Foundations of the theory of probability

PREFACE

The purpose of this monograph is to give an axiomatic

foundationforthetheoryofprobability.Theauthorsethimself

thetask of puttingin their natural place, among the general

notionsofmodernmathematics,thebasicconceptsofprobability

theory—conceptswhichuntilrecentlywereconsideredtobequite

peculiar.

Thistaskwouldhavebeenaratherhopelessonebefore

the

introductionofLebesgue'stheories ofmeasureandintegration.

However,after Lebesgue'spublicationofhisinvestigations,the

analogiesbetweenmeasureofasetandprobabilityofanevent,

andbetweenintegralofafunctionandmathematicalexpectation

ofarandomvariable,becameapparent.Theseanalogiesallowed

offurtherextensions; thus,forexample,variousproperties of

independent

randomvariableswere

seentobe

incompleteanalogy

withthecorrespondingproperties oforthogonal functions.But

ifprobabilitytheorywastobebasedontheaboveanalogies, it

stillwasnecessarytomakethetheoriesofmeasureandintegra-

tion independentof thegeometric elementswhichwerein the

foregroundwithLebesgue. ThishasbeendonebyFrechet.

Whileaconceptionofprobabilitytheorybasedontheabove

generalviewpointshasbeencurrentforsometimeamongcertain

mathematicians,there waslackingacompleteexpositionofthe

wholesystem, freeofextraneous complications. (Cf.,however,

thebookbyFrechet,

[2]

inthebibliography.)

Iwishtocallattentiontothosepointsofthepresentexposition

whichareoutsidetheabove-mentionedrangeofideasfamiliarto

thespecialist.They

arethe

following: Probabilitydistributions

ininfinite-dimensional

spaces (ChapterIII,§

4);

differentiation

andintegration ofmathematicalexpectationswithrespecttoa

parameter(ChapterIV,

§5);

andespeciallythetheoryofcondi-

tional probabilities and conditional expectations (Chapter V).

Itshouldbeemphasizedthatthesenewproblemsarose,ofneces-

sity,fromsomeperfectlyconcretephysicalproblems.

1

1

Cf.,e.g.,thepaperbyM.Leontovichquotedinfootnote 6 on

p.46;

alsothe

jointpaperbytheauthorandM.Leontovich,ZurStatistikderkontinuier-

lichen

Systemeunddeszeitlichen VerlaufesderphysikalischenVorgdnge.

Phys.Jour,oftheUSSR,Vol.3,1933,

pp.

35-63.