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Chapter 1．Measure Theory 1

1 Preliminaries 1

2 The exterior measure 10

3 Measurable sets and the Lebesgue measure 16

4 Measurable functions 27

4.1 Definition and basic properties 27

4.2 Approximation by simple functions or step functions 30

4.3 Littlewood's three principles 33

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 The space L1 of integrable functions 68

3 Fubini's theorem 75

3.1 Statement and proof of the theorem 75

3.2 Applications of Fubini's theorem 80

4 A Fourier inversion formula 86

5 Exercises 89

6 Problems 95

Chapter 3．Differentiation and Integration 98

1 Differentiation of the integral 99

1.1 The Hardy-Littlewood maximal function 100

1.2 The Lebesgue differentiation theorem 104

2 Good kernels and approximations to the identity 108

3 Differentiability of functions 114

3.1 Functions of bounded variation 115

3.2 Absolutely continuous functions 127

3.3 Differentiability of jump functions 131

4 Rectifiable curves and the isoperimetric inequality 134

4.1 Minkowski content of a curve 136

4.2 Isoperimetric inequality 143

5 Exercises 145

6 Problems 152

Chapter 4．Hilbert Spaces: An Introduction 156

1 The Hilbert space L2 156

2 Hilbert spaces 161

2.1 Orthogonality 164

2.2 Unitary mappings 168

2.3 Pre-Hilbert spaces 169

3 Fourier series and Fatou's theorem 170

3.1 Fatou's theorem 173

4 Closed subspaces and orthogonal projections 174

5 Linear transformations 180

5.1 Linear functionals and the Riesz representation the-orem 181

5.2 Adjoints 183

5.3 Examples 185

6 Compact operators 188

7 Exercises 193

8 Problems 202

Chapter 5．Hilbert Spaces: Several Examples 207

1 The Fourier transform on L2 207

2 The Hardy space of the upper half-plane 213

3 Constant coefficient partial differential equations 221

3.1 Weak solutions 222

3.2 The main theorem and key estimate 224

4 The Dirichlet principle 229

4.1 Harmonic functions 234

4.2 The boundary value problem and Dirichlet's principle 243

5 Exercises 253

6 Problems 259

Chapter 6．Abstract Measure and Integration Theory 262

1 Abstract measure spaces 263

1.1 Exterior measures and Carathéodory's theorem 264

1.2 Metric exterior measures 266

1.3 The extension theorem 270

2 Integration on a measure space 273

3 Examples 276

3.1 Product neasures and a general Fubini theorem 276

3.2 Integration formula for polar coordinates 279

3.3 Borel measures on ？ and the Lebesgue-Stieltjes in-tegral 281

4 Absolute continuity of measures 285

4.1 Signed measures 285

4.2 Absolute continuity 288

5 Ergodic theorems 292

5.1 Mean ergodic theorem 294

5.2 Maximal ergodic theorem 296

5.3 Pointwise ergodic theorem 300

5.4 Ergodic measure-preserving transformations 302

6 Appendix: the spectral theorem 306

6.1 Statement of the theorem 306

6.2 Positive operators 307

6.3 Proof of the theorem 309

6.4 Spectrum 311

7 Exercises 312

8 Problems 319

Chapter 7．Hausdorff Measure and Fractals 323

1 Hausdorff measure 324

2 Hausdorff dimension 329

2.1 Examples 330

2.2 Self-similarity 341

3 Space-filling curves 349

3.1 Quartic intervals and dyadic squares 351

3.2 Dyadic correspondence 353

3.3 Construction of the Peano mapping 355

4 Besicovitch sets and regularity 360

4.1 The Radon transform 363

4.2 Regularity of sets when d ≥ 3 370

4.3 Besicovitch sets have dimension 2 371

4.4 Construction of a Besicovitch set 374

5 Exercises 380

6 Problems 385

Notes and References 389

Bibliography 391

Symbol Glossary 395

Index 397