本书由在国际上享有盛誉普林斯大林顿大学教授Stein等撰写而成，作为一部优秀的教材，内容不仅涵盖了分析学的基本内容和技巧，还介绍了一些从事其他领域的研究工作所必需的基础知识。此外，教材中的大量习题，能够进一步拓展思维，从而易于更加深入地了解这些内容背后的真实想法。本书适用于理工类专业及相关专业的研究生。

本书由在国际上享有盛誉普林斯大林顿大学教授Stein等撰写而成，是一部为数学及相关专业大学二年级和三年级学生编写的教材，理论与实践并重。为了便于非数学专业的学生学习，全书内容简明、易懂,读者只需掌握微积分和线性代数知识。关于本书的详细介绍，请见“影印版前言”。

本书已被哈佛大学和加利福尼亚理工学院选为教材。与本书相配套的教材《傅立叶分析导论》和《复分析》也已影印出版。

Foreword

Introduction

1 Fourier series: completion

2 Limits of continuous functions

3 Length of curves

4 Differentiation and integration

5 The problem of measure

Chapter 1. Measure Theory

1 Preliminaries

2 The exterior measure

3 Measurable sets and the Lebesgue measure

4 Measurable functions

　4.1 Definition and basic properties

　4.2 Approximation by simple functions or step functions

　4.3 Littlewood's three principles

5* The Brunn-Minkowski inequality

6 Exercises

7 Problems

Chapter 2. Integration Theory

1 The Lebesgue integral: basic properties and convergence theorems

2 The space L1 of integrable functions

3 Fubini's theorem

　3.1 Statement and proof of the theorem

　3.2 Applications of Fubini's theorem

4* A Fourier inversion formula

5 Exercises

6 Problems

Chapter 3. Differentiation and Integration

1 Differentiation of the integral

　1.1 The Hardy-Littlewood maximal function

　1.2 The Lebesgue differentiation theorem

2 Good kernels and approximations to the identity

3 Differentiability of functions

　3.1 Functions of bounded variation

　3.2 Absolutely continuous functions

　3.3 Differentiability of jump functions

4 Rectifiable curves and the isoperimetric inequality

　4.1 Minkowski content of a curve

　4.2* Isoperimetrie inequality

5 Exercises

6 Problems

Chapter 4. Hilbert Spaces: An Introduction

1 The Hilbert space L2

2 Hilbert spaces

　2.1 Orthogonality

　2.2 Unitary mappings

　2.3 Pre-Hilbert spaces

3 Fourier series and Fatou's theorem

　3.1 Fatou's theorem

4 Closed subspaees and orthogonal projections

5 Linear transformations

　5.1 Linear flmetionals and the Riesz representation the-orem

　5.2 Adjoints

　5.3 Examples

6 Compact operators

7 Exercises

8 Problems

Chapter 5. Hilbert Spaces: Several Examples

1 The Fourier transform on L2

2 The Hardy space of the upper half-plane

3 Constant coefficient partial differential equations

　3.1 Weak solutions

　3.2 The main theorem and key estimate

4* The Dirichlet principle

　4.1 Harmonic functions

　4.2 The boundary value problem and Diriehlet's principle

5 Exercises

6 Problems

Chapter 6.Abstract Measure and Integration Theory

1 Abstract measure spaces

　1.1 Exterior measures and Carathdodory’s theorem

　1.2 Metric exterior measures

　1.3 The extension theorem

2 Integration on a measure space

3 Examples

　3.1 Product measures and a general Fubini theorem

　3.2 Integration formula for polar coordinates

　3.3 Borel measures on R and the Lebesgue-Stieltjes in.tegral

4 Absolute continuity of measures

　4.1 Signed measures

　4.2 Absolute continuity

5* Ergodic theorems

　5.1 Mean ergodic theorem

　5.2 Maximal ergodic theorem

　5.3 Pointwise ergodic theorem

　5.4 Ergodic measure—preserving transformations

6* Appendix：the spectral theorem

　6.1 Statement of the theorem

　6.2 Positive operators

　6.3 Proof of the theorem

　6.4 Spectrum

7 Exercises

8 Problems

Chapter 7.Hausdorff Measure and Fractals

1 Hansdorff measure

2 Hausdorff dimension

　2.1 Examples

　2.2 Self-similarity

3 Space-filling curves

　3.1 Quartic intervals and dyadic squares

　3.2 Dyadic cOrresDOndence

　3.3 Construction of the Peano mapping

4* Besicovitch sets and regularity

　4.1 The Radon transform

　4.2 Regularity of sets when d≥3

　4.3 Besicovitch sets have dimension 2

　4.4 Construction of a Besicovitch set

5 Exercise

6 Problems

Notes and References

Bibliography

Symbol Glossary

Index