1st Edition

Measure Theory and Fine Properties of Functions

    This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the fine properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and 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), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions).

    Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in 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 and Compactness for Radon Measures

    HAUSDORFF MEASURE
    Definitions and Elementary Properties; Hausdorff Dimension
    Isodiametric Inequality
    Densities
    Hausdorff Measure and Elementary Properties of Functions

    AREA AND COAREA FORMULAS
    Lipschitz Functions, Rademacher's Theorem
    Linear Maps and Jacobians
    The Area Formula
    The Coarea Formula

    SOBOLEV FUNCTIONS.
    Definitions And Elementary Properties. Approximation
    Traces. Extensions. Sobolev Inequalities
    Compactness. Capacity
    Quasicontinuity; Precise Representations of Sobolev Functions. Differentiability on Lines

    BV FUNCTIONS AND SETS OF FINITE PERIMETER
    Definitions and Structure Theorem
    Approximation and Compactness
    Traces. Extensions. Coarea Formula for BV Functions. Isoperimetric Inequalities.
    The Reduced Boundary
    The Measure Theoretic Boundary; Gauss-Green Theorem. Pointwise Properties of BV Functions
    Essential Variation on Lines
    A Criterion for Finite Perimeter. DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS.
    Lp Differentiability a.e.; 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

    NOTATION
    REFERENCES

    Biography

    LawrenceCraig Evans