Foundations of the theory of probability

38 IV.

MathematicalExpectations

morethanacountable

numberofnon-intersecting

sets

A

n

ofgf,

then

r

_

,

JxPXdE)=£jxP(dE).

A

nAn

III. Ifxisintegrable

r

|

a;

|

isalsointegrable,andinthatcase

\jxP(dE)\^j\x\P{dE),

A

A

IV. Ifineachevent

|,

theinequalities

^y

s^xhold,then

alongwith

x,y

isalsointegrable

3

,

andinthatcase

JyP(dE)

^fxP{dE)

A A

V. Ifm

^

as

g

MwheremandMaretwoconstants,then

m

P(A)^jxP(dE)

^MP{A).

VI. If£and

y

areintegrable,andKandLaretworealcon-

stants,thenKx

+

Lyis alsointegrable,

and

inthis

case

j(Kx+Ly)P(dE)

=

KJxP{dE)

+

LJyP(dE)

.

VII. Iftheseries

]?j\x

n

\P(dE)

nA

converges,then theseries

JmmiXfi

X

n

convergesateachpointofsetAwiththeexceptionofacertain

setBforwhichP(B)

—

0.Ifwesetx

=

everywhereexcepton

A

• B

t

then

jxP{dE)=^jx

n

P(dE).

n

A

VIII. If

x

and

y

are equivalent (P{*

4=

y)

~

0)»

then ^or

everysetAof

5

jxP(dE)=jyP(dE). (3)

3

Itisassumedthat

y

isarandomvariable,i.e.,

intheterminologyofthe

generaltheoryofintegration,measurablewith
respectto
%

.