Foundations of the theory of probability

20 II. InfiniteProbabilityFields

The investigation offields of

probability oftheabovetype

issufficientforallclassicalproblems
in

the

theoryof

probability

6

.

Inparticular,

aprobabilityfunctioninR

n

canbe definedthus:

Wetakeanynon-negativepointfunction

f(xu

x

2 ,

. ..
,

x

n

)

definedinR

n

,

suchthat

+00 +00 +90

j

j

...j

f(x

lt

x

2

,...,x

n

)dx

1

dx

2

...dx

n

=\

—00 —00

andset

P

(

A

)

=

//•••

ff(

x

i>

x

2>

••

.,x

n

)dx

1

dx

2

...dx

n

.

(5)

A

f(x
u

x

2 ,

...
,

x

n)

is,inthiscase,theprobabilitydensityatthe

point (x

u

x

2 ,

...

,

x

n

)

(cf.Chap.Ill,

§2).

AnothertypeofprobabilityfunctioninR

n

isobtainedinthe

followingmanner: Let
{£.}

be

a

sequence

of

points

ofR

n

,

and

let
{pi}

be asequence ofnon-negative realnumbers, suchthat

£pi

=

1 ;wethenset,aswedidinExampleI,

P(A)

=Z'Vi,

wherethesummation

2'

extendsoverall indicesiforwhich

£

belongstoA.Thetwo
typesof

probabilityfunctionsinR

n

men-

tionedheredonotexhaustallpossibilities,butare

usuallycon-

sideredsufficient for applications of the theoryof probability.

Nevertheless, wecan imagineproblems ofinterest forapplica-

tionsoutsideofthisclassicalregioninwhichelementaryevents

aredefinedbymeansofan infinitenumberofcoordinates. The

correspondingfieldsof probability weshallstudymore closely

afterintroducingseveralconceptsneededforthispurpose. (Cf.

Chap.Ill,

§3).

6

Cf.,forexample,R.v.Mises

[1],pp.

13-19.

Heretheexistenceof

proba-

bilities for "all practically possible"sets ofan n-dimensional space

is

required.