Leonard F. Richardson 
Measure and Integration 
A Concise Introduction to Real Analysis

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.

Table of Content

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.

Appendix A: The Banach-Tarski Theorem.

A.1 The Limits to Countable Additivity.

References.

Index.

About the author

Leonard F. Richardson, Ph D, is Herbert Huey Mc Elveen 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.