Foundations of the theory of probability

§

4. ProbabilitiesinInfinite-dimensionalSpaces 31

Letus

nowprovethatthefieldofprobability

(JP,

P) satisfies

allthe

AxiomsI

• VI.AxiomIrequiresmerelythat

g

M

beafield.

Thisfacthasalready

beenprovenabove.Moreover,foranarbi-

trary

/x:

P(E)

=

P

fl

(R*)

=

i,

whichprovesthatAxioms

II

and

IV

apply

inthiscase.Finally,

fromthedefinitionofP(A) it

follows

atonce

thatP(A) isnon-

negative (Axiom III).

Itis onlyslightlymorecomplicated toprovethatAxiomV

isalsosatisfied.Inordertodoso,we investigatetwo cylindersets

and
B

-«iV-*.<*>.

Weshallassumethatallvariablesx

h

.andx

N

belongtooneinclu-

sivefinite system (x^,x^,.

..,

x„

n

)

.IfthesetsA andB donot

intersect,therelations

[*/%'*/%'

-'"x

/Hk

)

(=:A

areincompatible.Therefore

?{A

+

B)=P**.;.*^,x

Hi

,

..

.,

*„.J

c:A'

or

(VS'-'SJ^J

=

P^,

fi2.••ftn

{

(^i

1

»

^i,

»*'*'

**fe)

C

^

}

+

P^^...^{(^.

,*„

v

• 
  
  
  
  • 
    
  
  
  
  

.,

*„,J

cB'}=P(^)

+

P(B)

,

whichconcludesourproof.

OnlyAxiomVIremains.Let

A

1

=>A

2

3

•••

idi4

w

z>

•••

beadecreasingsequenceofcylinder

setssatisfyingthecondition

limP(A

n)

=L>0.

• 
  

WeshallprovethattheproductofallsetsA

n

isnotempty. We

may

assume,withoutessentiallyrestrictingtheproblem,thatin

the

definitionofthefirstncylindersetsA

k,

onlythefirstnco-

ordinatesXp

k

inthesequence