Strong approximation of sets of finite perimeter in metric spaces

In the setting of a metric space equipped with a doubling measure that supports a Poincaré 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. The proof relies on a quasicontinuity-type result for BV functions proved by Lahti and Shanmugalingam (J Math Pures Appl 107(2): 150–182, 2017 ).