{
  "id": "1911.02433",
  "version": "v1",
  "published": "2019-11-06T15:22:34.000Z",
  "updated": "2019-11-06T15:22:34.000Z",
  "title": "AM-modulus and Hausdorff measure of codimension one in metric measure spaces",
  "authors": [
    "Vendula Honzlová Exnerová",
    "Jan Malý",
    "Olli Martio"
  ],
  "categories": [
    "math.FA"
  ],
  "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 $\\mathbb R^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-06T15:22:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B15",
      "28A78",
      "30L99"
    ],
    "keywords": [
      "metric measure space",
      "hausdorff measure",
      "codimension",
      "am-modulus",
      "suslin sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}