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

TL;DR

This paper extends Federer's characterization of finite perimeter sets from Euclidean spaces to complete metric spaces with doubling measures and Poincaré inequalities, establishing the equivalence of finite perimeter and finite measure-theoretic boundary measure.

Contribution

The paper proves the 'if' direction of Federer's characterization in metric spaces, completing the equivalence in a broader setting using fine potential theory.

Findings

The 'if' direction of Federer's characterization holds in metric spaces.

Finite perimeter sets correspond to finite measure-theoretic boundary measure in these spaces.

The result generalizes classical Euclidean theory to metric measure spaces.

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 one. In complete metric spaces that are equipped with a doubling measure and support a Poincar\'e 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.