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 Analysisthat is freely available online. For errata and updates, visit https://measure.axler.net/

Table of Contents

1 Riemann Integration

2 Measures

3 Integration

4 Differentiation

5 Product Measures

6 Banach Spaces

7 Lp Spaces

8 Hilbert Spaces

9 Real and Complex Measures

10 Linear Maps on Hilbert Spaces

11 Fourier Analysis

12 Probability Measures

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.