{
  "id": "1808.06747",
  "version": "v1",
  "published": "2018-08-21T03:11:35.000Z",
  "updated": "2018-08-21T03:11:35.000Z",
  "title": "On closedness of convex sets in Banach function spaces",
  "authors": [
    "Made Tantrawan",
    "Denny H. Leung"
  ],
  "comment": "27 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathcal{X}$ be a Banach function space. A well-known problem arising from theory of risk measures asks when the order closedness of a convex set in $\\mathcal{X}$ implies the closedness with respect to the topology $\\sigma(\\mathcal{X},\\mathcal{X}_n^\\sim)$ where $\\mathcal{X}_n^\\sim$ is the order continuous dual of $\\mathcal{X}$. In this paper, we give an answer to the problem for a large class of Banach function spaces. We show that under the Fatou property and the subsequence splitting property, every order closed convex set in $\\mathcal{X}$ is $\\sigma(\\mathcal{X},\\mathcal{X}_n^\\sim)$-closed if and only if either $\\mathcal{X}$ or the norm dual $\\mathcal{X}^*$ of $\\mathcal{X}$ is order continuous. In addition, we also give a characterization of $\\mathcal{X}$ for which the order closedness of a convex set in $\\mathcal{X}$ is equivalent to the closedness with respect to the topology $\\sigma(\\mathcal{X},\\mathcal{X}_{uo}^\\sim)$ where $\\mathcal{X}_{uo}^\\sim$ is the unbounded order continuous dual of $\\mathcal{X}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-21T03:11:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "46A55",
      "46A20"
    ],
    "keywords": [
      "banach function space",
      "order closedness",
      "order closed convex set",
      "risk measures asks",
      "unbounded order continuous dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}