Table of Contents

Foreword vii

Introduction xv
1Fourier series: completion xvi
Limits of continuous functions xvi
3Length of curves xvii
4Differentiation and integration xviii
5The problem of measure xviii

Chapter 1. Measure Theory 1
1Preliminaries 1
The exterior measure 10
3Measurable sets and the Lebesgue measure 16
4Measurable functions 7
4.1 Definition and basic properties 27
4.Approximation by simple functions or step functions 30
4.3 Littlewood's three principles 33
5* The Brunn-Minkowski inequality 34
6Exercises 37
7Problems 46

Chapter 2: Integration Theory 49
1The Lebesgue integral: basic properties and convergence
theorems 49
2Thespace L 1 of integrable functions 68
3Fubini's theorem 75
3.1 Statement and proof of the theorem 75
3.Applications of Fubini's theorem 80
4* A Fourier inversion formula 86
5Exercises 89
6Problems 95

Chapter 3: Differentiation and Integration 98
1Differentiation of the integral 99
1.1 The Hardy-Littlewood maximal function 100
1.The Lebesgue differentiation theorem 104
Good kernels and approximations to the identity 108
3Differentiability of functions 114
3.1 Functions of bounded variation 115
3.Absolutely continuous functions 127
3.3 Differentiability of jump functions 131
4Rectifiable curves and the isoperimetric inequality 134
4.1* Minkowski content of a curve 136
4.2* Isoperimetric inequality 143
5Exercises 145
6Problems 152

Chapter 4: Hilbert Spaces: An Introduction 156
1The Hilbert space L 2 156
Hilbert spaces 161
2.1 Orthogonality 164
2.2 Unitary mappings 168
2.3 Pre-Hilbert spaces 169
3Fourier series and Fatou's theorem 170
3.1 Fatou's theorem 173
4Closed subspaces and orthogonal projections 174
5Linear transformations 180
5.1 Linear functionals and the Riesz representation theorem 181
5.Adjoints 183
5.3 Examples 185
6Compact operators 188
7Exercises 193
8Problems 202

Chapter 5: Hilbert Spaces: Several Examples 207
1The Fourier transform on L 2 207
The Hardy space of the upper half-plane 13
3Constant coefficient partial differential equations 221
3.1 Weaksolutions 222
3.The main theorem and key estimate 224
4* The Dirichlet principle 9
4.1 Harmonic functions 234
4.The boundary value problem and Dirichlet's principle 43
5Exercises 253
6Problems 259

Chapter 6: Abstract Measure and Integration Theory 262
1Abstract measure spaces 263
1.1 Exterior measures and Carathèodory's theorem 264
1.Metric exterior measures 266
1.3 The extension theorem 270
Integration on a measure space 273
3Examples 276
3.1 Product measures and a general Fubini theorem 76
3.Integration formula for polar coordinates 279
3.3 Borel measures on R and the Lebesgue-Stieltjes integral 281
4Absolute continuity of measures 285
4.1 Signed measures 285
4.Absolute continuity 288
5* Ergodic theorems 292
5.1 Mean ergodic theorem 294
5.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.Positive operators 307
6.3 Proof of the theorem 309
6.4 Spectrum 311
7Exercises 312
8Problems 319

Chapter 7: Hausdorff Measure and Fractals 323
1Hausdorff measure 324
Hausdorff dimension 329
2.1 Examples 330
2.Self-similarity 341
3Space-filling curves 349
3.1 Quartic intervals and dyadic squares 351
3.Dyadic correspondence 353
3.3 Construction of the Peano mapping 355
4* Besicovitch sets and regularity 360
4.1 The Radon transform 363
4.Regularity of sets when d 3 370
4.3 Besicovitch sets have dimension 371
4.4 Construction of a Besicovitch set 374
5Exercises 380
6Problems 385

Notes and References 389
Bibliography 391
Symbol Glossary 395
Index 397