Foundations of the theory of probability

ChapterIII

RANDOM VARIABLES

§

1. Probability

Functions

GivenamappingofthesetEintoasetE'consistingofany-

typeofelements,i.e.,asingle-valuedfunctionu(£) definedonE,

whose valuesbelongtoE'.Toeach subsetA'ofE'weshallput

intocorrespondence,asitspre-imageinE,thesetu-

x

(A') ofall

elementsof Ewhich maponto elementsofA'. Let
%

(u)

bethe

system ofallsubsetsA' ofE', whose pre-imagesbelongtothe

fieldg.
%

(u)

willthenalsobeafield.If

5

happenstobea

Borel

field,
thesame

will

betrueof

5

(m)

• We

now

set

poo(A') =

P

K

1

^')}. (1)

Sincethisset-functionP

(m)

,

definedon

5

(M

\

satisfieswithrespect

tothefield
5

(m)

allofourAxioms I

• VI, itrepresentsaproba-

bilityfunctionon
%

(u)

.Beforeturningtotheproofofallthefacts

juststated,

we

shall

formulate

the

followingdefinition.

Definition.

Givenasingle-valuedfunctionu(£)ofa

random

event£.ThefunctionP

(M

>(A'),definedby

(1),

isthencalledthe

probabilityfunctionofu.

Remark 1 : Instudyingfieldsofprobability

(5,

P),wecallthe

function P(A) simplytheprobabilityfunction, but P^(A') is

calledtheprobabilityfunctionofu.Inthecaseu($)

=

£,

P

(m)

(A')

coincideswith P(A).

Remark2: Theevent

vr

x

(A') consistsofthefactthatu(£)

belongs
to

A'.

Therefore,P

(m)

(A') isthe

probability

ofu(£)c

A'.

Westillhavetoprovetheabove-mentionedpropertiesof

%

(u)

andP

(M

>.

Theyfollow,however,fromasinglefact,namely:

Lemma. Thesum, product,anddifference

of

anypre-image

setsw

-1

(A')arethepre-images

of

thecorrespondingsums,prod-

ucts,

anddifferences

of

theoriginalsetsA'.

The

proofofthislemmaisleftforthereader.

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