About the Book

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.

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 Caratheodory'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