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琼斯 (Frank Jones) 著 / 世界图书出版公司 / 2010-01 / 平装
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欧氏空间上的勒贝格积分(修订版)(英文版)
《欧氏空间上的勒贝格积分(修订版)(英文版)》简明、详细地介绍勒贝格测度和Rn上的积分。《欧氏空间上的勒贝格积分(英文版)》的基本目的有四个,介绍勒贝格积分;从一开始引入n维空间;彻底介绍傅里叶积分;深入讲述实分析。贯穿全书的大量练习可以增强读者对知识的理解。目次:Rn导论;Rn勒贝格测度;勒贝格积分的不变性;一些有趣的集合;集合代数和可测函数;积分;Rn勒贝格积分;Rn的Fubini定理;Gamma函数;Lp空间;抽象测度的乘积;卷积;Rn+上的傅里叶变换;单变量傅里叶积分;微分;R上函数的微分。
读者对象:《欧氏空间上的勒贝格积分(修订版)(英文版)》适用于数学专业的学生、老师和相关的科研人员。
PrefaceBibliographyAcknowledgments1IntroductiontoRnASetsBCountableSetsCTopologyDCompactSetsEContinuityFTheDistanceFunction2LebesgueMeasureonRnAConstructionBPropertiesofLebesgueMeasureCAppendix:ProofofP1andP23InvarianceofLebesgueMeasureASomeLinearAlgebraBTranslationandDilationCOrthogonalMatricesDTheGeneralMatrix4SomeInterestingSetsAANonmeasurableSetBABevyofCantorSetsCTheLebesgueFunctionDAppendix:TheModulusofContinuityoftheLebesgueFunctions5AlgebrasofSetsandMeasurableFunctionsAAlgebrasanda-AlgebrasBBorelSetsCAMeasurableSetwhichIsNotaBorelSetDMeasurableFunctionsESimpleFunctions6IntegrationANonnegativeFunctionsBGeneralMeasurableFunctionsCAlmostEverywhereDIntegrationOverSubsetsofRnEGeneralization:MeasureSpacesFSomeCalculationsGMiscellany7LebesgueIntegralonRnARiemannIntegralBLinearChangeofVariablesCApproximationofFunctionsinL1DContinuityofTranslationinL18FubinisTheoremforRn9TheGammaFunctionADefinitionandSimplePropertiesBGeneralizationCTheMeasureofBallsDFurtherPropertiesoftheGammaFunctionEStirlingsFormulaFTheGammaFunctiononR10LPSpaces,ADefinitionandBasicInequalitiesBMetricSpacesandNormedSpacesCCompletenessofLpDTheCasep=∞ERelationsbetweenLpSpacesFApproximationbyC∞c(Rn)GMiscellaneousProblems;HTheCase0[p[111ProductsofAbstractMeasuresAProductsof5-AlgebrasBMonotoneClassesCConstructionoftheProductMeasureDTheFubiniTheoremETheGeneralizedMinkowskiInequality12ConvolutionsAFormalPropertiesBBasicInequalitiesCApproximateIdentities13FourierTransformonRnAFourierTransformofFunctionsinL1(Rn)BTheInversionTheoremCTheSchwartzClassDTheFourier-PlancherelTransformEHilbertSpaceFFormalApplicationtoDifferentialEquationsGBesselFunctionsHSpecialResultsforn=iIHermitePolynomials14FourierSeriesinOneVariableAPeriodicFunctionsBTrigonometricSeriesCFourierCoefficientsDConvergenceofFourierSeriesESummabilityofFourierSeriesFACounterexampleGParsevalsIdentityHPoissonSummationFormulaIASpecialClassofSineSeries15DifferentiationATheVitaliCoveringTheoremBTheHardy-LittlewoodMaximalFunctionCLebesguesDifferentiationTheoremDTheLebesgueSetofaFunctionEPointsofDensityFApplicationsGTheVitaliCoveringTheorem(Again)HTheBesicovitchCoveringTheoremITheLebesgueSetofOrderpJChangeofVariablesKNoninvertibleMappings16DifferentiationforFunctionsonRAMonotoneFunctionsBJumpFunctionsCAnotherTheoremofFubiniDBoundedVariationEAbsoluteContinuityFFurtherDiscussionofAbsoluteContinuityGArcLengthHNowhereDifferentiableFunctionsIConvexFunctionsIndexSymbolIndex
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