{
  "id": "2110.05250",
  "version": "v3",
  "published": "2021-10-11T13:05:12.000Z",
  "updated": "2022-07-20T11:49:03.000Z",
  "title": "Haar Null Closed and Convex Sets in Separable Banach Spaces",
  "authors": [
    "Davide Ravasini"
  ],
  "comment": "11 pages. v2: Section 4 has been extensively rewritten and improved. Corollary 3.6 has a new proof. Fixed typos. v3: Accepted version for publication",
  "categories": [
    "math.FA"
  ],
  "abstract": "Haar null sets were introduced by J.P.R. Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, U.B. Darij defined a categorical version of Haar null sets, which he named Haar meagre sets. The present paper aims to show that, whenever $C$ is a closed, convex subset of a separable Banach space, $C$ is Haar null if and only if $C$ is Haar meagre. We then use this fact to improve a theorem of E. Matou\\v{s}kov\\'{a} and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation of separable Banach lattices whose positive cone is not Haar null.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-07-20T11:49:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46B42",
      "52A07"
    ],
    "keywords": [
      "separable banach space",
      "convex sets",
      "haar null sets",
      "named haar meagre sets",
      "zero haar measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}