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. Also available, the first two volumes in the Princeton Lectures in Analysis:

1 Preliminaries 1

The exterior measure 10

3 Measurable sets and the Lebesgue measure 16

4 Measurable 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

6 Exercises 37

7 Problems 46

Chapter 2: Integration Theory 49

1 The Lebesgue integral: basic properties and convergence theorems 49

2Thespace L 1 of integrable functions 68

3 Fubini'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

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. The Lebesgue differentiation theorem 104

Good kernels and approximations to the identity 108

3 Differentiability of functions 114

3.1 Functions of bounded variation 115

3. 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 L 2 156

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 theorem 181

5. 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 L 2 207

The Hardy space of the upper half-plane 13

3 Constant 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

5 Exercises 253

6 Problems 259

Chapter 6: Abstract Measure and Integration Theory 262

1 Abstract measure spaces 263

1.1 Exterior measures and Carath?dory's theorem 264

1. Metric exterior measures 266

1.3 The extension theorem 270

Integration on a measure space 273

3 Examples 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

4 Absolute 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

7 Exercises 312

8 Problems 319

Chapter 7: Hausdorff Measure and Fractals 323

1 Hausdorff measure 324

Hausdorff dimension 329

2.1 Examples 330

2. Self-similarity 341

3 Space-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

5 Exercises 380

6 Problems 385

Notes and References 389

Bibliography 391

Symbol Glossary 395

Index 397

Elias M. Stein & Rami Shakarchi  

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