{
  "id": "0803.1749",
  "version": "v1",
  "published": "2008-03-12T11:24:26.000Z",
  "updated": "2008-03-12T11:24:26.000Z",
  "title": "An isomorphism between the completion of an Algebra and its Caratheodory Extension",
  "authors": [
    "Jun Tanaka"
  ],
  "comment": "5 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Omega$ denote an algebra of sets and $\\mu$ a $\\sigma$-finite measure. We then prove that the completion of $\\Omega$ under the pseudometric $d(A,B)$ = $\\mu^{\\ast}(A \\triangle B)$ is $\\sigma$-algebra isomorphic and isometric to the Caratheodory Extension of $\\Omega$ under the equivalence relation $\\sim$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-03-12T11:24:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A12",
      "28B20"
    ],
    "keywords": [
      "caratheodory extension",
      "completion",
      "isomorphism",
      "equivalence relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.1749T"
    }
  }
}