Theory of Functions of a Real Variable (3rd Edition)

Name: Theory of Functions of a Real Variable (3rd Edition)
ISBN: 9787301276471

Zhou MinqiangPeking University Press · ChinaAug 2025

4.4

Word count: 109K words
Chapters: 62
ISBN: 9787301276471

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Summary

As a foundational course for learning modern analytical mathematics, real analysis has long had a relatively clear exposition and a mature system. From a teaching perspective, however, there is still much to be done in how to present its rich content and convey it smoothly to beginners. This revision mainly involves some adjustments to the content. First, some exercises that are too difficult have been deleted and some exercises better suited to students' understanding have been added. Second, some outdated content has been cut and some new content suited to current developments has been added. ...............................................................................................

Editor's review

Real Analysis (3rd Edition) is the latest revision of Professor Zhou Minqiang's classic textbook. While maintaining the rigorous system of modern analytical mathematics, it strives to improve readability for beginners. This revision deletes some overly difficult exercises, adds exercises better suited to students' level of understanding, cuts outdated content, and adds new content, achieving a better balance between theoretical depth and teaching friendliness. The book emphasizes the core ideas of real analysis, progressing layer by layer from measure theory to integration theory, and is suitable as a textbook for upper-level undergraduate mathematics majors or graduate foundation courses. For self-learners, the revised exercise gradient helps build analytical intuition step by step; for instructors, the adjusted content organization makes it easier to plan lectures and problem sessions. When reading, it is advisable to combine classroom notes with exercise work, pay attention to distinguishing definitions and theorem conditions, and avoid memorizing conclusions while ignoring proof ideas.

Who it's for

A good fit for

Upper-level undergraduate mathematics majors currently taking a real analysis course
Readers preparing for graduate entrance exams or graduate study in analysis who need a systematic review of measure and integration theory
University mathematics instructors who want a revised, easier-to-teach textbook for organizing courses
Science and engineering readers with some background in mathematical analysis who want to self-study real analysis

Not a good fit for

Readers who only know elementary calculus and have not yet had rigorous training in mathematical analysis
Engineering-oriented readers who want to quickly master computational techniques in real analysis while avoiding proofs
Readers who need many application cases and numerical experiments with a strong engineering focus

Reading tips

First review limits, continuity, and Riemann integration from mathematical analysis before entering the measure theory chapters.
For each section, first clarify definitions and theorem conditions, then prove key lemmas by hand, and finally do the revised exercises.
For more abstract chapters, make concept maps linking measure, measurable functions, and integration.
Use instructor problem sessions or study group discussions to resolve sticking points in proofs promptly.

About the publisher

Peking University Press China · Founded in 1979

Peking University Press is a comprehensive university press whose publishing areas cover classical literature, popular science, social culture, world history, Western philosophy, and culture. Its catalogue contains 6,031 titles, with representative books including A Dream of Red Mansions, Relativity: The Special and General Theory (15th Edition) (with Einstein's classic papers on unified field theory and other topics), Elements, Conics, The Aristocratic Families of the Eastern Jin, The Philosophy of Mythology (Volume I), The Philosophy of Mythology (Volume II), and The Origin of Continents and Oceans. Representative authors include Cao Xueqin, Zhou Ruchang, Albert Einstein, Euclid, Apollonius, Tian Yuqing, Schelling, and Xian Gang. The press has developed distinctive strengths in classical Chinese literature and in the translation of Western philosophical classics, while also covering scientific classics and social and cultural studies. As public information is limited, this description is based only on the books in the catalogue and offers no further extrapolation.

版权信息
内容简介
作者简介
第3版序言
第1版前言
积分论评述
第一章集合与点集
§1.1 集合与子集合
§1.2 集合的运算
§1.3 映射与基数
§1.4 Rn中点与点之间的距离·点集的极限点
§1.5 Rn中的基本点集：闭集开集·Borel集·Cantor集
§1.6 点集间的距离
习题 1
注记
第二章 Lebesgue测度
§2.1 点集的Lebesgue外测度
§2.2 可测集与测度
§2.3 可测集与Borel集的关系
§2.4 正测度集与矩体的关系
§2.5 不可测集
§2.6 连续变换与可测集
习题 2
注记
第三章可测函数
§3.1 可测函数的定义及其性质
§3.2 可测函数列的收敛
§3.3 可测函数与连续函数的关系
习题 3
注记
第四章 Lebesgue积分
§4.1 非负可测函数的积分
§4.2 一般可测函数的积分
§4.3 可积函数与连续函数的关系
§4.4 Lebesgue积分与Riemann积分的关系
§4.5 重积分与累次积分的关系
习题 4
注记
第五章微分与不定积分
§5.1 单调函数的可微性
§5.2 有界变差函数
§5.3 不定积分的微分
§5.4 绝对连续函数与微积分基本定理
§5.5 分部积分公式与积分中值公式
§5.6 R上的积分换元公式
习题 5
注记
第六章 Lp 空间
§6.1 Lp空间的定义与不等式
§6.2 Lp空间的结构
§6.3 L2内积空间
§6.4 Lp空间的范数公式
§6.5 卷积
∗§6.6 弱收敛
习题 6
注记
附录
(Ⅰ)Rn上不定积分的微分定理与积分换元公式
(Ⅱ) 勒贝格传
(Ⅲ) 部分思考题及习题的参考解答或提示
(Ⅳ) 人名表
参考书目

FAQ

What are the main adjustments in the third edition compared with the previous edition?

According to the publisher's note, this revision mainly deletes some overly difficult exercises and adds exercises better suited to students' understanding; it also cuts some outdated content and adds new content suited to current developments, making the textbook easier to use for teaching.

Can I read this book directly without having studied mathematical analysis?

Not really recommended. Real analysis builds on analytical concepts such as limits, continuity, and Riemann integration and requires some training in proofs. If your foundation in mathematical analysis is weak, it is best to fill in the relevant chapters first; otherwise you may struggle in the measure theory and measurable functions sections.

Is this book suitable for self-study?

It can be used for self-study, but it requires strong self-discipline and patience with proofs. It is advisable to proceed in chapter order, pay attention to distinguishing definitions and theorem conditions, practice gradually with the revised exercises, and consult other textbooks for comparison when encountering abstract concepts.

How does studying this book help with subsequent courses?

Real analysis is the foundation for courses such as functional analysis, probability theory, and harmonic analysis. After mastering measure and integration theory, you will find concepts such as spaces, convergence, and integrability easier to understand in these后续 courses.

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About the author

Zhou Minqiang Author · Chinese · 1933

Zhou Minqiang, born in December 1933 in Zhuji, Zhejiang, is a Chinese mathematician and professor at the School of Mathematical Sciences, Peking University. He served as director of the Functional Theory Teaching and Research Section in the Department of Mathematics at Peking University and as an editorial board member of Acta Mathematica Sinica and Mathematical Bulletin. His research focused on harmonic analysis and mathematical analysis, and he compiled or authored more than ten textbooks, including Functions of Real Variables and Mathematical Analysis, for which he received a teaching excellence award. His works, notably the textbook on real variable function theory and its accompanying problem-solving guide, emphasize rigorous theoretical structure and carefully integrated exercises, and they continue to influence instruction in mathematical analysis. Public information is limited, and further details of his life and academic activities remain insufficient.