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. 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. The book is clearly written with complete proofs, including all technicalities.
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That limit holds if h is an indicator function. But then it also holds for measurable simple functions. Then approximate any continuous compactly supported g-i with simple functions. Hope this makes it closer to your heart!
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Measure Theory and Fine Properties of Functions, Revised Edition
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Measure Theory and Fine Properties of Functions
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Book:Lawrence C. Evans/Measure Theory and Fine Properties of Functions