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1、Chapter9HaarMeasureWesawinChap.1thatLebesguemeasureonRdistranslationinvariant,inthesensethatλ(A+x)=λ(A)holdsforeachAinB(Rd)andeachxinRd.Furthermore,wesawthatLebesguemeasureisessentiallytheonlysuchBorelmeasureonRd:ifμisanonzeroBorelmeasureonRdthatisfiniteonthecompactsubsetsofRdandsatisfiesμ(A+x)
2、=μ(A)foreachAinB(Rd)andeachxinRd,thenthereisapositivenumbercsuchthatμ(A)=cλ(A)holdsforeveryBorelsubsetAofRd.Itturnsoutthatverysimilarresultsholdforeverylocallycompactgroup(seeSect.9.1forthedefinitionofsuchgroups);theroleofLebesguemeasureisplayedbywhatiscalledHaarmeasure.Thischapterisdevotedtoa
3、nintroductiontoHaarmeasure.Section9.1containssomebasicdefinitionsandfactsabouttopologicalgroups.Section9.2containsaproofoftheexistenceanduniquenessofHaarmeasure,andSect.9.3containsadditionalbasicpropertiesofHaarmeasures.InSect.9.4weconstructtwoalgebras,L1(G)andM(G),whicharefundamentalforthestu
4、dyofharmonicanalysisonalocallycompactgroupG.9.1TopologicalGroupsAtopologicalgroupisasetGthathasthestructureofagroup(saywithgroupoperation(x,y)→xy)andofatopologicalspaceandissuchthattheoperations(x,y)→xyandx→x−1arecontinuous.Notethat(x,y)→xyisafunctionfromtheproductspaceG×GtoGandthatwearer
5、equiringthatitbecontinuouswithrespecttotheproducttopologyonG×G;thusxymustbejointlycontinuousinxandyandnotmerelycontinuousinxwithyheldfixedandcontinuousinywithxheldfixed(seeExercise3).Alocallycompacttopologicalgroup,orsimplyalocallycompactgroup,isatopologicalgroupwhosetopologyislocallycompactand
6、Hausdorff.AcompactgroupisatopologicalgroupwhosetopologyiscompactandHausdorff.D.L.Cohn,MeasureTheory:SecondEdition,BirkhauserAdvanced¨279TextsBaslerLehrb¨ucher,DOI10.1007/978-1-4614-6956-89,©SpringerScience+BusinessMedia,LLC20132809HaarMeasureExamples9.1.1.(a)ThesetR,withitsusualtopologyandwit
7、hadditionasthegroupoperation,isalocallycompactgroup.(b)Likewise,Rd,Z,andZdarelocallycompactgroups.(c)ThesetR∗ofnonzerorealnumbers,withthetopologyitinheritsasasubspaceofRandwithmultiplicationasthegroupoperation,isalocallycompactgroup.(d)LetTbe
