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Full Description
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.
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.
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 Rn.
3.6 Jordan Measure in Rn.
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 L1 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.
9.2 The Dual of a Banach Space.
9.3 The Dual Space of Lp.
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.
Appendix A: The Banach-Tarski Theorem.
A.1 The Limits to Countable Additivity.
References.
Index.
