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So you're right. 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! Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Theorem 1. Asked 2 years, 9 months ago. Active 2 years, 9 months ago. Viewed times. L P L P 5 5 bronze badges. Will get back to you tomorrow! Active Oldest Votes. Behnam Esmayli Behnam Esmayli 3, 9 9 silver badges 20 20 bronze badges. Thanks for clearing that up!
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Measure theory and Fine Properties of Functions
Book:Lawrence C. Evans/Measure Theory and Fine Properties of Functions
Measure Theory and Fine Properties of Functions. Lawrence Craig Evans , Ronald F. 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. Area and Coarea Formulas.
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