Measure and integration : a concise introduction to real analysis / Leonard F. Richardson.

Format:

Book

Author/Creator:

Richardson, Leonard F.

Language:

English

Subjects (All):

Lebesgue integral.

Measure theory.

Mathematical analysis.

Physical Description:

1 online resource (255 p.)

Edition:

1st ed.

Place of Publication:

Hoboken, NJ : Wiley, c2008.

Language Note:

English

Summary:

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 meas

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 R3.3 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 Functions5.2.2 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 Integral7.3 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

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on metadata supplied by the publisher and other sources.

ISBN:

9786612237119

9781282237117

128223711X

9780470501153

0470501154

9780470501146

0470501146

OCLC:

457186540