Foundations of the theory of probability

§

4. ConditionalMathematicalExpectations
   53

E{u*A}(y)

=E

{uc^

}

E

tt

(y)

f

(1)

iscalled(if

it

exists)theconditionalmathematical

expectationof

thevariable

y

for

knownvalue

ofu.

If

wemultiply (1) byP

{u)

{A),weobtain

jy?(dE)=JE

u

(y)PM(dEW).

(2)

{ucA} A

Converselyfrom

(2)

followsformula (1).Incase P

(M)

(A)

—

0,

in whichcase

(1)

is meaningless,

(2)

becomes trivial. Inthe

samemannerasinthecaseofconditionalprobability

(§1)

we

canprovethatE„(y) isdetermineduniquely—exceptforequiva-

lence—

by

(2).

Thevalue

ofE

u(y)

for

w

=

awe

shalldenote

byE

u

(a;

y)

.Let

usalsonotethatE

u(y),

aswellas P

u(y),

dependsonlyuponthe

partition9l

M

andmaybedesignatedbyE

9ltt(y)

.

TheexistenceofE(y) isimpliedinthedefinitionofE

u

(y)

(if

wesetA

=#<»>,

thenE

{ucA}

(y)

=

E(y)).

Weshallnowprovethattheexistence

of

E

(y)

isalso

sufficient

for

theexistence

of

E

u(y)

.Forthisweonlyneedtoprovethatby

thetheoremofNikodym
(§1),

the

set

function

Q(A)=fyP(dE)

{ucA}

is completely additive on

5

(m)

and absolutely

continuous with

respectto P

(m)

(A). Thefirstpropertyis

provedverbatim asin

thecaseofconditionalprobability
(§1).

Thesecondproperty

—

absolutecontinuity—iscontainedinthefactthatfromQ(A)^0

the inequality P

U)

(A) >0 must follow. If we assume that

P

(M

>(A)

=

P{udA}

=

0,itisclearthat

Q(A)=fyP(dE)

=

f

{ucA}

andoursecond requirementisthus fulfilled.

Ifinequation

(1)

wesetA

—

E

(u

\

weobtaintheformula

E(y) =EE

U(V)

• (3)

Wecan

show

furtherthat

almostsurely

E

u

(ay+bz)

=aE

u(y)

+

bE

u

(z)

, (4)

whereaandbaretwoarbitraryconstants.

(Theproofisleftto

thereader.)