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 functio
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 functio
4.1 Definition and basic properties
4.2 Approximation by simple functio or step functio
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 functio
3 Fubini''s theorem
3.1 Statement and proof of the theorem
3.2 Applicatio of Fubini''s theorem
4* A Fourier inveion 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 approximatio to the identity
3 Differentiability of functio
3.1 Functio of bounded variation
3.2 Absolutely continuous functio
3.3 Differentiability ofjump functio
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 projectio
5 Linear traformatio
5.1 Linear functionals and the Riesz representation theorem
5.2 Adjoints
5.3 Examples
6 Compact operato
7 Exercises
8 Problems
Chapter 5. Hilbert Spaces： Several Examples
1 The Fourier traform on L2
2 The Hardy space of the upper half-plane
3 Cotant coefficient partial differential equatio
3.1 Weaak solutio
3.2 The main theorem and key estimate
4 The Dirichlet principle
4.1 Harmonic functio
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 exteion 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 traformatio
6* Appendix: the spectral theorem
6.1 Statement of the theorem
6.2 Positive operato
6.3 Proof of the theorem
6.4 Spectrum
7 Exercises
8 Problems
Chapter 7. Hausdorff Measure and Fractals
1 Hausdorff measure
2 Hausdorff dimeion
2.1 Examples
2.2 Self-similarity
3 Space-filling curves
3.1 Quartic intervals and dyadic squares
3.2 Dyadic correspondence
3.3 Cotruction of the Peano mapping
4* Besicovitch sets and regularity
4.1 The Radon traform
4.2 Regularity of sets when d ≥ 3
4.3 Besicovitch sets have dimeion 2
4.4 Cotruction of a Besicovitch set
5 Exercises
6 Problems
Notes and References
Bibliography
Symbol Glossary
Index
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