Description
Apple Academic Press Inc. Measure Theory And Fine Properties Of Functions Revised Edition 2015 Edition by Lawrence Craig Evans, Ronald F. Gariepy
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 n, 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 - 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 TheoryMeasures and Measurable FunctionsLusin's and Egoroff's TheoremsIntegrals and Limit TheoremsProduct Measures, Fubini's Theorem, Lebesgue MeasureCovering TheoremsDifferentiation of Radon MeasuresLebesgue Points, Approximate ContinuityRiesz Representation TheoremWeak ConvergenceReferences and NotesHausdorff MeasuresDefinitions and Elementary PropertiesIsodiametric Inequality, Hn=LnDensitiesFunctions and Hausdorff MeasureReferences and NotesArea and Coarea FormulasLipschitz Functions, Rademacher's TheoremLinear Maps and JacobiansThe Area FormulaThe Coarea FormulaReferences and NotesSobolev FunctionsDefinitions and Elementary PropertiesApproximationTracesExtensionsSobolev InequalitiesCompactnessCapacityQuasicontinuity; Precise Representatives of Sobolev FunctionsDifferentiability on LinesReferences and NotesFunctions of Bounded Variation, Sets of Finite PerimeterDefinitions, Structure TheoremApproximation and CompactnessTracesExtensionsCoarea Formula for BV FunctionsIsoperimetric InequalitiesThe Reduced BoundaryGauss-Green TheoremPointwise Properties of BV FunctionsEssential Variation on LinesA Criterion for Finite PerimeterReferences and NotesDifferentiability, Approximation by C1 FunctionsLp Differentiability; Approximate DifferentiabilityDifferentiability a.e. for W1,p (p>n)Convex Functions Second Derivatives a.e. for Convex FunctionsWhitney's Extension TheoremApproximation by C1 FunctionsReferences and NotesBibliography