《实分析》是美国数学家H.L.罗伊登（Halsey L. Royden）所著的数学教材，2010年8月由机械工业出版社出版，被斯坦福大学、哈佛大学等用作实分析课程教材。该书是《Real Analysis》第4版（2007年）的英文影印版，ISBN为9787111313052，共505页，定价49.00元，属于“经典原版书库”丛书。在豆瓣上，该书评分为8.7分（截至2023年9月）。

《实分析》（Real Analysis，第4版）是机械工业出版社出版的经典教材，包括英文影印版（2010年）、中文译本（2019年）及英文原版（2020年）等多个版本，是实分析课程的优秀教材，被国外众多著名大学（如斯坦福大学、哈佛大学等）采用。全书分为三部分：第一部分讨论一元实变量函数的Lebesgue测度与Lebesgue积分，包括一致可积分性和Vitali收敛定理等内容；第二部分讨论抽象空间——拓扑空间、度量空间、Banach空间以及Hilbert空间，新增了Hilbert空间算子章节；第三部分讨论一般测度空间上的积分，涵盖Lp空间的对偶表示及弱序列紧性。书中不仅包含数学定理和定义，而且还提出了富有启发性的问题，以便读者更深入地理解书中内容。与前一版相比，第4版新增了50%的习题，证明了一些基本结果，包括Egoroff定理和Urysohn引理，并正式引入了Borel-Cantelli引理、Chebychev不等式、快速Cauchy序列等概念。该书是实分析、测度论、泛函分析等领域的重要教材与参考书。

2010年8月由机械工业出版社出版英文原版第4版，收录于“经典原版书库”，ISBN为9787111313052，原作名《Real Analysis (4th ed., 2007)》。

2019年8月机械工业出版社出版中文译本《实分析(原书第4版)》，译者叶培新、李雪华，收录于“华章数学译丛”，ISBN为9787111630845。

2020年4月机械工业出版社出版英文影印版《实分析(英文版·原书第4版)》，收录于“华章数学原版精品系列”，ISBN为9787111646655。

本书的主要作者为美国数学家H. L. Royden（哈尔西·罗伊登）。其经典著作《实分析》前三版已帮助了几代学习数学分析的学生。本书的第4版由Patrick R. Fitzpatrick（帕特里克·R·菲茨帕特里克）参与修订。中文译本由叶培新、李雪华翻译。

《实分析》是实分析课程的优秀教材，被国外众多著名大学如斯坦福大学、哈佛大学等采用。第4版新增了50%的习题，证明了Egoroff定理和Urysohn引理等基本结果，并正式引入了Borel-Cantelli引理、Chebychev不等式等概念。

该书在豆瓣上的评分为8.7分，基于26人评价。学界对该书有不同观点，有观点指出部分定理的证明可能不够简洁，同时也有观点认为这种处理方式提供了灵活性。

Lebesgue Integration for Functions of a Single Real Variable

Preliminaries on Sets, Mappings, and Relations

Unions and Intersections of Sets

Equivalence Relations, the Axiom of Choice, and Zorn's Lemma

1 The Real Numbers: Sets. Sequences, and Functions

The Field, Positivity, and Completeness Axioms

The Natural and Rational Numbers

Countable and Uncountable Sets

Open Sets, Closed Sets, and Borel Sets of Real Numbers

Sequences of Real Numbers

Continuous Real-Valued Functions of a Real Variable

2 Lebesgne Measure

Introduction

Lebesgue Outer Measure

The o'-Algebra of Lebesgue Measurable Sets

Outer and Inner Approximation of Lebesgue Measurable Sets

Countable Additivity, Continuity, and the Borel-Cantelli Lemma

Noumeasurable Sets

The Cantor Set and the Cantor Lebesgue Function

3 LebesgRe Measurable Functions

Sums, Products, and Compositions

Sequential Pointwise Limits and Simple Approximation

Littlewood's Three Principles, Egoroff's Theorem, and Lusin's Theorem

4 Lebesgue Integration

The Riemann Integral

The Lebesgue Integral of a Bounded Measurable Function over a Set of

Finite Measure

The Lebesgue Integral of a Measurable Nonnegative Function

The General Lebesgue Integral

Countable Additivity and Continuity of Integration

Uniform Integrability: The Vifali Convergence Theorem

viii Contents

5 Lebusgue Integration: Fm'ther Topics

Uniform Integrability and Tightness: A General Vitali Convergence Theorem

Convergence in Measure

Characterizations of Riemaun and Lebesgue Integrability

6 Differentiation and Integration

Continuity of Monotone Functions

Differentiability of Monotone Functions: Lebesgue's Theorem

Functions of Bounded Variation: Jordan's Theorem

Absolutely Continuous Functions

Integrating Derivatives: Differentiating Indefinite Integrals

Convex Function

7 The Lp Spaces: Completeness and Appro~umation

Nor/ned Linear Spaces

The Inequalities of Young, HOlder, and Minkowski

Lv Is Complete: The Riesz-Fiseher Theorem

Approximation and Separability

8 The LP Spacesc Deailty and Weak Convergence

The Riesz Representation for the Dual of

Weak Sequential Convergence in Lv

Weak Sequential Compactness

The Minimization of Convex Functionals

II Abstract Spaces: Metric, Topological, Banach, and Hiibert Spaces

9. Metric Spaces: General Properties

Examples of Metric Spaces

Open Sets, Closed Sets, and Convergent Sequences

Continuous Mappings Between Metric Spaces

Complete Metric Spaces

Compact Metric Spaces

Separable Metric Spaces

10 Metric Spaces: Three Fundamental Thanreess

The Arzelb.-Ascoli Theorem

The Baire Category Theorem

The Banaeh Contraction Principle

H Topological Spaces: General Properties

Open Sets, Closed Sets, Bases, and Subbases

The Separation Properties

Countability and Separability

Continuous Mappings Between Topological Spaces

Compact Topological Spaces

Connected Topological Spaces

12 Topological Spaces: Three Fundamental Theorems

Urysohn's Lemma and the Tietze Extension Theorem

The Tychonoff Product Theorem

The Stone-Weierstrass Theorem

13 Continuous Linear Operators Between Bausch Spaces

Normed Linear Spaces

Linear Operators

Compactness Lost: Infinite Dimensional Normod Linear Spaces

The Open Mapping and Closed Graph Theorems

The Uniform Boundedness Principle

14 Duality for Normed Iinear Spaces

Linear Ftmctionals, Bounded Linear Functionals, and Weak Topologies

The Hahn-Banach Theorem

Reflexive Banach Spaces and Weak Sequential Convergence

Locally Convex Topological Vector Spaces

The Separation of Convex Sets and Mazur's Theorem

The Krein-Miiman Theorem

15 Compactness Regained: The Weak Topology

Alaoglu's Extension of Helley's Theorem

Reflexivity and Weak Compactness: Kakutani's Theorem

Compactness and Weak Sequential Compactness: The Eberlein-mulian

Theorem

Memzability of Weak Topologies

16 Continuous Linear Operators on Hilbert Spaces

The Inner Product and Orthogonality

The Dual Space and Weak Sequential Convergence

Bessers Inequality and Orthonormal Bases

bAdjoints and Symmetry for Linear Operators

Compact Operators

The Hilbert-Schmidt Theorem

The Riesz-Schauder Theorem: Characterization of Fredholm Operators

Measure and Integration: General Theory

17 General Measure Spaces: Their Propertles and Construction

Measures and Measurable Sets

Signed Measures: The Hahn and Jordan Decompositions

The Caratheodory Measure Induced by an Outer Measure

18 Integration Oeneral Measure Spaces

19 Gengral L Spaces:Completeness,Duality and Weak Convergence

20 The Construciton of Particular Measures

21 Measure and Topbogy

22 Invariant Measures

Bibiiography

index