Description
Real Analysis: Measure Theory, Integration, and Hilbert Spaces 1st Edition by Elias M. Stein, ISBN-13: 978-0691113869
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- Publisher: Princeton University Press; First Edition (April 3, 2005)
- Language: English
- 424 pages
- ISBN-10: 0691113866
- ISBN-13: 978-0691113869
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:
Cover
Half title
Title
Copyright
Dedication
Contents
Foreword
Introduction
1 Fourier series: completion
2 Limits of continuous functions
3 Length of curves
4 Differentiation and integration
5 The problem of measure
Chapter 1. Measure Theory
1 Preliminaries
2 The exterior measure
3 Measurable sets and the Lebesgue measure
4 Measurable functions
4.1 Definition and basic properties
4.2 Approximation by simple functions or step functions
4.3 Littlewood’s three principles
5* The Brunn-Minkowski inequality
6 Exercises
7 Problems
Chapter 2. Integration Theory
1 The Lebesgue integral: basic properties and convergence theorems
2 The space L1 of integrable functions
3 Fubini’s theorem
3.1 Statement and proof of the theorem
3.2 Applications of Fubini’s theorem
4* A Fourier inversion formula
5 Exercises
6 Problems
Chapter 3. Differentiation and Integration
1 Differentiation of the integral
1.1 The Hardy-Littlewood maximal function
1.2 The Lebesgue differentiation theorem
2 Good kernels and approximations to the identity
3 Differentiability of functions
3.1 Functions of bounded variation
3.2 Absolutely continuous functions
3.3 Differentiability of jump functions
4 Rectifiable curves and the isoperimetric inequality
4.1* Minkowski content of a curve
4.2* Isoperimetric inequality
5 Exercises
6 Problems
Chapter 4. Hilbert Spaces: An Introduction
1 The Hilbert space L2
2 Hilbert spaces
2.1 Orthogonality
2.2 Unitary mappings
2.3 Pre-Hilbert spaces
3 Fourier series and Fatou’s theorem
3.1 Fatou’s theorem
4 Closed subspaces and orthogonal projections
5 Linear transformations
5.1 Linear functionals and the Riesz representation theorem
5.2 Adjoints
5.3 Examples
6 Compact operators
7 Exercises
8 Problems
Chapter 5. Hilbert Spaces: Several Examples
1 The Fourier transform on L2
2 The Hardy space of the upper half-plane
3 Constant coefficient partial differential equations
3.1 Weak solutions
3.2 The main theorem and key estimate
4* The Dirichlet principle
4.1 Harmonic functions
4.2 The boundary value problem and Dirichlet’s principle
5 Exercises
6 Problems
Chapter 6. Abstract Measure and Integration Theory
1 Abstract measure spaces
1.1 Exterior measures and Carathéodory’s theorem
1.2 Metric exterior measures
1.3 The extension theorem
2 Integration on a measure space
3 Examples
3.1 Product measures and a general Fubini theorem
3.2 Integration formula for polar coordinates
3.3 Borel measures on r and the Lebesgue-Stieltjes integral
4 Absolute continuity of measures
4.1 Signed measures
4.2 Absolute continuity
5* Ergodic theorems
5.1 Mean ergodic theorem
5.2 Maximal ergodic theorem
5.3 Pointwise ergodic theorem
5.4 Ergodic measure-preserving transformations
6* Appendix: the spectral theorem
6.1 Statement of the theorem
6.2 Positive operators
6.3 Proof of the theorem
6.4 Spectrum
7 Exercises
8 Problems
Chapter 7. Hausdorff Measure and Fractals
1 Hausdorff measure
2 Hausdorff dimension
2.1 Examples
2.2 Self-similarity
3 Space-filling curves
3.1 Quartic intervals and dyadic squares
3.2 Dyadic correspondence
3.3 Construction of the Peano mapping
4* Besicovitch sets and regularity
4.1 The Radon transform
4.2 Regularity of sets when d ≥ 3
4.3 Besicovitch sets have dimension 2
4.4 Construction of a Besicovitch set
5 Exercises
6 Problems
Notes and References
Bibliography
Symbol Glossary
Index
Elias M. Stein is Professor of Mathematics at Princeton University. Rami Shakarchi received his Ph.D. in Mathematics from Princeton University in 2002.
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