1st Edition

Measure Theory and Fine Properties of Functions, Revised Edition

    314 Pages 15 B/W Illustrations
    by Chapman & Hall

    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
    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
    Sobolev Inequalities
    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
    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



    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