Foundations of the theory of probability

§

4. ProbabilitiesinInfinite-dimensionalSpaces 27

p<*.,*a

,...,**>(,4)=pttk.*.....-^^-!^)}.

(ii)

For

thecorresponding

distributionfunctions,weobtainfrom

(10) and(11) the

equations

:

/#*.•*«.•

—"Ufo,

a

ia

,

..

.,

a

in

)

=

F<*»**<••->^(a

1

,a

2

a

n)

,

(12)

pin,**....,**)

(a

lf

a

2t

...,a

k

)

=F

x

»•«••••*«>

(a

x

,...,a

ft

,+oo,...,+oo).(13)

§

4. ProbabilitiesinInfinite-dimensional

Spaces

In

§

3 ofthesecondchapterwehaveseenhowtoconstruct

variousfieldsofprobabilitycommoninthetheoryofprobability.

Wecan imagine, however, interesting problems in whichthe

elementaryevents are definedbymeans ofan infinite number

ofcoordinates.Let

ustakeasetMofindices/*

(indexing

set) of

arbitrary

cardinality

m

. Thetotalityofallsystems

ofrealnumbersx

M

,

where/xrunsthroughtheentiresetM,

we

shallcallthespaceR

M

(inordertodefineanelement

£

inspace

R

M

,

wemustputeachelement/xinsetMincorrespondencewith

a realnumber
%

or, equivalently, assign a real single-valued

function
x^

oftheelement

/*,definedonM)

3

.

Ifthe

setM

consists

of
the

first

nnaturalnumbers1,2,..

.

,n,thenR

M

isthe

ordinary

7i-dimensionalspaceR

n

.IfwechooseforthesetMallrealnum-

bersR

1

,

thenthecorrespondingspace R

M

=

R

R1

will consist of

allrealfunctions

((/*)

=x

tt

oftherealvariable
/*.

WenowtakethesetR

M

(withan arbitrary setM) as

the

basicsetE.Let I

={x^}

beanelementinE;weshalldenoteby

ft*

a...

>»:(£)

^ne Point {x

/tl

,x

iH9

..-.

t

x

fh

)' of the n-dimensional

spaceR

n

.AsubsetAofEweshall callacylinderset ifitcan

be representedintheform

whereA'isasubsetof#

w

.Theclassofallcylindersetscoincides,

therefore,withtheclassofallsetswhichcanbedefinedbyrela-

tionsoftheform

3

Cf.Hausdorff,Mengenlehre,

1927,p.

23.