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Chapter 1．Measure Theory1

1 Preliminaries1

2 The exterior measure10

3 Measurable sets and the Lebesgue measure16

4 Measurable functions27

4.1 Definition and basic properties27

4.2 Approximation by simple functions or step functions30

4.3 Littlewood's three principles33

5 The Brunn-Minkowski inequality34

6 Exercises37

7 Problems46

Chapter 2．Integration Theory49

1 The Lebesgue integral: basic properties and convergence theorems49

2 The space L1 of integrable functions68

3 Fubini's theorem75

3.1 Statement and proof of the theorem75

3.2 Applications of Fubini's theorem80

4 A Fourier inversion formula86

5 Exercises89

6 Problems95

Chapter 3．Differentiation and Integration98

1 Differentiation of the integral99

1.1 The Hardy-Littlewood maximal function100

1.2 The Lebesgue differentiation theorem104

2 Good kernels and approximations to the identity108

3 Differentiability of functions114

3.1 Functions of bounded variation115

3.2 Absolutely continuous functions127

3.3 Differentiability of jump functions131

4 Rectifiable curves and the isoperimetric inequality134

4.1 Minkowski content of a curve136

4.2 Isoperimetric inequality143

5 Exercises145

6 Problems152

Chapter 4．Hilbert Spaces: An Introduction156

1 The Hilbert space L2156

2 Hilbert spaces161

2.1 Orthogonality164

2.2 Unitary mappings168

2.3 Pre-Hilbert spaces169

3 Fourier series and Fatou's theorem170

3.1 Fatou's theorem173

4 Closed subspaces and orthogonal projections174

5 Linear transformations180

5.1 Linear functionals and the Riesz representation the-orem181

5.2 Adjoints183

5.3 Examples185

6 Compact operators188

7 Exercises193

8 Problems202

Chapter 5．Hilbert Spaces: Several Examples207

1 The Fourier transform on L2207

2 The Hardy space of the upper half-plane213

3 Constant coefficient partial differential equations221

3.1 Weak solutions222

3.2 The main theorem and key estimate224

4 The Dirichlet principle229

4.1 Harmonic functions234

4.2 The boundary value problem and Dirichlet's principle243

5 Exercises253

6 Problems259

Chapter 6．Abstract Measure and Integration Theory262

1 Abstract measure spaces263

1.1 Exterior measures and Carathéodory's theorem264

1.2 Metric exterior measures266

1.3 The extension theorem270

2 Integration on a measure space273

3 Examples276

3.1 Product neasures and a general Fubini theorem276

3.2 Integration formula for polar coordinates279

3.3 Borel measures on ？ and the Lebesgue-Stieltjes in-tegral281

4 Absolute continuity of measures285

4.1 Signed measures285

4.2 Absolute continuity288

5 Ergodic theorems292

5.1 Mean ergodic theorem294

5.2 Maximal ergodic theorem296

5.3 Pointwise ergodic theorem300

5.4 Ergodic measure-preserving transformations302

6 Appendix: the spectral theorem306

6.1 Statement of the theorem306

6.2 Positive operators307

6.3 Proof of the theorem309

6.4 Spectrum311

7 Exercises312

8 Problems319

Chapter 7．Hausdorff Measure and Fractals323

1 Hausdorff measure324

2 Hausdorff dimension329

2.1 Examples330

2.2 Self-similarity341

3 Space-filling curves349

3.1 Quartic intervals and dyadic squares351

3.2 Dyadic correspondence353

3.3 Construction of the Peano mapping355

4 Besicovitch sets and regularity360

4.1 The Radon transform363

4.2 Regularity of sets when d ≥ 3370

4.3 Besicovitch sets have dimension 2371

4.4 Construction of a Besicovitch set374

5 Exercises380

6 Problems385

Notes and References389

Bibliography391

Symbol Glossary395

Index397