Foundations of the theory of probability

ChapterIV

MATHEMATICAL

EXPECTATIONS

1

§

1. Abstract
   LebesgueIntegrals

Let#bearandomvariableandAasetofgf.Letusform,fora

positiveA,thesum

k=+00

S;.

^^H?{kk^f<

{k+i)X

t

(cA}. (1)

*=

-00

IfthisseriesconvergesabsolutelyforeveryA,thenasA

—

0,S

k

tendstowardadefinitelimit,whichisbydefinitiontheintegral

I-

xP(dE)

.

(2)

A

Inthisabstractformtheconceptofanintegralwasintroduced

by Frechet

2

;

it is indispensable for the theoryof probability.

(Thereaderwillseeinthefollowingparagraphsthattheusual

definition for the conditional mathematical expectation of the

variable x under hypothesisA coincides withthe definition of

the

integral

(2) exceptfor aconstantfactor.)

We shall give here a brief survey of the most important

propertiesoftheintegralsofform

(2)

.Thereaderwillfindtheir

proofsinevery textbookon real variables,althoughtheproofs

areusuallycarriedoutonlyinthecasewhereP(A)istheLebesgue

measureof

sets

in

R

n

.

The

extensionofthese

proofs

to

thegeneral

casedoesnotentailanynewmathematicalproblem;

forthemost

parttheyremainwordforwordthesame.

I. If arandomvariable xis integrableonA, thenit is in-

tegrateoneachsubsetA'ofA belongingto

g.

II. If x is integrable on A and A is decomposed into no

1

Aswasstatedin

§

5 ofthethirdchapter,weareconsideringinthis,aswell

asinthefollowingchapters,Borel

fieldsof

probabilityonly.

2

Frechet,

Sur

Vintegrale oVune

functionnelle

etendue a un ensemble

abstrait,

Bull. Soc.Math.Francev.43,1915,p.248.

37