A Federer-style characterization of sets of finite perimeter on metric spaces

TL;DR

This paper characterizes sets of finite perimeter in metric spaces with doubling measures and Poincaré inequalities, linking finite perimeter to the finiteness of the Hausdorff measure of the set's 1-fine boundary.

Contribution

It provides a Federer-style characterization of finite perimeter sets in metric spaces, connecting geometric measure theory with fine boundary analysis.

Findings

Finite perimeter sets are characterized by the finiteness of the Hausdorff measure of their 1-fine boundary.

The result extends classical Euclidean characterizations to metric measure spaces.

The characterization relies on the properties of the measure-theoretic interior and the 1-fine boundary.

Abstract

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that a set $E$ is of finite perimeter if and only if $\mathcal H(\partial^1 I_E)<\infty$, that is, if and only if the codimension one Hausdorff measure of the \emph{$1$-fine boundary} of the set's measure theoretic interior $I_E$ is finite.