A note on the Ionescu Tulcea theorem

A note on the Ionescu Tulcea theorem

Citation for published version (APA):

Simons, F. H., & Thiemann, J. G. F. (1978). A note on the Ionescu Tulcea theorem. (Memorandum COSOR; Vol. 7814). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 78-14

A note on the Ionescu Tulcea theorem

by

F.B. Simons and J.G.F. Thiemann

Eindhoven, June 1978 The Netherlands

A note on the Ionescu Tulcea theorem

by

F.H. Simons and J.G.F. Thiemann

Let (X,E ) be a sequence of measurable

n n

_n-

1 spaces, and let for every n pn

be a transition. probability from TI (X.,E.)

. i=O ~ ~ to (X ,E ). Roughly speaking, n n

the Ionescu Tulcea theorem ([3J, proposition V.l.l) states that for every x E Xo there exists a probability measure P x on the infi.nite product space

00

(Q,F)

=

TI (X IE ) representing the probability of a certain event during the n~ n n

action of the process, starting in the state x, and the probability P depends x

EO-measurable on x. In this note we shall show that if we weaken the condi-tion on pn by requiring only that the probability pn on (X IE ) satisfies

n n

n-l

pn{o,A) is universally measurable on TI (X.,E.) for every A E E , the

pro-i=O ~ ~ n

bability P on (Q,F) still exists, and depends universally measurable on x. x

Let (X,t) be a measurable space and ~et n(x) be the set of all probabi-lity measures on (X,t). For every ~ € n(X) we define the completion t of t

].I

with respect to ~ by

*

E

=

{A U N

I

A E E, N c: X and].l (N) = O} • 11

It is easily verified that E is a afield and that a function f on X is E

-~

measurable if and only if it equals Following Hinderer [2J, we now sur able sets by

E

=

n

llEn (X) E

11

11 modulo 11 a E-measurable function g on X. define the a-field of E-universally

mea-. -1

-and we shall call a mapping Ql: (X, E) -+- (y,8) universally measurable if Ql 8 c: L Obviously, a function f is universally measurable if and only if for every

].I e n(X) there exists a measurable function g with f

=

g [llJ. Note that g

may depend on 11. A simple but very useful property of the a-field of E-uni-versally measurable sets is the following one.

proposition 1. Let (X,E) and (y,8) be measurable spaces and Ql: X -+- Y. If

-1 -

2

-- i

Proof. Choose ~ € ~(X), and define PCB)

=

~(~ B) for every B €

a.

Since

-1

~ B €

EeL ,

this definition makes sense, and it is easily verified that ~

n

is a probability on a. Now for every A €

a

there exist sets Bl E a, B2 € a

-1 -1 -1 -1

with B

t

cAe B2 and P(B1)

=

P(B2), hence ~ Bl C ~ A c ~ B2, ~ Bl € E~,

-1 -1 -1 -1

~ B2 € Euand~(ql Bl)=~(CP B

2).ItfollowsthatQ) A€E~, and since ~ is

arbi-trary,

,-1

A € E.

0

Corollary. Let rabie, Then ~ 2

-1

(~2 0 Qll) Sl

=

Q)1: (X,E) ~ (y,9) and Ql2: (y,a) ~ (Z,Sl) be universally

measu-o CPl: (X,E) ~ (Z,Sl) is universally measurable, since -1 -1 1

-~1 Ql

2 Sl c CPl 9 c E by proposition 1.

proposition 2. Let (X1, El ) and (X2,L 2) be measurable spaces and (Xl x X2, El ® _L2) be the product space. Then El ® L2 c Ll ® _L2•

Proof. Obviously, the mapping (x

1,x2) ~ Xl is a measurable mapping (Xl x X2,

Ll

®

L

2

)

~ (Xl,L1)· Therefore by proposition 1 for every A € El we have A x X₂ € L1 ® E

2• Similarly for every B € E2 we obtain Xl x B € Ll ® L2, and therefore A x B E El ® L2 for every A € E

l, B € l:2· It follows that

r;

® L2

c Ll ® L₂•

0

Remark. The inclusion in this proposition can be strict. B.V. Rao [4J has shown that in the unit square there exists an analytic set which does not belong to the d-field

5

Q

5,

where

b

stands for the Borel sets of the unit

interval. Since analytic sets belong to

b

t3

b,

we have

5

t3

5

c

b

®

6

with

strict inclusion. Because of the isomorphy theorem, we now know that if (X

1,L1) and (X2,L2) are uncountable Borel spaces, we have strict inclusion in proposition 2.

For every set A c Xl x X

2 we define the x-section of A by Ax

=

{y

I

(x,y) € A}. It is well known that for every set in a product d-field the sections are mea-surable, but that the converse is not necessarily true.

Proposition 3. For every A € Ll ® L2 and every x € Xl we have Ax € L

2,

Proof, Choose x € Xl' and define the mapping~: (X

2,L2) ~ (Xl x X2,El ® L2)

by cp(y)

=

(x,y) , Obviously, this mapping is measurable. Therefore by proposi--1

tion 1 for every A € Ll ® L2 we have Ax

=

cP A € L

3

-Proposition 3 enables us to define for every x € Xl and V € ~(X2) a

probabi-lity on (Xl x X2'~l 0 ~2) by

!p (A)

=

V (A )

X1.I x

It follows from this definition that for every nonnegative universally mea-surable function f on Xl x X

2 and every V e ~(X2) we have

f

f(x,y) V (dy)

=

X 2

We shall investigate now how the first integral depends on x.

Let (X,~) be a measurable space and ~(X) the class of probabilities on

(X,~). As in [2J, we define the a-field ~* on ~(X) as the smallest a-field such that the mappings V ~ V (A) are measurable for every A €

E.

Note that a

mapping Q): (y,8) + (~(X) ,~*) is measurable if and only if for every A E E the function !p(.) (A) is measurable on (Y,8).

Theorem 1. Let (Xl,E

l) and (X2,E2) be measurable spaces. Let f be a nonnega-tive universally measurable function on Xl x X

2' and let x ~ Vx be a univer-sally measurable mapping from Xl to ~(X2)' Then the function

x

~

f

f(x,Y)Vx(dy) is universally measurable on Xl •

X 2

Proof. The existence of the integral f:or every x E Xl has already been noted

before. We shall show that the functi9n is a composition of three universally measurable mappings.

i) The mapping Q)l: (Xl/E_l) + (Xl x

~(~2)

,E

l 0

E;)

defined by Q)l (x)

=

(x,Vx) is measurable, since the composition]of Q) 1 with each of the projections in

I

Xl x ~ (X₂) is measurable. This means ithat Ql l' considered as mapping (Xl' E 1)

*

I

+ (Xl x ~(X2)'~1 0 E₂) is universall~ measurable.

ii) The mapping !P2: (Xl x

~(X2),El 0t~;)

+

(~(Xl

x X₂),(E_l 0. E

2)*) defined

by Q)2(x,~)

=

q>x~ (as above) is meaS1~able. In fact, the class of sets

A E El 0 E~ for which Ql2(·) (A) is

f

measurable function on Xl x ~(X2) is monotone, and contains the field ge/lerated by the rectangles Al x A2 (A_l E E

I, A2 E E₂) since

Qlx~

(AI x A₂)

=

lAI

(t

~

(A₂) is the product of two measurable

functions. It follows that Cf2 ( .) (A) lis measurable on Xl x ~ (X

2) for all

4

-iii) Let (X,L) be a measurable space and let f be a nonnegative universally measurable function on X. Then the mapping ~ ~

J

f d~ is universally measura-ble on 1T(X) •

In fact, i t suffices to show this statement for f

=

lA with AE

E.

Let

a

be

*

a probability on (1T(X),E ). Since for every BEL the mapping ~ ~ ~(B) is

*

E -measurable, the formula V(B} ::

J

~

(B) il

(d~)

1T (X)

defines a probability v on (X, E). Hence there exist sets Bl E E, B2 E E with Bl cAe B2 and V(B

1)

=

V(B2}. It follows that ~(Bl) ~ ~(A) ~ ~(B2) for all

~ E 1T{X) , and ~(Bl)

=

~(A) for ii-almost all ~; hence the mapping ~ ~ ~(A) is

*

Eil - measurable. Since il is arbitrary the statement follows. In particular we now have that the mapping Ql

3: (1T(X1 x X2):, (El ® E2) *) -+ (!R,B) defined by

is universally measurable.

To conclude the proof of the theorem, note that

f

f(x,y)

~x(dy)

:: (Ql3 0 "'2 I) Qll) (x)

X 2

and that by the corollary of proposition 1 the mapping Ql

3 I) "'2 I) Qll: (X

1,El)

-+ (!R,b) is universally measurable. 0

Corollary. A universally measurable mapping x -+ ~x from Xl to 1T(X

2) can be considered as a transition probability from (X1,El) to (X

2

,E

2). Indeed, for every x € Xl the probability ~x can uniquely be extended to a probability

E

2,

and for every A E L2 we have Xl x A E Ll ® L2 by proposition 2, and therefore

~x(A)

=

J

lXlxA(x'Y)~x(dY}

is a L

5

-Theorem 2 (Ionescu Tulcea). Let for n

=

0,1,2, ••• (X ,E ) be a measurable n n

space, and put (Q,F)

=

n

(X,E). Let for n

=

1,2, .•• n=O n n

pn be a universallY

*

measurable mapping from (X

O x ••• x X n-1,EO 0 .•• 0 E 1) n- to (~(X n ),E ). n Then for every finite sequence (xO, •.• ,x

t) there exists a unique probability P on (Q,F) such that for every cylinder set A E F depending on the

xo·· .xt

first (n + 1) coordinates only we have

f ... f

n l_A(x O'···'xn)P (xO,···,xn_1,dxn) ••• X n t+l P

(x

O

, ••• ,x

t 'dxt+

1)

Moreover the mapping xO, ••• ,x

t ~ P is universally measurable.

. _{xo·· .xt}

Proof. Let Fbe the sub a-field of F of the cylinder sets depending on the n

first (n + 1) coordinates. We may assume n > t. Then a repeated application of theorem 1 yields that P indeed isa probability on (Q,F ) such that

xo ••• x_t n

P (A) is universally measurable on Xo x ••• x X_t for every A E Fn"

xo·· .xt

The proof that P is a-additive on u Fn and therefore can be extended

xO···xt t+1

to a probability on F is identical to that part of the proof of the Ionescu Tulcea theorem as given in [3J, and therefore we omit it here. Finally, the class of sets of F for which P (A) is universally measurable on

xO···xt ~

Xo x ••• x X

t is monotone, and contains the algebra equalS F. u t+l F , and therefore n

o

In the classical form of the Ionescu Tulcea theorem the condition on the pn is measurability instead of universally measurability. Since in gene-ral there are more universally measurable sets than measurable sets, the condi tion in theorem 2 is weaker" However, if }: 1 0"." 0

f

n

=

E 1 0" •• 0E n for every n, then by the corollary of theorem 1, theorem 2 can be obtained from the classical Ionescu Tulcea theorem, applied to the spaces (X

,f )

instead

n n of (X IE ).

n n

If for instance for every n the space (X

,E )

is an uncountable Borel

n n

space, then i t follows from the remark after proposition 2 that theorem 2 is not a particular case of the Ionescu Tulcea theorem.

Finally we note that theorem 2 slightly simplifies and extends some of the results in [1J.

6

-References

[IJ Blackwell, D., D. Freedman and M. Orkin: The optimal reward operator

in

dynamic programming. The Annals of Probability! (1974),926-941. [2J Hinderer, R.: Foundations of Non-stationary Dynamic Programming with

Discrete Time Parameter. Lecture Notes in Operations Research and Mathematical Systems,

#-

33. Springer-Verlag. Berlin-Heidelberg-New York, 1970.

[3J

Neveu, J.: Mathematical foundations of the calculus of probability. Holden-Day, San Francisco 1965.

[4J Rao, B.V.: Remark on analytic sets. Fundamenta Mathematicae 66 (1970), 237-239.