Real Analysis (Hardcover)

PART I. AN INTRODUCTION TO GENERAL TOPOLOGYSET-THEORETIC AND ALGEBRAIC PRELIMINARIESSets and Basic NotationFunctionsSet Operations under MapsRelations and Well-Ordering PrincipleCartesian ProductCardinalityBasic Algebraic StructuresANALYSIS OF METRIC SPACESDefinitions and NotationsThe Structure of Metric Spaces5Convergence in Metric SpacesContinuous Mappings in Metric SpacesComplete Metric SpacesCompactnessLinear and Normed Linear SpacesELEMENTS OF POINT SET TOPOLOGYTopological SpacesBases and Subbases for Topological SpacesConvergence of Sequences in Topological Spaces and CountabilityContinuity in Topological SpacesProduct TopologyNotes on Subspaces and CompactnesFunction Spaces and Ascoli's TheoremStone-Weierstrass Approximation TheoremFilter and Net ConvergenceSeparationFunctions on Locally Compact SpacesPART II. BASICS OF MEASURE AND INTEGRATIONMEASURABLE SPACES AND MEASURABLE FUNCTIONSSystems of SetsSystem's GeneratorsMeasurable FunctionsMEASURESSet FunctionsExtension of Set Functions to a MeasureLebesgue and Lebesgue-Stieltjes MeasuresImage MeasuresExtended Real-Valued Measurable FunctionsSimple FunctionsELEMENTS OF INTEGRATIONIntegration on C -1(W,S)Main Convergence TheoremsLebesgue and Riemann Integrals on RIntegration with Respect to Image MeasuresMeasures Generated by Integrals. Absolute Continuity.OrthogonalityProduct Measures of Finitely Many Measurable Spaces andFubini's TheoremApplications of Fubini's TheoremCALCULUS IN EUCLIDEAN SPACESDifferentiationChange of VariablesPART III. FURTHER TOPICS IN INTEGRATIONANALYSIS IN ABSTRACT SPACESSigned and Complex MeasuresAbsolute ContinuitySingularityLp SpacesModes of ConvergenceUniform IntegrabilityRadon Measures on Locally Compact Hausdorff SpacesMeasure DerivativesCALCULUS ON THE REAL LINEMonotone FunctionsFunctions of Bounded VariationAbsolute Continuous FunctionsSingular FunctionsBIBLIOGRAPHY