Real Analysis : Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis, No. 3) (Hardcover)

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 Hilbert 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 tied 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.

Also available, the first two volumes in the Princeton Lectures in Analysis:

《實分析》是普林斯頓分析講座系列中的第三冊，該系列共有四本教科書，旨在以一種整合的方式呈現分析的核心領域。本書的焦點是測度和積分理論的發展、微分和積分、希爾伯特空間以及豪斯多夫測度和分形。這本書反映了整個系列的目標：明確顯示該學科各個部分之間的有機統一性，並說明分析思想對其他數學和科學領域的廣泛應用性。

在介紹測度理論、勒貝格積分和歐幾里得空間上的微分之後，作者們通過L2理論介紹希爾伯特空間的基本概念。接下來，他們從傅立葉分析、偏微分方程和復分析中基本的實例來展示這些概念。本書的最後一部分介紹了讀者對分數維集合這一迷人主題，包括豪斯多夫測度、自我復制集合、填充曲線和貝西科維奇集合。每章都有一系列練習題，從相對簡單到較複雜，與正文直接相關。大量的提示鼓勵讀者挑戰更具挑戰性的練習題。

與該系列的其他冊數一樣，《實分析》適合本科和研究生階段的數學、物理、工程和金融等不同學科的學生閱讀。

此外，普林斯頓分析講座系列還有前兩冊可供閱讀。