{
  "id": "1501.04400",
  "version": "v1",
  "published": "2015-01-19T06:17:27.000Z",
  "updated": "2015-01-19T06:17:27.000Z",
  "title": "A counterexample shows that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms",
  "authors": [
    "Mingzhi Wu",
    "Tiexin Guo"
  ],
  "comment": "11 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper constructs a counterexample showing that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms. Random convex analysis is the analytic foundation for $L^0$--convex conditional risk measures, this counterexample, however, shows that a locally $L^0$--convex module is not a proper framework for random convex analysis. Further, this paper also gives a necessary and sufficient condition for a locally $L^0$--convex topology to be induced by a family of $L^0$--seminorms. Finally, we give some comments showing that based on random locally convex modules, we can establish a perfect random convex analysis to meet the needs of the study of $L^0$--convex conditional risk measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-19T06:17:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46H25",
      "46A19",
      "46A16"
    ],
    "keywords": [
      "convex topology",
      "convex conditional risk measures",
      "necessarily",
      "counterexample",
      "perfect random convex analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150104400W"
    }
  }
}