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

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 $\mathcal{H}({\mathrm{\partial }}^1{I}_E)<\mathrm{\infty }$, 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.