Real Variable and Integration

Open Data:

1 Classical real variable -- 1.1 Set theory—a framework -- 1.2 The topology of R -- 1.3 Classical real variable—motivation for the Lebesgue theory -- 1.4 References for the history of integration theory -- Problems -- 2 Lebesgue measure and general measure theory -- 2.1 The theory of measure prior to Lebesgue, and preliminaries -- 2.2 The existence of Lebesgue measure -- 2.3 General measure theory -- 2.4 Approximation theorems for measurable functions -- Problems -- 3 The Lebesgue integral -- 3.1 Motivation -- 3.2 The Lebesgue integral -- 3.3 The Lebesgue dominated convergence theorem -- 3.4 The Riemann and Lebesgue integrals -- 3.5 Some fundamental applications -- Problems -- 4 The relationship between differentiation and integration on R -- 4.1 Functions of bounded variation and associated measures -- 4.2 Decomposition into discrete and continuous parts -- 4.3 The Lebesgue differentiation theorem -- 4.4 FTC-I -- 4.5 Absolute continuity and FTC-II -- 4.6 Absolutely continuous functions -- Problems -- 5 Spaces of measures and the Radon-Nikodym theorem -- 5.1 Signed and complex measures, and the basic decomposition theorems -- 5.2 Discrete and continuous, absolutely continuous and singular measures -- 5.3 The Vitali-Lebesgue-Radon-Nikodym theorem -- 5.4 The relation between set and point functions -- 5.5 Lp?(X), l?p?? -- Problems -- 6 Weak convergence of measures -- 6.1 Vitali’s theorems -- 6.2 The Nikodym and Hahn-Saks theorems -- 6.3 Weak convergence of measures -- Appendices -- I Metric spaces and Banach spaces -- I.1 Definitions of spaces -- I.2 Examples -- I.3 Separability -- I.4 Moore-Smith and Arzelà-Ascoli theorems -- I.5 Uniformly continuous functions -- I.6 Baire category theorem -- I.7 Uniform boundedness principle -- I.8 Hahn-Banach theorem -- I.9 The weak and weak topologies -- I.10 Linear maps -- II Fubini’s theorem -- III The Riesz representation theorem (RRT) -- III.1 Riesz’s representation theorem -- III.2 RRT -- III.3 Radon measures -- III.4 Radon measures and countably additive set functions -- III.5 Support and the approximation theorem -- III.6 Haar measure -- Index of proper names -- Index of terms

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