Strict interior approximation of sets of finite perimeter and functions of bounded variation

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It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate anopensetΩ\Omegaof finite perimeter inRn\mathbb {R}^nstrictlyfrom within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the(n−1)(n{-}1)-dimensional Hausdorff measure of the topological boundary∂Ω\partial \Omegaequals the perimeter ofΩ\Omega. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation ofBVBV-functions from a prescribed Dirichlet class.