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Abstract
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Federer’s characterization of sets of finite perimeter states (in Euclidean spaces) that
a set is of finite perimeter if and only if the measure-theoretic boundary of the set has
finite Hausdorff measure of codimension 1. In complete metric spaces that are
equipped with a doubling measure and support a Poincaré inequality, the “only if”
direction was shown by Ambrosio (2002). By applying fine potential theory in the
case
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we prove that the “if” direction holds as well.
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Keywords
metric measure space, set of finite perimeter, Federer's
characterization, measure-theoretic boundary, codimension-1
Hausdorff measure, fine topology
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Mathematical Subject Classification 2010
Primary: 30L99, 31E05, 26B30
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Milestones
Received: 27 April 2018
Revised: 3 January 2019
Accepted: 12 May 2019
Published: 27 July 2020
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