Foundations of the theory of probability

48 V. ConditionalProbabilitiesand

MathematicalExpectations

PM(A)

>

0.The

function

P

U

(B) ofuthusdeterminedtowithin

equivalence,wecallthe

conditionalprobability

of

Bwithrespect

tou(or,for

a

given

u).ThevalueofP

M

(Z?) whenu

=

awe

shall

designate

by

P

u

(a;B).

The

proof

of

the existenceanduniqueness

of P

U

(B). Ifwe

multiply

(1)

by

P{ucA} =

P<«>(A), we

obtain, on the left,

P{uczA}P

ucA

{B)=P(B{ucA})=

P\Bu-HAj)

and,ontheright,

P{ucA}E

{ucA}

P

u

(B)=

JP

U

(B)P(dE)

=JP

U

(B)P<*>(rf£(«))

;

{ucA}

A

leadingtotheformula

P(B«-

1

M))=/P

u

(B)PW(i£W).

(2)

A

andconversely

(1)

followsfrom

(2). Inthecase P

(u

HA)

=

0,

inwhich

case (1)ismeaningless,equation (2) becomestrivially

true.Condition

(2)

isthusequivalentto

(1).

Inaccordancewith

PropertyIXoftheintegral

(§1,

Chap.IV) therandomvariable

xis uniquelydefined (except forequivalence) bymeansofthe

valuesoftheintegral

fxPd(E)

A

forall setsof g. SinceP

U

(B) isa randomvariabledetermined

ontheprobabilityfield

(8f<*>,

P

(M

>),itfollowsthatformula

(2)

uniquelydeterminesthisvariable P

U

(B) exceptforequivalence.

Wemust

still

provetheexistenceof P

M

(J5). Weshall

apply

herethefollowingtheoremof Nikodym

1

:

Let

5

beaBorelfield,P(A)anon-negativecompletelyadditive

setfunctiondefinedon
5

(intheterminologyoftheprobability

theory,arandomvariableon
(5,

P)),andletQ(A) beanother

completely additive set function defined on
J$f>

such thatfrom

Q(A)4=0

follows

the

inequality P(A)

>

0. Thenthereexists
   a

function
/(£)

(intheterminology ofthe

theory

of

probability,

arandomvariable) whichismeasurablewith respectto
%,

and

whichsatisfies,foreachsetAof
5,

theequation

1

0.Nikodym,SurunegeneralisationdesintegratesdeM.J.Radon,Fund.

Math.v.
15,

1930

p.

168 (TheoremIII).