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