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 THEORYMeasures and Measurable FunctionsLusin's and Egoroff's TheoremsIntegrals and Limit TheoremsProduct Measures, Fubini's Theorem, Lebesgue MeasureCovering TheoremsDifferentiation of Radon MeasuresLebesgue PointsApproximate continuityRiesz Representation TheoremWeak Convergence and Compactness for Radon MeasuresHAUSDORFF MEASUREDefinitions and Elementary Properties; Hausdorff DimensionIsodiametric InequalityDensitiesHausdorff Measure and Elementary Properties of FunctionsAREA AND COAREA FORMULASLipschitz Functions, Rademacher's TheoremLinear Maps and JacobiansThe Area FormulaThe Coarea FormulaSOBOLEV FUNCTIONS. Definitions And Elementary Properties. ApproximationTraces. Extensions. Sobolev InequalitiesCompactness. CapacityQuasicontinuity; Precise Representations of Sobolev Functions. Differentiability on LinesBV FUNCTIONS AND SETS OF FINITE PERIMETERDefinitions and Structure TheoremApproximation and CompactnessTraces. Extensions. Coarea Formula for BV Functions. Isoperimetric Inequalities. The Reduced BoundaryThe Measure Theoretic Boundary; Gauss-Green Theorem. Pointwise Properties of BV FunctionsEssential Variation on LinesA Criterion for Finite Perimeter. DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS. Lp Differentiability a.e.; Approximate DifferentiabilityDifferentiability A.E. for W1,P (P > N). Convex FunctionsSecond Derivatives a.e. for convex functionsWhitney's Extension TheoremApproximation by C1 FunctionsNOTATIONREFERENCES