Elements of the Theory of Functions and Functional Analysis Vol II

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1、ELEMENTSOFTHETHEORYOFFUNCTIONSANDFUNCTIONALANALYSISVolume2MeasureTheLebesgueIntegralHilbertSpaceAll.KolmogorovandS.V.FominELEMENTSOFTHETHEORYOFFUNCTIONSANDFUNCTIONALANALYSISVOLUMEMEASURE.THELEBESGUEINTEGRAL.HILBERTSPACEOTHERGRAYLOCKPUBLICATIONSKHINCHIN:ThreePearlsofNumberTheoryMath

2、ematicalFoundationsofQuantumStatisticsPONTRYAGIN:FoundationsofCombinatorialTopologyNOVOZHILOV:FoundationsoftheNonlinearTheoryofElasticityKOLMOGOROVandFOMIN:ElementsoftheTheoryofFunctionsandFunctionalAnalysis.Vol.1:MetricandNormedSpacesPETROVSKIT:LecturesontheTheoryofIntegralEquatio

3、nsALEKSANDROV:CombinatorialTopologyVol.1:Introduction.Complexes.Coverings.DimensionVol.TheBettiGroupsVol.3:HomologicalManifolds.TheDualityTheorems.CohomologyGroupsofCompacta.ContinuousMappingsofPolyhedraElementsoftheTheoryofFunctionsandFunctionalAnalysisVOLUME2MEASURE.THELEBESGLTEI

4、NTEGRAL.HILBERTSPACEBYA.N.KOLMOGOROVANDS.V.FOMINTRANSLATEDFROMTHEFIRST(1960)RUSSIANEDITIONbyHYMANKAMELANDHORACEKOMMDepartmentofMathematicsRensselaerPolytechnicInstituteGI?AYLOCKPRESSALBANY,N.Y.1961Copyright©1961byGRAYLOCKPRESSAlbany,N.Y.SecondPrinting—January1963Allrightsreserved.T

5、hisbook,orpartsthereof,maynotbereproducedinanyform,ortranslated,withoutpermis-sioninwritingfromthepublishers.LibraryofCongressCatalogCardNumber57—4134ManufacturedintheUnitedStatesofAmericaCONTENTSPrefaceviiTranslators'NoteixCHAPTERVMEASURETHEORY33.Themeasureofplanesets134.Collectio

6、nsofsets1535.Measuresonsemi-rings.Extensionofameasureonasemi-ringtotheminimalringoverthesemi-ring2036.ExtensionofJordanmeasure2337.Completeadditivity.Thegeneralproblemoftheextensionofmeasures2838.TheLebesgueextensionofameasuredefinedonasemi-ringwithunity3139.ExtensionofLebesguemeas

7、uresinthegeneralcase36CHAPTERVIMEASURABLEFUNCTIONS40.Definitionandfundamentalpropertiesofmeasurablefunctions..3841.Sequencesofmeasurablefunctions.Varioustypesofconvergence.42CHAPTERVIITHELEBESGUEINTEGRAL42.TheLebesgueintegralofsimplefunctions4843.Thegeneraldefinitionandfundamentalp

8、ropertiesoftheLebesgueinte