Real variable and integration

Inhaltsverzeichnis1 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.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.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.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??.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 Linearmaps.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.