{
  "id": "1109.5309",
  "version": "v1",
  "published": "2011-09-24T21:08:37.000Z",
  "updated": "2011-09-24T21:08:37.000Z",
  "title": "Borel sets which are null or non-$σ$-finite for every translation invariant measure",
  "authors": [
    "Márton Elekes",
    "Tamás Keleti"
  ],
  "journal": "Adv. Math. 201 (2006), 102-115",
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that the set of Liouville numbers is either null or non-$\\sigma$-finite with respect to every translation invariant Borel measure on $\\RR$, in particular, with respect to every Hausdorff measure $\\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\\RR$ that are not $F_\\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-24T21:08:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28C10",
      "28A78",
      "43A05"
    ],
    "keywords": [
      "translation invariant measure",
      "translation invariant borel measure",
      "hausdorff measure",
      "gauge function",
      "liouville numbers"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.5309E"
    }
  }
}