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1、AnIntroductiontoMeasureTheoryTerenceTaoThisisapreliminaryversionofthebookAnIntroductiontoMeasureTheorypublishedbytheAmericanMathematicalSociety(AMS).ThispreliminaryversionismadeavailablewiththepermissionoftheAMSandmaynotbechanged,edited,orrepostedatanyotherwebsitewithoutexplicitwrittenpermissi
2、onfromtheauthorandtheAMS.Author'spreliminaryversionmadeavailablewithpermissionofthepublisher,theAmericanMathematicalSocietyAuthor'spreliminaryversionmadeavailablewithpermissionofthepublisher,theAmericanMathematicalSocietyToGarthGaudry,whosetmeontheroad;Tomyfamily,fortheirconstantsupport;Andtot
3、hereadersofmyblog,fortheirfeedbackandcontributions.Author'spreliminaryversionmadeavailablewithpermissionofthepublisher,theAmericanMathematicalSocietyAuthor'spreliminaryversionmadeavailablewithpermissionofthepublisher,theAmericanMathematicalSocietyContentsPrefaceixNotationxAcknowledgmentsxviCha
4、pter1.Measuretheory1x1.1.Prologue:Theproblemofmeasure2x1.2.Lebesguemeasure17x1.3.TheLebesgueintegral46x1.4.Abstractmeasurespaces79x1.5.Modesofconvergence114x1.6.Di erentiationtheorems131x1.7.Outermeasures,pre-measures,andproductmeasures179Chapter2.Relatedarticles209x2.1.Problemsolvingstrategie
5、s210x2.2.TheRademacherdi erentiationtheorem226x2.3.Probabilityspaces232x2.4.In niteproductspacesandtheKolmogorovextensiontheorem235Bibliography243viiAuthor'spreliminaryversionmadeavailablewithpermissionofthepublisher,theAmericanMathematicalSocietyviiiContentsIndex245Author'spreliminaryversionm
6、adeavailablewithpermissionofthepublisher,theAmericanMathematicalSocietyPrefaceInthefallof2010,Itaughtanintroductoryone-quartercourseongraduaterealanalysis,focusinginparticularonthebasicsofmea-sureandintegrationtheory,bothinEuclideanspacesandinabstractmeasurespaces.Thistextisbasedonmylecturenot
7、esofthatcourse,whicharealsoavailableonlineonmyblogterrytao.wordpress.com,togetherwithsomesupplementarymaterial,suchasasectiononprob-lemsolvingstrategiesinrealanalysis(Section2.1)whichevolvedfromdiscussionswithmystudents.Thistextisintend
