Measure and Integration (PDF)

Preface. Acknowledgments. Introduction. 1 History of the Subject. 1.1 History of the Idea. 1.2 Deficiencies of the Riemann Integral. 1.3 Motivation for the Lebesgue Integral. 2 Fields, Borel Fields and Measures. 2.1 Fields, Monotone Classes, and Borel Fields. 2.2 Additive Measures. 2.3 Carathéodory Outer Measure. 2.4 E. Hopf's Extension Theorem. 3 Lebesgue Measure. 3.1 The Finite Interval [-N,N). 3.2 Measurable Sets, Borel Sets, and the Real Line. 3.3 Measure Spaces and Completions. 3.4 Semimetric Space of Measurable Sets. 3.5 Lebesgue Measure in R¯n. 3.6 Jordan Measure in R¯n. 4 Measurable Functions. 4.1 Measurable Functions. 4.2 Limits of Measurable Functions. 4.3 Simple Functions and Egoroff's Theorem. 4.4 Lusin's Theorem. 5 The Integral. 5.1 Special Simple Functions. 5.2 Extending the Domain of the Integral. 5.3 Lebesgue Dominated Convergence Theorem. 5.4 Monotone Convergence and Fatou's Theorem. 5.5 Completeness of L¯1 and the Pointwise Convergence Lemma. 5.6 Complex Valued Functions. 6 Product Measures and Fubini's Theorem. 6.1 Product Measures. 6.2 Fubini's Theorem. 6.3 Comparison of Lebesgue and Riemann Integrals. 7 Functions of a Real Variable. 7.1 Functions of Bounded Variation. 7.2 A Fundamental Theorem for the Lebesgue Integral. 7.3 Lebesgue's Theorem and Vitali's Covering Theorem. 7.4 Absolutely Continuous and Singular Functions. 8 General Countably Additive Set Functions. 8.1 Hahn Decomposition Theorem. 8.2 Radon-Nikodym Theorem. 8.3 Lebesgue Decomposition Theorem. 9. Examples of Dual Spaces from Measure Theory. 9.1 The Banach Space L¯p. 9.2 The Dual of a Banach Space. 9.3 The Dual Space of L¯p. 9.4 Hilbert Space, Its Dual, and L². 9.5 Riesz-Markov-Saks-Kakutani Theorem. 10 Translation Invariance in Real Analysis. 10.1 An Orthonormal Basis for L²(T). 10.2 Closed Invariant Subspaces of L²(T). 10.3 Schwartz Functions: Fourier Transform and Inversion. 10.4 Closed, Invariant Subspaces of L²(R). 10.5 Irreducibility of L²(R) Under Translations and Rotations.

Leonard F. Richardson, PhD, is Herbert Huey McElveen Professor and Director of Graduate Studies in Mathematics at Louisiana State University, where he is also Assistant Chair of the Department of Mathematics. Dr. Richardson's research interests include harmonic analysis, homogeneous spaces, and representation theory. He is the author of Advanced Calculus: An Introduction to Linear Analysis, also published by Wiley.

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