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

Abstract

In the setting of a metric space equipped with a doubling measure that supports a Poincaré 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.