Measure Theory and Fine Properties of Functions, Revised Edition
CRC Press (Verlag)
978-1-138-58249-1 (ISBN)
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Topics covered include a quick review of abstract measure theory, theorems and differentiation in ℝn, Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions as well as functions of bounded variation.
The text provides complete proofs of many key results omitted from other books, including Besicovitch's covering theorem, Rademacher's theorem (on the differentiability a.e. of Lipschitz functions), area and coarea formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Aleksandrov's theorem (on the twice differentiability a.e. of convex functions).
This revised edition includes countless improvements in notation, format, and clarity of exposition. Also new are several sections describing the π-λ theorem, weak compactness criteria in L1, and Young measure methods for weak convergence. In addition, the bibliography has been updated.
Topics are carefully selected and the proofs are succinct, but complete. This book provides ideal reading for mathematicians and graduate students in pure and applied mathematics.
Lawrence Craig Evans, University of California, Berkeley, USA Ronald F. Gariepy, University of Kentucky, Lexington, USA
General Measure Theory
Measures and Measurable Functions
Lusin’s and Egoroff’s Theorems
Integrals and Limit Theorems
Product Measures, Fubini’s Theorem, Lebesgue Measure
Covering Theorems
Differentiation of Radon Measures
Lebesgue Points, Approximate Continuity
Riesz Representation Theorem
Weak Convergence
References and Notes
Hausdorff Measures
Definitions and Elementary Properties
Isodiametric Inequality, Hn=Ln
Densities
Functions and Hausdorff Measure
References and Notes
Area and Coarea Formulas
Lipschitz Functions, Rademacher’s Theorem
Linear Maps and Jacobians
The Area Formula
The Coarea Formula
References and Notes
Sobolev Functions
Definitions and Elementary Properties
Approximation
Traces
Extensions
Sobolev Inequalities
Compactness
Capacity
Quasicontinuity; Precise Representatives of Sobolev Functions
Differentiability on Lines
References and Notes
Functions of Bounded Variation, Sets of Finite Perimeter
Definitions, Structure Theorem
Approximation and Compactness
Traces
Extensions
Coarea Formula for BV Functions
Isoperimetric Inequalities
The Reduced Boundary
Gauss-Green Theorem
Pointwise Properties of BV Functions
Essential Variation on Lines
A Criterion for Finite Perimeter
References and Notes
Differentiability, Approximation by C1 Functions
Lp Differentiability; Approximate Differentiability
Differentiability a.e. for W1,p (p>n)
Convex Functions
Second Derivatives a.e. for Convex Functions
Whitney’s Extension Theorem
Approximation by C1 Functions
References and Notes
Bibliography
| Erscheint lt. Verlag | 31.12.2023 |
|---|---|
| Reihe/Serie | Textbooks in Mathematics |
| Zusatzinfo | 15 Illustrations, black and white |
| Verlagsort | London |
| Sprache | englisch |
| Maße | 152 x 229 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| ISBN-10 | 1-138-58249-2 / 1138582492 |
| ISBN-13 | 978-1-138-58249-1 / 9781138582491 |
| Zustand | Neuware |
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