《实分析教程(第2版)》编著者麦克唐纳。

《实分析教程》是一部备受专家好评的教科书，书中用现代的方式清晰论述了实分析的概念与理论，定理证明简明易懂，可读性强，全书共有200道例题和1200例习题。《实分析教程》的写法像一部文学读物，这在数学教科书很少见，因此阅读本书会是一种享受。

Preface

PART ONE Set Theory，Real Numbers，and Calculus

1 SET THEORY

Biography: Georg Cantor

1.1 Basic Definitions and Properties

1.2 Functions and Sets

1.3 Equivalence of Sets； Countability

1.4 Algebras，σ-Algebras，and Monotone Classes

2 THE REAL NUMBER SYSTEM AND CALCULUS

Biography: Georg Friedrich Bernhard Riemann

2.1 The Real Number System

2.2 Sequences of Real Numbers

2.3 Open and Closed Sets

2.4 Real-Valued Functions

2.5 The Cantor Set and Cantor Function

2.6 The Riemann Integral

PART TWO Measure，Integration，and DifFerentiation

3 LEBESGUE THEORY ON THE REAL LINE

Biography: Emile Felix-Edouard-Justin Borel

3.1 Borel Measurable Functions and Borel Sets

3.2 Lebesgue Outer Measure

3.3 Further Properties of Lebesgue Outer Measure

3.4 Lebesgue Measure

4  THE LEBESGUE INTEGRAL ON THE REAL LINE

Biography: Henri Leon Lebesgue

4.1 The Lebesgue Integral for Nonnegative Functions

4.2 Convergence Properties of the Lebesgue Integral for

Nonnegative Functions

4.3 The General Lebesgue Integral

4.4 Lebesgue Almost Everywhere

5 ELEMENTS OF MEASURE THEORY

Biography: Constantin Carath~odory

5.1 Measure Spaces

5.2 Measurable Functions

5.3 The Abstract Lebesgue Integral for Nonnegative Functior

5.4 The General Abstract Lebesgue Integral

5.5 Convergence in Measure

6 EXTENSIONS TO MEASURES AND PRODUCT MEASURE

Biography: Guido Fubini

6.1 Extensions to Measures

6.2 The Lebesgue-Stieltjes Integral

6.3 Product Measure Spaces

6.4 Iteration of Integrals in Product Measure Spaces

7 ELEMENTS OF PROBABILITY

Biography: Andrei Nikolaevich Kolmogorov

7.1 The Mathematical Model for Probability

7.2 Random Variables

7.3 Expectation of Random Variables

7.4 The Law of Large Numbers

8 DIFFERENTIATION AND ABSOLUTE CONTINUITY

Biography: Giuseppe Vitafi

8.1 Derivatives and Dini-Derivates

8.2 Functions of Bounded Variation

8.3 The Indefinite Lebesgne Integral

8.4 Absolutely Continuous Functions

9 SIGNED AND COMPLEX MEASURES

Biography: Johann Radon

9.1 Signed Measures

9.2 The Radon-Nikodym Theorem

9.3 Signed and Complex Measures

9.4 Decomposition of Measures

9.5 Measurable Transformati6ns and the General

Change-of-Variable Formula

PART THREE

Topological, Metric, and Normed Spaces

10 TOPOLOGIES, METRICS, AND NORMS

Biography: Felix Hausdorff

10.1 Introduction to Topological Spaces

10.2 Metrics and Norms

10.3 Weak Topologies

10.4 Closed Sets, Convergence, and Completeness

10.5 Nets and Continuity

10.5 Separation Properties

10.7 Connected Sets

11 SEPARABILITY AND COMPACTNESS

Biography: Maurice Frechet

11.1 Separability, Second Countability, and Metrizability

11.2 Compact Metric Spaces

11.3 Compact Topological Spaces

11.4 Locally Compact Spaces

11.5 Function Spaces

12 COMPLETE AND COMPACT SPACES

Biography: Marshall Harvey Stone

12.1 The Baire Category Theorem

12.2 Contractions of Complete Metric Spaces

12.3 Compactness in the Space C(□, A)

12.4 Compactness of Product Spaces

12.5 Approximation by Functions from a Lattice

12.5 Approximation by Functions from an Algebra

13 HILBERT SPACES AND BANACH SPACES

Biography: David Hilbert

13.1 Preliminaries on Normed Spaces

13.2 Hilbert Spaces

13.3 Bases and Duality in Hilbert Spaces

13.4 □-Spaces

13.5 Nonnegative Linear Functionals on C(□)

13.5 The Dual Spaces of C(□) and C0(□)

14 NORMED SPACES AND LOCALLY CONVEX SPACES

Biography: Stefan Banach

14.1 The Hahn-Banach Theorem

14.2 Linear Operators on Banach Spaces

14.3 Compact Self-Adjoint Operators

14.4 Topological Linear Spaces

14.5 Weak and Weak* Topologies

14.5 Compact Convex Sets

PART FOUR

Harmonic Analysis, Dynamical Systems, and Hausdorff Measure

15 ELEMENTS OF HARMONIC ANALYSIS

Biography: Ingrid Daubechies

15.1 Introduction to Fourier Series

15.2 Convergence of Fourier Series

15.3 The Fourier Transform

15.4 Fourier Transforms of Measures

15.5 □-Theory of the Fourier Transform

15.5 Introduction to Wavelets

15.7 Orthonormal Wavelet Bases; The Wavelet Transform

15 MEASURABLE DYNAMICAL SYSTEMS Biography: Claude E/wood Shannon

16.1 Introduction and Examples

16.2 Ergodic Theory

16.3 Isomorphism of Measurable Dynamical Systems; Entropy

16.4 The Kolmogorov-Sinai Theorem; Calculation of Entropy

17 HAUSDORFF MEASURE AND FRACTALS Biography: Benoit B. Mandelbrot

17.1 Outer Measure and Measurability

17.2 Hausdorff Measure

17.3 Hausdorff Dimension and Topological Dimension

17.4 Fractals

Index