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1、Chapter5ProductMeasuresIncalculuscoursesonedefinesintegralsovertwo-(orhigher-)dimensionalregionsandthenevaluatestheseintegralsbyapplyingtheusualtechniquesofintegration,onevariableatatime.InthischapterweshowthatsimilartechniquesworkfortheLebesgueintegral.Moregenerally,givenσ-finitemeasure
2、sμandνonspacesXandY,wefirstdefineanaturalproductmeasureontheproductspaceX×Y(Sect.5.1).ThenwelookathowintegralswithrespecttothisproductmeasurecanbeevaluatedintermsofintegralswithrespecttoμandνoverXandY(Sect.5.2).Thechapterendswithafewapplications(Sect.5.3).5.1ConstructionsLet(X,A)and(Y,B)
3、bemeasurablespaces,and,asusual,letX×YbetheCartesianproductofthesetsXandY.AsubsetofX×YisarectanglewithmeasurablesidesifithastheformA×BforsomeAinAandsomeBinB;theσ-algebraonX×Ygeneratedbythecollectionofallrectangleswithmeasurablesidesiscalledtheproductoftheσ-algebrasAandBandisdenotedbyA×B
4、.Example5.1.1.ConsiderthespaceR2.Thisis,ofcourse,aCartesianproduct,theproductofRwithitself.Letusshowthattheproductσ-algebraB(R)×B(R)isequaltotheσ-algebraB(R2)ofBorelsubsetsofR2.RecallthatB(R2)isgeneratedbythecollectionofallsetsoftheform(a,b]×(c,d](Proposition1.1.5).ThusB(R2)isgenerated
5、byasubfamilyoftheσ-algebraB(R)×B(R)andsoisincludedinB(R)×B(R).Weturntothereverseinclusion.Theprojectionsπ1andπ2ofR2ontoRdefinedbyπ1(x,y)=xandπ2(x,y)=yarecontinuousandhenceBorelmeasurable(Example2.1.2(a)).ItfollowsfromthisandtheidentityA×B=(A×R)∩(R×B)=π−1(A)∩π−1(B)12D.L.Cohn,MeasureTheor
6、y:SecondEdition,BirkhauserAdvanced¨143TextsBaslerLehrb¨ucher,DOI10.1007/978-1-4614-6956-85,©SpringerScience+BusinessMedia,LLC20131445ProductMeasuresthatifAandBbelongtoB(R),thenA×BbelongstoB(R2).SinceB(R)×B(R)istheσ-algebrageneratedbythecollectionofallsuchrectanglesA×B,itmustbeincludedi
7、nB(R2).ThusB(R)×B(R)=B(R2).Letusintroducesometerminologyandnotation.SupposethatXandYaresetsandthatEisasubsetofX×Y.ThenforeachxinXandeachyinYthesectionsExandEyarethesubsetsofYandXgivenbyEx={y∈Y:(x,y)∈E}andEy={x∈X:(x,y)∈E}.IffisafunctiononX×Y,thenthesectionsfxandfyarethefunctionsonYan
