AM‐modulus and Hausdorff measure of codimension one in metric measure spaces. Issue 1 (10th January 2022) - Honzlová‐Exnerová, Vendula; Malý, Jan; Martio, Olli
Abstract: Let Γ ( E ) $Γ (E)$ be the family of all paths which meet a set E in the metric measure space X . The set function E ↦ A M ( Γ ( E ) ) $E \mapsto AM(Γ (E))$ defines the A M $AM$ ‐modulus measure in X where A M $AM$ refers to the approximation modulus [22]. We compare A M ( Γ ( E ) ) $AM(Γ (E))$ to the Hausdorff measure c o H 1 ( E ) $co\mathcal {H}^1(E)$ of codimension one in X and show that c o H 1 ( E ) ≈ A M ( Γ ( E ) ) \begin{equation*}\hskip6pc co\mathcal {H}^1(E) ≈ AM(Γ (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 A M $AM$ ‐modulus. We also study the level sets of B V $BV$ functions and show that for a.e. t these sets have finite c o H 1 $co\mathcal {H}^1$ ‐measure. Most of the results are new also in R n $\mathbb {R}^n$ .