1st Edition
Measure Theory and Fine Properties of Functions, Revised Edition
Measure Theory and Fine Properties of Functions, Revised Edition provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions.
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.
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
Biography
Lawrence Craig Evans, University of California, Berkeley, USA
Ronald F. Gariepy, University of Kentucky, Lexington, USA
"This is a new revised edition of a very successful book dealing with measure theory in Rn and some special properties of functions, usually omitted from books dealing with abstract measure theory, but which a working mathematician analyst must know. … The book is clearly written with complete proofs, including all technicalities. … The new edition benefits from LaTeX retyping, yielding better cross-references, as well as numerous improvements in notation, format, and clarity of exposition. The bibliography has been updated and several new sections were added … this welcome, updated, and revised edition of a very popular book will continue to be of great interest for the community of mathematicians interested in mathematical analysis in Rn."
—Studia Universitatis Babes-Bolyai Mathematica, 60, 2015
