Table of contents

  1. Front Matter
    Pages i-xviii
  2. Sheldon Axler
    Pages 1-12 Open Access
  3. Sheldon Axler
    Pages 13-72 Open Access
  4. Sheldon Axler
    Pages 73-100 Open Access
  5. Sheldon Axler
    Pages 101-115 Open Access
  6. Sheldon Axler
    Pages 116-145 Open Access
  7. Sheldon Axler
    Pages 146-192 Open Access
  8. Sheldon Axler
    Pages 193-210 Open Access
  9. Sheldon Axler
    Pages 211-254 Open Access
  10. Sheldon Axler
    Pages 255-279 Open Access
  11. Sheldon Axler
    Pages 280-338 Open Access
  12. Sheldon Axler
    Pages 339-379 Open Access
  13. Sheldon Axler
    Pages 380-399 Open 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.


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

Authors and affiliations

  • Sheldon Axler
    • 1
  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA

Bibliographic information