Foundations of the theory of probability

50 V. ConditionalProbabilitiesandMathematicalExpectations

allowustocarryovermanyotherbasicpropertiesofthe

absolute

probabilityP(B)totheconditionalprobabilityP

U

(B).However,

wemustnotforgetthatP
U

(B) is,for

a

fixed

set

B,

arandomvari-

abledetermineduniquely onlytowithinequivalence.

ProofofTheorem

I.Ifwe

assume—contrarytotheassertion

tobeproved—thatonasetM

s

a

E

(M

> with

P

(M

>

(M) >0,thein-

equalityP

U

(B)

g

1

+e,

e>0, holdstrue,thenaccordingtofor-

mula
(1)

P{uc:M}{B)

=E

{ucM}

P

u

(B)^i

+

e,

whichis obviously impossible.In thesame wayweprove that

almostsurelyP

U

(B)

^

0.

Proofof

TheoremII. Fromtheconvergence oftheseries

ZE\P

u

(B

n)\

=2E(P

u

(fi

fl

))

=2P(£

n

)

=

P(B)

n n n

itfollowsfromPropertyVofmathematicalexpectation (Chap.

IV,

§

2) thattheseries

2P.(BJ

n

almostsurelyconverges.Sincetheseries

ZE

{

uoA}\Pu(B

n

)\=Z

E

{u<:A}(Pu(Bn))

=

£

P{

UC

A}(B

n)

=

P{uCA}(B)

n n n

converges foreverychoice ofthe setA suchthat P

(u>

*(A)

>0,

thenfromPropertyVofmathematicalexpectationjustreferred

toitfollowsthatforeachAoftheabovekindwehavetherelation

E

{

uc^}(|;P„(£

n

))

=|E(,

ei)(W)

=

P

{uca}(B)

=E

{ucA}

(P

u

(B

n))

f

andfromthis,equation

(5)

immediatelyfollows.

Toclosethissectionweshallpointouttwoparticularcases.

If, first,

u(i)

=

c (a

constant), then P

C

(A)

=

P(A) almost

surely. If, however, we

set

u(i)

=

£,

thenwe obtain at once

thatP$\A) isalmostsurelyequaltooneon

Aandisalmostsurely

equaltozeroonA. P${A)isthusrevealedtobe

thecharacteristic

functionofsetA.

§

2. ExplanationofaBorelParadox

LetuschooseforourbasicsetE the setofall pointsona

sphericalsurface. Our

5

wil1betheaggregateofallBorel

sets

ofthe

sphericalsurface.Andfinally, ourP(A) istobepropor-

tional

to

themeasureofsetA.Letusnowchoosetwodiametrically