Elias M.Stein、Rami Shakarchi所著的《实分析》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。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 Prelhninaries

 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 ofintegrable 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 ofjump functions

 4 Rectifiable curves and the isoperimetric inequality

4.1 Minkowski content of a curve

4.2 Isoperimetric 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 subspaces and orthogonal projections

 5 Linear transformations

5.1 Linear functionals and the Riesz representation theorem

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 Weaak solutions

3.2 The main theorem and key estimate

 4 The Dirichlet principle

4.1 Harmonic functions

4.2 The boundary value problem and Dirichlet's principle

 5 Exercises

 6 Problems

Chapter 6. Abstract Measure and Integration Theory

 1 Abstract measure spaces

1.1 Exterior measures and Carathodory's theorem

1.2 Metric exterior measures

1.3 The extension theorem

 2 Integration o 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 and the Lebesgue-Stieltjes integral

 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 Hausdorff 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 correspondence

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 Exercises

 6 Problems

Notes and References

Bibliography

Symbol Glossary

Index