Real Variable and Integration: with Historical Notes: Mathematische Leitfäden

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 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.