AM-modulus and Hausdorff measure of codimension one in metric measure spaces

Abstract

Let $\Gamma (E)$ be the family of all paths which meet a set $E$ in the metric measure space $X$. The set function $E\mapsto AM(\Gamma (E))$ defines the $AM$--modulus measure in $X$ where $AM$ refers to the approximation modulus. We compare $AM(\Gamma (E))$ to the Hausdorff measure $co{\mathcal{H}}^1(E)$ of codimension one in $X$ and show that $$co{\mathcal{H}}^1(E)\approx AM(\Gamma (E))$$ for Suslin sets $E$ in $X$. This leads to a new characterization of sets of finite perimeter in $X$ in terms of the $AM$--modulus. We also study the level sets of $BV$ functions and show that for a.e. $t$ these sets have finite $co{\mathcal{H}}^1$--measure. Most of the results are new also in ${\mathbb{R}}^n$.