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

2019/11/06 by Vendula Honzlová Exnerová, Exnerová, Vendula Honzlová, Jan Malý +4
Mathematics · #Advanced Topology and Set Theory #math.FA #msc:28A78 #msc:30L99 #msc:31B15

paper · pdf · doi:10.48550/arxiv.1911.02433

arxiv created 2019/11/06 · arxiv updated 2019/11/07

Abstract

Let Γ(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)) defines the AM--modulus measure in X where AM refers to the approximation modulus. We compare AM(Γ(E)) to the Hausdorff measure co\mathcal H¹(E) of codimension one in X and show that co\mathcal H¹(E) ≈ AM(Γ(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 H¹--measure. Most of the results are new also in \mathbb Rⁿ.

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