Foundations of the theory of probability

§

2. AbsoluteandConditionalMathematicalExpectations 39

IX. If

(3)

holds for every set A of

gf,

then x and

y

are

equivalent.

Fromthe foregoingdefinitionof anintegralwealsoobtain

thefollowingproperty,whichisnotfoundintheusualLebesgue

theory.

X. Let

Pi

(A)andP

2

(A) betwoprobabilityfunctionsdenned

onthesamefield
%,

P

(

A)

=

P

x

(

A

)

+ P

2

(

A

\

andletxbeintegrable

onA

relative

toP

1

(A)

andP

2

(A).Then

jxP(dE)=^jxP

x

(dE)

+

jxP

2

{dE).

AAA

XL Everyboundedrandomvariableisintegrable.

§

2. AbsoluteandConditional

MathematicalExpectations

Leta;bearandomvariable.Theintegral

E(x)

=

JxP(dE)

E

iscalledinthetheoryofprobabilitythemathematicalexpectation

ofthevariablex.FromthepropertiesIII,IV,V,VI,VII,VIII,

XI,itfollowsthat

I. |.E(*)|£E(|*|);

II.

E(y) gE(x) if ^

y

^xeverywhere;

III. inf

(x)^E(x)^sup (x)

;

IV. E(Kx+Ly)

=

KE(x)

4-

LE(y)

;

V. E

(2

x

n)

=

2

E

(*n)

»

iftheseries

2

E

(

I

*»l

)

converges

;

\n I

n n

VI. Ifxand

y

areequivalentthen

E(z) =E(2/).

VII. Every bounded random variable has

a mathematical

expectation.

Fromthe

definitionofthe integral, wehave

k=+oo

E(x)==lim^£raP{&m:^

#

<

(jfe.-f

1)

w}

&=—OO

=lim^rm{F((^+

l)m)

• F(£m)}.