{
  "id": "1109.5310",
  "version": "v1",
  "published": "2011-09-24T21:12:55.000Z",
  "updated": "2011-09-24T21:12:55.000Z",
  "title": "Hausdorff measures of different dimensions are isomorphic under the Continuum Hypothesis",
  "authors": [
    "Márton Elekes"
  ],
  "journal": "Real Anal. Exchange 30 (2004/05), no. 2, 605-616",
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that the Continuum Hypothesis implies that for every $0<d_1\\leq d_2<n$ the measure spaces $(\\RR^n,\\iM_{\\iH^{d_1}},\\iH^{d_1})$ and $(\\RR^n,\\iM_{\\iH^{d_2}},\\iH^{d_2})$ are isomorphic, where $\\iH^d$ is $d$-dimensional Hausdorff measure and $\\iM_{\\Hd}$ is the $\\sigma$-algebra of measurable sets with respect to $\\Hd$. This is motivated by the well-known question (circulated by D. Preiss) whether such an isomorphism exists if we replace measurable sets by Borel sets. We also investigate the related question whether every continuous function (or the typical continuous function) is H\\\"older continuous (or is of bounded variation) on a set of positive Hausdorff dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-24T21:12:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "26A16",
      "26A45",
      "28A05",
      "03E50"
    ],
    "keywords": [
      "isomorphic",
      "dimensional hausdorff measure",
      "continuum hypothesis implies",
      "continuous function",
      "measure spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.5310E"
    }
  }
}