Measure Theory and Fine Properties of Functions, Revised Edition: 1st Edition (Hardback) book cover

Measure Theory and Fine Properties of Functions, Revised Edition

1st Edition

By Lawrence Craig Evans, Ronald F. Gariepy

Chapman and Hall/CRC

313 pages | 15 B/W Illus.

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Description

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.

Reviews

"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

Table of Contents

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

About the Authors

Lawrence Craig Evans, University of California, Berkeley, USA

Ronald F. Gariepy, University of Kentucky, Lexington, USA

About the Series

Textbooks in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT003000
MATHEMATICS / Applied
MAT037000
MATHEMATICS / Functional Analysis