| Lebesgue Integration for Functions of a Single Real VariablePreliminaries on Sets, Mappings, and RelationsUnions and Intersections of SetsEquivalence Relations, the Axiom of Choice, and Zorn's Lemma 1 The Real Numbers: Sets. Sequences, and FunctionsThe Field, Positivity, and Completeness AxiomsThe Natural and Rational NumbersCountable and Uncountable SetsOpen Sets, Closed Sets, and Borel Sets of Real Numbers Sequences of Real NumbersContinuous Real-Valued Functions of a Real Variable2 Lebesgne MeasureIntroductionLebesgue Outer MeasureThe o'-Algebra of Lebesgue Measurable SetsOuter and Inner Approximation of Lebesgue Measurable Sets Countable Additivity, Continuity, and the Borel-Cantelli LemmaNoumeasurable SetsThe Cantor Set and the Cantor Lebesgue Function3 LebesgRe Measurable FunctionsSums, Products, and CompositionsSequential Pointwise Limits and Simple ApproximationLittlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem4 Lebesgue IntegrationThe Riemann IntegralThe Lebesgue Integral of a Bounded Measurable Function over a Set ofFinite MeasureThe Lebesgue Integral of a Measurable Nonnegative FunctionThe General Lebesgue IntegralCountable Additivity and Continuity of IntegrationUniform Integrability: The Vifali Convergence Theoremviii Contents5 Lebusgue Integration: Fm'ther TopicsUniform Integrability and Tightness: A General Vitali Convergence TheoremConvergence in MeasureCharacterizations of Riemaun and Lebesgue Integrability6 Differentiation and IntegrationContinuity of Monotone FunctionsDifferentiability of Monotone Functions: Lebesgue's TheoremFunctions of Bounded Variation: Jordan's TheoremAbsolutely Continuous FunctionsIntegrating Derivatives: Differentiating Indefinite IntegralsConvex Function7 The Lp Spaces: Completeness and Appro~umationNor/ned Linear SpacesThe Inequalities of Young, HOlder, and Minkowski Lv Is Complete: The Riesz-Fiseher TheoremApproximation and Separability8 The LP Spacesc Deailty and Weak ConvergenceThe Riesz Representation for the Dual ofWeak Sequential Convergence in Lv Weak Sequential CompactnessThe Minimization of Convex FunctionalsII Abstract Spaces: Metric, Topological, Banach, and Hiibert Spaces9. Metric Spaces: General PropertiesExamples of Metric SpacesOpen Sets, Closed Sets, and Convergent SequencesContinuous Mappings Between Metric Spaces Complete Metric SpacesCompact Metric SpacesSeparable Metric Spaces10 Metric Spaces: Three Fundamental ThanreessThe Arzelb.-Ascoli TheoremThe Baire Category TheoremThe Banaeh Contraction PrincipleH Topological Spaces: General PropertiesOpen Sets, Closed Sets, Bases, and SubbasesThe Separation PropertiesCountability and SeparabilityContinuous Mappings Between Topological SpacesCompact Topological SpacesConnected Topological Spaces12 Topological Spaces: Three Fundamental TheoremsUrysohn's Lemma and the Tietze Extension Theorem The Tychonoff Product TheoremThe Stone-Weierstrass Theorem13 Continuous Linear Operators Between Bausch SpacesNormed Linear SpacesLinear OperatorsCompactness Lost: Infinite Dimensional Normod Linear SpacesThe Open Mapping and Closed Graph TheoremsThe Uniform Boundedness Principle14 Duality for Normed Iinear SpacesLinear Ftmctionals, Bounded Linear Functionals, and Weak TopologiesThe Hahn-Banach TheoremReflexive Banach Spaces and Weak Sequential ConvergenceLocally Convex Topological Vector SpacesThe Separation of Convex Sets and Mazur's TheoremThe Krein-Miiman Theorem15 Compactness Regained: The Weak TopologyAlaoglu's Extension of Helley's TheoremReflexivity and Weak Compactness: Kakutani's TheoremCompactness and Weak Sequential Compactness: The Eberlein-mulianTheoremMemzability of Weak Topologies 16 Continuous Linear Operators on Hilbert SpacesThe Inner Product and OrthogonalityThe Dual Space and Weak Sequential ConvergenceBessers Inequality and Orthonormal BasesbAdjoints and Symmetry for Linear OperatorsCompact OperatorsThe Hilbert-Schmidt TheoremThe Riesz-Schauder Theorem: Characterization of Fredholm Operators Measure and Integration: General Theory17 General Measure Spaces: Their Propertles and ConstructionMeasures and Measurable SetsSigned Measures: The Hahn and Jordan Decompositions The Caratheodory Measure Induced by an Outer Measure18 Integration Oeneral Measure Spaces19 Gengral L Spaces:Completeness,Duality and Weak Convergence20 The Construciton of Particular Measures21 Measure and Topbogy22 Invariant MeasuresBibiiographyindex |
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