Foundations of the theory of probability

§

4. ProbabilitiesinInfinite-dimensionalSpaces
   29

tionfunction ^^...^(fli,a

2 ,

...

,

a

w).

Itisobviousthatfor

everyBorelcylinderset

the followingequationholds:

p^=

p

w,...,.w,

where A' is aBorel setof

/?".

Inthismanner, theprobability

functionPisuniquelydeterminedonthefield
%

M

ofallcylindersets

bymeansofthevaluesofallfiniteprobabilityfunctions P^^...
^

for allBorelsets ofthecorrespondingspaces R

n

.However, for

Borelsets,thevaluesoftheprobabilityfunctions P^,...^ are

uniquelydeterminedbymeansofthecorrespondingdistribution

functions.Wehavethusprovedthefollowingtheorem

:

T.he set of all finite-dimensional

distribution

functions

F/hih

—

i

1

*

uniquelydeterminestheprobabilityfunctionP(A) for

allsetsin
$

M

.

If

P(A) isdefinedon

%

M

,

then (accordingtothe

extension theorem) it is uniquely determined onB%

M

by the

values
of

thedistribution

f

unctionsF^^...^.

Wemaynowaskthefollowing.Underwhatconditionsdoesa

systemofdistribution functions
F^^,,.^

givenapriori define

afield ofprobabilityon

%

M

(and, consequently, onB%

M

)

?

We

mustfirst

note

thateverydistributionfunction

F^/h.../**

mustsatisfytheconditions givenin
§

3,

III

ofthesecondchap-

ter;
indeedthis iscontainedinthe veryconcept

of

distribution

function.Besides,asaresultofformulas (13) and (14) in
§2,

wehavealsothefollowingrelations

:

F

fHifHt

...

Hn

{a

il

,

a

it

,

..

.,

a

in

)

=F

/<l/<2

.../ttt

K,

a

2

,

..

.,

a

n)

,

(2)

*V*...**(«i.

a

2

>

->

a

k)

=^W,...^K,

«

2

.

...,**,+<»,...,

+oo),(3)

where
k<n and
[/ /

"'

n

)

is an

arbitrary permutation.

\*1>*2» •

• •»
  W

These necessary conditions prove alsoto be sufficient, as

will

appear fromthe following theorem.

FundamentalTheorem:Everysystem

of

distributionfunc-

tions

F

fll

HM

...pH

,

satisfyingtheconditions

(2)

and

(3),

definesa

probabilityfunctionP(A) on
%

M

,

whichsatisfiesAxiomsI

• VI.

ThisprobabilityfunctionP(A) canbe extended (bytheexten-

siontheorem) toB%

M

also.