AM‐modulus and Hausdorff measure of codimension one in metric measure spaces. Issue 1 (10th January 2022)

• Record Type: Journal Article
• Authors: Honzlová‐Exnerová, Vendula; Malý, Jan; Martio, Olli
• Is Part Of: Mathematische Nachrichten. Volume 295:Issue 1(2022)
• Page Start: 140
• Page End: 157
• Publication Date: 2022-01-10
• Subjects: AM$AM$‐modulus -- level sets of BV$BV$‐functions -- metric measure spaces -- perimeter -- sets of co‐dimension one; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.202000059
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: 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 ↦ A M ( Γ ( E ) ) $E \mapsto AM(\Gamma (E))$ 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 ) ) $AM(\Gamma (E))$ 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) \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 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 $\mathbb {R}^n$ .