

2) DEFINITION. A colleetien A of sets is called a Dynkin atass when

i) vA,BEA [A~ B ~A\ B € Al

ii) if (A ) 1N is an increasing sequence in A then U A E A. n nE nE1N n

For Dynkin classes we have the following proposition due to Dynkin, a

proof of which can be found in [Ash] Theorem 4.1.2, and in [Cohn]

Theorem I. 6. I.

PROPOSITION l.I. Let A be a eotleetion of subsets of a set E sueh that

Ad = A and E E A. Then cr(A) is the smallest Dynkin alass eontaining A.

3) Next we collect some facts on mappings. Let ~ be a mapping of a set E

4)

into a set F. For every subset A of E we define ~A := {~x I x E A}, which

is a subset of F. By ~-I we denote the mapping of the colleetien of all

subsets of F into the colleetien of all subsets of E defined by

~-IB := {x E E I ~x E B}. In particular we have ~-lF = E and ~-l0 = 0. Note that we did not suppose ~ to be injective or surjective. Note also

-1 -1 that ~ does not map points of F on points of E, so ~ should not be

considered as an inverse of ~ in the usual sense.

For every colleetien B of subsets of F wedefine ~-JB := {~-IB I B E 8},

which is a colleetien of subsets of E. Mappings like ~-I will be used

extensively in the sequel, so we list some of their properties.

-I The mapping ~ commutes with the set-theoretic operations union,

intersectien and complementation, i.e. for any pair A,B of subsets of F

we have ~-I(A u B) =(~-IA) u (~-IB), ~-I(A n B) =(~-IA) n (~-tB) and -1 c -1 c

~ (A ) = (w A) • More generally, we have, for every colleetien B of

subsets of F, the equalities ~- 1 (u8) = u(~- 1 8) and ~- 1 (nB) = n(w- 1B).

As a simple consequence we have, for every colleetien Bof subsets of F,

that ~- 1 (BP) (w-1B)P, where p stands for any of the subscripts s, cr,

d, o or c.

Th . f -I 'd d f . 1 f e propert~es o ~ cons~ ere so ar are s~mp e consequences o

the definition of ~-l. Slightly more involved is the proof of the

following property: when Bis a colleetien of subsets of F, then -1 -1

~ (crB) = o(~ B). To prove this we observe that crB is closed under the

formation of countable unions and under complementation and that the

5)

same holds for the collection <p- 1(crB), due to the properties of qJ-I

mentioned before. So <p- 1(crB) is a cr-algebra which, moreover, contains

!Jl-18. Hence <p- 1(crB) ~ cr(<p- 18). On the other hand, the collection

{Cc F I <p-IC ~ cr(<p- 18)} contains 8 and it is closedunder the formation

of countable unions and under complementation. It therefore contains -1 -1 cr(B) and, consequently, we have qJ (crB) c cr(<p 8).

Each of the properties of qJ- 1 mentioned above expresses the fact that -I <p commutes with a certain set-theoretic operation on collections of

sets.

A particular case of the foregoing is forming the trace of a collec-

tion of sets. Let F be a set, let B be a collection of subsets of F, and

let E be a subset of F. The truce BIE of B on E is defined to be the

3

collection {B n E I B E B} of subsets of E. Obviously we have I -1 B E = !Jl B,

where <p: E + F is the identity on E. From this it follows for instanee

that the trace of a cr-algebra is again a cr-algebra.

6) The main object of interest in this monograph will be measurable

spaces, i.e. pairs (E,E) where E is a set and f is a cr-algebra of subsets

of E. Subsets of E that belang to f will be called meaau~le subsets of

(E,f). To simplify the notation we shall, as a rule, notmention the

cr-algebra of a measurable space when confusion is unlikely. A subset E

of a measurable space (F,F) will always be supposed to be endowed with

the cr-algebra FIE; when we want to stress the measurable-space structure

of E we call E a subspaoe of F rather than a subset of F.

7) A product of measurable spaces will always be supposed to be equipped

with the product cr-algebra. We reeall some facts on this subject.

Let ((F.,F.)). I be a family of measurable spaces, let F be the 1 1 1E

Cartesian product IT. I F. of the family (F.). I' and, for each i E I, 1€ 1 1 1E

let wi be the i-th ooo~dinate on F, i.e. the mapping ~i: F + Fi defined

by ~.(x) :=x .• The p~oduat ®, I F. of the family (F1·). 1 of cr-algebras

1 1 1E 1 1E

is by definition the smallest cr-algebra F on F such that, for each i E I,

the mapping ~.: (F,F) + (F.,F.) is measurable. 1 1 1

The measurable space (IT. I F.,®. I F.) will be 1€ 1 1E 1

or simply by rr. I F. and it is called the p~oduat 1E 1

((F.,F.)). I of measurable spaces. 1 1 l.E

denoted by IT. I(F.,F.) 1E 1 1

of the family

4

8)

9)

As is clear from its definition, the product cr-algebra @. I F. is -1 ~€ ~

generated by the colleetien B := U. I(~. F.) and therefore by the collec-~€ ~ ~

tion Bd. The memhers of the collection Bd are called meaaurabte cytinders;

they are characterized by the fact that they can be written as n. I A., ~€ ~

where Ai € Fi for all i € I and Ai Fi for all but a finite number of

i € I.

Let (E,E) be another measurable space and let~: E + n. I F .• Then ~€ ~

-1 ~ 0 F.

~

-1 -1 q1 o U (11. F.)

iEl ~ ~ = 0 u

i€!

-1 -1 cp (11. F.)

~ ~ = 0

-1 u ( 11. o q1) F. i€! ~ ~

so (j)-l 0. I F. c E iff V. I ((~. o IJ))-1F. c El, hence IJ) is measurable iff

~€ 1 1€ 1 ~

V i€ I [1! i 0 <P is measurable]. Th is property can be phrased as: "A mapping

into a product space is measurable iff its coordinates are measurable".

JO) When all

the product IN spaces IN

spaces in a family (Fi)içi are identical to a space F, then I

space ni€I Fi may be denoted by F • In particular we have the

of all sequences of positive integers; each n ç 1N1N equals

the sequence (n 1,n2,n3, ••• ) of its coordinates.

When only two measurable spaces, say (F1,F1) and (F2,F2), are involved

the product space is denoted by (F 1,F1) x (F2,F2) and the product cr­

algebra by F1 @ F2• Cylinders are usually called reatangles in this case.

For any two colleedons A1 and A2 of subsets of F1 and F2, respectively,

the collection {A1 x A2 I A1 € A1 and A2 € A2} is denoted by A1 x A2•

IJ) Let (E,E) be a measurable space. A probabitity on (E,E) is a mapping

p: E + [0,1] such that ~(E) = 1 and such that ~(U 1NA) = L 1Nv(A) n€ n n€ n for each sequence (A ) IN of mutually disjoint merobers of E. When v is n nE a probability on (E,E) and when Ij) is a measurable mapping of (E,E) into

a measurable space (F,F), then ~ o <P-l is a probability on. (F,F). As a

special case of this we have the following. Let ~

product space n. I(F.,F.). Then, for each I' c I, ~€ 1 1

n. I,(F.,F.) is defined to be the probability v o 1E ~ 1

projection of niel Fi onto niel' Fi.

be a probability on a

the marginal of v on -1

~ , where 11 is the

Frequently used probabilities are those concentrated at one point.

Let (E,E) be a measurable space and let x E E. Then the probability öx

on E is defined by ox{A) = IA{x) and it is called the probabiZity aon­

aentrated at x. Note that {x} is not supposed to be a measurable subset

of E. Obviously, for each measurable function f on E we have

Effdox =f(x).

5

12) A subset A of a set E is said to separate a pair x,y of distinct

points of E when either x E A and y t A or x t A and y € A. A colleetien

A of subsets of E is called separating if each pair of distinct points

of E is separated by a member of A. A measurable space (E,E) is called

separated when E is separating and it is called aountabZy separated when

some countable subclass of E is separating. When the a-algebra E is

generated by a subclass A, then a pair x,y of distinct points of E is

separated by some memher of E iff it is separated by some member of A,

because the colleetien of subsets of E that do not separate x and y is a

a-algebra and it therefore contains E as soon as it contains A.

Identification of points of a measurable space that are not separated

by measurable sets yields a separated measurable space whose a-algebra

is isomorphic to the a-algebra of the original space. Such an identifica­

tion therefore is inessential in many situations. We do not, however,

restriet ourselves to separated spaces as this turns out to be incon­

venient.

13) A measurable space (E,E) is called aountabZy generated when the cr­

algebra E is generated by some countable subclass. When the cr-algebra E of a measurable space is generated by a (not necessarily countable)

subclass A and when E0 is a countably generated sub-cr-algebra of E, then

there' exists a countable subclass A0 of A such that E0 c cr(A0). This is

a simple consequence of the fact ,that the colleetien U{cr(A0) I A0 count­

able subclass of A} is a cr-algebra which contains A and therefore E.

14) By IN we denote the set {1,2,3, ••• } of positive integers equipped

with the a-algebra of all its subsets. IR denotes the set of real

numbers endewed with the cr-algebra generated by the colleetien of all

intervals; the membars of this a-algebra are called BoreZ sets. ÏR is

the set IR u {-m,oo} equipped with the usual ordering and with the cr­

algebra generated by the colleetien of intervals [a,b] (a,b E ÏR).

Addition and multiplication of IR are extended on IR in the usual way

subject to the convention that oo + (-oo) = -oo and 0•(-oo) = 0•® = 0.

6

Note that in,IR multiplication by -1 is not distributive over addition:

(-l)•(w + (-oo)) ~ (-l)•oo + (-1)•(-w).

By ajUnetion on a set E we mean a mapping of E into IR. A function

will be called positive when its values belong to [O,oo]. For any set A

the function IA is defined by: IA(x) = I when x € A and IA(x) = 0 when

x ~ A. Mappings will often be denoted by their argument-value pairs 2 separated by the symbol ~. Example: x~ x + I (x € IR) denotes the

mapping defined on IR that assigns to x the value x2 + I.

§ 2. Spaces of probabilities

In this section the set of all probabiliti~s on a measurable space is

equipped with a certain structure, which makes it a measurable space with

some desirabie properties. These spaces of probabilities play a predominant

role in the sequel; not only do many notions and results find their most

natural formulation in terms of this structure, but also results hearing

upon individual probabilities often can be derived more easily when these

probabilities are considered as memhers of the measurable space of all

probabilities on a certain space.

There are several ways to provide the set M of all probabilities on a

measurable space (E,E) with a cr-algebra. One way is to endow this set with

the metric corresponding to the total-variation norm, which turns M into a

metric space (see [Neveu] section IV. I). Next, one may equip M with the

cr-algebra generated by the topology of this metric space. A second metbod

applies when the cr-algebra E itself is generated by some topology T on E.

In this case one may consider the smallest topology on M such that the

mapping ~ ~ Ef fd~ is continuous on M for every bounded continuous function

f on (E,T}. This, so-called weak, topology again can be used to generate a

'cr-algebra on M. In the latter procedure continuity may be replaced by upper

semicontinuity; this results in a third construction of a cr-algebra on M

(see [Tops~e]}.

In our approach, however, we directly define a cr-algebra on M without

first constructins a topology.

DEFINITION. Let (E,E) be a measurable space. The set of all probabilities

on E will be denoted by E(E). For every A € E the function ~~~(A) maps

E(f) into the measurable space IR and Ë is defined to be the smallest cr­

algebra on E(E) with respect to which all these functions are measurable.

7

The measurable space (Ë(E),Ë) will also be denoted by (E,E)~ When no

confusion can arise both the set E(E) and the space (E(E),Ë) will be denoted

by just E. The definition of Ë can also be phrased: Ë is the smallest cr-algebra

on Ë(E) such that, for every A E E, the function ~ ~ EJ !Ad~ is measurable.

We shall see later on (Proposition 6.1) that this is equivalent to the

measurability of p ~ EJ fdp for all bounded measurable functions f on (E,E).

So, our construction of the cr-algebra Ë bears some resemblance to th~ con­

structions discussed above and, in fact, coincides with them in some special

cases (see [Bertsekas & Shreve] Proposition 7.25).

The set {p ~ p(A) I A E E} of functions on E(E) in terms of which the

cr-algebra Ë is defined can be considerably reduced:

Let (E,E) be a measurable spaae and Zet A be a subalass of

E that generatea E and is alosed under formation of finite interseations.

Then Ë is the smallest cr-algebra on E suah that for evePy A E A the fUnation

p ~ p(A) on Ë is measurable.

PROOF. Let B be the smallest cr-algebra on E such that for each A € A the

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

Ë and from A c E that B c Ë. Now the sets A € E for which the function p ~ p(A) is measurable on

(E,B) constitute a Dynkin class V containing A u {E}. As the collection

A u {E} is closed under the formation of finite intersections it follows

from Proposition 1.1 that V~ o(A u {E}) = E. So for every A E E the func-

tion p ~ p(A) is measurable on (E,B), which implies that B ~ Ë. D

The following proposition is a frequently used tool to check measur­

ability of mappings into spaces of probabilities.

PROPOSITION 2.2. Let (E,E) be a measurable spaae and Zet A be a subalass of

E that generatea E and is elosed under formation of finite interseations.

Then a mapping ~= F 7 Ë ofan arbitraPy measurable spaae F into Ë is

measurable iff for evePy A E A the funation x~ ~(x)(A) is measurable on F.

8

~

PROOF. The "only if" part is a trivial consequence of the definition of E. -So, for every A € A, let ~A: E ~ IR be the mapping p ~ p(A) and suppose

that the mapping ~A o ~ is measurable. Denoting by f and R the cr-algebras

of F and IR, respectively, by Proposition 2.1 we have

and, consequently,

-1 because (~A o ~) R c f for every A € A. So ~ is measurable. D

In particular, the collection A may be the whole of E. This will be

the case in most applications of Proposition 2.2. Note the similarity to the

measurability property of mappings into product spaces mentioned in the

preliminaries. In fact, a similar result holds for mappings into any measur­

able space whose cr-algebra is generated by a set of mappings.

We now give some applications of the foregoing proposition, which will

be used in the sequel.

EXAMPLES.

~

1) Let (E,E) be a measurable space and let ó: E ~ E be the mapping that

maps each point x of E onto the probability óx concentrated at that

point. Then ó is measurable, because for each A € E the number 6 (A) x

equals IA(x), which is a measurable function of x.

2) Let lP be a measurable mapping of a measurable space (E,E) into a

measurable space {F,f). Then, for each probability pon E, p o ,-1 is a

probability on F. Moreover, p o ~-I depends measurably on p, i.e.

p ~po IP-l is a measurable mapping of Ë into F, because for each B € F h (" o m-l)(B) -- u(m-IB) d i -lB E h 1 . we ave ~ T ~ T an , s nee lP ~ , t e ast express~on

is a measurable function of p.

3) When, in example 2), ~ is taken to be a projection in a product space

(see preliminaries 11) then it follows that the marginals of a probabil­

ity on a product space depend measurably on that probability.

4) When, in example 2), (F,f) is taken to be (E,E0) with E0 c E, and

when ~ is taken to be the identity on E, then for each probability ~ on

9

E the probability ~ o ~- 1 is the restrietion of ~ to the sub-a-algebra E0 of E. So the restrietion of a probability to a sub-a-algebra depends

measurably on that probability.

5) Another example is the product of probabilities. Let E and F be

measurable spaces. Then the product ~ x v of a probability ~ on E and a

probability v on F depends measurably on ~ and v simultaneously, i.e.

(~,v) ~ ~ x v is a measruable mapping of E x F into (E x F)~. To prove

this we apply Proposition 2.2, taking forA the collection of measurable

reetangles of the product space E x F.

6) The last example we consider is the transition probability. A transi-

tion probability from a measurable space (E,E) to a measurable space

(F,f) is a function pon E x F such that for every B E F the function

x~ p(x,B) is measurable and for every x E E the function B ~ p(x,B) is

a probability on F. Clearly, p can be identified with a measurable

mapping of E into F. In fact, this will be the way by which transition

probabilities will be introduced in section 9.

Certain properties of a measurable space E are inherited by the space

E, as is illustrated by the following proposition. Other examples will be

given later.

PROPOSITION 2.3. Let (E,E) be a measurable space. Then E separates the

points ofE. When (E,E) is countably generated, then (E,E) is countably

generated and countably separated.

PROOF. Let ~ 1 .~ 2 E Ë and ~I~ ~2 • Then ~ 1 (A) < ~ 2 (A) forsome A E E. Hence

{~ E E' I ~(A) ~ ~2 (A)} is a member of E that separates ~ 1 and ~ 2 • Now let'

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

of E. Then by Proposition 2.1 the a-algebra Eis generated by the furictions

~ ~ ~(A) (A E Cd) and, hence, by the countable collection

10

{{p E Ë I p(A) ~ r} I r e Q, A e Cd}' where Q denotes the set of rationat

numbers. As (Ë,Ë) is separated, this countable generating colteetion must

be separating as well. D

Finally we remark that the theory which was dealt with in this section

can easily be extended to bounded measures, not necessarily probabilities.

§ 3. Universa! measurability

Let (E,f) be a measurable space and let p be a probability on E. By

the completion EP of E with respect to p we mean the collection of subsets

A of E for which there exist sets B1

,B2

E E (depending on A) such that

B1

cA c B2

and p(B1) c p(B

2). It is well known (see [Cohn] Proposition

1.5.1) that E is a cr-algebra containing E, and that p can be extendedtoa p

probability on E • So, as far as p is concerned, there is not much differ­P

ence between the spaces (E,f) and (E,EP).

Now, the completion E of E depends on the probability p and one may ~

ask whether there exists a cr-algebra which can be looked upon as a kind of

completion of E for all probabilities on E simultaneously.

The subject of this section is to prove that such a completion does

indeed exist, and that the measurability notion associated with it bas some

nice stability properties.

The usefulness of this generalized measurability concept will become

evident in the subsequent sections where certain, not necessarily measur­

able, sets and mappings emerge which turn out to be measurable in this

generalized sense.

DEFINITION. Let (E,E) be a measurable space and for each probability v on E

let"EP be the completion of E with respect to ~.

A subset of E is called universaZly measurabZe if it belongs to E V

for every probability v on E. The collection of all universally measur-

able subsets of (E,E) is denoted by U(E) and is called the universat com­pletion of E.

11

PRDPOSITION 3.1. Let (E,E) be a measurabLe spaae. Then U(E) is a ~-aLgebra

aontaining E. When A is a cr-aLgebra of subsets of E suah that E c A c U(E), then every probabiLity on E aan be uniqueLy extended to a probabiLity on A.

PROOF. We have U(E) = n E , where the intersection is taken over all -- 1.11.1 ' probabilities 1.1 on E. Consequently U(E) is the intersectien of a collection

of cr-algebras and therefore it is a cr-algebra itself. Obviously, E c U(E).

Now let A be a a-algebra such that E c A c U(E) and let u be a

probability on E. Then 1.1 can be extended to E and, hence, to the sub-a­ll

algebra A of E • Let u' be an extension of 1.1 to A and let A E A. Then 1.1

A E Ell, so there exist B1,B2 E E such that B1 cA c B2 and u(B1) = u(B2).

This, however, implies u(B 1) = u'(B 1) ~ u'(A) ~ u'(B2) = u(B2), so u'(A) is

uniquely determined by p. From the arbitrariness of A it fellows that 1.1' is

the unique extension of 1.1 on A. 0

The inclusion E c U(E) can be strict as will be seen later (see

Proposition 4.11).

When confusion is unlikely we shall denote both a probability on E and its extension to U(E) by the same symbol.

The term "completion" for the colleetien U(E) is justified by the

following proposition.

PROPOSITION 3.2 •. Let (E,E) be a measurabLe spaae. Then U(U(E)) = U(E), i.e.

in the measurabLe spaae (E,U(E)) every universaLLy measurabLe subset is

measurabLe.

PROOF. Let A E U(U(E)). We shall prove that A E U(E), i.e. that A E E for -- ' 1.1 every probability 1.1 on E. So, let 1.1 be a probability on E and let its

extensions to U(E) and to U(U(E)) be denoted by 1.1 as well. Since

A E U(U(E)) c (U(E))Il, there exists a set B1 E U(E) such that B1 cA and

u(B 1) = v(A). From B1 E U(E) c E it follows that there exists a set c1 E E 1.1 .

such that c1 c B1 and u(C 1) = v(B 1), and hence such that c1 cA and

u(C 1) = v(A). By an analogous argument there exists a set c2 E E such that

Ac c2 and v(A) = u(C2). SoA E Ell. D

As has been argued earlier, identification of points of a measurable

space that are not separated by measurable sets is inessential in many

cases. This is also the case with regard to universa! completion. In fact,

12

the following proposition implies that the result of unive'rsal completion

and identification of points does not depend on the order in which these

two operations have been performed.

PROPOSITION 3.3. A pair of points of a measurabte spaae is se'!_)ai'ated by the

measu~te sets if it is separated by the universatry measu~te sets.

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

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

ties 6 and ö , which are defined on the o-algebra of all subsets of E, do x y not coincide on U(E). By Proposition 3.1 this implies that these probabili-

ties do not coincide on E either and that therefore the points x and y are

separated by E.

We now turn to the measurability of mappings with respect to cr­

algebras of universally measurable sets.

DEFINITION. A mapping ~ of a measurable space E into a measurable space F

is called universa"L"Ly measu~Ze when for each measurable subset B of F the

set ~-IB is a universally measurable subset of E.

D

So, universa! measurability with respect to the cr-algebras E and F is

the same as measurability with respect to U(E) and F. Consequently a mapping

of a measurable space into a product space is universally measurable iff its

coordinates are universally measurable. Also in Proposition 2.2 the words

'1measurable" can be replaced by "universally measurable". Universa! measur­

ability with respect to E and F is equivalent also with measurability with

respect to U(E) and U(f), as is stated in the following proposition.

~ROPOSITION 3.4. Let E and F be measurabte spaaes, Zet ~= E + F be univer--I . saUy measurabte and B a universatry measurable subset of F. Then q1 B 1-s a

universatry measurabte subset of E.

PROOF. Let E be the cr-algebra of E. Due to Proposition 3.2 it is sufficient

to prove that ~-IB belongs to U(U(E)) or, equivalently, that ql-IB belongs

to (U(E)) for every probability v on U(E). V

Let therefore v be the probability on U(E). Then v o q1-J is a

probability on F. Now, B is a universally measurable subset of F, so there

13

exist measurable subsets B1 and B2 of F such that B1 c B c B2 and -1 -1 -1 -1 -1 (v 0 q> )B 1 = (v<>q> )B2, and hence such that q> B1 c q> B c q> .B2 and

-1 -1 -1 -1 v(q> B1) = v(q> B2). Since q> B1 and q> B2 belong to U(E) by the universa!

measurability of q>, this implies that q>-IB € (U{E)) • 0 V

COROLLARY 3.5. A aomposition of universally measurable mappings is univer­

sally measurable.

As a particular kind of universally measurable mappings we have the

universally measurable functions. Let (E,E) be a measurable space, ~ a

probability on E, and f a positive universally measurable function on E.

Since f is measurable with respect to the cr-algebra U(E) and as ~ is

uniquely extendible to U(E), we can define J fd~ to be the integral of f

with respect to the measure space (E,U(E),p). This integral is the unique

extension, as a a-additive functional, of the integral of positive measur­

able functions.

The last proposition in this section bears upon univeraal measur~

ability in spaces of probabilities. Reeall that the cr-algebra Ë on the

space E of all probabilities on a measurable space (E,E) has been defined

such that ~ ~ ~(A) is a measurable function on Ë for every measurable

subset A of E.

PROPOSITION 3.6. Let E be a measurable spaae and A a universally measurable

subset of E. Then the fUnction ~~~(A) is universally measurable on Ë.

PROOF. We have to show that p ~~(A) is measurable with respect to (Ë)v for

every probability v on Ë. So let v be a probability on Ë and let À be

defined on the cr-algebra E of E by À(B) := ËJ p(B)v(d~). Then À is easily

seen to be a probability. As A is universally measurable and therefore

belongs to EÀ, there exist sets Bl'B2 E E such that B1 c Ac B2 and

À(B2 \ B1) = 0. Consequently, by the definition of À we. have

0 À(B2 \ B1) = _f ~(B2 \ B 1 )v(d~) = _f[~(B2 ) - ~(B 1 )]v(dp) •

'E E

We also have for all p € Ë the inequalities p(B1) s ~(A) s ~(B2 ). So

p(A) = ~(B2 ) for v-almost all~ € Ë. Now B2 E E, so p(B2) is a measurable

function of ~· Therefore the function ~ ~ ~(A) is measurable with respeét

to (Ë\· 0

14

As is easily seen, Proposition 3.6 is equivalent to the inclusion

[U(E)]- c U(Ë). When the cr-algebra E consistsof ~ and E only, then Ë consists of only one probability and, consequently, this inclusion in fact

is an equality. In all other cases, however, the inclusion is strict, as

will be proved insection 4, (see the remark following Propaaition 4.10).

§ 4. Souslin sets and Souslin functions

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

construction principle by which the memhers of cr(A) can be obtained from

those of A. In this section a construction principle, the Souslin operation,

is considered which, when applied to a class A meeting certain conditions,

yields a class of sets which contains o(A). Moreover, the class obtained is

not too large, since it is itself contained in the cr-algebra of universally

measurable sets derived from cr(A).

Another instanee where the Souslin eperation appears concerns projee­

tions in product spaces. Let S be a measurable subset of a product space

E x F. Then the projection {x E E I 3 F (x,y) E S} of S on E is in general yE not a measurable subset of E. It can, however, be obtained from the measur-

able subsets of E by application of the Souslin eperation in many cases.

For the interestins history of the Souslin oparation we refer to

[Hoffmann-J~rgensen] Chapter II, § 11.

Reeall that JNJN is the space of all (infinite) sequences of positive IN integers and that each n E 1N equals the (infinite) sequence

(n1,n2,n3

, ••• ) of its coordinates.

DEFINITIÓN. Let A be a collection of sets and let ~ be the set of all finite

sequences of positive integers. A SousZin saheme on A is a family (Alfl)tjiE'I''

such that Vlf!E'I' Atjl E A. The ke~et of a Souslin scheme (Alfl)tjiE'I' is the set

The collection consisting of the kernels of all Souslin schemes on a collee­

tion-A is called the Soustin ctaas generated by A; it is denoted by S(A).

The eperation S, i.e. the mapping A~ S(A), is called the Soustin ope~ation.

When (E,E) is a measurable space then the memhers of S(E) will be called

SousZin (sub)sets of (E,E).

15

' It follows directly from the definition of S that for each collection

Á and each S E S(A) there exists a countable subclass A0 of A such that

SE S(Ao).Also the implication Ac B • S(A) c S(B) and the inclusion

A c S(A) are obvious, but more can be said:

PROPOSITION 4.1. Let A be a aolleation of subsets of some set: Then

i)

ii)

PROOF.

i)

A0

c S(A) and A0

c S(A),

when A c S(A) (in p~tiau~r when A is alosed under aomplementation) c

then cr(A) c S(A).

Let (B~)~E1N be a sequence in A and let A ~ ~ nl•". ,nk

Th en

U 1N (l A U B. nE1N kE1N nt• • • • ·~ ie1N 1

So A0

c S(A) and A0 c S(A).

(or (l B.) ie1N 1

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

{S e S(A) I Sc e S(Á)} is a sub-cr-algebra of S(A) which contains A. 0

It follows from ii) in the foregoing proposition that cr(A) c S(Ac)

for every collection A of subsets of some set. So, the memhers of a cr­

algebra can be obtained from the memhers of a generating class by comple­

mentation foliowed by the Souslin operation.

Like most of the set-theoretic operations considered up to now the

Souslin operation is idempotent:

PROPOSITION 4.2. Let A be a colteation of sets. Then S(S(A)) = S(A).

PROOF. The inclusion S(A) c S(S(A)) is obvious. To prove the reverse inclu-'

sion, let A e S(S(A)). Then A can be written as

1N where Au1, ••• ,nk E S(A) for all n and k. Also, for each n E lN and k e 1N,

we can write

16

u lN n mElN JI.ElN

with A E A for all m and Jl.. Consequently, n 1, ••• ,~;m 1 , ••• ,mR.

where all indexed sets belong to A. This equality is equivalent to

A= um u lN n n nElN f:lN+lN kElN JI.ElN

because for every family (B ) lN of sets we have k,m kElN, mElN

xE n ulNBk * kElN mElN ,m

x €

Next, in the expression for A we combine the two unions into one. To this

, end let a: lN + lN and 8: lN + lN be such that the mapping n ~ (a(n),S(n))

is a surjection of lN onto lN x lN. Then for each sequence n E lN lN and for

each mapping f: lN + lNlN there exists a (non-unique) mapping g: lN x lN + lN,

such that

ni = a(g(i,l)) and f(i). = 8(g(i,j)) J

Therefore we can write A in the form

(i,j E lN) •

A • g:lN~lN+lN (k,JI.)~lNxlN Aa(g(l,l)) ,. •. ,a(g(k,l)) ;8(g(k,l)),. • • ,8(g(k,R.))

where we have also combined the two intersections.

Our next step is the transformation of the set lN x lN, appearing

twice, into lN. Let y: lN x lN + lN be a bijeetion such that

(I) V ... ,., lN[i+j < i'+j' .. y(i,j) < y(i',j')]. ~.J.~ ,J E:

Since y is injective, for each mapping g: lN x lN + IN there exists a

sequence h E: lNIN such that g(i,j) = h (' ') (i,j E IN). As a consequence y ~.J

we have

It follows from (1) that the numbers y(l,l), ••• ,y(k,~) appearing in the

multi-index do not exceed y(k,~). So, the multi-index is a function F of

the coordinates hl'h2, ••• ,hy(k,t) of h only~ '7he function Fitself depends

on k and t only and therefore on y(k,t) only, because y is injective. As y

is bijective, we can replace (k,~) oy y(k,t) in the intersectien thus ob­

taining

A

for suita~ly defined index functions Fj on INj (j E: IN), Consequently, A

17

belongs to S (A) • 0 .

As a simple consequence of the two foregoing propositions a Souslin

class is closed under the formation of countable unions and countable

intersections. In general however, a Souslin class is not closed under

complementation and therefore is not a a-algebra.

The following two propositions express the fact that the Souslin

eperation commutes with certain other operations.

PROPOSITION 4.3. Let~ be a mapping of a set E into a set F and Zet B be a . -1 .. -~

aoUeation of subsets of.F. Then Ql (S(B)) = ,S(q~ B).

PROOF. Since ~p-I commutes with unions and intersections, for each Souslin

scheme B on B we have

Hence the result. 0

18

PROPOSITION 4.4. Let F be a set, E a subset of F and B a aoZZeation of

subsets of F. Then

i) S(B) IE = S(BIE)

ii) when, in addition, E ~ S(B), then

S(B)IE- {S € S(B) I s c E}

PROOF.

i) Let ~: E + F be the identity on E. Then the result follows from the

preceding proposition.

ii) LetS be a subset of E. When S € S(B)IE, then forsome S' € S(B) we ·

have S = S' n E ~ [S(B)]d = S(B). When, on the other hand, S € S(B),

then S = S n E E S(B) IE. 0

The last set-theoretic property of Souslin classes we mention concerns

product a-algebras.

PROPOSITION 4.5. Let (E,E) and (F,F) be meas~abZe spaaes, let A be a

· aoZZeation of subsets of E suah that E c S(A), and Zet B be a aolZeation of subsets of F auch that F c S(B). Then E 8 F c S(A x B). In pa:I'tiauZa:r,

E 8 F c S(E x f).

PROOF. As a simple consequence of the definition of the Souslin operation

we have, for each B E B,

S(A) x {B} = {S x B I s € S(A)} = S({A x B I A € A}) c S(A x B)

Consequently, S(A) x B c S(A x B).

Interchanging the role of A and B in the foregoing argument we get

A x S(B) c S(A x B) and, applying this argument to the collections A and

S(B) insteadof A and B, we get S(A) x S(B) c S(A x S(B)). The last two

inclusions yield:

E x F c S(A) x S(B) c S(A x S(B)) c SS(A x B) = S(A x B) •

Now the complement of any measurable rectangle in E x F is the union of two

such rectangles, so

(E x F) c (E x f) c [S(A x B)] = S(A x B) • c s s

19

It now follows from the secoud part of Proposition 4.1 that

E 0 F = cr(E x F) c S(A x B) •

Beside the set-theoretic properties mentioned before, Souslin classes

have some useful measure-theoiietic features. To start with, we have the

following relation to universally measurable sets.

PROPOSITION 4.6. Let (E,E) be a measurable space. Then each SousZin set is

universally measurable and the cr-algebra of universally measurable sets is

closedunder the SousZin operation, i.e. S(E) c U(E) = SU(E).

The proof of this proposition will be combined with the proof of

Proposition 5.2.

From the foregoing sections we know that for every (universally)

measurable set A in a measurable space (E,E) the function ~ ~ ~(A) is

(universally) measurable on Ë. What can he said of this function when A is

a Souslin set? Of course, in general measurability with respect to S(E) is

not defined, as S(E) may fail to he a cr-algebra.

0

In order to describe the dependenee of ~(A) on ~ when A is a Souslin

set, we introduce functions which closely resemble the measurable functions~

DEFINITION. Let (E,E) be a measurable space. A function f: E + IR is called

a SousZin function if {x E E I f(x) > a} E S(E) for each a E IR.

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 be one, because the com­

plement of a Souslin set may not be a Souslin set.

Note that in the following proposition we use the conventions

co - co = - co and ± co• 0 = 0 (see Preliminaries 14) •

PROPOSITION 4.7.

i) Let (fn)nEIN be a sequence of SousZin functions on a measurable

space. Then sup fn, inf fn, limsup fn and liminf fn ave SousZin .func-n n n n

ti ons as 7:/Je ll.

~0

ii) Let f and g be Soustin jUnations on a measurable spaae. Then f + g

also is a Soustin fUnation. When in addition f and g are positive~

then fg is a Soustin jUnation as ~lt.

PROOF. Let the functions be defined on a measurable space (E,E).

i) For every a e IR we have

{x e E I sup f (x) > a} n

U {x e E I f (x) > a} e S(E) S(E) nem n cr

and

n

{x E E I inf f (x) > a} n

n

U n {x E E I fn(x) ~ a + ~} e S(E)acr S(E) • mElli nem

So supn fn and infn fn are Souslin functions. From this and the

equalities limsupn fn = infn supm~ fm and liminfn fn supn infm~ fm the remainder of i) follows.

ii) For every a e IR we have

{f+g > a} = U {f > r} n {g > a-r} e S(E)dcr S(E) , reQ

where Q is the set of rational numbers. So f + g is a Souslin function.

For f and g positive and a ~ 0 we have

{fg > a} = U reQ r>O

{f > r} n {g > ~} e S(E) r

which implies that fg is a Souslin function.

Reeall that we introduced Souslin functions in order to describe the

function ~»~(A) for Souslin sets A.

PROPOSITION 4.8. Let A be a Soustin set of a measurable spaae E. Then

~~~(A) is a Soustin jUnation on E.

The proof of this proposition will be combined with the proof of

Proposition 5.2.

0

21

We conclude this section with some results that will not be used in

the rest of this monograph. We first consider the inclusions f c S(f) c U(f)

and [U(f)]- c U(E), as given in Proposition 4.6 and the remark following

Proposition 3.6, and in particular the question whether these inclusions

are strict.

PROPOSITION 4.9. Let (E,f) be a countably generated measurable space that

can be ma:pped measu:rub ly onto the product space lN lN. Then S ( f) is not

closed under complementation.

PROOF. Let A = {A1

,A2

, ••• } be a countable generating subclass of E that is

closedunder complementation. Then by Proposition 4.1, E o(A) c S(A) c

c S(f), so S(E) = S(A).

Let ' be the set of all finite sequences of positive integers and let w F be the product space lN . Moreover, let the subset U of E x F be defined

by

u := u m n u [A. x {y € F 1 Yc > nElN kEJN tEJN "' n) '•" '~

R.}] •

Then U is a Souslin subset of E x F. Also, for each y E F, we find for the

"y-section" of U:

{x € E I (x,y) E U} u lN n n<::lN k<::lN

So for each S <:: S(A) we have S = {x E E I (x,y) E U} for some y E F. lN Now W is countably infinite, so F is isomorphic to IN and, there-

fore, there exists a measurable surjection ~: E ~ F. The mapping

x: E + E x F, defined by x(x) = (x,~(x)), is measurable, because its

coordinates are, and x-1u is a Souslin subset of E by Proposition 4.3. -I It will be sufficient to prove, that the complement C of x U is not

a Souslin set of E. We reason by contradiction. Suppose C E S(E). Then

C <:: S(A) and therefore C = {x E E I (x,y0) E U} for some y0 E F. Since ~ is

surjective, Yo = ~(x0) for some x0 € E. Now we have the following equiva­

lences:

which is a contradiction. D

22

PROPOSITION 4.10. Let (E,E) be a measurable spaae and let E; {0,E}. Then

S(E) is not alosed under aomplementation.

PROOF. Let A E E \ {-,E}, and let x E A and y i A. Then ö (A) = I ; 0 = ó (A), x y

so ó ; ó • Consequently, x y

I := {Aó + (1-A)ó I À E (0,1]} x y

is a subspace of E which is isomorphic to the interval (0,1].

Next consider the mapping

n»-Ï 2-(nl+ ••• +~)

k=l

of ININ onto (0,1]. As this mapping is an isomorphism, the space (0,1] can IN be mapped measurably onto IN •

The foregoing implies that I can be mapped measurably onto ININ and,

as a consequence of the foregoing proposition, the Souslin class of I is

not closed under complementation. As the Souslin class of I is the trace on

I of the Souslin class S(E) of E, the class S(E) is not closed under com-

plementation either.

Now let (E,E) be a measurable space such that E; {0,E}. Then

U(E) f {0,E} and it follows from the foregoing proposition applied to the

space (E,U(E)) that S(U(E)~) is not closedunder complementation and that

the inclusion U(E)~ c S(U(E)-} is therefore strict. From the inclusion

U(E)- c U(E) mentioned after Proposition 3.6 and from Proposition 4.6 we

deduce

S(U(E)~) c S(U(E)) U(E)

The inclusion U(E)~ c U(E) is therefore strict.

As an interesting consequence of Proposition 4.9 we have:

PROPOSITION 4.11. Let B be the a-algebra of Borel subsets of IR. Then the

inelusione B c S(B) c U(B) are striet and the a-algebva U(B) is not

eountably generated.

PROOF. The space (IR,B) can be mapped measurably onto its subspace (0,1], -- 2-1 . e.g. by the mapping x»- (1 +x) , and the space (0,1] can be mapped

0

23

lN measurably onto IN (see the proof of Proposition 4.10). So, (IR,B) can be lN mapped measurably onto IN and, of course, the same holds for (IR,U(B)).

Since B is countably generated, by Proposition 4.9 the collection S(B)

is not closed under complementation and therefore it is not a a-algebra.

Consequently, the inclusions B c S(B) c U(B) are strict.

Next suppose that U(B) is countably generated. Then it follows from

Proposition 4.9 applied tothespace (IR,U(B)) that S(U(B)) is not closed

under complementation, which contradiets Proposition 4.6. So U(B) is not

countably generated. 0

Beside the a-algebra of universally measurable sets there is another

cr-algebra of subsets of a measurable space which has equally nice properties.

The remainder of this section is devoted to this cr-algebra. The result will

not be used in the sequel.

For any measurable space (E,E) let L(E) be the smallest cr-algebra of

subsets of E that contains E and is closed under the Souslin operatien S. Following Bertsekas and Shreve we call L(E) the limit-cr-algebra of (E,E)

and we call its memhers limit measurable subsets of (E,E) (see [Bertsekas & Shreve] p. 292).

All properties of the cr-algebra of universally measurable sets derived

up to now are ahared by the cr-algebra of limit measurable sets, as will be

proved presently.

Since the role of universally measurable sets and universally measur­

able mappings in the remainder of this monograph is based entirely on these

common properties, the adjective "universally measurable" can be replaced

everywhere by "limit measurable" without affecting the validity of the

results.

A property of limit o-algebras that is possibly not shared by all

univeraal completions is the equality

L(f) = U L(E0

) , Eo

where the union is over all countably generated sub-a-algebras E0 of E. To prove this equality we merely note that the righthand side is a cr-algebra

which is closed under S.

24

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

Proposition 4.6, for every measurable space (E,E) we have L(E) c U(E). From

this inclusion the analogues of the Propositions 3.1 and 3.3 for the cr­

algebra of limit measurable sets easily follow. As to Proposition 3.4, we

remark that for a limit measurable mapping ~: (E,E) ~ (F,F) the colleetien

I -1 {B c F ~ B € L(E)} is a cr-algebra which is closedunder S and which

contains F. The analogue of Corollary 3.5 is a simple consequence of the

analogue of Proposition 3.4 again, while the analogue of Proposition 3,2

follows directly from the definition of L.

The proof of the analogue of Proposition 3.6 is slightly more

laborious. Let (E,E) be a measurable space and B the collection consisting

of those memhers A of U(E) for which the function ~ ~ ~(A) on E is limit

measurable, i.e. measurable with respect to L(f). B is easily seen to be a

Dynkin class. Zorn's lemma implies that among the subclasses of B that

contain E and are closed under the formation of finite intersections, there

is a maximal one, say A. By Dynkin's theorem (Proposition 1.1) we have

cr(A) c B and the maximality of A therefore implies that cr(A) = A, i.e. that

A is a cr-algebra.

We now consider the space (E,A). Since E cA c U(E), it fellows from

Proposition 3.1 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 A c B, so for every A € A the function p » ~(A)

on E is measurable with respect to L(E) and consequently A c L(f). · It

follows from Prposition 4.8, applied to the space (E,A), that for every

A € S(A) we have V JR {pEE I p(A) > a} E S(A) and, as S(A) c SL(f) = L(E), aE ~

also that p » p(A) is limit measurable on E.

Due to the maximality of A again we conclude from this that S(A) = A. So A is a cr-algebra which contains E and which is closed under S and it

therefore contains L(E). We thus have proved the analogue of Proposition 3.6, that for every

limit measurable subset A of E the function ~ ~ p(A) on Ë is limit measur­

able.

The proof of Proposition 4.11 applies also to the limit-cr-algebra of

the space IR and this cr-algebra is therefore not countably generated,

§ 5. Semicompact classes

In this section we introduce the concept of a semicompact class.

"Countably compact" might have been a more suitable adjective for these

classes, because their defining property is precisely the set-theoretic

feature exhibited by the class of closed subsets of a countably compact

topological space. However, the term "semicompact" is the most usual one.

25

The usefulness of semicompact classes lies in the fact that cr-additi­

vity of functions defined on algebras of sets can be deduced from certain

approximation properties of semicompact classes (see [Neveu] Proposition

1.6.2; note that in this reference the term "compact" is used instead of

"semicompact").

DEFINITION. A colleetien A of sets is said to possess the finite inter­

Beation pPoperty if every finite subcollection of A has a nonempty inter­

section. A colleetien A of sets is called semiaompaat if every countable

subcollection of A which possesses the finite intersectien property bas a

nonempty intersection.

~~~~~~~'· When Cis a semiaompaat aoZZeation of sets, then Csö is semiaompaat as weU.

PROOF. Let {B I n e IN} be a subcollection of C having the finite inter---- n s section property. We first prove that there exists a subcollection

{Cn I n E IN} of C having the finite intersectien property and such that

VneiN Cn c Bn. We praeeed by recursion. Let n e IN and suppose that c1, ••• ,cn-l e C

have been defined such that the collection B' := {C1, ••• ,Cn-I'Bn,Bn+l''''}

bas the finite intersectien property.

As B E C, we have B =uP 1 C' forsome p E IN and c 1•, ••• ,C' E C. n s n m= m p Now forsome mE {I, ••• ,p} the collection B' u {C'} has the finite inter­m section property: if not, then for each mE {J, ••• ,p} there exists a finite

subcollection B' of B' such that (n B') n C' = ~. Consequently, uP 1 B' is m m m m= m a finite subcollection of B' whose intersectien does nat meet UP 1 C' and m= m this contradiets the finite intersectien property of B'. So there exists a

setenE C such that B' u {Cn}' and therefore also {c1, ••• ,cn-I'Cn,Bn+l''''},

bas the finite intersectien property.

26

The sequence (Cn)n<!lN constructed in this way has the desired proper­

ties. These properties and the semicompactness of C now imply that

n lN B ~ n lN C ~ 0. Finally it follows from the arbitrariness of the nE n nE n sequence (B ) lN that C is semicompact.

n nE s Next, let B be a countable subclass of Cso such that nB = ~. Then

B = {n lN C I n E IN} for a suitable choice of C E C . Hence mE run run s n( ) 1N2 C = nB = ~ and, as C is semicompact, n( ) I C = 0 for some n,m E nm s n,m E run finite subset I of 1N2 • This implies thatB has a finite subclass the

intersection of which is empty. From the arbitrariness of B it follows that

C50 is semicompact.

Let À be Lebesgue measure on IR. It is well known that every

Lebesgue-measurable subset A of IR can be approximated from the inside by

compact sets in the following sense:

À(A) = sup {À(B) I B c A.and B compact}.

This proparty of the Lebesgue measure is called inner regularity. The

following proposition shows that all probabilities on a measurable space

are inner regular with respect to suitably chosen collections of subsets of

the space.

PROPOSITION 5.2. Let (E,f) be a meaeurabZe epaae and let C be a aolleation

of subsets of E suah that C c E c S(C). Then

~(A) = sup {p(B) I B c A and B € cso}

foP evePy pPobability ~ on E and foP evePy univePsally measurable subset A

of E.

PROOF. As announced, the proofs of the Propositions 4.6 and 4.8 and of the

present proposition will be given simultaneously. For pairs

(n1, ••• ,nt),(m1, ••• ,mt) of sequences (of length t) of positive integere we

define (np ... ,nt) s (mi, ... ,mt) to mean nj s mj (j = 1, ... ,t).

Let A E S(f), Then by the definition of S we can write

A nl' .•• ·~

where the sets A belong tof, For each finite sequence (m1, ••• ,mt) nl, • • • '~

0

27

of positive integers we define

Since both the intersection and the union occurring in this definition are

finite, the sets Bm mo belong to E. Also for each sequence m of positive I • ... ' "'

integers the sequence (Bm m ) 0 1N of sets is decreasing and, as we shall I'. • •' i/, "'€

show, no 1N R m c A. To prove this inclusion, let x € n R "'€ -mi • • • • • !/, telN "11ll • • • • ,m.~~, •

Then by the definition of the sets B we have m1 , ••• ,m_q_

and, consequently,

is a decreasing sequence of nonempty subsets of ININ, each of which belongs

to the collection Ads' where

A := {{n € lNIN I n p

q} I p,q e: m} •

Now A is semicompact, as is easily seen, and Ads is semicompact as well,

because Ads = Asd c Aso' The sequence (*) therefore has a nonempty inter-

. · h · n € INlN such that sect1on, 1.e. t ere ex1sts an

and therefore such that

So x € A, and the inclusion has been proved.

Now let p be a probability on E and let a € m. Suppose that

(I) p*(A) ;:: a ,

where * is the outer measure corresponding to p. Then, as will be p

demonstrated, for every i E IN there exists an m E ININ such that for every

~ € IN

I -..,..

so

(2)

The existence of such an m for every i follows, by induction on ~. from the

* continuity of ~ on increasing sequences of sets, and from the relations

Statement (2) is equivalent to

~

(2') ~ E 11 U IN 11 {v E E iEIN mEIN R.EJN

v(B ) > a - ~} m1, ••• ,m~ J.

For each mE lNIN the sequence (B ) is decreasing and m1, ••• ,m.t R.ElN

therefore

Together with (2) this implies

so

sup mEINlN

;;:: a •

I ;:: a - ..,...] • l.

As A => 11 B for every m E lN IN, we can conclude R-EIN mi' • • • ,mt

l.

(3) Jl*(A) 2!: sup me: IN IN

29

"' a '

where \l* is the inner measure corresponding to Jl•

Now (3) is easily seen to imply (1). So, for every \l e: E and a e: IR,

the statements (1), (2), (2') and (3) are equivalent. From these equiva­

lences we shall now derive the proofs for the Propositions 4.6, 4.8, and

5.2.

i) Taking a := \l*(A) in the equivalent statements (1) and (3), we get

\l (A) ~ \l*(A). SoA belongs toE • As \l is arbitrary, this implies * \l

that A is universally measurable. The arbitrariness of A, in turn,

implies that S(E) c U(E). Applying this inclusion to the space

(E,U(E)) instead of (E,E) we get SU(E) c UU(E), so SU{f) = U(f). Thus

the proof of Proposition 4.6 is complete. Note that the existence of

a colleetien C, as mentioned in Proposition 5.2, does not restriet

the space (E,f), since we always have E c E c S(E).

i i) The equivalence of ( 1) and (2 1 ) tagether with the universa! measur­

ability of A implies that the set {v e: Ë I v(A) 2!: a} equals the right­

hand side of (2'), which is a Souslin subset of E. From the arbitrari­

ness of a we conclude that

{v e: Ë I v(A) > b} = U {v e: E I v(A) ie: IN

b + ~} e: S(Ë) ].

for every b e: IR. So v ~ v(A} is a Souslin function, thus proving

Proposition 4.8.

iii) To prove Proposition 5.2 we take a := Jl*(A) once more. It then

fellows from the equivalence of (I} and (3) that

\l(A) sup mElNIN

Now C is a subclass of E such that E c S(C), and hence, such that

Ac S(E) c SS(C) = S(C). The sets Au1, •.• ,nk may therefore be supposed

to belang to C, and the sets n. lN R m can be assumed to belang "'" "1111 • • • • • 11.

to Cdsö' Since Cdsö = Csdö = C90 , the equality appearing in the pro-

position has been proved for Souslin sets A and, consequently, for

measurable sets A. For universally measurable sets A the equality

follows from this and the mere definition of universa! measurability. D

30

DEFINITION. A positive function ~ defined on a collection A of sets is

called additive (a-additive) when, for each finite (countable) subclass A0 of A consisting of pairwise disjoint sets, and such that U A0 € A, the

equality ~(U A0) ~ I ~(A) holds. AEA0

When ~ is additive then obviously ~(0) = 0.

PROPOSITION 5.3. Let A be a semicompaat algebra of subsets of a set. Then

every positive additive fUnction on A is a-additive.

PROOF. Let {A I n E IN} be a countable collection of pairwise disjoint --- n memhers of A such that U 1N A € A. Then the collection {U

00

A I m € lN} nE n n=m n

has empty intersection. As a subcollection of A the collection is semicom-

pact, and consequently Uw A = 0 for some m E IN, which in turn implies n=m n

that only finitely many members of {A I n E IN} are nonempty. The fore­n

going implies that additivity and cr-additivity are equivalent for positive

functions defined on A.

Next we introduce the auxiliary space ID, the so-called Cantor space.

0

The role played by ID strongly resembles the role of the space IR in many

measure-theoretic arguments. In fact, ID and IR can be shown to be isomorphic

measurable spaces ([Bertsekas & Shreve] Proposition 7.16). The space ID

however is better suited to our needs.

DEFINITION. The measurable space ID is the product space TI lN(D ,V), n€ n n where, for each n E IN, Dn := {0,1} and Vn is the cr-algebra consisting of

all (four) subsets of Dn.

I Note that for any countable set,I the spaces ID and ID are isomorphic.

As another useful property of ID we have:

PROPOSITrON 5.4. The a-algebra of ID is generated by a aountable semiaampact

algebra.

PROOF. Let C :={{x E ID I xi = j} I i E IN, j = 0,1}. Then Cis generating,

countable and semicompact. By Proposition 5.1 the same holds for Csd' As C

is closed under complementation, the collection Csd is an algebra. 0

31

§ 6. Measurability of integrals

A well-known theorem in measure theory says that the integral

J f(x,y)p(dy) of a measurable function of two variables is a measurable

function of x. In this section we shall prove that this integral depends

measurably on ll as well, ,and that similar results are valid for universally

measurable functions and Souslin functions. Before proving these measur­

ability properties we introduce a generalization of the integral, which

will turn out to be convenient later on.

A universally measurable function f defined on a measurable space E

can be written in the form f+- f-, where f+ and f- are the positive

universally measurable functions on E defined by

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

When ll is a probability on E then f is called quasi-integrable with respect

to ll when at least one of the integrals J f+ dp and J f- d)l is finite. In

this case one defines

As is easily seen, a universally measurable function defined on a

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 consider bounded functions only or to demand quasi-integrabil­

ity in advance every time, we generalize the integral concept such as to be

applicable even to functions that are not quasi-integrable.

DEFINITION. For a universally measurable function f and a probability ll on

a measurable space we define J f d)l to be equal to J f dp if f is quasi­

integrable with respecttolland equal to -ro in the other case.

Using the convention ~ - oo:::-oo and ± <»•0 = 0 (see Preliminaries 14)

we conclude that J fdp depends on f in an additive and positively homo-

geneaus manner:

I (f+g)du J fdp + I gdp and I Hdp À J fd)l for À > 0 •

32

In particular we have

J f dJJ

In general however, J -f dJJ differs from - J f dJJ. Al so, Fubini' s theorem

cannot be generalized.

We now turn to the measurability properties of the integral defined

above.

PROPOSITION 6.1. Let f be a universally measurable (or measurable, or

Souslin) funotion on a meaaurable apace E. Then the function Jl 1+ [ fdJJ on E is universally meaaurable (or measurable, or Souslin, respectively).

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

case that f is positive. Then f is the limit of the increasing sequence

of positive functions, so

J fdJJ : lim 2-n ~ Jl{f>m 2-n}

n m=l for each Jl € E

For each m,n E lN the set {f>m 2-n} is a Souslin subset of E, which implies

by Proposition 4.8 that JJ{f>m 2-n} is a Souslin function of Jl, Applying

Proposition 4.7 we conclude that J fdJJ also is a Souslin function of Jl•

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 for each m,n € lN let

the function fn be defined on E by m

f 0 (x) :=min {n,max{-m,f(x)}} m

Note that fn is the function f truncated at the values + n and - m. For m each m,n € IN f 0 is a bounded Souslin function and

' m

33

It now follows from Proposition 4.7 that 1 fdv is a Souslin function of v.

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

in a similar way: instead of Proposition 4.8 use the fact that v(A) depends

(universally) measurable on v for every (universally) measurable set A,

i.e. Proposition 3.6 and the definition of E. 0

PROPOSITION 6.2. Let E and F be measurable spaces and f a universally

measurabZe (or measurable, or SousZin) funation on E ~ F. Then the funations

y ~ f(x,y) (x E E) on F and the funation (x,v) ~ [ f(x,y)v(dy) on E x F are

universally measurable (or measurable, or Souslin, respeatively).

PROOF. We prove the statement on Souslin functions; the other cases can be

treated in a similar way.

Let x E E. Then the mapping y ~ (x,y) of F into E x F is measurable,

because its coordinates y ~ x and y ~ y are measurable. Now y ~ f(x,y) is

the composition of this measurable mapping and the Souslin function f and

therefore it is a Souslin function itself.

For every x E E, p E F we have

and similarly for f-. Consequently J f(x,y)v(dy) equals J fd(ö x u), - - x

which is a Souslin tunetion of öx x v by Proposition 6.1. The result now

follows from the fact that ox x v depends measurably on (x,v) (see the

examples 1 and 5 following Proposition 2.2), and that the composition of a

measurable mapping and a Souslin function is a Souslin function. 0

34

CHAPTER 11 ANAL YT IC SPACES

We have now arrived at the main topic of this monograph: a measure­

theoretic treatment of analytic spaces. The importance of analytic topo­

logical spaces (for a definition see [Hoffmann-J~rgensen], Ch. III § 1) in

dynamic programming is due to certain properties of the ~-algebras generated

by their topologies. In our measure-theoretic approach we have taken these

properties as a starting point, and, in fact, we have chosen one of them as

the defining prÓperty of the class of measurable spaces to be studied. This

class, therefore, is an extension of the class of analytic topological

spaces as far as the measure-theoretic structure is concerned.

The facts on analytic spaces collected in the first section of this

chapter should give the reader a good impression of such spaces. But, since

we have always kept in mind the applications of chapter III, the treatment

of the subject can only be considered complete in conneetion with these

applications. In the second section, the one on separating classes, count­

ably generated analytic spaces are considered, and their relation to topo­

logical analytic spaces is made clear. In the third and final section of

this chapter probabilities on analytic spaces are treated, and, in particu­

lar, sets of these probabilities that will play a role in the applications.

§ 7. 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:

~) The projection SE of S on E is a Souslin subset of E,

ii) S contains the graph of a universally measurable mapping of the

subspace SE of E into F.

The reader should note that, in the above definition, it is not the

graph of the mapping that is supposed to be universally measurable but the

mapping itself.

A cr-algebra E of subsets of a set E is called analytic, when the

space (E,E) is analytic.

The usefulness of analytic spaces for dynamic programming is mainly

due to the property expressed in the following proposition, as will become

evident in Chapter III.

PROPOSITION 7.1 (Exact selection theorem). Let E be a meaau~abZe spaae, F

an analytia spaae~ S a SousZin subspaee of E x F eve~ seetion

35

Sx := {y E F I (x,y) E S} of whieh is nonempty~ and f a SousZin jUnetion on

s. F~the~ let g: E + ÏR be defined by g(x) = sup f(x,y) and let yESX

T := {x E E I 3yESx g(x) f(x,y)} •

Then g is a SousZin jUnetion and T is universally measurabZe.

Moreover, for every universaZZy measurable jUnation h on E sueh that

< g(x) if -oo < g(x), h(x)

g(x) else,

a universally measurable mapping ~= E + F exists the graph of whieh is

eontained in S and suah that for eaeh x E E

• g(x) f(x,~(x))

> h(x) else.

PROOF. Let E and F be the o-algebras of E and F, respectively. For every

a E IR the set {x E E I g(x) > a} is the projection on E of the Souslin

subset {(x,y) ES I f(x,y) >a} of Ex F, and therefore it is a Souslin

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

Next, let·

and

Th en

and

A:= {(x,y) ES I f(x,y) g(x)}

B := {(x,y) ES I f(x,y) > h(x)} •

A= s n [ n ({r<f} u ({g~r} x F))] rEQ

36

B = S n [ U ({r<f} n ({hsr} x F))] , rEQ

where Q is the set of rational numbers. Now, for each r E Q we have

{r<f} € S(E ® F) c S(U(E) ® F) ,

{gsr} x F E U(E) x F E S(U(E) 0 F) ,

and similarly for {hSr} x F. Consequently, A and B belong to S(U(E) ® F). We now consider the product space (E,U(l~)) x (F, F). From the analyti·­

city of F we conclude that the projection ~ of A on E is universally

measurable with respect to U(E) and that there exists a mapping a of ~

into F that is universally measurable with respect to U(E)IAE and whose

graph is contained in A. Similarly, the projection BE of B on E is univer­

sally measurable with respect to U(E) and there exists a mapping B of BE

into F that is universally measurable with respect to U(E)I~ and whose

graph is contained in 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

therefore with respect to E by Proposition 3.2.

Finally, the mapping ~: E ~ F defined by

= a(x) ifx ~ ~(x)

= B(x) else

is easily seen to have the desired properties.

For f a Souslin function defined on E x F taking the values 0 and 1

only, the exact selection theorem reduces to the definition of analyticity

of F. Consequently, the validity of the exact selection theorem character­

izes analytic spaces.

We shall show that the class of analytic spaces is closed under the

constructions for measurable spaces most commonly used and that it there­

fore contains many of the measurable spaces encountered in practice. To

begin with, we shall prove that the space ID, introduced in section 5, is

analytic.

PROPOSITION 7.2. The spaae ID is analytia.

D

PROOF. Let E be a measurable space and S a Souslin subset of E x ID. For

each k e: IN let the partition Pk of ID be defined by

I . k

l, ... ,k)! n € {0,1 } }

and let P ~= Uk€INPk. Then, for each k E IN and for each P E Pk' Pc is

union of 2 - I memhers of Pk. So P c P c S(P). Since, moreover, the cr-c s .

the

algebra V of ID is generated by P, we have V c S(P) as a consequence of

Proposition 4.1. When Eis the cr-algebra of E, then, by Proposition 4.5,

E ®V c S(E x P) and, therefore, S(E ®V) c S(E x P). Consequently, by the

definition of S the set S can be written as

s

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

37

For each n and k the set Bn n is contained in a memher of Pk or ]• .... k equal toa finite union of memhers of Pk; in any case Bn

1, ••• ,~ can be

written as a countable union, say Ut€IN B!1

, ••• ,nk' of memhers of P each of

which is contained in some memher of Pk. Hence we have

s

From this we deduce:

u IN n U€IN ke:IN

where n I+ (a (n), B (n)) is some surj eetion of IN onto lN x IN • Hence

s

where, for each n and k, Ari1

, ••. ,nk is a measurable subset of of E, and

B' is a memher of P that is contained in some memher of Pk. This nl, • • • ,nk

38

implies that for each n E 1N IN 0 R' contains at most one point kEJNllt• • • • ,nk

of ID.

For each n and k we now define

A' nl ' ••• ·~

Th en

k if n

i= I

el se.

B' r 0 , nl, .•• ,nt

Moreover, as P is semicompact, for each n E ININ we have the implications:

n kEJN

We now show that the projection SE of S on E is a Souslin subset of E.

Let x E SE. Then there is some y E lD such that (x,y) E S and hence

On the other hand, when x belongs to the last set, then there exists n E ININ

such that x E Ok IN ~ n , which implies Ok IN ~ n r 0, and E I•····k E l>"'•k

hence, the existence of y E 1D with y E nk IN R' n • Then E 11) ''' •' k

(x,y) E 0 kEJN

(A" x BI ) c s • nl '• •• ·~ nl • ••• ·~

so the point x belongs to SE. The foregoing implies that SE equals

U- nk A" n and, consequently, SE is a Souslin subset of E. n "llt••••• k

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

mapping q1: SE + ID. For each n € IN IN and k E IN we define

:= < u IN n mEIN R,€IN

A" ) o A" nl'''''~'ml, ••• ,mt nl, ••• ,~

Th en

* A. l.

39

and c

for each n E 1N 1N and k E lN. Now let x E SE and let p E 1N 1N be such that,

for each k E lN, pk is the smallest positive integer for which

* x E A • Then pl' ••• ,pk

n kElN

A" => fl Pp ... ,pk kE:lN

and hence nk ~T Bp' p ~ 0. This implies that the last intersectien E.u.• I• •••• k

contains precisely one point, say ~(x), of 1D. Moreover, we have

(x,~ (x)) € n kElN

The graph of the mapping ~: SE + 1D defined above is therefore contained

in S.

To prove the universal measurability of~. let k E 1N and B'E Pk.

Let N be the set of those n E 1Nk for which B' c B. Then the n 1 , •.• ,nk following four statements are equivalent:

~(x) E B • B' c B , (pi •••• ,pk) E N ' Pp .• • ,pk

k nt-I

(A* \ IJ * ) X E u n A • • nEN ~=I nl, ... ,nt i=l nl, ••• ,n~-1'1.

* -1 As each A is a Souslin subset of E and N is countable, ~ B belongs to the

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

measurable. Now the result follows from V= cr(UkElN Pk). D

The analyticity of a large class of measurable spaces will be deduced

from the analyticity of lD. Material in these deductions is Proposition 7.5,

where mappings of analytic spaces into lD are considered. First we give some

preparatory results on mappings in general.

40

PROPOSITION 7.3. Let E and F be measurable spaces, let F be countably

separated and let ~= E + F be measurable. Then the graph of ~ is a measur­

able subset of E x F.

PROOF. Let A be a countable collection of measurable subsets of F that

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

(x,y) i graph ~ ~ y I ~(x) ~

3AEÁ [(~(x) € A A y t A) V (~(x) /A A y € A)] ~

(x,y) E U [(~-IA x Ac) u ((~-lA)c x A)} . AEA

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

subsets of E x F, and therefore is measurable itself.

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

A right inverse of ~ is a mapping ~: ~E + E such that ~ o 4 is the identity

on ~E. The mapping ~ is called strictly measurable if E = ~- 1 F. The mapping

~ is called an isomorphia embedding when it is strictly measurable and

injective.

Note that every mapping has a right inverse, but that this inverse is

unique only when the mapping is injective.

PROPOSITION 7.4.

i) Let (E,E) and (F,F) be measurable spaces and Zet ~= E + F be striatly

measurable. Then S(E) = ~- 1 S(F) and every right inverse of~ is

measurable ~th respect to FI(~E).

ii) Let E be a aountabZy generated measurabZe spaae. Then there exists a

striatly measurable mapping ~= E + ID.

PROOF.

i) S(E) = S(~- 1 F) = ~- 1S(F) because of Proposition 4.3. Let ~he a right

inverse of ~· For every measurable subset A of E there exists a -I measurable subset B of F such that A = ~ B and, hence, such that

D

41

-1 -1 -1 -l ~ A = ~ (~ B) = (~o~) B = B n (~E) •

Consequently, ~is measurable with respect to FI(~E).

ii) Let {C I n E lN} be a countable colleetien generating i::he a-algebra n

E of E. Moreover, let V be the a-algebra of ID, and for each n E IN

let D := {y E lD I y • I}. Define q>: E ..,. lD by (~(x)) := Ie (x) n n _1

n n (n E IN,x E E). Then for every n E lN we have C = ~ D and con-

-I n n sequently o{C n E lN} = w cr{D n E IN} or equivalently _

1 n n

E = w V. So ~ is strictly measurable. 0

The characterization of analytic spaces given in the following

proposition is a useful alternative to the defining one. It will be used

frequently in the rest of this chapter.

PROPOSITION 7. 5. A measurahZe spaae F is anaZytia for every measurahZe

mapping w: F ..,. ID the range of ~ is a SousZin subset of lD and ~ has a

universaZZy measurahZe Y'ight inverse.

Let F be analytic and w: F ->- ID measurable. As lD is countably

separated, the graph of w is a measurable subset of F x ID by Proposition

7.3 and, hence, a Souslin subset of F x ID. The range of~ is the

projection on lD of the graph of (j) and therefore is a Souslin subset of ID.

Moreover, the graph of w contains the graph of a universally measurable

mapping of q>F into F, which obviously is a right inverse of~·

Conversely, let (F,F) be a measurable space having the property

stated in the proposition. We shall show that F obeys the definition of

analyticity. To this end let (E,E) be a measurable space, and let S be a

Souslin subset of E x F. Then by Propositions 4.2 and 4.5 we have

SE S(E ® F) c SS(E x F) = S(E x F), i.e. S is the kemel of a Souslin

scheme on E x F. As each Souslin scheme involves only countably many sets,

we even have S E S(E x C) for some countable subclass C of F. By Proposition 7.4 a mapping w: F->- ID exists that is strictly

measurable with respect to o(C), and, therefore, measurable with respect to

F. When ~: E x F + E x 1D is defined by ~(x,y) = (x,q>(y)), then ~ is

easily seen to be strictly measurable with respect toE® cr(C). It now

follows from Proposition 7.4 that a Souslin subsetS' of Ex 1D exists such -I that S = ~ S', and hence, such that ~S = S' n (Ex q>F). The properties of

F imply that ~F is a Souslin subset of ID. So E x q>F is a Souslin subset of

42

E x ID and this in turn implies that ~S is the intersectien of two Souslin

subsets of E x ID and, consequently, a Souslin set itself.

From the definition of ~ it now fellows that the projection SE of S

on E equals the projection of ~S on E. The latter set, however, is a

Souslin subset of E, due to the analyticity of ID.

It also fellows from the analyticity of lD that there exists a

universally measurable mapping a: SE + ID the graph of which is contained

in ~S. Further, by the properties of F, there exists a universally measur­

able right inverse 8 of ~· The mapping S o a: SE + F therefore is univer­

sally measurable, and for every x E SE we have

Hence,

-I graph(Soa) c 4 S' ~ S •

(x,ax) e ~s c S' •

The foregoing implies that F satisfies the defining conditions for

analyticity. 0

We are now able to prove the stability properties of the class of

analytic spacec announced earlier.

PROPOSITION 7.6. The aZass of anaZytia spaces is aZosed under the formation

of i) ~asurabZe images~

ii) SousZin subspaaes,

iii) product spaaes.

PROOF.

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

measurable mapping ~. We shall prove that the cha~acterization of

analytic spaces given in the preceeding proposition applies toF.

So, let ~ he a measurable mapping of F into ID. Then ~ o 4 is a

measurable mapping of the analytic space E into ID and, by the

preceeding proposition, its range is a Souslin subset of ID, while it

has a universally measurable right inverse X· As the range of ~ equals

the range of ~ o ~ and as ~ o x is a right inverse of ~. the range of~

is a Souslin subset of ID and ~ bas a universally measurable right

inverse. From this and the arbirariness of ~ it follows that F is

analytic.

43

ii) To prove the statement on Souslin subspaces we use the definition of

analyticity. Let F be a Souslin subspace of an analytic space F'.

Further, let E be a measurable space and let S be a Souslin subset of

E x F. Then, by Proposition 4.4, S is the intersectien of E x F and

some Souslin subset of Ex F'. Now Ex Fitself is a Souslin subset of

Ex F', so Sis a Souslin subset of Ex F' as well. From the analyti­

city 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 of SE into F'. The range of this mapping is contained in F and

it can,therefore, be considered as a universally measurable mapping of

SE into F.

The foregoing implies that F is analytic.

iii) Let (Fi,Fi)iEI be a family of analytic spaces and let F := niEI Fi.

For each measurable mapping ~: F + 1D we shall show that the range of

~ is a Souslin subset of 1D and that ~ has a universally measurable

right inverse. These facts imply that F is analytic.

Since the cr-algebra V of 1D is countably generated, so is the sub-cr­

algebra ~-IV of F. So ~-lv is contained in a sub-cr-algebra of F that

is generated by countably many reetangles in niEI Fi. Consequently,

there exists a countable subset 0 ably generated sub-cr-algebra F.

-1 0 l. and such that ~ V c ®. I F .•

l.E l.

J of I and, for each i E I, a count-0

of Fi such that viEI\J Fi

We shall now show that the mapping ~: F + 1D can be decomposed into a

mapping Ijl: F + lDJ and a mapping x: (j!F + lD. To this end let, for each

j E J, the mapping Ijl.: F. + 1D be strictly measurable with respect to J J

~ (such mappingsexist by Proposition 7.4) and let Ijl: F + lDJ be J

defined by ljl(x). := q,.(x.) (j E J, x E F). Now suppose that x and x' J J J

are points of F such that ljl(x) = ljl(x'). Then for each j E J we have

(j!J·(x.) = q,.(x!), so x. and x! are not separated J J J J J

F9. Since FQ is trivia! for i i J, x and x' are J 0 l.

®. I F. and therefore they are not separated by l.E l.

-I by (jij V, hence not by

not separated by -I

the smaller class ~ V either. As V separates the point of lD, this implies that ~(x) = ~(x'). The arbitrariness of x and x' now implies that ~ = x o Ijl for some

mapping x: (j!F + lD.

44

Next we show that the range of ~ is an analytic space and that ~ has

a universally measurable right inverse. Let j e J. Since (F.,F.) is J J

analytic and ~. is measurable with respect to F., the range of~. is J J J

a Souslin subset, s. say, of ID and ~. has a right inverse, say 6., J J J

that is universally measurable with respect to F .. Moreover, when u.

is the j-th coordinate of IDJ, then u: 1s. is a S~uslin subset of J J J J

1D by Proposition 4.3. The foregoing now implies that

lf.F n 4.F. jeJ J J

TT S. jEJ J

n jEJ

-I (Jt. s.}

J J

so q,F is a countable intersection of Souslin sets and, therefore, a

Souslin set itself. Since J is countable, the space IDJ is isomorphic

to ID and therefore it is analytic. Consequently 4F, being a Souslin J subset of ID , is analytic by the result ii) proved above.

To construct a universally measurable right inverse 8: q.F + F of 4 we

merely choose a point a E F and define

1\ (yi} if i E J [ B(y)] i (i E I, y € 4F) .

a. if i € I \ J l.

The universal measurability of B follows from the fact that for each

coordinate Jtl of nid Fi we have l.

u! o B l. a. if i E I \ J

l.

and from the univeraal measurability of the Si's (see remark pre­

ceeding Proposition 3.4).

Finally, we turn to the mapping x: 4F + ID. It follows from the

definition of 4: F + IDJ that 4 is strictly measurable with respect 0 to ®. I F .• Since we also have

l.€ l.

-I (xo4) V r=?

1

it follows that x- 1v is contained in the a-algebra of q,F. So x is

measurable. As the domain q,F of x has been proved to be analytic, the

45

range of x is a Souslin subset of ID and x has a universally measur­

able right inverse, say y.

Now the range of ~ equals the range of x, and B o y is easily seen to

be a universally measurable right inverse of ~· From the arbitrariness

of~ it now follows that the space n. (F.,F.) is analytic. 0 1 1 1

COROLLARY 7. 7. Let (F, F) be an analytia spaae and let G be a sub-a-algebra

of F. Then the spaae (F,G) is analytia as well.

The construction principles mentioned in Proposition 7.6, and the

analyticity of ID enable us to obtain a large variety of analytic spaces.

We give an example.

PROPOSITION 7.8. The spaae IR is analytia.

PROOF. Let~: 1D ~ [O,i] be defined by

~(x) = L n=l

Then ~ is a surjection. Moreover, ~ is measurable, because for each n € IN

the number xn depends measurably on x. So, [0,1] is a measurable image of

ID and, therefore, it is analytic. The same holds for the measurable sub­

space (0,1) of [0,1] and, consequently, forthespace IR, which is the

image of (0,1) under the measurable mapping

1 I XI+-+---,

x x- I 0

When the cr-algebras E and F of two measurable spaces (E,E) and (F,F)

are isomorphic, the spaces (E,E) and (F,F) themselves are not necessarily

isomorphic. In many situations, however, this distinction is immaterial.

Note, for instance, that the spaces (E,Ë) and (F,F) are isomorphic if the

a-algebras E and F are isomorphic. It follows from these considerations and

the following proposition that it is often possible to consider a countably

generated analytic space to be a Souslin subspace of ID.

46

PROPOSITION 7.9. Let (F,F) be a eountably gene~ated analytia spaae. Then F

is isomo~phia to the a-algeb~a of some SousZin subspaae of ID.

PROOF. It fellows from Proposition 7.4 that there is a strictly measurable

mapping ~: F +ID. Proposition 7.5 implies that the range of~ is a Souslin

subset of ID. Hence the result. D

If (F, F) is a measurable space, G a sub-cr-algebra of F and IJ a

probability on G, then IJ is not necessarily extendable to a probability on

F. 'When such an extension happens to exist, it need not he unique. For

analytic spaces, however, things are not too bad:

PROPOSITION 7 .10. Let (F, F) be an analytie spaae and let G be a aountably

gene~ated sub-a-algebra of F. Then the~ exists a unive~saZZy measu~le 'mapping ~: (F,G)- + (F,F)- suah that fo~ eve~y p~obabiZity u on G the

probabiZity ~(u) is an extenaion of u to F.

PROOF. By Proposition 7.4 there exists a mapping ~: F + ID that is (strict­

ly) measurable with respect to G, and hence, measurable with respect to F. Since (F,F) is analytic, by Proposition 7.5 there is a right inverse x of~ that is universally measurable with respect to F.

- - -1 -1 Now let ~: (F,G) + (F,F) be defined by ~(IJ) :=IJ o ~ o x . ' -1 For every A € F the set x A is a universally measurable subset of the

space ~F. Consequently, the set 4- 1(x-1A) is a universally measurable

subset of (F,G) by Proposition 3.4, which in turn implies, by Proposition -1 -1 -3.6, that u(4 (x A)) depends universally measurably on IJ e F(G). So, from

the definitions of (F,F)- and universa! measurability it follows that the

mapping ~ is universally measurable.

Finally, we shall show that for every probability u on G the proba­

bility ~(u) is an extension of u to F. From the strict measurability of 4

it follows that G = 4- 1V, where V is the cr-algebra of ID. So what we have

to prove is that ~(u) and u coincide on 4- 1v. Therefore, let B e V. Then

because 4 o x is the identity on ~F. D

47

The mapping ~ in the foregoing proposition is a (nonunique) univer­

sally measurable right inverse of the measurable mapping p: (F, F) ~ + (F ,G)­

defined by (p~)(G) ~ ~(G) (G € G), i.e. the mapping corresponding to

restrietion to G (see example 4 following Proposition 2.2).

The remaining part of this section consists of the proof of Proposi­

tion 7.13, which statea that analyticity of a space F implies analyticity -of the space F. We start with two lemmas.

LEMMA 7. IJ • ID is analytia.

PROOF. Let A be a countable semicompact algebra generating the cr-algebra of

ID, let the mapping ~: ID -+ [0 ,I] A be defined by (~~)A = J.1 (A), and let M be

the subspace of [O,J]A consisting of all additive functions v: A+ [0,1]

for which v(ID) = I.

Since A is a semicompact algebra, every memher of M is cr-additive by

Proposition 5.3, and therefore is uniquely extendable to a probability on

cr(A). This, however, implies that ~ is injective and that the range of ~

equals M. The mapping ~ even is an isomorphic embedding because the cr­

algebra of ID is generated by the mappings J.1 ~ p(A) (A E A), that is, by

the mappings J.1 ~ (~J.!)A (A E A). It is therefore sufficient to prove that M

is analytic.

Now [O,l]A is analytic, because [0,1] is, and Mis a measurable

subset of [O,I]A, as it can be written as a countable intersectien of

measurable sets, e.g.

(l {v E [0,1 ]A I v(A) + v(B) = v(A + B) and v(ID) = 1} , (A,B)

where the intersectien is over the countable set of disjoint pairs (A,B) of

merobers of A. Hence M is analytic. D

LEMMA 7 .12. Let F be a aountably generated analytie spaae. Then F is

analytia.

PROOF. By Proposition 7.9 the cr-algebra of Fis isomorphic to the cr-algebra

of some Souslin subspace, S say, of ID, and, consequently, the spaces F and

S are isomorphic. It will therefore be sufficient to prove that S is

analytic.

48

~

To this end let us define the mapping ~: S + ID by (~~)(B) := ~(B n S)

(BE V), where V is the a-algebra of ID. When ~.~' ES are such that

~~ ~~·, then pand~· coincide on the a-algebra VIs of S and, therefore,

are identical. So ~ is injective. Moreover, since the cr-algebras of S and

ID are generated by the functions p ~ ~(B n S) and v ~ v(B) (BE V),

respectively, the mapping ~ is an isomorphic embedding. So S is isomorphic

to the range of ~ and what remains to be proved is the analyticity of this

range.

Now let ~ E S. For every B E V such that Sc B we have (~~)(B)

p(B n S) = ~(S) = 1, so (~p)(S) = I. On the other hand, let v E 1D be

such that v(S} = I. When wedefine ~on VIs by ~(A) := v(A), then ~ is a

probability and ~P = v. So the range of ~ is èqual to the subset

{v E ID I v(S) = I} of ID. Since this subset can be written as

I -1 0 {v E 1D v(S) > I - n } , nElN

and since v ~ v(S) is a Souslin function on ID by Proposition 4.8, the

range of ~ is a Souslin subset of ID. From the analyticity of 1D tagether

with Proposition 7.6 we now deduce that the range of~ is analytic.

PROPOSITION 7.13. If a measurabZe spaee Fis anaZytie, then Fis anaZytie

as weU.

We shall prove that F has the characterizing property of analytic

,spaces given in Proposition 7.5. Therefore, let~: (F,F)~ + (ID,V} be -I

measurable. Since V is countably generated, ~ V is a countably generated

sub-a-algebra of F. From the definition of F it follows that F is genrated

by the collection C of subsets of F(f) of the form {~ E F(f) I p(A) E B},

where A E F and where B is a measurable subset of IR. As -I

there exists a countable subclass C0 of C such that ~ V of C, there exists a countable subclass A

a consequence,

c a(C0). Hence, by

of F such that the definition -I

~ V is contained in the a-algebra generated by the functions p ~ p(A)

(A E A) on F(F).

Now let G be the sub-cr-algebra of F generated by A. Suppose that v' and p" are probabilities on F that coincide on G. Then ~ and ~" coincide on

A and, therefore, they are not separated by the a-algebra on F(f) generated

by the functions ~ »- p(A) (A E A). As a consequence u' and v" are not -1 .

~>eparated by the smaller a-algebra q> V either. So, ~v' and ll>ll" are not

0

49

separated by V and, therefore, are identical, since V separates the points

of ID.

The foregoing implies that we can write ~ = x o q,, where

~: (F,F)- + (F,G)- is the mapping corresponding to restrietion to G, i.e.

(~~)(A) :=~(A) for all A EG and ~ E F(F), and where x maps (F,G)~ into

(ID,V). As Gis a sub-cr-algebra of F and since (F,F) is analytic, the space

(F,G) is analytic as well by Corollary 7.7. Moreover, Gis countably

generated, so by Proposition 7.10 and the remark following it the mapping q,

is surjective and has a universally measurable right inverse.

From the definitions of G and q, it follows that q,- 1G is the cr-algebra

on F(f) that is generated by the functions ~~~(A) (A EG); it therefore· -1 -1 -l -1

contains the cr-algebra ~ V. Consequently, we have ~ x V = (xoq,) V = -1 -)~ -1 ~

~ V c ~ G and, since q, is surjective, x V c G. So x is m~asurable.

Due to Lemma 7.12, the space (F,G)- is analytic, because (F,G) is a

countably generated analytic space. Tagether with the measurability of x this implies, by Proposition 7.5, that the range of x is a Souslin subset

of 1D and that x has a universally measurable right inverse.

Since q, is surjective, the range of ~ equals the range of x and hence

is a Souslin subset of ID. Moreover, the composition of the right inverses

of x and q, mentioned above is a universally measurable right inverse of ~· 0

§ 8. Separating classes

We shall now prove the so-called first separation theorem for analytic

spaces. From this theorem it follows that the two conditions in the defini­

tion of analytic spaces are not independent when countably generated or

countably separated spaces are considered. Moreover, with the aid of this

theorem it is possible to characterize those analytic spaces whose cr­

algebras are generated by an analytic topology.

DEFINITION. Let A and B be collections of sets. Then B is said to sevarate

A, if for every disjoint pair A1,A2 E A there exists a disjoint pair

B1,B2 E B such that Al c B1 and A2 c B2•

This terminology is consistent with the notion of a separating

collection of subsets of a set introduced earlier, since such a colleetien

is characterized by the fact that it separates the collection of singletons.

50

For the proof of the first separation theorem we need:

LEMMA 8.1. Let (E,E) be a measurable space, {s! I i €. lN} and {S~ I i E lN} I

1 I 2

1 2 countable collections of subsets of E, S :• U. lNS. and S :• U. lNsi.

1 2 1E 1 1€. When the pair {S ,s } is not separated by E, then for some i,j E 1N the

I 2 pair {S. ,s.} is not separated by E either. 1 J

PROOF. We argue by contradiction. Let the pair {s!,s:} be separated by E 1 J

for every i,j E lN. Then for every i,j E 1N there exists M .. €. E such that 2 ~

s! cM .. and S. c M7 .• Consequently, s 1 is contained in the set 1 1] J l.J

UiEIN njEIN Mij and S2 in the compliment of this set. So the pair {SI ,s2

}

is separated by E. 0

PROPOSITION 8.2 (First separation theorem). Let (E,f) be an analytic space.

Then E separates S(E).

PROOF.

i) We first consider the case that E equals ID. Let V be the cr-algebra

of lD and let A be a countable semicompact algebra generating V (cf.

Proposition 5.4). As A is closedunder complementation, by Proposition

4.1 we have cr(A) c S{A). Moreover, V • cr(A), so S(V) c S(S(A)) S(A). I 2 Now let S ,s E S(V). Then

u 1N n OElN kEIN

(r 1,2)

for certain sets Ar . n belonging toA. For r E {1,2} and for ''np ... ' k

every finite sequence m1, ••• ,mp of positive integers wedefine

Sr := u { n Ar I n E INlN and V 9.< nR. • mR.} m1, ••• ,mp kEIN nl ' ••• ·~ -P

Th en

Sr u s~ and Sr u Sr . ÜlN

]. m1 , ••• ,mp ÎEIN m1, ••. ,mp,i

'' L~~ks suppose that the pair {S

1 ,s2J is not separated by V. It then

fo~lows by induction on p, and by use of Lemma 8.1 at each step,

t'h:at there exist m1 ;m2 E 1N IN such that for each p the pair

.

is not separated by V. As

Sr ·C n Ar e: V r r kSp r r m1, ••• ,mp mi, ••• ·~

for every p and r, this implies that for every p the pair

{ n Ar I r 1,2} k< r r

-P mt•····~

is not disjoint. It now follows from the semicompactness of A that

the set

51

is nonempty, and hence, that s1 and s2 àre not disjoint. The fore­

going implies that any disjoint pair of memhers of S(V) is separated

by V.

ii) Next let (E,E) be a countably generated analytic space. Then, by

Proposition 7.9, the space (E,E) may be supposed to be a Souslin

subspace of (ID,V) and consequently S(E) c S(V) by Proposition 4.4.

As S(V) is separated by V, so is S(E). Since each memher of S(E) is

contained in E, the collection S(E) is séparated by the trace E of V on E as well.

iii) I 2 Finally, let (E,E) be an arbitrary analytic space and let S ,S E S(E). I 2 Then there exists a countable subclass C of E such that S ,S € S(C) c

c S(cr(C)) (see the remark following the definition of S). Applying

ii) tothespace (E,cr(C)) we conclude that the pair s1,s2 is separated

by cr(C), and hence, also by the larger class E. 0

COROLLARY 8. 3. (E,E) is an analytie spaee, then

E = {AcE I A E S(E) and Ac E S(E)} ,

and henae, Eis the largesta-algebra aontained in S(E).

\

52

The following proposition is a generalization of the "only if"-part

of Proposition 7.5.

PROPÖSITION 8.4. Let ~ be a measurable mapping of an analytie spaee F into

a aountably separated measurable spaae G. Then the range of ~ is a SousZin

subset of G and ~ has a universally measurable right inverse. When~ in

addition, ~ is injeative, then ~ is an isomorphia embedding.

PROOF. To prove the first part of the proposition we can repeat the first

part of the proof of Proposition 7.5, with 1D replaced by G.

Next, let ~ be injective. Let A be a measurable subset of F. We

shall prove that ~A is a measurable subset of ~F. By Proposition 7.6, A is

an analytic subspace of F, which is mapped by ~ into the countably sepa­

rated space ~F. So, by what has already been proved above, ~A is a Souslin

subset of the space ~F. A similar reasoning holds for the set Ac. Since ~

is injective, ~(A) and ~(Ac) are complementary Souslin subsets of ~F. From

Proposition 7.6 wededuce that ~Fis, an analytic space and, hy Corollary

8.3, this implies that ~A is measurable. As A is arbitrarily chosen, it

follows that ~ is an isomorphic embedding. D

PROPOSITION 8.5. Let (E,E) be an analytia space. Then any aountable

separating subalass of E generatee E.

PROOF. Let C be a countable separating subclass of E. Then (E,o(C)) is

countably separated and the identity on E is a measurable injection of

(E,E) onto (E,o(C)). The preceding proposition now implies that E equals

a~C). 0

Proposition 8.5 implies an extrema! property of the a-algebra of a

countably separated analytic space: no smaller a-algebra is countably

separated, no larger one is analytic.

In the definition of analytic spaces as well as in some propositions

of the last two sections and in their proofs we can distinguish two parts:

one hearing on Souslin sets and another one having to do with universally

measurable mappings. Moreover, inspeetion of the proofs mentioned reveals

that the first part is in a sense independent of the second one and, after

'skipping everything hearing on universally measurable mappings in defini­

tions as well as in propositions and proofs, we are left with a theory

53

about a class of spaces that is possibly larger than the class of analytic

spaces.

As has been remarked at the end of Section 4, another possible

modification of the theory is obtained by substituting limit measurable

sets and mappings for.universally measurable ones. The following proposi­

tion implies that these three theories coincide for measurable spaces that

are countably generated or countably separated.

PROPOSITION 8.6. Let F be a measurabZe space that is countabZy generated or

countabZy separated. Then the foZZowing statements are equivalent:

i) F is anaZytic.

ii) For every measurabZe space E and every SousZin subset S of E x F, the

projection of S on E is a SousZin subset of E.

iii) For every measurabZe mapping cp: F -+ ID the range of cp is a SousZin

subset of ID.

PROOF. The proof consists of two parts.

Part I. We first consider the case that F is countably generated. We shall

prove the implications i)~ ii) ~ iii) ~i).

i)~ ii). This follows from the definition of analyticity.

ii) ~ iii). Let cp: F-+ ID be a measurable mapping. Since ID is ~ountably

separated, the graph of cp is a measurable subset of F x ID by Proposition

7.3. As a consequence of ii) the projection on ID of this graph, being the

range of cp, is a Souslin subset of ID.

iii) ~i). Let cp: F-+ ID be measurable. By Proposition 7.5 it is sufficient

to prove that cp has a universally measurable right inverse.

Let ~: F -+ ID be strictly measurable and let n: ~F -+ F be a measurable

right inverse of ~; the existence of such mappings follows from Proposition

7.4. Moreover, it follows from iii), applied to ~. that the range ~F of~

is a Souslin subspace of lD and hence, that ~F is analytic.

Now we show that cp cp o n o ~. To this end let x and x' be points of

F such that ~x= ~x'. Then ~x and ~x' are not separated by the cr-algebra V -I

of ID and, therefore, x and x' are not separated by the cr-algebra~ V of F.

As cp-1v c ~- 1 v, by the measurability of cp, the points x and x' are not

-I separated by cp V either, and therefore cpx and cpx' are not separated by V.

54

So ~x= ~x'. In particular we can take x' := (no~)x. Then ~x'= 4x, and

hence, ~x ~x' = (~ono~)x. The arbitrariness of x now implies ~ = ~ o n o 4·

Next consider the mapping ~ o n: 4F ~ ID. It follows from the equality

last obtained that the range of ~ o n equals the range of ~· Moreover, ~ o n

is measurable, si~ce ~ and nare, and ~Fis analytic. So, by Proposition 7.5

~ o n has a universally measurable right inverse, ~: ~F ~ ~F say. It follows

that the mapping n o ~ is a universally measurable right inverse of ~·

Part II. Next let F be countably separated, and suppose that at least one

of the statements i), ii) or iii) is true.

Let F be the cr-algebra of F, C a countable separating subclass of F,

A E F, and A the cr-algebra generated by the collection C u {A}. Then A is a

sub-a-algebra of F, so each (hence, at least one) of the statements i), ii)

or iii) that holds for (F,F) also holds for (F,A). As A is countably gener­

ated, it follows from Part I that the three statements are equivalent for

(F,A). So each of them (in particular i)) holds for (F,A), i.e. (F,A) is

analytic. It now follows from Proposition 8.5 that the countable separating

class C generatea A and therefore that A E cr(C).

Since A is an arbitrarily chosen memher of F, we have F c cr(C), and

hence, F cr(C). So (F,F) is countably generated and, by Part I, the three

statements are equivalent for (F,F). As at least one of them is true, they

all are. 0

The measurable spaces appearing in many applications are countably

generated or, when not, one often is interested only in a function defined

on 'such a space, to which again there corresponds a countably generated

cr-algebra. So there would have been no serious loss of applicability had we

incorporated in the definition of analytic spaces the condition that such a

space were countably generated. This however would have been at the cost of

stability of the class of analytic spaces under the formation of measurable

images and product spaces. A similar remark can be made about separation:

a nonseparated measurable space is not essentially different from a sepa­

rated one, but the property of being separated is not conserved under the

formation of measurable images.

We conclude this section with a characterization of those measurable

spaces whose cr-algebra is generated by an analytic topology. This charac­

terization will not be used in the sequel. The cr-algebra of a measurable

55

space is generated by an analytic topology iff that space is isomorphic to

a Souslin subset of the measurable space [0,1] (cf. [Hoffman-J-rgensen]

Ch. III §2, theorems 3 and 4). Since the measurable spaces [0,1] and ID

are isomorphic ([Bertsekas & Shreve] Proposition 7.16) the latter condition

is equivalent to the space being isomorphic to a Souslin subspace of ID. It

now follows from Propositions 7.2, 7.6 and 7.9 that this in turn is equiva­

lent to the condition that the space is countably separated and analytic.

Most of our results concerning countably generated analytic spaces

are known to be valid for analytic topological spaces. So these results are

new only in that they have been derived by measure-theoretic means only.

§ 9. Probabilities on analytic spaces

This sectionis devoted to what .may be called Kolmogorov's theerem

for analytic spaces and to the decomposition of probabilities defined on

product spaces into a marginal probability and a transition probability.

In both topics semicompact classes play a central role and we sta~t with a

proposition concerning them.

PROPOSITION 9.1. Let (F,F) be a aountably genePated analytic spaae. Then

thePe exists a semiaonrpact subeZ.aas C of F such that foP eT)ery 1l E F and

A E F:

l.l(A) = sup {l.l(C) I C E C and C ~ A} •

PROOF. As F is isomorphic to the cr-algebra of some Souslin subspace of lD

(cf. Proposition 7.9), (F,f) itself may be supposed to be a Souslin sub­

space of ID. By Proposition 5.4 the cr-algebra V of ID is generated by a

semicompact algebra A and, as A is closed under complementation, we have

V c S(A) by Proposition 4.1. It now follows from Proposition 5.2 that, for

every Souslin subset S of ID and for every 1l E ID, we have

l.l{S) = sup {l.l(C) I C E A8ö and C c S} •

Specializing to sets S contained in F and probabilities 1l concentrated on

F we get

l.l(S) sup {l.l(C) I C E C and C c S} ,

56

where C :~ {C E Asö I C c F}. Moreover C is a subclass of Asö and As& is

semicompact because A is (Proposition 5.1). Consequently Cis semicompact

as well. Finally, we observe that every merober of F is the intersectien of

F with some memher of V, and hence, is a Souslin subset of 1D that is

'contained in F. 0

PROPOSITION 9.2. Let (Fi}iEI be a family of analytia spaaes and for every

finite subset I' of I let ~ 1,be a probability on n. I' F.~ suah that~ ~E ~

for every pair I', I" of fini te subsets of I with I' c I", the probabiU1;y lli,

is the marginal of III" ( aorresponding to the projeation of n iE I" F i onto

n. I' F.). Then there exists a unique probability on n. I F. having the l.E ~ l.E l.

probabiUties JJ1

, (I' finite subset of I) as its marginale.

PROOF. Let F1

, :~ n. I' F. for each subset I' of I, and let A be the ----- l.E l.

colleetien of measurable cylinders of FI. We now define JJ! A + IR as

fellows. For every A E A there exists a finite subset I' of I and a measur­

able cylinder A' of FI' such that A= A' x FI\I'' Next wedefine

JJ(A} := III 1 (A'). The definition of JJ(A) is easily seen to be independent of

the particular choice of I'. Moreover, the function JJ thus defined is the

only function on A whose marginals coincide with the probabilities 11I,. As

A generates the cr-algebra of F1

, all that remains to be proved is cr-addi­

tivity of JJ on A (see [Neveu] Proposition I.6.1).

Let C be a subclass of A consisting of a countable colleetien pair­

wise disjoint sets together with their union. Then there exists a family

(C.). I' with C. a countable colleetien of measurable subsets of F1. for

l. lE 1

each i E I, and such that C is contained in the semialgebra B of measurable

cylindersof the space n. I(F.,cr(C.)). Bis contained in A and the restric-l.E ~ l.

tion of 11 to B is the unique function on B whose marginals coincide with

the restrietion of the measures 111 , to the spaces n. 1

,(F.,cr(C.)). 1€ . l. l.

Now for each iE I the space (F.,cr(C.)) is countably generated and l. l.

analytic, being a measurable image of the space Fi with the original cr-

algebra. Therefore each of the cr-algebras cr(Ci) contains a semicompact

class as described in Proposition 9.1. These facts, however, imply ([Neveu]

Theorem III.3) that 11 is cr-additive on B, and hence, on C. From the arbi-

trariness of C it now follows that JJ is cr-additive on A. 0

DEFINITION. Le~ E and F be measurable spaces. A transition probabiZity

from E to F is a mapping of E into F.

57

When p is such a transition probability, x E E, and B a universally measur­

able subset of F, then we shall usually write p(x;B) insteadof p(x){B) i~

order to enhance readability.

PROPOSITION 9.3. Let E and F be measurabZe spaaes, Zet v be a probabiZity

on E and Zet p be a universaZZy measurabZe transition probabiZity from E to

F. Then there exists a unique probabiZity on E x F, denoted by v x p, suah

that

J fd(vxp)

EXF

J J f(x,y)p{x;dy)v(dx)

E F

for every universaUy measurabZe f: E x F + IR+

PROOF. By Proposition 6.2, FJ f(x,y)p{x){dy) depends universally measurably

on (x,p{x)) and therefore on x, because p(x) depends universally measurably

dn x. So the repeated integral is well defined. Moreover it is easily seen

to be nonnegative, to depend cr-additively on f, and to assume the value I

for f = IExF" From this the result follows.

The notation introduced in Proposition 9.3 is consistent with the

notation of product probabilities when constant transition probabilities

are identified with their value. Note that for constant transition proba­

bilities Proposition 9.3 reduces to the theorem of Tonelli (see [Cohn]

Proposition 5.2.1).

In general, not every probability ].1 on a product space E x F can be

written as the product of its marginal on E and a transition probability

from E toF. When, however, the space F meets certain conditions, then

such a decomposition of ].1 is possible, as is stated in the following

proposition. This proposition, together with the "Exact selection theorem"

(Proposition 7.1) forms the basis for the applications in Chapter III.

PROPOSITION 9.4. Let E be a measurabZe spaae, F a aountabZy generated

anaZytia spaae, ].1 a probabiZity on E x F and v the marginaZ probabiZity of

ll on E. Then there exists a measurabZe transition probabiZity p from E to

F suah that ].1 = v x p.

D

58

As, by Proposition 7.9, the cr-algebra of F is isomorphic to the cr­

algebra of some Souslin subspace of lD, we can suppose F itself to be a

Souslin subspace of ID. Let E and F be the cr-algebras of E and F, respec­

tively, and let B be a countable semicompact algebra that generates the cr­

algebra V of ID.

For every B E B the mapping A~ ~(A x (BnF)) is a bounded nonnegative

measure on E that is absolutely continuous with respect to v. Hence by the

Radon-Nikodym theerem ([Neveu] Proposition IV.I.4) there exists a measur­

able function fB: E + IR+ such that for every A E E

~(A x (BnF)) = J fB(x)v(dx) •

A

Now for every A E E ~(A x (BnF)) depends additively on B and equals v(A)

for B = ID. As a consequence of (*) the same holds for the integral in (*).

Due to the countability of B this implies that there is a v-null subset N

of E such that for every x E E\N the function B ~ fB(x) is additive, and

fiD(x) =I. Redefining the functions fB on N by fB(x) = fB(x0), where x0 is

some point in E \ N, we can suppose N = 0 and still have (*) for every

A E E, B E B. As B is a semicompact algebra, for each x E E the function

B ~ fB(x) is even cr-additive, and therefore it is uniquely extendible to a

probability B ~ fB(x) on V. The functions fB' with B E V, are measurable

again, because the sets B E V for which fB is measurable constitute a

Dynkin class (see Preliminaries 2) containing the algebra B. Moreover we

have ~(A x (BnF)) = Af fB(x)v(dx) for all A E E and B E V, because both

memhers of this equality, taken as functions of B, are measures on V, and

by (*) these measures coincide on the generating algebra B. Next consider the probability À on V defined by À(B) := ~(E x (BnF)).

* Then À (F) = 1 and, as F is universally measurable, À*(F) = I as well.

Consequently, there exists a set GE V such that G c F and À(G) = 1. From

the definition of À it now fellows that ~(E x (F\G)) = 0.

Finally, for each x E E and each B E F we define p(x)(B) := fBnG(x).

It is easily seen that p is a measurable transition probability from E toF.

Moreover, for every A E E and B E F we have

~(AxB) = ~(A x BnG) + ~(A x Bn(F\G))

J fBnG(x)v(dx) = J p(x)(B)v(dx) = (vxp)(AxB) ,

A A

because p(A x Bn(F\G)) ~ p(E x F\G) = 0. So p and v x p coincide on the

measurable reetangles of E x F, which implies that they are equal.

59

In the preceding proposition, the analyticity of F may be replaced by

the weaker condition that a semicompact approximating subclass of F exists

for the marginal of pon F (in the sense of Proposition 9.1; see [Pfanzagl

& Pierlol Section 7).

The decomposition of a probability on a product space into a marginal

probability and a transition probability is, in general, not unique:

PROPOSITION 9.5. Let E be a measurabte spaae, F a aountabty generated

measurabte spaae, v E Ë, and p,p': E ~r universally measurabte. Then

v x p = v x p' iff p(x) = p'(x) for v-atmost att x E E.

PROOF. The "if"-part is a trivial consequence of the definition of v x p

and v x p', so let v x p = v x p'. Let B be a countable algebra generating

the o-algebra of F. For every B E B and every universally measurable subset

A of E we have

J p(x;B)v(dx)

A

(vxp)(AxB) = (vxp')(AxB) J p'(x;B)v(dx) ,

A

0

and hence, p(x;B) = p'(x;B) for v-almost all x E E. It follows from the

countability of B that for v-almost x € E the measures p(x) and p'(x)

coincide on B and therefore that they are equal. 0

The nonuniqueness of the transition probability appearing in the

decomposition of a probability p enables us to .make this transition proba­

bility depend on ~ in a decent way:

PROPOSITION 9.6. Let E be a aountabty generated measurabte spaae anà F a

aountabty generateà anatytic spaae. Then there exists a universatty measur­

abte mapping p: (ExF)- x E ~ F suah that every probabitity u on E x F is equat to the product v x q ofits marginatv on E and a transition proba­

bitity q ~m E to F defined by q(x) := p(~,x). When F = lD, then p may be

supposed to be measurabte.

60

PROOF. As the cr-algebra of F is isomorphic to the cr-algebra of some Souslin

subspace of ID, we may suppose F itself to be a Souslin subspace of ID. Let

E and F be the cr-algebras of E and F, respectively, let A= {A I n € IN} n

be a countable algebra generating E and let B be a countable semicompact

algebra generating the cr-algebra V of ID. For each n e IN let G be the n

partition of E generated by {A1,A2, ••• ,An}.

Now let ~he a probability on E x F, v its marginal on E, and B € B. We define a sequence (fn)nelN of functions on E by

Let m,n € IN and m $ n. Then Gn is a refinement of G , so for each A e cr(G ) m m we have

L { J fn dv I G <: Gn and G c: A} G

= I {~(G x BnF) I G <: G and G c A} u(A x BnF) • n

Moreover each of the functions fn is bounded by 1. This implies that the

sequence (fn) is a martingale relative to the cr-algebras cr(Gn) and the

measure v, and that the martingale convergence theorem applies (see [Chow &

Teicher] §7.4 Theorem 2 ii)). So, if wedefine f := li~up fn' then

f f dv = J fn dv ll (A x (BnF))

A A

for every n € IN and A € cr(G ). Consequently, for every A<: U INcr(G) =A, n H n we have Af f dv = ~(A x (BnF)) and, as bath memhers of this equality depend

cr-additively on A, this equality holds for every A <: cr(A) = E. The function f introduced above depends on u and B, and we now make

this dependenee explicit by defining f: (ExF)~ x E x B +IR as

f(p,x,B) := limsup L ~(G x (BnF))h(u(GxF))lG(x) , n GeGn

where h is the measurable function on IR defined by h(t) = t-I if t + 0 and

h(O) = 0. The foregoing then implies that, for every u € (ExF)~, A e E and

B e B, we .have

f f(t~,x,B)v(dx) A

J..1 (A x (BnF)) ,

61

where v is the marginal of J..l on E. Moreover, it follows directly from the

definition of f, that f(p,x,B) is a measurable function of (p,x) for every

B E B.

We now turn to the dependenee on B. In general f(u,x,B), taken as a

function of B, is not a probability on B, but a nonessential redefinition

of f suffices to make it one. Let N be the set of pairs (p,x) E (ExF)~ x E

for which the function B ~ f(p,x,B) is not additive on B or for which

f (p,x, ID) {- 1. Then

or f(p,x,ID) .; 1} ,

where the union is over the countable set of disjoint pairs (B1,B2) of

memhers of B. The measurability of f mentioned above now implies that N is

a measurable subset of (ExF)- x E. Moreover, for every J..1 E (ExF)- the

section {x E E I (p,x) E N} of N is a v-null set, where v is the marginal

of J..l on E: Let B1,B2 EBbe disjoint. It then follows from (*), that

J [f(u,x,B1) + f(u,x,B2) - f(u,x,B 1uB2)]v(dx) 0

A

for every A E E, and hence, that the integrand is a v-null function.

Similarly f(p,x,ID) - 1 is a v-null function of x. Together with the

definition of N this implies the result on the sections of N.

Now let (u0,x0) iN and, for each BE B, redefine f(•,•,B) on N by

setting it equal to f(u0,x0,B). Then, for every (p,x) E (ExF)~ x E,

f(p,x,B) is an additive function of BonBand f(u,x,ID) = I. Moreover, for

every B E B, f(u,x,B) still is a measurable function of (u,x), and for

every u, A and B the equality (*) still holds.

As B is a semicompact algebra, for every (p,x) the function

B ~ f(u,x,B) is even cr-additive on B by Proposition 5.3, and it therefore

is uniquely extendible to a probability p(u,x) on U(V). It follows from

Proposition 2.2 applied to the subclass B of V, and from Proposition 3.~,

that p(u,x,B) depends (universally) measurable on (p,x) for every (uni­

versally) measurable subsetBof ID. Also,it follows from (*), that for

62

every ~ E (ExF)~, A E E and B E B we have:

J p(~,x)(B)v(dx) ~(A x (BnF))

A

and, as both memhers of this equality depend cr-additively on A as well as

on B, the equality holds for every (universally) measurable subset A of E

and B of ID.

Whèn F = ID, then the foregoing implies that p'is a measurable mapping

as mentioned in the proposition. When F r lD for each (~,x) E (ExF)~ x E we

consider the restrietion of p(~,x) to the cr-algebra F of F. Such a restrie­

tion is a cr-additive function on F, but it need not be a probability on F, since it may fail to attain the value I at F. Now let

M := {(~,x) E (ExF)~ x E I p(~,x)(F) f I}

Then Mis universally measurable and, taking (~ 1 ,x 1 ) i M, we redefine p onM

by settingit equal to p(~ 1 ,x 1 ), which makesp a universally measurable

mapping of (ExF)~ x E into F. Moreover, this redefinitien of p does not

affect the validity of (**), because for every ~ E (ExF)~ we have

J [p(~,x)(F) - l]v(dx) = ~(AxF)- v(A) = 0

A

for every universally measurable subset A of E, and hence, the integrand is

a v-null function. All this now implies, that the (redefined) mapping p

meets the requirements of the proposition. 0

In the following two propositions we consider conditional expectations

of functions defined on analytic spaces. Let p be a measurable transition

probability from a measurable space E toa measurable space F, and for

every positive measurable function f on F let the function ~f on E be

defined by (~f)(x) := f f(y)p(x;dy). Then, by Proposition 6.1, ~fis

measurable. Moreover, for each x E E, (~f)(x) depends cr-additively on f.

These facts, together with the decomposition of probabilities given in

Proposition 9.4, enable us to construct regular versions of conditional

expectations. For general information see [Chow & Teicher] Section 7.2 and,

in particular, Theerem 3; in fact, our next proposition is a generalization

of this theorem.

63

PROPOSITION 9.7. Let (E,E) be a measurable space, ~a probabitity on .E and

F, G sub-o-aZgebras of E suah that (E,G) is a aountabZy generated analytia

space. Then there exists an F-measurabte transition probability p from

(E,F) to (E,G) suah that, for every positive G-measurabZe funation f on E,

the funation y ~ f f(x)p(y;dx) is a version of the aonditionat expeatation

of f with respeat to F and ~.

PROOF. Let the measurable mapping w: (E,f) + (E,F) x (E,G) be defined by

w(x) := (x,x). Then ~ o $-1 is a probability on F ® G whose marginal on F

equals u. As (E,G) is countably generated and analytic, there exists a

measurable transition probability p from (E,F) to (E,G) such that

u o $-1 = ~ x p. Now for every A E F and B E G we have

A

= J p(y;B)u(dy) = J J lB(x)p(y;dx)~(dy) •

A A E

So, for every A e: F and for every positive G-measurable function f on E,

the equality

f fd~ I J f(x)p(y;dx)u(dy)

A A E

holds.

PROPOSITION 9.8. Let (E,f) be a aountably generated anatytia svaae3 ~ a

probabitity on E, and F a sub-o-atgebra of E. Then there exists an F­measurable transition probabitity p tpom (E,F) to (E,E) suah that, for

every positive E-measurabZe funation f, the funation y 1+ f f(x)p(y;dx) is

a version of the aonditionaZ expeatation of f with respeat to F and u.

PROOF. Apply Proposition 9.7 taking G := E.

Some properties of probabilitie.s on a measurable space give rise to

measurable, or Souslin, subsets of the space of all probabilities. In the

rest of this section we shall discuss some of these. The reader may think

the properties considered somewhat peculiar; their significanee will become

evident in Chapter III.

0

0

64

PROPOSITION 9.9. Let E and F be aountabZy generated measurabZe spaaes, Zet

F be anaZytia and Zet G be a measurabZe (SousZin) subset of E x F. Then the

probabiZities on E x F that aan be deaompoaed into their marginaZ on E and

a universaZZy measurabZe transition probabiZity whose graph is aontained in

G aonstitute a measurabZe (Sous Zin) subset of (E x F) ~.

PROOF: Let E, F and V he the o-algebras of E, F and ID, respectively. As F

is a countably generated analytic space, F is isomorphic to the cr-algebra

of some Souslin subspace of ID by Proposition 7.9. SoF itself may be

supposed to be a Souslin subspace of ID.

For the proof we need some mappings, and we start by introducing

them. Let rp: F + ID be defined by ( q>U) (B) : = J1 (BnF) (B € V) • Si nee

F = {B n F I B € V}, the mapping rp is strictly measurable. Consequently,

the mapping (x,u) ~ (x,rpu) of E x F into E x ID is strictly measurable as

well, and, therefore, there exists a measurable (Souslin) subset G' of -E x ID such that

(I) V( ) E -F [(x, u) € G ~ (x,rpv) E G' J x, )1 E X

Next let~: (ExF)~ + (ExJD) be defined by (~Jl)(C) := u(C n (ExF))

(C E E ®V). As E ® F = {C n (ExF) I C E E ®V}, the mapping ~ is (strictly)

measurable. Also, for each v E E and for every universally measurable

q: E + F, we have

[~(vxq)](AxB) = (vxq)[(AxB) n (ExF)] =

(vxq)[A x (BnF)] = J q(x)(BnF)v(dx)

A

= J rp(q(x))(B)v(dx)

A

[v x (rpoq)](AxB)

for every A E E and B € V, and hence

(2)

Further let p: (ExiD) x E + ID be measurable and such that for each

v E E and for every universally measurable q: E + ID

(3) p(vxq,x) = q(x) for v-almost every x E E.

65

Such a mapping p exists by Propositions 9.5 and 9.6.

Finally, let x: (E x ID) + (E x ID) ~ be defined by

(4) L f d(X\l) := J f(x,p(\l,x))\l(d(x,y)) • ExiD ExiD

for each positive measurable function f on E x ID. In (4), the integral on

the right hand side depends measurably on ll by Proposition 6,2 arid this

implies that the mapping x is measurable. Note that (4) is equivalent to

(5) J~ fd(x!l) J f(x,p(p,x))v(dx) , ExiD E

where v is the marginal of ll on E.

Now let u be a probability on E x F and let v x q be a decomposition

of ll into its marginal v on E and a universally measurable transition

probability from E to F. Then we have

[x(~(vxq))](G') (~)

[x(v x (~oq))](G')

E

f~ IG,dx(v x (~oq)) (~) ExiD

= J IG,(x,(~oq)x)v(dx) (!) J IG(x,q(x)}v(dx) •

E E

So, when the graph of q is contained in G, then IG(x,q(x)) = I for v­almost all x c E, and hence, [(xo~)p](G') =I. Suppose. on the other hand,

that the latter equality holds. Then (x,q(x)) c G for v-almost all x E E.

Now we may assume that a universally measurable mapping q0 : E + F exists

whose graph is contained in G; otherwise the proposition is trivially true.

When we redefine q(x) := q0(x) for those x E E for which (x,q(x)) t G, then

the graph of q will be contained in G and the equality p = v x q will still

hold.

66

The foregoing implies that the probabilities ~ on E x F that can be

decomposed into their marginal on E and a universally measurable transition

probability whose graph is contained in G, can be characterized by the

equality [(xo~)~](G') = 1. Therefore, by the measurability of x o ~. they

constitute a measurable (Souslin) subset of (E x F)~. D

A particularly useful example of the measurable (Souslin) subset of

E x F occurring in the foregoing proposition is expressed in

PROPOSITION 9.10. Let E and F be measUPable spaaes, Sa meas~ble (Souslin)

subset of E x F, and for every x E E let Sx := {y E F I (x,y) E S}. Then

{(x,À) E Ex F I À(S) = I} is a measurable (Souslin) subset of Ex F. x

PROOF. For every (x,À) E E x F we have

À(S ) x J 15(x',y')(óx x À)(d(x',y')) •

EXF

It follows from Proposition 6.1 that the integral is a measurable (Souslin)

function of óx x À, and the examples I and 5 following Proposition 2.2

imply that óx x À depends measurably on (x,À). From this the result fol-

lows.

In the applications of analytic spaces in Chapter III we shall

encounter convex linear combinations of probabilities. The following

propositions express the fact that certain measurability properties of a

set M of probabilities are preserved when we add to M all convex linear

combinations of its members.

DEFINITION. Let M be a set of probabilities on a measurable space E. A

·probability ~ on E is called a aountable convex combination of memhers of

M if there exist a countable family (~ ) I of probabilities in M, and a m mE countable famÎly (a) I of numbers in (0,1], such that E

1 a =I and

m• • m EmEI am~m = ~. The set of countable convex combinations of memhers of M is

called the a-convex huZZ of M. M is called a-convex, if M equals its cr­

convex hull.

0

67

PROPOSITION 9.11. Let E and F be measurable spaces, Ga subset of Ex F and

M the set of probabilities on E x F that aan be decomposed into their

marginal on E and a universally measurable transition probability whose

graph is contained in G .. When each of the sections {À E F I (x,À) E G}

(x E EJ of G is cr-convex, then M is cr-convex as well.

PROOF. Let ~ = L I a ~ be a countable convex combination of probabilities --- mE m m on E x F such that for each m E I the probability ~m can be decomposed into

its marginal vm on E and a universally measurable transition probability pm

whose graph is contained in G. Sinae the marginal v of ~ on E equals

LmEI amvm, for each m E I the probability vm is absolutely continuous with

respect to v, and by the Radon-Nikodym theorem there exists a positive

measurable function f on E such that dv = f dv. Clearly L I a f (x) = m m m mE m m for v-almost all x E E and, as the functions fm are determined up to a v-

null function, this equality may be supposed to hold for each x E E.

Now let the transition probability p: E + F be defined by

p(x) := ~ a f (x)p (x) • mEI m m m

Then the graph of p is contained in G, since the sections of G are cr-convex.

Moreover, for every measurable rectangle A x B c E x F we have

so ~

~(A x B) a ~ (A x B) mm ~ mEI

am J pm(x;B)vm(dx)

A

J ~ a p (x;B)f (x)v(dx) I m m m

A mE J p(x;B)v(dx)

A

= (v x p)(A x B) ,

v x p. Hence ~ E M.

PROPOSITION 9.12. Let E be a measurable space, Fa countably generated

analytic space and S a SousZin subset of E x F. Moreover let S be the subset

of E x F such that for every x E E the section {À E F I (x,À) E S} of S equals the cr-convex hull of {À E F I (x,À) E S}. Then S is again a SousZin

subset of Ex F.

D

68

PROOF

i) We first consider the case that E ID. Let

H [O,l]IN ~IN

:= x ID x F

and

Ho := {(a,x,y) E H I ~ a I and V IN (x,y ) E S} • mE IN

m mE m

It follows from the properties of product spaces (see Preliminaries 8

and 9) that, for each m E IN, a and (x,y ) depend measurably on m m (a,x,y) E H. So, by Proposition 4.3, H0 is a Souslin subspace of H.

As H is a product of analytic spaces, it follows from Proposition 7.6

(applied twice) that H0 is analytic. Now let x: H0 + 1D x F be defined

by

x(a,x,y) (x, ~ mE IN

a y ) • mm

Then x is measurable and the range of x is S. Since by Proposition 2.3

F is countably separated, ID x F is countably separated as well. So,

by Proposition 8.4, S is a Souslin subset of lD x F.

ii) Next let E be an arbitrary measurable space and let E and F he the

a-algebras of E and F, respectively. By Propositions 4.2 and 4.5 we

have S E S(E ® F) c SS(E x F) = S(E x F), and from this it follows

(see the remark following the definition of the Souslin operation)

that S E S(A x F) for some countable subclass A of E. So S is a

Souslin subset of (E,E0) x F, where E0 := a(A), which is a countably

generated sub-a-algebra of E. Let~: (E,E0) + ID be strictly measurable (see Proposition 7.4) and

let~: (E,E0) x F + 1D x F be defined by ~(x,À) = (~(x),À). Then ~ is

strictly measurable as well, and by Proposition 7.4 S = ~-IT for ~

some Souslin subset Tof lD x F.

Finally, let T be the subset of 1D x F such that for every x E lD the

section {À E F (x,À) E T} equals the a-convex hull of

{À E F I (x,À) E T}. Then S = ~-lT by the definition of ~. and T is a

Souslin subset of 1D x F by i). So Sis a Souslin subset of Ex F by

Proposition 4.3. D

69

COROLLARY 9.13. Let F be a eountably genepated analytie spaee and let s be

a Soustin subset of F. Then the aonvem hult of S is a Soustin subset of F.

In the foregoing (Example 1, following Proposition 2.2) we introduced

probabilities o concentrated at a point x. A generalization of this kind x of probability is given in:

DEFINITION. A probability is called deterministia when it attains the

values 0 and 1 only.

In general, a deterministic probability need not be concentrated at

one point. Also, neither the set of deterministic probabilities nor the set

of pro~abilities concentrated at a point need be a measurable subset of the

space of all probabilities. However,for countably generated spaces we have:

PROPOSITION 9.14. Let E be a aountably generated meaBUPable spaee. Then

i) eaah detePministia probability on E is eonaentPated at one point;

ii) the set of detePministia probabilities on E is a measurable subset

~~

PROOF. Let A be a countable algebra generating the cr-algebra E of E.

i) Let p E E be deterministic. The collection A' := {A E A I p(A) = 1}

is countable, so ~(nA') I, hence nA' f 0. Now let x E nA'. Then the

probabilities ~ and o coincide on A and therefore on E. So p = o • x x ~

ii) Let~ e E be such that p(A) E {0,1} for every A E A. Then

{Be E I p(B) E {0,1}} is a cr-algebra that contains A, and hence,

equals E; so ~ is deterministic. The set of all deterministic

probabilities therefore equals

n {peE 1 ~(A) € {O,I}} , AeA

which is a countable intersectien of measurable subsets of E and

therefore measurable itself. D

70

PROPOSITION 9.15. Let E and F be eountably gene~ated measurable spaees, and

let F be analytie. Then the probabilities on E x F that aan be deaomposed

into thei~ marginal on E and a universally measurable transition probability

that attains deterministie values onl~ eonstitute a measurable subset of (E x F) -.

PROOF. Let D be the set of deterministic probabilities on F, and let

G :=Ex D. By Proposition 9.14 Dis a measurable subset of F, so Gis a

measurable subset of E x F. Application of Proposition 9.9 now gives the

desired result.

The final subject of this section is Ionescu-Tulcea's theorem; in

fact a generalization of that theorem, since the transition probabilities

appearing in it need not be measurable. Our proof is a modification of the

proofs in [Neveu] sectien V.I.

Let E and F be measurable spaees and let p be a (unive~sally)

measu~able t~ansition probability from E to F. Then v ~ v x p is a (univer­

sally) measurable mapving ofË into (Ex F)-.

PROOF. Let A x B be a measurable rectangle in E x F. Then

( v x p) (A x B) J IA(x)p(x;B)v(dx) •

E

0

Since the integrand is a (universally) measurable function of x, by Proposi­

tien 6.1 the integral depends (universally) measurably on v. The result

fellows from Proposition 2.2 applied to the colleetien of measurable reet-

angles of E x F.

LEMMA 9.17. Let E and F be measurable spaaes, Zet~ be a probability on E x F and Zet ~(I) and ~(2 ) be the marginals of~ on E and F, ~speatively. When ~(I) or ~( 2 ) is deterministia. then ~ ~(l) x ~(2 ).

PROOF. Suppose that ~(l) is deterministic. Let A x B be a measurable rect­

angle of Ex F. When ~(l)(A) = 0, then ~(A x B) s ~(A x F) • ~(l)(A) = O,

so

~(A x B)

D

When V(I)(A) =I, then p(Ac x B) ~ v(Ac x F)

v(Ac x B) = 0 and therefore

0, so

71

So v = v(l) x v(2), because this equality holds on the class of measurable

rectangles.

PROPOSITION 9.18 (Ionescu-Tulcea's theorem). Let (E 1,E2, ••. ) be a finite or

infinite sequenae ofmeasurabZe spaaes and for eaah n Zet pn be a (univer­

saZZy) measurabZe transition probabiZity from E1 x ••• x En to En+l' Then

i) for eaah probabiZity v on E1

there is a unique probabiZity vv on

nn~l En suah that

(1) \1\) v and, for eaoh n,

where v(n) denotes the marginaZ of 11 on E1 x •.• x E V V n

ii) there is a unique (universally) measurabZe transition probability p

from E1 to nn~Z En suah that, for eaah probabiZity v on E1,

\lv = v x p.

PROOF. We prove the statement for infinite sequences; the case of finite

sequences is contained in this.

Let A be the algebra of measurable subsets of the space nn~l En that

depend on only finitely many coordinates. Let x1 t E1 and let us define the

function P on A as fellows: For every m E JN and for every A t A depending

on the first m coordinates only

(I)

where cp is the projection of n >I E onto TI < 1 Note that for each A m n- n n-m+ this definition does not depend on the particular choice for m. Clearly, P

is additive on A; we shall show that P is cr-additive on A, or, equivalently,

that P is continuous at 0. We argue by contradiction.

0

72

So let (A ) lN be a decreasing sequence in A satisfying n lN A = ~. m mE mE m and suppose that lim ,_ P(A ) > 0. Without loss of generality we suppose

m~ m that for each m E lN the set Am depends on the first m coordinates only, so

(2) lim ( ••• (ó x p1) x ••• x p )(~A) = lim P(A) > 0 • m-T<O x 1 m m m m_,.... m

We shall prove, by induction on n, that an x E nn~l En exists such

that for each n E lN

(3) limsup ( ••• (6( ) x p ) x ••• x p )(~A) > 0. m+oo· x 1, ••• ,xn n m m m

For n = 1, (3) is a consequence of (2). Also, for each n E IN by the

definition of 6 x p the inequality (3) can be written as (x1, .. , ,xn) n

limsup m+oo I [( ... (6( )xp +l)x, .• xp )(~A )]p (x 1 , ••• ,x ;dx +I)> 0,

x 1 , ••• ,xn+l n m m m n n n En+ I

and it fellows from Fatou's lemma (see [Ash] 1.6.8) that this inequality

remains valid when the order of integration and formation of the limit is

changed. Hence

forsome xn+l EEn+!' Let n E lN. As (A ) lN is a decreasing sequence and since for each m m mE

the set Am depends on the first m coordinates only, for each m ~ n we have

( ••• (ó( ) x p) x ••• x p )(~A) ~ x1, ••• ,xn n m m m

~ ( ••• (6( ) x p) x ••• x p )(~A) x 1, ••• ,xn n m m n

Tagether with (3) this implies that IAn(x) > 0, and hence, x E An. Since n

is arbitrary, we conclude that x E nnEIN An' which contradiets our assump­

tion n IN A = ~. nE n

So P is cr-additive on A. Since the algebra A generatas the cr-algebra

of the space n >J E , P can be extended to a probability on that space, n- n

denoted again by P.

We now make explicit the dependenee of Pon x 1 by P(x1

)

73

instead of P. It fellows from (I) by repeated application of Lemma 9.16 and

by the measurability of o that P(x1)(A) depends (universally) measurably on

x1 for each A E A. Tagether with Proposition 2.2 this implies that the

probability P(x1

) depends (universally) measurably on x1

, and hence, by ex­

ample 3 of section 2 that the marginal p(x1) of P(x 1) on E := nn~Z En depends

(universally) measurably on x 1 as well. So p is a (universally) measurable

transition probability from E1

toE. Moreover, it fellows from (I) that öx

is the marginal of P(x) on EP so by Lemma 9.17 P(x) = óx x p(x) (x E B1).

Now let v be a probability on E1

• Then for each mE lN and for each

A E A depending on the first m coordinates only

(v x p)(A) I J IA(x,y)p(x;dy)v(dx)

E1

E

J J I IA(x',y)p(x;dy)ox(dx')v(dx)

E1

E1

E

J (6x x p(x))(A)v(dx)

El

I P(x)(A)v(dx)

El

I ( ... (ö x p1) x ••• x p )(T A)v(dx) x m m

so the marginal of v x pon E1 x ••• x Em+l is ( ••. (v x p1) x ..• x pm).

Hence, the probability v x p satisfies i) and, since v is arbitrary, the

transition probability p satisfies ii).

Finally, the uniqueness of pv in i) fellows from the fact that pv is

completely defined by its restrietion to the generating algebra A, hence,

by its marginals. The uniqueness of p in ii) follows from this by the

identity p0 ox x p = ox x p(x) (x e E1). x

0

74

The main idea of Ionescu-Tulcea's theorem is that the transition

probabilities p1,p2 , ••• can be comhined into a single transition probability

p. Applying the above proposition to the sequence (p ,p 1, .•. ) and the m m+ probability v := o (xl x_) on E1 x ••• x E (x E E , m ~ I) we obtain

·····-~ m n n 'the usual formulation of Ionescu-Tulcea's theorem, e.g. the one given in

[Neveu].

CHAPTER 111 DYNAMIC PROGRAMMING

75

In this final chapter we shall show how analytic measurable spaces

can be used to solve measurability and selection problems occurring in

dynamic programming. As an analytic measurable space is a generalization of

an analytic topological space (see the remark at the end of section 8), our

treatment of the subject has many points in common with the formalism of

dynamic programming based an analytic topological spaces and many of our

propositions and proofs are mere adaptations of those met with in the

topological theory. Two topics should, however, be considered as new, viz.

our definition of decision models, which is a generalization of the usual

one, and the existence of a, what may he called uniformly optimal, strategy

(see Proposition 11.4).

§ 10. Decision models

For beuristics concerning decision models, which provide the

mathematica! structure dynamic progrpmming is based on, we refer to

[Hinderer] and [Bertsekas & Shreve].

DEFINITION. A deeision model is a quintuple (S,A,G,p,u), the elementsof

which can be characterized and interpreted as follows.

i)

U)

S is a sequence of countably generated analytic spaces (Sn)nElN•

Sn is called the state spaee at time n. Memhers of s1 are called

initiaZ states.

A also is a sequence of countably generated analytic spaces (A ) m· U UE

A is called the spaee of aations that are available at time n. n

For each n E IN wedefine Hn := s1 x A1 x s2 x ••• x Sn and

H := rr:=I (Sn x A0). H is called the spaee of realizations and H

0 the spaee

of histories at time n.

iii) G is a sequence (G0

)nEIN' where Gn is a Souslin subset of H0

x A0

•

Moreover for each n E IN and h E Hn the set

76

G h := {À € A I (h,À) E G } is nonempty; G h is called the set of n, n n · n, admissible probabilities at time n given history h.

iv) p is a sequence (pn)nEIN' where for each n E IN pn is a measurable

transition probability from Hn x An to Sn+J' pis called the

transition laJ.;J.

v) u is a Souslin function on H. It is called the utility.

A strategy for the decision model defined above is a sequence (~)nEIN'

where qn is a universally measurable transition probability from Hn to An

whose graph is contained in G • For every decision model at least one n

strategy exists, because for every n E lN it follows from the analyticity

of An that there exists a universally measurable mapping qn of Hn into An

whose graph is contained in Gn.

By an initial probability we simply mean a probability on s 1• When

for each n E IN and every h E Rn the set Gn,h of admissible probabilities

is cr-convex, then the decision model will be said to allow aombination of

strategies.

In the usual definition of a decision model, instead of G a sequence

(Dn)nEJN is introduced, where Dn is a Souslin subset of Hn x An. For each

n E IN and h E H the set D h := {y E A I (h,y) E D } is called the set n n, n n of admissible actions at time n given history h. A strategy is then defined

as a sequence (~)nEIN' where qn is a universally measurable transition

probability from Hn to An such that VhEliu ~(h;Dn,h) = I.

It is easily deduced from Pro,osition 9.10 that our definition of a

decision model is a generalization of the usual one. The generalized

decision model bas the advantage that one can prescribe not only which

actions may be used at each step of a decision process, but also how they

may be combined into a probability on the action space. For example, the

property that a strategy q is deterministic (i.e. that each of the proba­

hilities ~(h) is deterministic) is equivalent to the condition that for

each n E IN the graph of ~ is contained in Hn x 8n' where 8n is the

measurable subset of An consisting of all deterministic probabilities on An

(see Proposition 9.14).

To every initial probability of a decision model and to each strategy

there corresponds, in a natural way, a probability on the space of

77

realizations. Moreover, the set of all such probabilities turns out to have

useful algebraic and measure-theoretic properties.

DEFINITION. For every probability v on the space of realizations of a

decision model v(Zn-J) and ~(Zn) denote its marginals on H and H x A , n n n

respectively (n E 1N) (where H and A are the space of histories and the n n space of actions, respectively, both at timen).

PROPOSITION 10.1. Let v be an initial probability and q a strategy fora

deaision model with transition law p. Then there exists a unique probabil­

ity v on the spaee of realizations satisfying

(I) ~ \) '

(2n) (2n-1) x ~ ~ ll ~ nd (2n+l)

a ~ (2n)

V x Pn

PROOF. This is a particular case of Proposition 9.18.

(n E 1N) •

0

Note that the above proposition is valid for a more general decision

model, viz. one with arbitrary state and action spaces, and a transition

law that is merely universally measurable.

DEFINITION. For any decision model the probability on the space of

realizations that is generated by an initial probability v and a strategy ~

in the sense of Proposition 10.1 will be denoted by !l(v,q). Probabilities

of this kind will be called strategie probabilities.

A strategie probability depends universally measurably on the initial

probability. This property, and a kind of reverse of it, are treated in the

next three propositions. Also, the set o! all strategie probabilities, as

well as a set related to it, turn out to be Souslin sets; this is the sub­

ject of Propositions 10.5 and 10.6.

PROPOSITION 10.2. For every strategy q the strategie probability !l(v,q)

depends universally measurably on v.

PROOF. Due to Propositions 10.1 and 9.18 there exists a universally measur­

able transition probability p such that ~(v,q) ~ v x p for each initia!

probability v. The result now fellows from Lemma 9.16. 0

78

As a particular case of Proposition 10:2, for every strategy q the mapping

x ~ ~(ó ,q) of the space of initial states into the space of strategie x probabilities is universally measurable. The converse also holds:

PROPOSITION 10.3. Let ~= s1 + b be a universally measurable mapping of the

space s1

of initial states into the set E of strategie probabilities of a

decision model, and Zet ~(x)(l) = ox for every x € s1. Then there is a

strategy q such that for every x E s1 one has ~(x) = ~(öx,q).

PROOF. Let (S,A,G,p,u) be the decision model and let n E JN. When we apply

Proposition 9.6 to the space of histories ~ and the action space An' both

at timen, and to the probabilities ~(x)(2n) (x E SJ) on Hn x An' we get a

universally measurable mapping p: (H x A )~x H +A such that n n n n

(I) (2n) J (2n) (2n-l) ~(x) (A x B) = p(~(x) ,h;B)~(x) (dh)

A

for each measurable rectangle A x B c Hn x~ and for each x E SJ. Now for

each x E SJ we have (~(x)(Zn-I))(J) = ~(x)(l) = öx and therefore by Lemma

9.17 equality (I) is equivalent to

(2) ~(x)(Zn)(A x B) = J p(~(h 1 )(Zn),h;B)~(x)(2n-l)(dh) •

A

Let qn: Hn + An be defined by ~(h) = p(~(hJ)(Zn),h). Then ~is universally measurable because p and ~ are, and (2) can be written as

(3) ( ) (2n) ( )(2n-l) ~ x = ~ x x qn

The set {h E Hn I (h,~(h)) E Gn} is the inverse image of the Souslin

subset G of H x A under the universally measurable mapping h ~ (h,o (h)), n n n -n and as a consequence this set is a universally measurable subset of Hn.

When therefore q' is an arbitrary strategy and ~ is redefined by

q (h) n

:= ~(h)

:= ~ {h) else,

then ~ becomes a universally measurable mapping whose graph is contained

in Gn. Moreover we still have (3): Let x E s1• Since ~(x) is strategie • x (2n) (2n-l) x

there ex~sts a strategy, q say, such that ~(x) = w(x) x ~· By

79

(3) and Proposition 9.5 this implies that qn(h) = q~(h), and hence; that

(h,q (h)) EG for w(x)(Zn-l)_almost all hE H • So the redefinition of a n n (2 I) n ~

does affect its values on a ~(x) n- -null set only and as a consequence

(3) remains valid.

Now letting n run through IN, we get a sequence (qn) which obviously

is a strategy, q say. Now for each x E s1 we have ~(x)(l) = öx and

( ) (2n+l) ( ) (2n) f h lN b ( ) • • w x = w x x pn or eac n E ecause ~ x ~s strateg~c.

Tagether with (3) and Proposition 10.1 this implies that w(x) = p(óx,q)

for each x E s1•

An equivalent formulation of the preceding proposition runs as fol-

lows:

COROLLARY 10.4. FoP evecy initiaZ. state x of a decision model Zet a

stPategy qx be given, and Zet u(öx,qx) depend univePsally measu:Pably on x.

Then a stPategy q exists such that u(ö ,qx) = u(ö ,q) foP each initial x x . state x.

In general, the strategy q in Corollary 10.4 cannot be obtained by

simply combining the strategies qx in the following way:

In fact the ~·s thus obtained may fail to be universally measurable, as

the following example-shows.

For all n E IN let S A [0,1] and pn ~ À, where À is Lebesgue n n measure on [0,1]. Moreover let, for all x E s1, ~~À (n ~ 2) and

OandxEK,

À else,

where Kis a subset of [0,1] which is not universally measurable. A simple x computation gives p(o ,q) = ó x n >Z À, which depends measurably on x. x x n_

On the other hand

D

80

À else,

hl/ so q2 (h 1,h2,h3) does notdepend universally measurably on (hJ,h2,h3)•

PROPOSITION 10.5. The st~ategic p~babiZities on the space H of realiza­

tions of a decision model constitute a SousZin subset of H. When the

decision model allows combination of strategies~ then this subset is

a-convex.

PROOF. Let the decision model be (S,A,G,p,u). For each n E lN let H be the n

space of histories at time n, Mn the set of probabilities on Hn x An that

can he decomposed into a marginal on H and a universally measurable trans-n

ition probability whose graph is contained in Gn' and M~ the probabilities

on (Hn x An) x Sn+l that can he decomposed into a marginal on Hn x An and

the transition probability Pn• Then by Proposition 10.1 the set E of

strategie probabilities can be written as

n nElN

[{~EH I ~(Zn) E M} n {p EH n

(2n+l) E M' }] • ~ n

Let n E IN. Proposition 9.9 implies that Mn is a Souslin subset of

(Hn x An) and it follows from Proposition 9.11 that Mn is a-convex when

combination of strategies is allowed. Since ~ ~ ~(2n) is a measurable

linear mapping of H into (H x A)~, the set {p EH I ~( 2n) E M } is a n n n

Souslin subset of H which, in addition, is a-convex provided that combina-

tion of strategies is allowed.

Next let Gn' he the graphof p • Then, by Proposition 7.3, G' is a ~n n

measurable subset of (Hn x An) x Sn+l and each ?f the sets

{Ä ES +I I (x,Ä) EG'} (x EH x A) is a singleton and therefore is n n n n a-convex. Moreover, M' can be described as the set of probabilities on

n (Hn x An) x Sn+l that can he decomposed into a marginal on Hn x An and a

transition probability whose graph is contained in G'. As a consequence, we n~ I (2n+I) .

can repeat the argument above and conclude that {u E H p E M~} 1s a

Souslin (in fact measurable) subset of H that is a-convex.

From the foregoing it is easily deduced that Z has the desired

properties. 0

81

PROPOSITrON 10.6. Let s1 be the space of initial states and H the srace of

Pealizations of a decision model. Let TI be the subset of s1 x H consisting

of the pairs (v,~) foP which a stPategy q exiats such that ~ ; ~(v,q). Then

rr ia a Soustin subset of s1

x H.

PROOF. Let E be the set of strategie probabilities on H and let

Then IT = (S1 x E) n 8. Now E is a Souslin subset of H by Proposition 10.5

and n is a measurable subset of 51 x H, being the graph of the measurable

mapping ~~~(I) of H into the countably separated space s1

(see Proposi­

tions 2.3 and 7.3). The foregoing implies that IT is a Souslin subset of

s1

x H.

§ 11. The expected utility and optima! strategies

D

The object of interest in dynamic programming is the expected utility.

We first define this quantity and prove its measurability.

DEFINITION. Let a decision model be given with space of realizations H and

utility u. For every initial probability v and for each strategy q we

de fine

w(v,q) := J ud~(v,q) H-

The function w is called the expected utility. Also, for every initial

probability v we define

v(v) := sup w(v,q) , q

where the supremum is taken over all strategies. The function v is called

the maximal e:.r:pected utility.

PROPOSITION I l.I. The maximal expected utilityvis a Soustin funation~ and

for each strategy q the expected utility w(v,q) is a universally measurable

function of v.

82

~

PROOF. Let the Souslin subset IT of s1 x H be defined as in Proposition 10.6

ànd let f: rr ~IR be defined by f(v,~) = L udp, where u is the utility.

Then, by Proposition 6.2, f is a Souslin function because u is, and for

each v € s1 we have v(v) = sup {f(v,~) I ~: (v,~) € IT}. Application of the

exact selection theorem (Proposition 7.1) yields the desired result.

Also it follows from the definition of wand Proposition 6.1 that

w(v,q) is a universally measurable (in fact, Souslin) function of ~(v,q).

By Proposition 10.2 this implies that w(v,q) depends universally measurably

on v. 0

In the rest of this section we consider optimal strategies, i.e.

strategies for which the expected utility equals the maximal expected

utility. In particular the question is raised whether optimality, or a

prescribed optimality defect, can be realized simultaneously for all initia!

probabilities by one and the same strategy. A secoud topic will be the

linear dependenee of the (maximal) expected utility on the initial proba­

bility. We conclude with a derivation of the optimality equation.

DEFINITION. Let v be an initia! probability and q a strategy for a decision

model. Then q is called v-optimaZ if w(v,q) = v(v).

It may happen that for an initia! probability v no v-optima! strate­

giesexist (see the example following Proposition 11.4).

Let h be a universaZZy measurabZe funetion on the spaae

s1

of initiaZ states of a deaision model satisfying

h(x) < v(ö )

x

-"" etse.

Then there exists a strategy q sueh that

i) w(öx,q) ~ h(x) for eaeh x € s1;

ii) q is öx-optimaZ for eaeh x € s1 for whieh a öx-optimaZ strategy

exists.

Let IT c SI x H be defined as in Proposition 10,6 and let rr• c sl x H

be the inverse image of the Souslin set IT under the measurable mapping

83

(x,~) ~ (óx.~) of s1 x H into s1 x H. Then IT' is a Souslin subset of s1 x H

and for all (x,~) E s1

x H we have (x,~) E rr• iff ~ = ~(6x,q) for some

strategy q.

Moreover, let f: s1

x H + ÏR be defined by f(x,~) = J ud~, where u is

the utility. Then f is a Souslin function and for each x E s1

we have

v(ó) = sup {f(x,~) u: (x,u) E IT'}. Now from the exact selection theorem x

(Proposition 7.1) the existence follows of a universally measurable mapping

~: s1 + H the graphof which is contained in IT', and such that

f(x,~(x)) ~ h(x) for each x E s1

, and f(x,~(x)) = v(óx) for each x E s1

for

which a ox-optimal strategy exists.

Now by Proposition 10.3 a strategy q exists such that for each x E s1 ~(x) = p(óx,q), and consequently f(x,~(x)) = J ud~(x) = w(ó ,q). The defini-

- x tion of ~ now implies that q has the properties stated in the proposition. 0

At every instant of time in a decision process one knows both the

state the system is in, and (given the strategy) the probability distribu­

tion of the state the system will enter at the next instant of time. Due to

this alternation of states and probabilities, either a quantity is most

naturally expressed as a function of states or as a function of probabili­

ties. The following proposition gives the conneetion between the two repre­

sentations for the (maximal) expected utility. Note that representing a

state x by the probability ó amounts merely to an identification of non-x

separated points.

PROPOSITION 11.3. Let v be an initial probability fora deaision model.

i)

i i)

iii)

When -~ < v(v), then x~ v(óx) is quasi-integrable with resveat to v

and J v(ó )v(dx) = v(v). x

f v(o )v(dx) = sup , f w(ó ,q')v(dx), where the supremum is taken - x q - x over all strategies q'.

When q is a strategy suah that the utility is quasi-integrable with

reapeet to ~(v,q), in partiaular, when w(v,q) > -oo, then x ~ w(óx,q)

is quasi-integrable with respeat to v and f w(ox,q)v(dx) = w(v,q).

iv) When the utility is quasi-integrable with resveat to every strategie

probability, then J v(ó )v(dx) = v(v). - x

84'

PROOF. The order in which we shall prove the various parts of the proposi­

tien is: ii), iii), iv), and i).

ii) Suppose that I v(ó )v(dx) >-~.As by Proposition 11.1 v(ó) is a - x x

universally measurable function of x, it fellows from Proposition 11.2

that a sequence (qm)mEIN of strategies exists such that for each

initia! state x and m E IN

-1 - m if v(ó)

else.

< "'

Then the sequence (x t+ w( ó ,qm)) IN of functions converges to the x mE

function x t+ v(ó ), while the integrable function x t+ min {v(ó) -1,0} x x is a common lower bound. So by Fatau's lemma (see [Ash] 1.6.8)

s liminf J w(óx,qm)v(dx) s m

sup J w(óx,q')v(dx) q'

This inequality is satisfied also when the left hand member equals -oo,

The reverse inequality on the other hand is a direct consequence of

v(ó ) ~ w(ó ,q'). x x

iii) From the definition of ~(v,q) it follows that for every measurable

cylinder C we have ~(v,q)(C) =I ~(ó ,q)(C)v(dx). This results extends x to integrals of positive functions in the usual way. In particular,

one has for the utility u

w(v,q)

As u is quasi-integrable with respect to ~(v,q), at least one of the

repeated integrals is finite, say the one of u- for definiteness.

Then J u-d~(ó ,q) is finite for v-almost all x, and, as a function of x

x, is integrable with respect to v. Hence the last expression in (*)

can be written as

which in turn equals I w(ox,q)v(dx).

iv) It follows from ii) and iii) that

sup w(v,q') = v(v) • q'

85

i) For ea.ch n E JN let the function un be defined on the space of reali­

zations by un(h) =min {u(h),n} and let vn be the maximal expected

utility corresponding to the utility un. Then for every n the function

un is quasi-integrable with respect to every strategie probability

and it follows from iv) that vn(v) = f vn(o )v(dx). Also for every - x

initial distribution À we have

v(\) sup w(\,q) = sup J ud].l(À,q) q q -

sup sup J und].l(À,q) q n

sup sup J und].l(À,q) = sup vn(Ä) n q n

So

-oo < v(v)

J sup vn(ox)v(dx) n

where the third equality follows from the fact that vn(ó ) is in­x

creasing in n and that -ro < f vn(ö )v(dx) for sufficiently large n. 0 - x

The condition on the utility in iii) cannot be omitted. As an example

we consider a decision model for which s1 = (0,1), A1 =IR and

u(x,y, ••. ) = y. Take a strategy q for which q1(x)({x-l}) ! and

q1(x)({-x- 1}) = ~ (x E s1) and for the initial probability v take Lebesgue

measure on (0,1). Then w(öx,q) = 0 for all x and w(v,q) = -ro, so

w(v,q) ~ I w(ó ,q)v(dx). x

86

With the aid of Proposition 11.3 we can now extend the results in

Proposition 11.2 to arbitrary initia! probabilities.

PROPOSITION 11.4. Let the utility of a decision model be quasi-integrable

with respect to every strategie probability and let combination of strate­

gies be allowed. Then the expected utility w and the maximal expected

utility v are bounded on at least one side. When moreover E > 0 and

n: s1 + (O,oo) is a universally measurable function on the space s1 of

initial states, then there is a strategy q such that

i) w{ox,q) ~ v(ox) - n(x) for each initial state x;

ii) 'w(v ,q) ~ v(v) - E for each initial probabiUty v;

iii) q is v-optimal for each initial probability v for which a v-optimal

strategy exists.

PROOF. Suppose that the function w is unbounded on both sides. Then for

each m E IN there exists a strategie probability

and therefore such that f u+ d]J ~ 2m, where u is m

]J such that J ud]J ~ 2m m _ m the utility. When

00 -m ]J := Lm=l 2 ]Jm' then by Proposition 10.5 ]J is a strategie probability and

.In the same way one can show that a strategie probability ]J 1 exists such

that f u-d]J 1 = oo, Consequently, the utility u is not quasi-integrable with

respect to the strategie probability !(lJ + ]J 1), which contradiets our

assumptions. The function w therefore is bounded on at least one side. The

definition of v now implies that v is bounded on the same side.

In the proof of the remaining part of the proposition we suppose,

withoutlossof generality, that n(x) < E for every x E s1• Let h: s1 +IR be defined by

h(x) el se.

Application of Proposition 11.2 with this function h yields a strategy q

such that w(ox,q) ~ h(x) for each x E s 1 and such that q is ox-optimal for

every x E s 1 .for which a ox-optimal strategy exists. We shall prove that q

satisfies i), ii), and iii).

87

To prove i) we merely have to show that a ox-optimal strategy exists

for every x E s1

satisfying v(ox) = Moreover, for each m E lN let qm be

""· Therefore, let x E s1 and v(ox) = ""·

a strategy such that w(o ,qm) ~ 2m and x -m m let JJ : = I: lN 2 JJ ( o , q ) • mE x Then IJ is a strategie probability and

(I) jJ ' 2-m (o m) (I)

!.. IJ x'q m

0 x

so JJ JJ(ox,q') forsome strategy q'. With utility u we now have

L 2-m I u+ dJJ(ox,qm) ~ L 2-m I udJJ(ox,qm) m m

and since u is quasi-integrable with respect to JJ(ox,q') this implies that

w(ox,q') = ""• So q' is ox-optimal and thereby i) has been proved.

Next let v be an initia! probability. For every x E s1

we have

n(x) < e and therefore w(o ,q) ~ v(o ) - e by i). From this inequality ii) x x can be obtained by integration with respect to v and application of Propo-

sition 11.3.

To prove iii) we first consider an initia! probability v such that

v(v) is infinite. From ii) it then follows that w(v,q) = v(v) and hence

that q is v-optimal. Next let v(v) be finite and suppose that a v-optima!

strategy q' exists. Then by Proposition 11.3

I w(ox,q' )v(dx) w(v,q') v(v)

From the finiteness of these integrals together with the inequalities

w(ox,q') ~ v(öx) (x E s1) we conclude that w(ox,q') = v(ox) for v-almost

all x E s1• So for v-almost all x E s1

a ox-optimal strategy exists and

hence the equality w(ox,q) = v(ox) holds, as follows from the definition of

q. This in turn implies by Proposition 11.3 that

w(v,q) = I w(ax,q)v(dx) = I v(ax)v(dx) v(v) ,

so q is v-optima!. D

88

The preceding proposition cannot be essentially improved by allowing

E to depend on v, as the following example shows.

Consider a decision model for which s1 := {0,1}, A1 := (0,1) and

u(x,y,z, ••• ) := y. Then for every initial probability v and for every

strategy q we have

w(v,q) v({O}) I yq 1(0)(dy) + v({l}) J yq 1(1)(dy) ~

s; max {J yq 1 (O) (dy), J yq 1 (I) (dy)} ,

and the last expression is smaller than I and independent of v. Moreover

v(v) = I. So, for every strategy q, w(v,q) is bounded away from v(v)

uniformly in v.

We conclude this section with a derivation of the optimality equation.

For the formulation we need a transformation of decision models.

DEFINITION. The aontraation of a decision model (S,A,G,p,u) is the decision

model (S,A,G,p,û) defined by

i) sl = SI x Al x s2

ii) Sn+ I = 8n+2' A An+l' a = Gn+l' Pn Pn+l (n E lN) ; n n

ii) û u •

For every strategy q for the given model the strategy q for the contracted

model is defined by ~ = qn+l (n E JN).

Note that in the above definition of p, û and q w~ have tacitly

identified domains of definitions like (s 1xA1xs2) x A2 x s3 x ••• and

s 1 x Al x s2 x Az x s3 x ••••

The contraction operation can be repeated a number (say n) of times;

this results in a decision model having as its initial states the histories

up to time n of the original model. To all these contracted models the

results derived so far apply.

89

PROPOSITION 11.5. Let v be an initiaZ vrobability and Zet q be a strategy

for a deeision model with transition law p. Moreover~ let w (and v) be the

(maximaZ) expeeted utility for the eontraetion of the model. Then

i) w(v,q) = w(vxq1 xpl ,(i) '

ii)

PROOF.

(Optimality equation) v(v)

over all strategies q',

i) As a consequence of the definition of q and of Propaaition 10.1 we

have ~(v,q) = U(vxq 1xp 1,q), where D denotes the strategie probabili­

ties of the contracted model. Consequently, for utility u,

ii) From i) we deduce

v(v) sup w(v,q') = sup w(vxqjxpt,q') = q' q'

sup s~1p w(vxqjxp1,q') qj q

sup v(vxqjxpl) • qj

iii) From ii), the v-optimality of q, and i) we conclude

v(vxqlxpl) ~ sup v(vxqjxpl) = v(v) = w(v,q) = q'

Hence the result.

As in the case of Proposition 10.1 the result above is valid fora

more general decision model: arbitrary state and action spaces, and a

transition law and utility that are merely universally measurable.

D

90

REFERENCES

ASH, R.B., Measure~ Integration and Functional Analysis, Academie Press,

New York etc., 1972.

BERTSEKAS, D.P. and S.E. SHREVE, Stochastic

New York etc., 1978.

Control, Academie Press,

BLACKlYELL, D., Discounted dynamic programming, Ann. Math. Statist. 36 (1965)

226-235.

BLACKWELL, D., D. FREEDMAN, and M. ORKIN, The optimal reward operator in

dynamic programming, Ann. Prob. ~ (1974} 926-941.

CHOW, Y.S. and H. TEICHER, pPobability Theory, Springer-Verlag, New York

etc., 1978.

CHRISTENSEN, J.P.R., Topology and BoreZ Structure, North-Holland Mathernaties

Studies No. 10, North Holland Publishing Company, Amsterdam-London,

1974.

COHN, D.L., Measure Theory, Birkhäuser, Boston etc., 1980.

HINDERER, K., Foundations of Non-stationary Dynamic pPogramming with Dis­

crete Time Parameter, Lecture Notes in Operatiens Research and

Mathematica! Systems No. 33, Springer-Verlag, Berlin etc., 1970.

HOFFMANN-J~RGENSEN, J., The theory of Analytic Spaces, Various Publication

Series No. 10, Dept. of Mathematics, University of Aarhus, Denmark,

1970.

NEVEU, J., MathematicaZ Foundations of the Calculus of Probability,

Holden-Day, San Francisco etc., 1965.

PFANZAGL, J. and W. PIERLO, Compact Systems of Sets, Lecture Notes in

Mathernaties No. 16, Springer-Verlag, Berlin etc., 1966.

SHREVE, S.E., Dynamic Programming in Complete Separable Spaaes, University

of Illinois, 1977.

TOPS~E, F., Topology and Measure, Lecture Notes in Mathernaties No. 133,

Springer-Verlag, Berlin etc., 1970.

INDEX

action space 75

additive 30

admissible probability 76

analytic ~-algebra 34

Borel set

class

space 34

topological space 54

5

coneetion

combination of strategies 76

completion 10

concentrated at a point 5

conditional expectation 62

contraction 88

coordinate 3

countable

convex combination 66

countably generated 5

separated 5

cylinder 4

decision model 75

decomposition of a probability 57

deterministic probability 69

Dynkin class 2

Dynkin•s theorem 2

Exact selection theerem 35

expected utility 81

finite intersectien property 25

First separation theerem 50

function 6

history space 75

initial probability 76

state 75

inner regularity 26

integral 31

Ionescu-Tulcea•s theerem 70

isomorphic embedding 40

kernel 14

limit measurable 23

cr-algebra 23

marginal (probability)

maximal expected utility

measurable cylinder 4

(sub)set 3

optimal strategy 82

optimality equation 89

positive function 6

probability 4

product probability 9

cr-algebra 3

space 3

quasi-integrable 31

realization space 75

rectangle 4

4,9

81

restrietion of a probability 9,47

right inverse 40

semicompact 25

algebra

separated 5

separating 5,49

cr-additive 30

a-convex 66

Souslin class 14

function 19

eperation 14

scheme 14

(sub)set 1~

30

91

92

state space 75

strategie probability 77

strategy 76

strictly measurable 40

subspace 3

trace 3

A d r , A8

r , crA c, ,s,u,cr u

EJJ

(E,E)

Ë , Ë , (E,E)~

Ë(E)

E x F , E 0 F , A x B

ni€1 Fi' 0 icl Fi

Fr

L

s u Q x

3

10

3

7

6

4

3

4

23

14

10

5

transition law 76

probability 57

univeraal completion 10

universally measurable mapping 12

set 10

utility 76

-I Ijl

-1 lJ 0 q>

(n) lJ • lJ (v,q)

n:. ].

(nl'n2 , ... )

p(x;B) , p(x) (B)

V , W

IN,m.,iR

ID

f

2

8

77

3

57

31

4

57

81

5

4

30

6

6

31

93

SAMENVATTING

In de dynamische programmering beschouwt men een systeem dat een

reeks toestanden doorloopt en waarbij men elke overgang naar een nieuwe

toestand kan beÏnvloeden. De nieuwe toestand hangt dan af van de toestand

waarin het systeem zich bevond en van de ondernomen actie, maar niet een~

duidig: slechts de kansverdeling van de nieuwe toestand wordt bepaald door

oude toestand en actie. Verder is voor elke ontwikkeling van het systeem in

de loop der tijd, d.w.z. voor elke reeks van toestanden en acties die kan

optreden, een opbrengst gedefinieerd; dit is een getal dat aangeeft hoe

gewenst de betreffende ontwikkeling is. Men tracht nu de acties telkens zo

te kiezen, dat de verwachte opbrengst zo groot mogelijk is. In het algemeen

zal dit tot gevolg hebben dat de op elk tijdstip te kiezen actie zal afhan­

gen van de toestand waarin het systeem zich op dat tijdstip bevindt. Een

rij van dergelijke keuzevoorschriften, één voor elk tijdstip, noemt men een

strategie.

Wanneer het aantal toestanden waarin het systeem zich kan bevind~n en

het aantal mogelijke acties beide eindig of aftelbaar oneindig zijn en

wanneer bovendien slechts eindig veel overgangen van het systeem beschouwd

worden, dan is de verwachte opbrengst zonder meer gedefinieerd. In andere

gevallen is dit niet noodzakelijk het geval en kunnen zich meetbaarheids­

problemen voordoen. Men heeft echter ontdekt dat deze problemen niet optre­

den als men meetbare strategieën en een meetbare opbrengst heeft en als de

verzamelingen van toestanden en acties analytische topologische ruimten zijn.

Een onbevredigende kant van het gebruik van analytische ruimten is

dat topologische middelen worden aangewend om maattheoretische moeilijkheden

te vermijden. In dit proefschrift nu, ontwikkelen we een maattheoretisch

alternatief voor de analytische ruimte en laten zien hoe het kan worden

toegepast op dynamische programmering. Het daarbij beschouwde model voor

dynamische programmering is van zeer algemene aard.

Het proefschrift is verdeeld in drie hoofdstukken. In het eerste

hoofdstuk wordt de benodigde maattheorie ontwikkeld; slechts elementaire

voorkennis is vereist voor lezing van dit gedeelte. Hoofdstuk II bevat de

theorie van analytische ruimten voor zover deze van belang is voor de

toepassing op dynamische programmering in Hoofdstuk III. Ook voor lezing

.van deze hoofdstukken is geen speciale voorkennis nodig, hoewel enig begrip

van dynamische programmering de appreciatie van het laatste hoofdstuk zal

bevorderen.

95

CURRICULUM VITAE

De schrijver van dit proefschrift werd op 19 mei 1939 geboren te

Hoorn (N.H.) en in 1956 behaalde hij het diploma H.B.S.-b aan het Sint

Werenfridus-lyceum aldaar. Vervolgens studeerde hij aan de Universiteit van

Leiden en slaagde er in januari 1961 voor het candidaatsexamen "natuur- en

wiskunde met sterrenkunde". De studie werd voortgezet aan de Universiteit

van Amsterdam en in maart 1969 werd hier (cum laude) het doctoraal examen

afgelegd met hoofdvak theoretische natuurkunde en bijvakken wiskunde en

mechanica. Hierna was hij gedurende vier jaar werkzaam bij het Mathematisch

Instituut van de Universiteit van Amsterdam en is sinds augustus 1973 als

wetenschappelijk medewerker verbonden aan de onderafdeling der wiskunde en

informatica van de Technische Hogeschool te Eindhoven, in het bijzonder

belast met de verzorging van onderwijs in de wiskundige modelvorming.

STELLINGEN

I. De vraag of in elke Blackwell-ruimte de klasse der Souslin-verzame­

lingen gescheiden wordt door die der meetbare verzamelingen dient

bevestigend te worden beantwoord.

Litt.: Hoffman-J~rgensen, "The theory of analytic spaces", II.I1.3B.

II. In een aftelbaar voortgebrachte meetbare ruimte kunnen versies van de

Radon-Nikodym-afgeleide van begrensde maten zodanig gekozen worden dat

(x)

een meetbare functie is van ~. v en x tezamen.

III. Het formalisme van de niet-standaard analyse sluit aan bij de intuïtie

waar het oneindig grote en oneindig kleine getallen betreft. Het

onderscheid tussen interne en externe objecten mist echter elke in­

tuïtieve achtergrond.

IV. Het al dan niet consistent zijn van het geheel van definities en

stellingen in een wiskundige tekst mag niet afhankelijk zijn van de

aanvullende tekst.

V. Het idee dat nulverzamelingen en equivalentieklassen van meetbare

functies slechts voor complete maatruimten bruikbaar zijn is onjuist.

Het op grond van dit idee completeren van een maatruimte is dan ook

onnodig en bovendien ongewenst, daar bij completering van een cr­

algebra nuttige eigenschappen verloren kunnen gaan.

VI. Een goed leraar baseert zijn onderricht mede op de denkfouten van

zijn pupillen. Bij gebruik van een computer voor onderwijs is dit in

te geringe mate mogelijk.

VII. Toelichting bij educatieve films dient te worden uitgesproken door

iemand die het behandelde onderwerp goed kent.

VIII. De strategie genoemd in Propositie IJ.4.iii van dit proefschrift is

zo optimaal mogelijk.