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  • Textbook
  • Open Access
  • © 2020

Measure, Integration & Real Analysis


  • Electronic version is free to the world via Springer’s Open Access program

  • Provides student-friendly explanations with ample examples and exercises throughout

  • Includes chapters on Hilbert space operators, Fourier analysis, and probability measures

  • Prepares students for further graduate studies by promoting a deep understanding of key concepts

  • Includes supplementary material:

Part of the book series: Graduate Texts in Mathematics (GTM, volume 282)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xviii
  2. Riemann Integration

    • Sheldon Axler
    Pages 1-12Open Access
  3. Measures

    • Sheldon Axler
    Pages 13-72Open Access
  4. Integration

    • Sheldon Axler
    Pages 73-100Open Access
  5. Differentiation

    • Sheldon Axler
    Pages 101-115Open Access
  6. Product Measures

    • Sheldon Axler
    Pages 116-145Open Access
  7. Banach Spaces

    • Sheldon Axler
    Pages 146-192Open Access
  8. Lp Spaces

    • Sheldon Axler
    Pages 193-210Open Access
  9. Hilbert Spaces

    • Sheldon Axler
    Pages 211-254Open Access
  10. Real and Complex Measures

    • Sheldon Axler
    Pages 255-279Open Access
  11. Linear Maps on Hilbert Spaces

    • Sheldon Axler
    Pages 280-338Open Access
  12. Fourier Analysis

    • Sheldon Axler
    Pages 339-379Open Access
  13. Probability Measures

    • Sheldon Axler
    Pages 380-399Open Access
  14. Back Matter

    Pages 400-411

About this book

This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics.

Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn.

Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability.

Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online. For errata and updates, visit


  • Measure theory textbook
  • Graduate real analysis textbook
  • Open Access
  • Riemann integration
  • Lebesgue integration
  • Product measures
  • Signed and complex measures
  • Abstract measure
  • Lebesgue Differentiation Theorem
  • Banach spaces
  • Hilbert spaces
  • Hahn–Banach Theorem
  • Hölder’s Inequality
  • Riesz Representation Theorem
  • Spectral Theorem
  • Singular Value Decomposition
  • Fourier analysis
  • Fourier series
  • Fourier transform
  • Open Access math textbook


“This textbook is addressed to students with a good background in undergraduate real analysis. Students are encouraged to actively study the theory by working on the exercises that are found at the end of each section. Definitions and theorems are printed in yellow and blue boxes, respectively, giving a clear visual aid of the content.” (Marta Tyran-Kamińska, Mathematical Reviews, May, 2021)

“The book will become an invaluable reference for graduate students and instructors. Those interested in measure theory and real analysis will find the monograph very useful since the book emphasizes getting the students to work with the main ideas rather than on proving all possible results and it contains a rather interesting selection of topics which makes the book a nice presentation for students and instructors as well.” (Oscar Blasco, zbMATH 1435.28001, 2020)

Authors and Affiliations

  • Department of Mathematics, San Francisco State University, San Francisco, USA

    Sheldon Axler

About the author

Sheldon Axler is Professor of Mathematics at San Francisco State University. He has won teaching awards at MIT and Michigan State University. His career achievements include the Mathematical Association of America’s Lester R. Ford Award for expository writing, election as Fellow of the American Mathematical Society, over a decade as Dean of the College of Science & Engineering at San Francisco State University, member of the Council of the American Mathematical Society, member of the Board of Trustees of the Mathematical Sciences Research Institute, and Editor-in-Chief of the Mathematical Intelligencer. His previous publications include the widely used textbook Linear Algebra Done Right.

Bibliographic Information

  • Book Title: Measure, Integration & Real Analysis

  • Authors: Sheldon Axler

  • Series Title: Graduate Texts in Mathematics

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Sheldon Axler 2020

  • License: CC BY-NC

  • Hardcover ISBN: 978-3-030-33142-9Published: 24 December 2019

  • eBook ISBN: 978-3-030-33143-6Published: 29 November 2019

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XVIII, 411

  • Number of Illustrations: 21 b/w illustrations, 20 illustrations in colour

  • Topics: Measure and Integration

Buy it now

Buying options

Hardcover Book USD 59.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access