Vol. 13, No. 5, 2020

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Federer's characterization of sets of finite perimeter in metric spaces

Panu Lahti

Vol. 13 (2020), No. 5, 1501–1519
Abstract

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 p = 1, we prove that the “if” direction holds as well.

Keywords
metric measure space, set of finite perimeter, Federer's characterization, measure-theoretic boundary, codimension-1 Hausdorff measure, fine topology
Mathematical Subject Classification 2010
Primary: 30L99, 31E05, 26B30
Milestones
Received: 27 April 2018
Revised: 3 January 2019
Accepted: 12 May 2019
Published: 27 July 2020
Authors
Panu Lahti
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing
China