Foundations of the theory of probability

§

3.ExamplesofInfiniteFieldsofProbability 19

P(A) on

3

which satisfies

thecondition P(E)

=

1. As is well

known

3

,suchafunctionis

uniquelydeterminedbyits values

P[-oo;x)

=F(x) (1)

forthespecialintervals

[-<*>;

x).The

functionF(x) iscalledthe

distributionfunction

of

£. Furtheron

(Chap.Ill,

§2)

weshall

shownthatF(x) is non-decreasing,continuous

ontheleft,and

hasthefollowinglimitingvalues

:

limF(x)

=i^-oc)=

6,

limF(x)=F(

+

oo)=

1

.

(2)

*—

—

oo a;

->-»-oo

Conversely,if agiven functionF(x) satisfiesthese

conditions,

thenitalwaysdeterminesanon-negativecompletelyadditiveset-

functionP(A) forwhichP(E)

=

l

4

.

IV. LetusnowconsiderthebasicsetE asann-dimensional

Euclidianspace R

n

,

i.e.,thesetofallorderedn-tuples

£

=

{x

u

x

2

,

...,x

nj

ofrealnumbers.Let

$

consist,inthiscase,ofall

Borel

point-sets

5

ofthe spaceR

n

. Onthebasisofreasoninganalogous

tothatusedinExampleII,weneednotinvestigatenarrowersys-

temsofsets,forexamplethesystemsofn-dimensionalintervals.

Theroleofprobabilityfunction P(A) will be playedhere,

as always, by any non-negative and completely

additive set-

functiondefinedon

$

and

satisfyingtheconditionP(E)

=1.

Such

aset-functionisdetermineduniquelyifweassignitsvalues

P{L

aiai...an)

=F{a

lt

a

2

,...,a

n)

(3)

forthe special setsL

aia%

„,

an

,

where L

aia,...an

represents the

aggregateofall

£

forwhich

Xi<Oi

(i

=

1,2,

...

,

n).

ForourfunctionF(a

lf

a

2

,

..

.

,a

n)

wemaychooseanyfunction

whichforeachvariableisnon-decreasing

andcontinuousonthe

left,andwhichsatisfies

the

followingconditions

:

lim F(a

v

a

2>

...,«„)

=

F(a

v

.

.

.,«i_i,

—oo,a

i+1

,...,#„)

=0,

"—~

t=

4,2i

....,»

f

lim F(a

v

a

2

,..

.,a

n

)

=F(+oo,+00,...,-foo)

=

1.

Oi

->

+00,Oj

->

+00,...,o»

—

-t-00

F(a

u

a

2 ,

..

.

,a

n)

iscalledthedistributionfunctionofthevari-

ables

a?i,x

2 ,

..

.

,x

n

.

3

Cf

.,

forexample,Lebesgue, LegonssurVintegration, 1928,

p.

152-156.

*

Seethepreviousnote.

8

ForadefinitionofBorelsetsinR

n

seeHausdorff,

Mengenlehre,1927,

pp.177-181.