Foundations of the theory of probability

§

4. ConditionalMathematicalExpectations 55

isthenecessary

conditionfortheexistenceofE(y) (seeChap.IV,

§

1). Fromthis

convergenceitfollowsthattheseries

(10)

con-

vergesalmostcertainly (seeChap.

IV,

§2,V).

Wecanfurther

show,exactlyasinthetheoryofthe

Lebesgueintegral,thatfrom

theconvergenceof (10) forsomeA,itsconvergenceforeveryA

follows,andthatinthecasewhereseries(10)converges,
S

x

M

tendstoadefinitelimitasA

—

3

. Wecanthendefine

E

u(y)

=limS

;».

(U)

ToprovethattheconditionalexpectationE
u(v)

definedbyrela-

tion
(11)

satisfiestherequirementssetforthabove,weneedonly

convince ourselvesthat E
M(y),

asdeterminedby

(11),

satisfies

equation
(1).

Weprovethisfactthus:

E{ueA}Eu(y)

=hmE

{Mc^

}

S;.(w)

=lim

2

kXp

{u<=A}{k*

^y<(k+l)X}=E

{ucA}(y)

.

'/.

->

k

——oo

Theinterchangeofthemathematicalexpectationsignwiththe

limitsign is admissibleinthis computation,since S

x

(u) con-

vergesuniformlytoE

M

(y)

asA

—

(asimpleresultofProperty

V

of mathematical expectation in
§2).

The interchange

of the

mathematical expectationsign
and

the

summation sign is also

admissiblesincetheseries

=

^{u,A}{\kX\

?

u

[kl

^y

<

(k

+1)A]}

k=

—oo

=

ZW

?{uC

A}[kl

^y<(k

+

\)X\

converges (animmediateresultofPropertyVofmathematical

expectation)

.

Insteadof (11) wemaywrite

E.(y)=/yP.

(<*£). (12)

E

Wemust

notforgethere,however,that

(12)

isnotanintegral

3

In

thiscaseweconsideronlyacountablesequenceofvaluesofA;then

all

probabilities

P

u

{kl<Zy<(k

+

i)X\ arealmostcertainlydefinedforall

thesevaluesofA.