Chapter 1. Measure Theory Page 1 1. Preliminaries Page 1 2. The exterior measure Page 10 3. Measurable sets and the Lebesgue measure Page 16 4. Measurable functions Page 27 4.1. Definition and basic properties Page 27 4.2. Approximation by simple functions or step functions Page 30 4.3. Littlewood's three principles Page 33 5*. The Brunn-Minkowski inequality Page 34 6. Exercises Page 37 7. Problems Page 46 Chapter 2. Integration Theory Page 49 1. The Lebesgue integral: basic properties and convergence theorems Page 49 2. The space L^1 of integrable functions Page 68 3. Fubini's theorem Page 75 3.1. Statement and proof of the theorem Page 75 3.2. Applications of Fubini's theorem Page 80 4*. A Fourier inversion formula Page 86 5. Exercises Page 89 6. Problems Page 95 Chapter 3. Differentiation and Integration Page 98 1. Differentiation of the integral Page 99 1.1. The Hardy-Littlewood maximal function Page 100 1.2. The Lebesgue differentiation theorem Page 104 2. Good kernels and approximations to the identity Page 108 3. Differentiability of functions Page 114 3.1. Functions of bounded variation Page 115 3.2. Absolutely continuous functions Page 127 3.3. Differentiability of jump functions Page 131 4. Rectifiable curves and the isoperimetric inequality Page 134 4.1*. Minkowski content of a curve Page 136 4.2*. Isoperimetric inequality Page 143 5. Exercises Page 145 6. Problems Page 152 Chapter 4. Hilbert Spaces: An Introduction Page 156 1. The Hilbert space L^2 Page 156 2. Hilbert spaces Page 161 2.1. Orthogonality Page 164 2.2. Unitary mappings Page 168 2.3. Pre-Hilbert spaces Page 169 3. Fourier series and Fatou's theorem Page 170 3.1. Fatou's theorem Page 173 4. Closed subspaces and orthogonal projections Page 174 5. Linear transformations Page 180 5.1. Linear functionals and the Riesz representation theorem Page 181 5.2. Adjoints Page 183 5.3. Examples Page 185 6. Compact operators Page 188 7. Exercises Page 193 8. Problems Page 202 Chapter 5. Hilbert Spaces: Several Examples Page 207 207. 2 The Hardy space of the upper half-plane Page 213 3. Constant coefficient partial differential equations Page 221 3.1. Weak solutions Page 222 3.2. The main theorem and key estimate Page 224 4*. The Dirichlet principle Page 229 4.1. Harmonic functions Page 234 4.2. The boundary value problem and Dirichlet's principle Page 243 5. Exercises Page 253 6. Problems Page 259 Chapter 6. Abstract Measure and Integration Theory Page 262 1. Abstract measure spaces Page 263 1.1. Exterior measures and Carathéodory's theorem Page 264 1.2. Metric exterior measures Page 266 1.3. The extension theorem Page 270 2. Integration on a measure space Page 273 3. Examples Page 276 3.1. Product measures and a general Fubini theorem Page 276 3.2. Integration formula for polar coordinates Page 279 3.3. Borel measures on R and the Lebesgue-Stieltjes integral Page 281 4. Absolute continuity of measures Page 285 4.1. Signed measures Page 285 4.2. Absolute continuity Page 288 5*. Ergodic theorems Page 292 5.1. Mean ergodic theorem Page 294 5.2. Maximal ergodic theorem Page 296 5.3. Pointwise ergodic theorem Page 300 5.4. Ergodic measure-preserving transformations Page 302 6*. Appendix: the spectral theorem Page 306 6.1. Statement of the theorem Page 306 6.2. Positive operators Page 307 6.3. Proof of the theorem Page 309 6.4. Spectrum Page 311 7. Exercises Page 312 8. Problems Page 319 Chapter 7. Hausdorff Measure and Fractals Page 323 1. Hausdorff measure Page 324 2. Hausdorff dimension Page 329 2.1. Examples Page 330 2.2. Self-similarity Page 341 3. Space-filling curves Page 349 3.1. Quartic intervals and dyadic squares Page 351 3.2. Dyadic correspondence Page 353 3.3. Construction of the Peano mapping Page 355 4*. Besicovitch sets and regularity Page 360 4.1. The Radon transform Page 363 4.2. Regularity of sets when d >= 3 Page 370 4.3. Besicovitch sets have dimension 2 Page 371 4.4. Construction of a Besicovitch set Page 374 5. Exercises Page 380 6. Problems Page 385 Notes and References Page 389 Bibliography Page 391 Symbol Glossary Page 395 Index Page 397