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Zusatztext "As one would expect from these authors, the exposition is, in general, excellent. The explanations are clear and concise with many well-focused examples as well as an abundance of exercises, covering the full range of difficulty. . . . [I]t certainly must be on the instructor's bookshelf as a first-rate reference book." ---William P. Ziemer, SIAM Review Informationen zum Autor Elias M. Stein & Rami Shakarchi Klappentext "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: Zusammenfassung Intended for the students interested in the disciplines of mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels, this third volume in a series of titles focuses on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. Inhaltsverzeichnis Foreword vii Introduction xv 1 Fourier series: completion xvi Limits of continuous functions xvi 3 Length of curves xvii 4 Differentiation and integration xviii 5 The problem of measure xviii Chapter 1. Measure Theory 1 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 ...
