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Lebesgue Integration

J.H. Williamson
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a concise development of the Lebesgue integral and measure on \(\mathbb{R}^n\), using outer and inner measure. It also includes some development of Borel sets and more general measures. The present book is an unaltered reprint of the 1962 edition from Holt Rinehart & Winston. There hasn’t been much change in Lebesgue integration since then, and the book is still up-to-date.

The book  assumes a minimal knowledge of analysis, and develops all the needed topology and set theory in the first chapter (although very concisely). There are also useful sections showing that all the familiar formulas from calculus, such as integration by parts, also work for this Lebesgue integral.

The book is easy to follow and has everything you would want to know about Lebesgue integration in a first course. The end-of-chapter exercises are numerous but not very hard.

The biggest weakness, which is probably inherent in such a concise presentation, is that the subject appears in almost total isolation from the rest of mathematics (there are some short sections on Fourier analysis and on functionals). We develop a theory of measure without knowing why we need one, then we use that to develop a theory of integration without knowing why we need that either. It’s valuable to have a brief, complete development such as this one, especially at the bargain Dover price, but I am happier with more comprehensive real analysis books that show the subject in context. Another concise book, that does a much better job of showing how Lebesgue integration fits in with the rest of analysis, is Boas’s Carus Monograph A Primer of Real Functions.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

  • Chapter 1. Sets and Functions
    • 1.1. Generalities.
    • 1.2. Countable and Uncountable Sets
    • 1.3. Sets in Rn
    • 1.4. Compactness
    • 1.5. Functions
  • Chapter 2. Lebesgue Measure
    • 2.1. Preliminaries
    • 2.2. The Class J
    • 2.3. Measurable Sets
    • 2.4. Sets of Measure Zero
    • 2.5. Borel Sets and Nonmeasurable Sets
  • Chapter 3. The Integral I
    • 3.1. Definition
    • 3.2. Elementary Properties
    • 3.3. Measurable Functions
    • 3.4. Complex and Vector Functions
    • 3.5. Other Definitions of the Integral
  • Chapter 4. The Integral II
    • 4.1. Convergence Theorems
    • 4.2. Fubini's Theorems
    • 4.3. Approximations to Integrable Functions
    • 4.4. The Lp Spaces
    • 4.5. Convergence in Mean
    • 4.6. Fourier Theory
  • Chapter 5. Calculus
    • 5.1. Change of Variables
    • 5.2. Differentiation of Integrals
    • 5.3. integration of Derivatives
    • 5.4. Integration by Parts
  • Chapter 6. More General Measures
    • 6.1. Borel Measures
    • 6.2. Signed Measures and Complex Measures
    • 6.3. Absolute Continuity
    • 6.4. Measures, Functions, and Functionals
    • 6.5. Norms, Fourier Transforms, Convolution Products
  • Index