A note on some topological properties of sets with finite perimeter

Some well-known results about the $2$-density topology on $ $ (in par-ti-cu-lar in the context of the Lusin-Menchoff property) are extended to $ au_{b_m}$, i.e. the $m$-density topology on $ ^n$ with $min (n,+infty)$. Every set of finite perimeter in $ ^n$ is equivalent (in measure) to a set in $ au_{b_{m_0}}$, where $m_0=n+1+{1over n-1}$. There exists a set of finite perimeter in $ ^n$ which is not equivalent (in measure) to any member in the a.e.-modification of $ au_{b_m}$, whatever $min [n,+infty)$.