测度、积分与概率的勒贝格理论（英文版—paul loya）

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1、Lebesgue’sremarkabletheoryofmeasureandintegrationwithprobabilityPaulLoya(Pleasedonotdistributethisbooktoanyoneelse.Pleaseemailpaul@math.binghamton.edutoreporterrorsorgivecriticisms)ContentsPrologue:Thepaperthatchangedintegrationtheory...forever1Suruneg´en´eralisationdel’int´egraled´efinie1Someremark

2、sonLebesgue’spaper3Part1.Finiteadditivity7Chapter1.Measure&probability:finiteadditivity91.1.Introduction:Measureandintegration91.2.Probability,events,andsamplespaces161.3.Semirings,ringsandσ-algebras261.4.TheBorelsetsandthePrincipleofAppropriateSets361.5.Additivesetfunctionsinclassicalprobability441

3、.6.Moreexamplesofadditivesetfunctions59NotesandreferencesonChapter167Chapter2.Finitelyadditiveintegration692.1.Integrationonsemirings692.2.Randomvariablesand(mathematical)expectations752.3.Propertiesofadditivesetfunctionsonsemirings862.4.Bernoulli’sTheorem(TheWLLNs)andexpectations972.5.DeMoivre,Lap

4、laceandStirlingstarinTheNormalCurve108NotesandreferencesonChapter2127Part2.Countableadditivity131Chapter3.Measureandprobability:countableadditivity1333.1.Introduction:Whatisameasurableset?1333.2.Countablyadditivesetfunctionsonsemirings1373.3.Infiniteproductspacesandpropertiesofcountablyadditivity144

5、3.4.Outermeasures,measures,andCarath´eodory’sidea1583.5.TheExtensionTheoremandregularitypropertiesofmeasures168NotesandreferencesonChapter3178Chapter4.ReactionstotheExtension&RegularityTheorems1814.1.Probability,BernoullisequencesandBorel-Cantelli1814.2.Borel’sStrongLawofLargeNumbers1904.3.Measurab

6、ilityandLittlewood’sFirstPrinciple(s)1984.4.Geometry,Vitali’snonmeasurableset,andparadoxes2104.5.TheCantorset227NotesandreferencesonChapter4239iiiivCONTENTSPart3.Integration245Chapter5.BasicsofIntegrationTheory2475.1.Introduction:Interchanginglimitsandintegrals2475.2.MeasurablefunctionsandLittlewoo

7、d’ssecondprinciple2525.3.SequencesoffunctionsandLittlewood’sthirdprinciple2645.4.Lebesgue’sdefinitionoftheintegralandtheMCT2755.5.FeaturesoftheintegralandthePrincipleofAppropriateFunctions2855.6.TheDCTandOsg