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

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。