Lebesgue Integration for Functions of a Single Real Variable Preliminaries on Sets, Mappings, and Relations Unions and Intersections of Sets Equivalence Relations, the Axiom of Choice, and Zorn's Lemma 1 The Real Numbers: Sets. Sequences, and Functions The Field, Positivity, and Completeness Axioms The Natural and Rational Numbers Countable and Uncountable Sets Open Sets, Closed Sets, and Borel Sets of Real Numbers Sequences of Real Numbers Continuous Real-Valued Functions of a Real Variable 2 Lebesgne Measure Introduction Lebesgue Outer Measure The o'-Algebra of Lebesgue Measurable Sets Outer and Inner Approximation of Lebesgue Measurable Sets Countable Additivity, Continuity, and the Borel-Cantelli Lemma Noumeasurable Sets The Cantor Set and the Cantor Lebesgue Function 3 LebesgRe Measurable Functions Sums, Products, and Compositions Sequential Pointwise Limits and Simple Approximation Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem 4 Lebesgue Integration The Riemann Integral The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure The Lebesgue Integral of a Measurable Nonnegative Function The General Lebesgue Integral Countable Additivity and Continuity of Integration Uniform Integrability: The Vifali Convergence Theorem viii Contents 5 Lebusgue Integration: Fm'ther Topics Uniform Integrability and Tightness: A General Vitali Convergence Theorem Convergence in Measure Characterizations of Riemaun and Lebesgue Integrability 6 Differentiation and Integration Continuity of Monotone Functions Differentiability of Monotone Functions: Lebesgue's Theorem Functions of Bounded Variation: Jordan's Theorem Absolutely Continuous Functions Integrating Derivatives: Differentiating Indefinite Integrals Convex Function 7 The Lp Spaces: Completeness and Appro~umation Nor/ned Linear Spaces The Inequalities of Young, HOlder, and Minkowski Lv Is Complete: The Riesz-Fiseher Theorem Approximation and Separability 8 The LP Spacesc Deailty and Weak Convergence The Riesz Representation for the Dual of Weak Sequential Convergence in Lv Weak Sequential Compactness The Minimization of Convex Functionals II　Abstract Spaces: Metric, Topological, Banach, and Hiibert Spaces 9. Metric Spaces: General Properties Examples of Metric Spaces Open Sets, Closed Sets, and Convergent Sequences Continuous Mappings Between Metric Spaces Complete Metric Spaces Compact Metric Spaces Separable Metric Spaces 10 Metric Spaces: Three Fundamental Thanreess The Arzelb.-Ascoli Theorem The Baire Category Theorem The Banaeh Contraction Principle H Topological Spaces: General Properties Open Sets, Closed Sets, Bases, and Subbases The Separation Properties Countability and Separability Continuous Mappings Between Topological Spaces Compact Topological Spaces Connected Topological Spaces 12 Topological Spaces: Three Fundamental Theorems Urysohn's Lemma and the Tietze Extension Theorem The Tychonoff Product Theorem The Stone-Weierstrass Theorem 13 Continuous Linear Operators Between Bausch Spaces Normed Linear Spaces Linear Operators Compactness Lost: Infinite Dimensional Normod Linear Spaces The Open Mapping and Closed Graph Theorems The Uniform Boundedness Principle 14 Duality for Normed Iinear Spaces Linear Ftmctionals, Bounded Linear Functionals, and Weak Topologies The Hahn-Banach Theorem Reflexive Banach Spaces and Weak Sequential Convergence Locally Convex Topological Vector Spaces The Separation of Convex Sets and Mazur's Theorem The Krein-Miiman Theorem 15 Compactness Regained: The Weak Topology Alaoglu's Extension of Helley's Theorem Reflexivity and Weak Compactness: Kakutani's Theorem Compactness and Weak Sequential Compactness: The Eberlein-mulian Theorem Memzability of Weak Topologies 16 Continuous Linear Operators on Hilbert Spaces The Inner Product and Orthogonality The Dual Space and Weak Sequential Convergence Bessers Inequality and Orthonormal Bases bAdjoints and Symmetry for Linear Operators Compact Operators The Hilbert-Schmidt Theorem The Riesz-Schauder Theorem: Characterization of Fredholm Operators Measure and Integration: General Theory 17 General Measure Spaces: Their Propertles and Construction Measures and Measurable Sets Signed Measures: The Hahn and Jordan Decompositions The Caratheodory Measure Induced by an Outer Measure 18 Integration Oeneral Measure Spaces 19 Gengral L Spaces:Completeness,Duality and Weak Convergence 20 The Construciton of Particular Measures 21 Measure and Topbogy 22 Invariant Measures Bibiiography index