{
  "id": "0806.1247",
  "version": "v1",
  "published": "2008-06-06T22:48:18.000Z",
  "updated": "2008-06-06T22:48:18.000Z",
  "title": "O-segments on topological measure spaces",
  "authors": [
    "Mohammad Javaheri"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $X$ be a topological space and $\\mu$ be a nonatomic finite measure on a $\\sigma$-algebra $\\Sigma$ containing the Borel $\\sigma$-algebra of $X$. We say $\\mu$ is weakly outer regular, if for every $A \\in \\Sigma$ and $\\epsilon>0$, there exists an open set $O$ such that $\\mu(A \\backslash O)=0$ and $\\mu(O \\backslash A)<\\epsilon$. The main result of this paper is to show that if $f,g \\in L^1(X,\\Sigma, \\mu)$ with $\\int_X f d\\mu=\\int_X g d\\mu=1$, then there exists an increasing family of open sets $u(t)$, $t\\in [0,1]$, such that $u(0)=\\emptyset$, $u(1)=X$, and $\\int_{u(t)} f d\\mu=\\int_{u(t)} g d\\mu=t$ for all $t\\in [0,1]$. We also study a similar problem for a finite collection of integrable functions on general finite and $\\sigma$-finite nonatomic measure spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-06T22:48:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A25",
      "46G10"
    ],
    "keywords": [
      "topological measure spaces",
      "open set",
      "finite nonatomic measure spaces",
      "o-segments",
      "nonatomic finite measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1247J"
    }
  }
}