Foundations of the theory of probability

§

1.ConditionalProbabilities 49

0(A)

=

//(f)

P(dE).

A

Inorderto

applythistheoremtoourcase,weneedtoprove

1°

that

Q(A) =

P(Bu-HA))

isacompletely

additivefunctionon

Jp>,

2°,

thatfromQ(A)+0

followsthe

inequalityP

(M

>(A)

>

0.

Firstly,

2°

follows

from

^P{Bu-HA))^

P(u-HA))

=

P<

m

HA)

.

Fortheproofof

1°

weset

A=

Z

A

n-

then

u-

l

(A)=%u-HA

n

)

n

and

B«->(^)=2B«-

l

(4).

n

SincePiscompletelyadditive,itfollowsthat

P{BurKA$=2P{Bu-HAj)

%

n

whichwastobeproved.

Fromtheequation (1) followsanimportant

formula (ifwe

setA

=#<«>)

:

P(B)=

E(P

U

(B)).

(3)

Nowweshallprovethefollowingtwofundamentalproperties

ofconditionalprobability.

Theorem I. Itisalmostsure that

0^P

u

(B) gl.

(4)

Theorem II.

//

B is decomposed into at most a countable

number

of

setsB

n

:

B=

ZB'nt 9

n

thenthefollowing equality holdsalmostsurely:

,

P«(£)=ZP»(£»)-

(5)

n

These

twopropertiesof P

U

(B) correspondtothetwo char-

acteristic properties

of the probability function P(B): that

g

P(B)

^

1 always,andthatP(B)iscompletelyadditive.These