Princeton Lectures in Analysis系列中的第三本，Stein的力作。

Princeton Lectures in Analysis这套丛书，是作者们为普林斯顿大学本科生写的分析学系列教材，旨在对读过大学微积分以及线性代数的学生或科研工作者，提供一套分析学的核心教材。他们依次为：(I)傅立叶分析导论，(II)复分析，(III)实分析，(IV)泛函分析。第四本还没有出，如果需要我可以把前面两本一起传上来。从2000年起，普林斯顿大学的分析学就开始使用这套教材，而现在越来越多的北美高校开始采用这套教材。

就不多说了，需要的人自然知道这本书，不知道的google一下也就知道这套教材的影响了。下面是书的目录：

Elias M. Stein & Rami Shakarchi  | 2005 | 392 pp.

TABLE OF CONTENTS:

Foreword vii

Introduction xv
Chapter 1. Measure Theory 1
     1 Preliminaries 1
     2 The exterior measure 10
     3 Measurable sets and the Lebesgue measure 16
     4 Measurable functions 7
     5* The Brunn-Minkowski inequality 34
     6 Exercises 37
     7 Problems 46

Chapter 2: Integration Theory 49
      1 The Lebesgue integral: basic properties and convergence theorems 49
      2 Thespace L 1 of integrable functions 68
      3 Fubini's theorem 75
      4* A Fourier inversion formula 86
      5 Exercises 89
      6 Problems 95

Chapter 3: Differentiation and Integration 98
      1 Differentiation of the integral 99
      2 Good kernels and approximations to the identity 108
      3 Differentiability of functions 114
      4 Rectifiable curves and the isoperimetric inequality 134
      5 Exercises 145
      6 Problems 152

Chapter 4: Hilbert Spaces: An Introduction 156
      1 The Hilbert space L 2 156
      2 Hilbert spaces 161
      3 Fourier series and Fatou's theorem 170
      4 Closed subspaces and orthogonal projections 174
      5 Linear transformations 180
      6 Compact operators 188
      7 Exercises 193
      8 Problems 202

Chapter 5: Hilbert Spaces: Several Examples 207
      1 The Fourier transform on L 2 207
      2 The Hardy space of the upper half-plane 13
      3 Constant coefficient partial differential equations 221
      4* The Dirichlet principle 9
      5 Exercises 253
      6 Problems 259

Chapter 6: Abstract Measure and Integration Theory 262
      1 Abstract measure spaces 263
      2 Integration on a measure space 273
      3 Examples 276
      4 Absolute continuity of measures 285
      5* Ergodic theorems 292
      6* Appendix: the spectral theorem 306
      7 Exercises 312
      8 Problems 319

Chapter 7: Hausdorff Measure and Fractals 323
      1 Hausdorff measure 324
      2 Hausdorff dimension 329
      3 Space-filling curves 349
      4* Besicovitch sets and regularity 360
      5 Exercises 380
      6 Problems 385

Notes and References 389
Bibliography 391
Symbol Glossary 395
Index 397

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