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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 an open set Ω of finite perimeter in R strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the (n−1)-dimensional Hausdorff measure of the topological boundary ∂Ω equals the perimeter of Ω. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of BV-functions from a prescribed Dirichlet class.

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