.... for the space of random variables A proof of the representation theorem

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1967)

ANN ALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

E. Sik o rsk i and B. Zno jk iew icz (Warszawa)

A proof of the representation theorem for the space of random variables

If / is a real function on the w-dimensional Euclidean space P n, then Ą f (h ? • • • ? hi) ~ f (h ? - * * ? h‘— i jh' } h'+i j • • • ? hi)

f (h j • • • ? h-11Ц1 h'+iч • • • j hi) and

bi — a

( - / J° n ~ l ~ an —l

n — 1 (Abn-anf(a i , . . . , a n)) . .. ).

By a space of random variables, shortly an EY-space, we mean a pair (Ж, P) where Ж is a set of elements called random variables, and P is a mapping that assigns, to every finite sequence of different elements CO j j • • • у OOtft в Ж, a real function P Xl,...,xn defined on B n, in such a way that

(i) if i x, . . . , i n is a permutation of numbers 1, then

^xjll ,...,Xj (hi > • • • ln 1 ? hw) Рх1,...,х1)1(1ц •'••jin), (ii) the function PXl,...,xn is continuous from the left;

(iii) if a?- < b}- for i = 1, ., (iv) lim P x (t) = 1;

(—>oo

.. , n , then P^,.. ^x > 0

(v) lim PXl.... %,*n+1(* i,...

t—>oo , tn, t) = Pxj,...,xn(h? • • • ?hi) J (vi) lim PXl>...,xn(h,

tj-*~ oc II о H-h О i-i II ,,n .

If (Q, ju) is a normed measure space (i.e., such that p(Q) = 1 ) let Фц be the set of all //-measurable a.e. finite real functions on Q. If F is any mapping from a set Ж into Ф^ and

( ! ) Pxly...,xn{h, • • • ? *n) = /u{coeQ: FcDi(co) < t i for i = 1 ,

then (Ж, P) is an EY-space. As it is well known, the converse theorem due to A. N. Kolmogorov is also true:

(*) I f (31, P) is an EY-space, then there exists a normed measure space (О,/и) and a mapping F from M into Ф^ such that (1) holds.

1 7 6 E . S i k o r s k i a n d B . Z n o j k i e w i c z

The purpose of this note is to give a new proof of this theorem based on an idea used earlier by the first of the authors to another problem of measure theory.

Let 8 be the set obtained from the set E' of all finite or infinite real numbers by splitting every finite point into two points. More precisely, S is the set composed of all the pairs

(r, —1), {r, + 1 ), where — oo < r < oo and of the pairs

(+ ° ° ) —l h ( —°°? + 1).

8 is an ordered set with respect to the relation < defined as follows

< (r 2, V*)

if and only if either rx < r2 or rx = r2 and rjx < r\2. S is conceived as a topological space with the topology determined by the ordering. It is easy to see that $ is a compact space.

For every interval I = (a,

by

b

{a, + 1 ), (b, —1) and (r, —1), (r, +1) for a < r < b . A set A c 8 is open-closed if and only if A = w ... w I* for some disjoint intervals I x, , I n.

Let Q = 8 M, i.e. Q is the set of all mappings со from M into 8.

Otherwise speaking, Q is the Cartesian product Q = p S x ,

XeM

where 8X = 8 for every x <? M. By the Tychonoff theorem, Q is a compact space.

By an Q-interval we mean any set J c= Q of the form

(2) J = p K x

X<lM

such that there exists a finite sequence of different elements x x, ..., xn e M and intervals

such that

I x • , bxy , K x = 8 for K x. = I t for For every jQ-interval (2), let v(J) - A

j In —■ (.^n j &пУ

/V» /у» /V»

iV -f— tV} у • • • J wtyi j

i = 1, ..., n.

bn P , ^{Д/ j j}.,, у ^* (3)

Proof of the representation theorem 1 7 7

In the case where some щ are equal to — oo and some bj are eqnal to + oo, the right side of (3) denotes the corresponding limit when those щ and bj tend to — oo or +oo respectively. It follows from (i), (v) and (vi) that

1 ,+oo,£>£+i>..4&№ p

al> 1>— °°>ai + l’ •••*an x\<> ■ i>xi>xi + i> ■ ■•>xn

— /1

— si + 1 , . . . , x n -

This implies together with (i) that the value of v{J) does not depend on the choice of x x, . . . , x n in the representation (2). By (iii), v is a non­

negative function. By (iv) and (vi),

(4) v ( Q ) = 1.

If a,i = — oo and bi < оо for i = 1, . . . , n, we shall denote the in­

terval (2) by

.. . , bn) . By definition (3) and (vi),

(5) v {Jx1,...,xn( ^ U ’' ^ b n)^ — Pxlf ...,xn{b\ 1 . . . J bn) .

Let 9ft be the field of all open-closed subsets of Q. By definition, А e9ft if and only if A is the union

(6) A — J i ... \^> J n

of disjoint £>-intervals J 1, . . . , J n. We extend the set function over 9ft by defining

(7) v( A) — v (J x) + . . . Av(Jn)

for every set (6). It follows from additive properties of A that number (7) does not depend on the representation of 1 as a disjoint union (6). By definition, v is an additive function on 9ft.

Since Q is compact, the non-negative set-function v is a-additive and, consequently, it can be extended to a measure p, on the smallest cr-field 9fta containing 9ft. It follows from (4) that jn{Q) = 1 .

Let ж be the mapping from S onto R' defined by 7l({r, fj)) = Г.

For any x e M let x = F x be the function x(co) — л{а>(ж)) for co e f i .

The measure space (D, ц ) and the mapping F (from I f into just defined have the properties required.

!2 — Prace Matematyczne XI (1967)

178 R . S i k o r s k i a n d B . Z n o j k i e w i c z

In fact, let bi>k/ t i for к -* oo and i = 1, n, and let Jk = JXi,...,xniP\,ki • • • f bn,k) •

It is easy to check that

{со: %i(co) < ti for i = 1, n}

OO

= {со: 7l(co(%)) < ti for i = 1, . . . , n} — U Jk- k= 1 Since J 1 c J 2 c= ...,

/г ({со: ^г(со) < ^ for i = 1, n}\ = lim fi{Jk)

oo

= Пш. Px1,...,xn{^l,kj-” j^n,k) ~ Pxl,...,xn(^l i i tn) k-^-oo

by (5) and (ii). This proves (1).