Measure and Integration : A Concise Introduction to Real Analysis.
Title:
Measure and Integration : A Concise Introduction to Real Analysis.
Author:
Richardson, Leonard F.
ISBN:
9780470501146
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (255 pages)
Contents:
Measure and Integration: A Concise Introduction to Real Analysis -- CONTENTS -- 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 -- 2.4.1 Fields, σ-Fields, and Measures Inherited by a Subset -- 3 Lebesgue Measure -- 3.1 The Finite Interval [-N, N ) -- 3.2 Measurable Sets, Borel Sets, and the Real Line -- 3.2.1 Lebesgue Measure on R -- 3.3 Measure Spaces and Completions -- 3.3.1 Minimal Completion of a Measure Space -- 3.3.2 A Nonmeasurable Set -- 3.4 Semimetric Space of Measurable Sets -- 3.5 Lebesgue Measure in Rn -- 3.6 Jordan Measure in Rn -- 4 Measurable Functions -- 4.1 Measurable Functions -- 4.1.1 Baire Functions of Measurable Functions -- 4.2 Limits of Measurable Functions -- 4.3 Simple Functions and Egoroff's Theorem -- 4.3.1 Double Sequences -- 4.3.2 Convergence in Measure -- 4.4 Lusin's Theorem -- 5 The Integral -- 5.1 Special Simple Functions -- 5.2 Extending the Domain of the Integral -- 5.2.1 The Class L+ of Nonnegative Measurable Functions -- 5.2.2 The Class L of Lebesgue Integrable Functions -- 5.2.3 Convex Functions and Jensen's Inequality -- 5.3 Lebesgue Dominated Convergence Theorem -- 5.4 Monotone Convergence and Fatou's Theorem -- 5.5 Completeness of L1(X, Afr, μ ) 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 Lp(X, Afr, μ ) -- 9.2 The Dual of a Banach Space -- 9.3 The Dual Space of Lp(X, Afr, μ ) -- 9.4 Hilbert Space, Its Dual, and L2( X, Afr, μ ) -- 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) -- 10.2.1 Integration of Hilbert Space Valued Functions -- 10.2.2 Spectrum of a Subset of L2(T) -- 10.3 Schwartz Functions: Fourier Transform and Inversion -- 10.4 Closed, Invariant Subspaces of L2(R) -- 10.4.1 The Fourier Transform in L2(R) -- 10.4.2 Translation-Invariant Subspaces of L2(R) -- 10.4.3 The Fourier Transform and Direct Integrals -- 10.5 Irreducibility of L2(R) Under Translations and Rotations -- 10.5.1 Position and Momentum Operators -- 10.5.2 The Heisenberg Group -- Appendix: The Banach-Tarski Theorem -- A.1 The Limits to Countable Additivity -- References -- Index.
Abstract:
A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis Measure and Integration: A Concise Introduction to Real Analysis presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis. The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems Lebesgue measure on the line and in Euclidean space Measurable functions, Egoroff's theorem, and Lusin's theorem Convergence theorems for integrals Product measures and Fubini's theorem Differentiation theorems for functions of real variables Decomposition theorems for signed measures Absolute continuity and the Radon-Nikodym theorem Lp spaces, continuous-function spaces, and duality theorems Translation-invariant subspaces of L2 and applications The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, Measure and Integration: A Concise Introduction to Real Analysis is a useful book for courses in real analysis at the graduate level. It is also a valuable
reference for practitioners in the mathematical sciences.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Electronic Access:
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