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

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.