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 in metric measure space X. The function ↦ A M $E \mapsto AM(\Gamma (E))$ defines $AM$ -modulus X where refers to approximation modulus [22]. We compare $AM(\Gamma Hausdorff c o H 1 $co\mathcal {H}^1(E)$ codimension one and show that ≈ \begin{equation*}\hskip6pc co\mathcal {H}^1(E) \approx (E))\hskip-6pc \end{equation*} for Suslin sets This leads new characterization finite perimeter terms -modulus. also study level B V $BV$ functions a.e. t these have {H}^1$ -measure. Most results are R n $\mathbb {R}^n$ .