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1、MarekCapinskiandEkkehardKoppMeasure,IntegralandProbabilitySpringer-VerlagBerlinHeidelbergNewYorkLondonParisTokyoHongKongBarcelonaBudapestToourchildren;grandchildren:Piotr,Maciej,Jan,Anna;LukaszAnna,EmilyPrefaceThecentralconceptsinthisbookareLebesguemeasureandtheLebesgueinteg
2、ral.TheirroleasstandardfareinUKundergraduatemathematicscoursesisnotwhollysecure;yettheyprovidetheprincipalmodelforthedevelopmentoftheabstractmeasurespaceswhichunderpinmodernprobabilitytheory,whiletheLebesguefunctionspacesremainthemainsourceofexamplesonwhichtotestthemethodsoff
3、unctionalanalysisanditsmanyapplications,suchasFourieranalysisandthetheoryofpartialdi erentialequations.Itfollowsthatnotonlybuddinganalystshaveneedofaclearunderstandingoftheconstructionandpropertiesofmeasuresandintegrals,butalsothatthosewhowishtocontributeseriouslytotheapplica
4、tionsofanalyticalmethodsinawidevarietyofareasofmathematics,physics,electronics,engineeringand,mostrecently, nance,needtostudytheunderlyingtheorywithsomecare.Wehavefoundremarkablyfewtextsinthecurrentliteraturewhichaimexplicitlytoprovidefortheseneeds,atalevelaccessibletocurrent
5、under-graduates.Therearemanygoodbooksonmodernprobabilitytheory,andincreasinglytheyrecognizetheneedforastronggroundinginthetoolswedevelopinthisbook,butalltoooftenthetreatmentiseithertooadvancedforanundergraduateaudienceorelsesomewhatperfunctory.Wehopethereforethatthecurrenttex
6、twillnotberegardedasonewhich llsamuch-neededgapintheliterature!Onefundamentaldecisionindevelopingatreatmentofintegrationiswhethertobeginwithmeasuresorintegrals,i.e.whethertostartwithsetsorwithfunctions.Functionalanalystshavetendedtofavourthelatterapproach,whiletheformerisclea
7、rlynecessaryforthedevelopmentofprobability.Wehavedecidedtosidewiththeprobabilistsinthisargument,andtousethe(reasonably)systematicdevelopmentofbasicconceptsandresultsinproba-bilitytheoryastheprincipal eldofapplication{theorderoftopicsandtheviiviiiPrefaceterminologyweusere
ectt
8、hischoice,andeachchapterconcludeswithfurtherdevelopmentoftherelevant
