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

Let Γ(E)$\Gamma (E)$ be the family of all paths which meet a set E in the metric measure space X. The set function E↦AM(Γ(E))$E \mapsto AM(\Gamma (E))$ defines the AM$AM$‐modulus measure in X where AM$AM$ refers to the approximation modulus [22]. We compare AM(Γ(E))$AM(\Gamma (E))$ to the Hausdorff measure coH1(E)$co\mathcal {H}^1(E)$ of codimension one in X and show that coH1(E)≈AM(Γ(E))\begin{equation*}\hskip6pc co\mathcal {H}^1(E) \approx AM(\Gamma (E))\hskip-6pc \end{equation*}for Suslin sets E in X. This leads to a new characterization of sets of finite perimeter in X in terms of the AM$AM$‐modulus. We also study the level sets of BV$BV$ functions and show that for a.e. t these sets have finite coH1$co\mathcal {H}^1$‐measure. Most of the results are new also in Rn$\mathbb {R}^n$.