  We study fine potential theory and in particular partitions of unity in
quasiopen sets in the case $p=1$. Using these, we develop an analog of the
discrete convolution technique in quasiopen (instead of open) sets. We apply
this technique to show that every function of bounded variation (BV function)
can be approximated in the BV and $L^{\infty}$ norms by BV functions whose jump
sets are of finite Hausdorff measure. Our results seem to be new even in
Euclidean spaces but we work in a more general complete metric space that is
equipped with a doubling measure and supports a Poincar\'e inequality.
