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1、AnintroductiontomeasuretheoryTerenceTaoDepartmentofMathematics,UCLA,LosAngeles,CA90095E-mailaddress:tao@math.ucla.eduToGarthGaudry,whosetmeontheroad;Tomyfamily,fortheirconstantsupport;Andtothereadersofmyblog,fortheirfeedbackandcontributions.ContentsPrefaceixNotationxAcknowledgmentsxviChapter1.Measur
2、etheory1x1.1.Prologue:Theproblemofmeasure2x1.2.Lebesguemeasure17x1.3.TheLebesgueintegral46x1.4.Abstractmeasurespaces79x1.5.Modesofconvergence114x1.6.Di erentiationtheorems131x1.7.Outermeasures,pre-measures,andproductmeasures179Chapter2.Relatedarticles209x2.1.Problemsolvingstrategies210x2.2.TheRadama
3、cherdi erentiationtheorem226x2.3.Probabilityspaces232x2.4.In niteproductspacesandtheKolmogorovextensiontheorem235Bibliography243viiviiiContentsIndex245PrefaceInthefallof2010,Itaughtanintroductoryone-quartercourseongraduaterealanalysis,focusinginparticularonthebasicsofmea-sureandintegrationtheory,bot
4、hinEuclideanspacesandinabstractmeasurespaces.Thistextisbasedonmylecturenotesofthatcourse,whicharealsoavailableonlineonmyblogterrytao.wordpress.com,togetherwithsomesupplementarymaterial,suchasasectiononprob-lemsolvingstrategiesinrealanalysis(Section2.1)whichevolvedfromdiscussionswithmystudents.Thiste
5、xtisintendedtoformaprequeltomygraduatetext[Ta2010](henceforthreferredtoasAnepsilonofroom,Vol.I),whichisanintroductiontotheanalysisofHilbertandBanachspaces(suchasLpandSobolevspaces),point-settopology,andrelatedtop-icssuchasFourieranalysisandthetheoryofdistributions;together,theyserveasatextforacomple
6、te rst-yeargraduatecourseinrealanalysis.Theapproachtomeasuretheoryhereisinspiredbythetext[StSk2005],whichwasusedasasecondarytextinmycourse.Inparticular,the rsthalfofthecourseisdevotedalmostexclusivelytomeasuretheoryonEuclideanspacesRd(startingwiththemoreelementaryJordan-Riemann-Darbouxtheory,andonly
7、thenmovingontothemoresophisticatedLebesguetheory),deferringtheabstractaspectsofmeasuretheorytothesecondhalfofthecourse.IfoundixxPrefacethatthisapproachstrengthenedthestudent'sintuitionintheearlystages
