{
  "id": "2010.15902",
  "version": "v1",
  "published": "2020-10-29T19:33:46.000Z",
  "updated": "2020-10-29T19:33:46.000Z",
  "title": "A decomposition for Borel measures $μ\\le \\mathcal{H}^{s}$",
  "authors": [
    "Antoine Detaille",
    "Augusto C. Ponce"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove that every finite Borel measure $\\mu$ in $\\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\\mathcal{H}^s$ can be split in countable many parts $\\mu\\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\\mathcal{H}_\\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We also investigate the case where $\\mu$ is not necessarily finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-29T19:33:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A12"
    ],
    "keywords": [
      "decomposition",
      "finite borel measure",
      "finite hausdorff measure",
      "straight sets",
      "borel set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}