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Graduate Texts in Mathematics #282
Measure, Integration & Real Analysis
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.
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.
429 pages, Kindle Edition
Published November 29, 2019
12 people are currently reading
187 people want to read
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Displaying 1 - 5 of 5 reviews
December 16, 2024
This is a great intermediate analysis book: perfect for someone who's had an introduction to real analysis (see the Supplement on Axler's site, where a pdf of the book is also available for free, for the introductory real analysis you should know going in) and wants to be very prepared for a more advanced analysis course (like Stein & Shakarchi's "Princeton Lectures in Analysis" or Reed & Simon's "Methods of Modern Mathematical Physics").
As always with Axler, the exposition is very clear, and there are many illuminating examples and exercises. The last chapter, on probability, provides an excellent "dictionary" between probability-theoretic terminology and the measure-theoretic terminology used in the rest of the book, and I found this especially useful.
As always with Axler, the exposition is very clear, and there are many illuminating examples and exercises. The last chapter, on probability, provides an excellent "dictionary" between probability-theoretic terminology and the measure-theoretic terminology used in the rest of the book, and I found this especially useful.
May 9, 2022
I am a big fan of 'Linear Algebra Done Right' by the same author. This book is more advanced theory of real analysis of twentieth century: Measure theory, Banach and Hilbert spaces, etc with applications to the Fourier Analysis and Probability Theory. This is such a wonderful and concise exposition of these ideas with a very decent problems set for self studying. Also kudos to the author for making this book public.
August 1, 2022
Read the first six chapters. The book is clear and pedagogical and succinct at the same time, so it is both a good intro and later reference. I need a break with something more practical before tackling the rest of the book.
May 27, 2025
Call me an LP space the way I’d let this book Lick my Pussy
February 3, 2024
I think this book is excellent for what it does: Speak in a familiar language, and at a comfortable pace, to a graduate mathematics student. That is to say, the audience must already be quite mathematically mature -- but if they are, then they will get a lot out of this textbook, with less struggle than they would from other textbooks on similar topics.
That said, I'm not sure I would recommend this as a teach-yourself kind of textbook. I think this text would require, for most readers with an undergraduate background, an instructor to explain and smooth over some of the presentation.
That said, I'm not sure I would recommend this as a teach-yourself kind of textbook. I think this text would require, for most readers with an undergraduate background, an instructor to explain and smooth over some of the presentation.
Displaying 1 - 5 of 5 reviews
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