{
  "id": "1210.5804",
  "version": "v5",
  "published": "2012-10-22T04:58:58.000Z",
  "updated": "2013-06-10T20:03:34.000Z",
  "title": "Syndetic submeasures and partitions of $G$-spaces and groups",
  "authors": [
    "Taras Banakh",
    "Igor Protasov",
    "Sergiy Slobodianiuk"
  ],
  "comment": "8 pages",
  "categories": [
    "math.GR",
    "math.GN"
  ],
  "abstract": "We prove that for every number k each countable infinite group $G$ admits a partition $G=A\\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\\subset G$ the sets $KA$ and $KB$ are not thick. The proof is based on the fact that $G$ possesses a syndetic submeasure, i.e., a left-invariant submeasure $\\mu:\\mathcal P(G)\\to[0,1]$ such that for each $\\epsilon > 1/|G|$ and subset $A\\subset G$ with $\\mu(A)<1$ there is a set $B\\subset G\\setminus A$ such that $\\mu(B)<\\epsilon$ and $FB=G$ for some finite subset $F\\subset G$.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-06-10T20:03:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A07",
      "05B40",
      "20F24",
      "28C10",
      "54H11"
    ],
    "keywords": [
      "syndetic submeasure",
      "countable infinite group",
      "finite subset",
      "left-invariant submeasure",
      "element subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.5804B"
    }
  }
}