A NOTE ON SOME TOPOLOGICAL PROPERTIES OF SETS WITH FINITE PERIMETER

Abstract

AbstractSome well-known results about the 2-density topology on ${\mathcal R}$ (in particular in the context of the Lusin–Menchoff property) are extended to τbm, i.e. the m-density topology on ${\mathcal R}$n with m ∈ (n,+∞). Every set of finite perimeter in ${\mathcal R}$n is equivalent (in measure) to a set in τbm0, where m0=n+1+${1\over n-1}$. There exists a set of finite perimeter in ${\mathcal R}$n which is not equivalent (in measure) to any member in the a.e.-modification of τbm, whatever m ∈ [n,+∞).