Real Analysis Measure Theory, Integration, and Hilbert Spaces

ISBN-10: 0691113866
ISBN-13: 9780691113869
Edition: 2005
List price: $99.95 Buy it from $63.10
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Description: Real Analysisis 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,  More...

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Book details

List price: $99.95
Copyright year: 2005
Publisher: Princeton University Press
Publication date: 4/3/2005
Binding: Hardcover
Pages: 424
Size: 6.50" wide x 9.25" long x 1.25" tall
Weight: 1.848
Language: English

Real Analysisis 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 Analysisis 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:

Foreword
Introduction
Fourier series: completion
Limits of continuous functions
Length of curves
Differentiation and integration
The problem of measure
Measure Theory 1 1 Preliminaries
The exterior measure
Measurable sets and the Lebesgue measure
Measurable functions
Definition and basic properties
Approximation by simple functions or step functions
Littlewood's three principles
The Brunn-Minkowski inequality
Exercises
Problems
Integration Theory
The Lebesgue integral: basic properties and convergence theorems
Thespace L 1 of integrable functions
Fubini's theorem
Statement and proof of the theorem
Applications of Fubini's theorem
A Fourier inversion formula
Exercises
Problems
Differentiation and Integration
Differentiation of the integral
The Hardy-Littlewood maximal function
The Lebesgue differentiation theorem
Good kernels and approximations to the identity
Differentiability of functions
Functions of bounded variation
Absolutely continuous functions
Differentiability of jump functions
Rectifiable curves and the isoperimetric inequality
Minkowski content of a curve
Isoperimetric inequality
Exercises
Problems
Hilbert Spaces: An Introduction
The Hilbert space L 2
Hilbert spaces
Orthogonality
Unitary mappings
Pre-Hilbert spaces
Fourier series and Fatou's theorem
Fatou's theorem
Closed subspaces and orthogonal projections
Linear transformations
Linear functionals and the Riesz representation theorem
Adjoints
Examples
Compact operators
Exercises
Problems
Hilbert Spaces: Several Examples
The Fourier transform on L 2
The Hardy space of the upper half-plane
Constant coefficient partial differential equations
Weaksolutions
The main theorem and key estimate
The Dirichlet principle
Harmonic functions
The boundary value problem and Dirichlet's principle
Exercises
Problems
Abstract Measure and Integration Theory
Abstract measure spaces
Exterior measures and Carathegrave;odory's theorem
Metric exterior measures
The extension theorem
Integration on a measure space
Examples
Product measures and a general Fubini theorem
Integration formula for polar coordinates
Borel measures on R and the Lebesgue-Stieltjes integral
Absolute continuity of measures
Signed measures
Absolute continuity
Ergodic theorems
Mean ergodic theorem
Maximal ergodic theorem
Pointwise ergodic theorem
Ergodic measure-preserving transformations
Appendix: the spectral theorem
Statement of the theorem
Positive operators
Proof of the theorem
Spectrum
Exercises
Problems
Hausdorff Measure and Fractals
Hausdorff measure
Hausdorff dimension
Examples
Self-similarity
Space-filling curves
Quartic intervals and dyadic squares
Dyadic correspondence
Construction of the Peano mapping
Besicovitch sets and regularity
The Radon transform
Regularity of sets whend3
Besicovitch sets have dimension
Construction of a Besicovitch set
Exercises
Problems
Notes and References

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