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

TL;DR

This paper establishes a relationship between the $AM$-modulus measure and Hausdorff measure of codimension one in metric measure spaces, providing new insights into sets of finite perimeter and level sets of BV functions.

Contribution

It introduces a new characterization of finite perimeter sets via the $AM$-modulus and compares it to Hausdorff measure, extending results to general metric measure spaces.

Findings

$co\mathcal H^1(E) \approx AM(\Gamma(E))$ for Suslin sets

Finite perimeter sets characterized by $AM$-modulus

Level sets of BV functions have finite $co\mathcal H^1$-measure for a.e. $t$

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 $\mathbb R^n$.