Real Analysis Measure Theory, Integration, & Hilbert Spaces

Princeton Lectures in Analysis #3

by Elias M. Stein

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hibert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tired directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. About The Author: Elias M. Stein is Professor of Mathematics at Princeton University. Rami Shakarchi received his Ph.D. in Mathematics from Princeton University in 2002. Table of Contents Introduction Measure Theory Integration Theory

Genres: Mathematics, Textbooks, Science, Technical, Reference

426 pages, Hardcover

First published March 14, 2005

Reviews

[Jacob · Rating 4 out of 5]

This book is a much more pleasant and approachable introduction to Measure theory than the usual grad school texts (Rudin, Royden, etc.) The description is clear and detailed, and there are plenty of illustrations to augment the proofs. My two complaints are:

(i) It often references earlier volumes in Stein's series with little or no explanation of what the reference is - since many readers don't own volumes I-III, it would be useful to at least have the referenced result stated in some summary form

(ii) The book looks at integrable (L^1) and square-integrable (L^2) functions, but completely omits discussion of general L^p-spaces, which arguably underlie much of the Harmonic/Functional Analysis and PDE work of the last fifty years or more.

[Harris · Rating 4 out of 5]

Some weird, small gaps (mostly in the Fourier transform chapter) as a result of this being part of a series, but overall a good text that, for me, has been more beneficial than Rudin's or Royden's text on the same material. Sad that Lp-spaces are missing, but the chapter on Hausdorff measure and fractals more than made up for it.

[Wei Ye · Rating 5 out of 5]

Good book but so difficult for me, especially the exercises..What a book! Need to read and review it in the future

[Ron · Rating 5 out of 5]

best book on the subject, very clear and thorough explanations, enjoyable to read

[tiffanie · Rating 5 out of 5]

didn't use for real analysis sequence...used for measure & integration theory. wasn't amazing at it but enjoyed anyways