Ebook
Measure and Integration: A Concise Introduction to Real AnalysisISBN: 9780470501146
256 pages
July 2009

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 RadonNikodym theorem
 Lp spaces, continuousfunction spaces, and duality theorems
 Translationinvariant 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.
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 RadonNikodym 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 L^{2}.
9.5 RieszMarkovSaksKakutani Theorem.
10 Translation Invariance in Real Analysis.
10.1 An Orthonormal Basis for L^{2}(T).
10.2 Closed Invariant Subspaces of L^{2}(T).
10.3 Schwartz Functions: Fourier Transform and Inversion.
10.4 Closed, Invariant Subspaces of L^{2}(R).
10.5 Irreducibility of L^{2}(R) Under Translations and Rotations.
Appendix A: The BanachTarski Theorem.
A.1 The Limits to Countable Additivity.
References.
Index.

Provides an organized and logical selection of topics that prepares readers for subsequent courses in functional analysis and other core topics in pure and applied mathematics

Serves as a concise, yet thorough, introduction to measure and integration, and the complete text can be covered in a onesemester course in real analysis.

Presents the essentials of measure, integration, differentiation, and L^{p} spaces and clearly develops their interrelations

Provides coverage of the BanachTarski theorem and culminates in the famous RieszMarkovSaksKakutani theorem

Contains numerous relevant and interesting exercises as well as supplementary topical coverage intended to stimulate and motivate further learning

Strikes a balance between both fundamental and specialized topics and successfully provides an introduction to measure and integration that does not focus too much attention on the possible embellishments or ramifications of various topics