Strong approximation of sets of finite perimeter in metric spaces

Abstract

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that any set of finite perimeter can be approximated in the BV norm by a set whose topological and measure theoretic boundaries almost coincide. This result appears to be new even in the Euclidean setting. The work relies on a quasicontinuity-type result for BV functions proved by Lahti and Shanmugalingam (2016).