Foundations of the theory of probability

22 III.RandomVariables

LetA'andB'betwosetsof

$

(M

>.Theirpre-imagesAandB

belongthento
J.

Since

%

isafield,thesetsAB,A

+

B,

andA

• B

alsobelong
to
g

;butthesesetsarethepre-imagesofthesetsA'B\

A'

• 
  

B\and

A'

-B',

whichthusbelongto

^

u

\Thisprovesthat

5

(u)

isa field.Inthesamemanneritcanbeshownthatif

g

isa

Borelfield,sois
%

(u

\

Furthermore,itisclearthat

PM(E') = P^-

1

^)}

= P(#) =1.

ThatP

U)

isalwaysnon-negative,isself-evident. Itremainsonly

tobeshown,therefore,

that

P

(m)

is

completely

additive (cf.

the

endof
§

1, Chap. II).

LetusassumethatthesetsA'

n,

andthereforetheirpre-images

u-

1

(A\)

}

a,Yedisjoint.Itfollowsthat

n n

n

n n

whichprovesthecomplete additivityof P

u)

.

Inconclusionletus alsonotethefollowing. Letu

x(g)

be a

functionmappingEonE',and
u
2 (t)

beanotherfunction,map-

ping

£"

onE".Theproductfunctionu

2

uA£) mapsEonE".

We

shallnowstudytheprobabilityfunctionsP

(Ml)

(A') andP

(u

HA")

forthefunctions u
rU)

andu(()

=

UzUiU).

It iseasy toshow

thatthesetwoprobabilityfunctionsareconnectedbythefollow-

ingrelation:

?^(A

,f

)^?^){u^(A

ff

)}.

(2)

§

2. DefinitionofRandomVariablesandof

DistributionFunctions

Definition.Arealsingle-valuedfunction

*(£),

definedonthe

basicsetE,iscalledarandomvariable ifforeachchoiceofareal

numberatheset{x

<

a}ofall

|

forwhichtheinequalityx<a

holdstrue,belongstothesystemofsets
$•

Thisfunctionx(£) mapsthebasicsetEintothesetR

1

ofall

realnumbers.Thisfunctiondetermines,

as

in

§1,a

field

%

(x)

of

subsetsofthe

setR

1

.We

mayformulateourdefinitionofrandom

variable

inthismanner:Arealfunctionx

(£)

isa

randomvariable

ifandonlyif

g

U)

containseveryintervalofthe

form (-ooj

a)

.