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Birkhäuser

Real Analysis

  • Textbook
  • © 2016

Overview

Authors:
  1. Peter A Loeb

    1. University of Illinois, Urbana, USA

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  • Written by one of the leading scholars in the field

  • Includes a novel presentation of differentiation and absolute continuity using a local maximum function, resulting in an exposition that is both simpler and more general than the traditional approach

  • Theorems are stated for Lebesgue and Borel measures, with a note indicating when the same proof works only for Lebesgue measures

  • Appendices cover additional material, including theorems for higher dimensions and a short introduction to nonstandard analysis

  • Includes supplementary material: sn.pub/extras

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Set Theory and Numbers

    • Peter A. Loeb
    Pages 1-23

  3. Measure on the Real Line

    • Peter A. Loeb
    Pages 25-43

  4. Measurable Functions

    • Peter A. Loeb
    Pages 45-56

  5. Integration

    • Peter A. Loeb
    Pages 57-77

  6. Differentiation and Integration

    • Peter A. Loeb
    Pages 79-93

  7. General Measure Spaces

    • Peter A. Loeb
    Pages 95-108

  8. Introduction to Metric and Normed Spaces

    • Peter A. Loeb
    Pages 109-125

  9. Hilbert Spaces

    • Peter A. Loeb
    Pages 127-145

  10. Topological Spaces

    • Peter A. Loeb
    Pages 147-178

  11. Measure Construction

    • Peter A. Loeb
    Pages 179-190

  12. Banach Spaces

    • Peter A. Loeb
    Pages 191-219

  13. Back Matter

    Pages 221-274
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Keywords

About this book

This textbook is designed for a year-long course in real analysis taken by beginning graduate and advanced undergraduate students in mathematics and other areas such as statistics, engineering, and economics. Written by one of the leading scholars in the field, it elegantly explores the core concepts in real analysis and introduces new, accessible methods for both students and instructors.


The first half of the book develops both Lebesgue measure and, with essentially no additional work for the student, general Borel measures for the real line. Notation indicates when a result holds only for Lebesgue measure. Differentiation and absolute continuity are presented using a local maximal function, resulting in an exposition that is both simpler and more general than the traditional approach.


The second half deals with general measures and functional analysis, including Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To correctly discuss weak limits of measures, one needs the notion of a topological space rather than just a metric space, so general topology is introduced in terms of a base of neighborhoods at a point. The development of results then proceeds in parallel with results for metric spaces, where the base is generated by balls centered at a point. The text concludes with appendices on covering theorems for higher dimensions and a short introduction to nonstandard analysis including important applications to probability theory and mathematical economics. 


Reviews

“This is a very well written book. Its chapters are no more than 20 pages each, which allows students to easily work through them. The proofs are sharp, lively and rigorously written. … I recommend it, not only, to any student who wants to study or do research on measures and integration or who will use these notions in studying other subjects; but, also to every mathematics department’s library.” (Salim Salem, MAA Reviews, July, 2018)

Authors and Affiliations

  • University of Illinois, Urbana, USA

    Peter A Loeb

About the author

Peter Loeb is an emeritus Professor of Mathematics at the University of Illinois in Champaign-Urbana. His research is centered on problems of real analysis and applications of model theory to real analysis.

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