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
Series: Studies in Advanced Mathematics
Number Of Pages: 288
Published: 5th September 2000
Publisher: Taylor & Francis Inc
Country of Publication: US
Dimensions (cm): 23.5 x 15.9
Weight (kg): 0.54
Edition Number: 1