{
  "id": "1910.14555",
  "version": "v1",
  "published": "2019-10-31T15:54:27.000Z",
  "updated": "2019-10-31T15:54:27.000Z",
  "title": "On the range of a vector measure",
  "authors": [
    "José Rodríguez"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $(\\Omega,\\Sigma,\\mu)$ be a finite measure space, $Z$ be a Banach space and $\\nu:\\Sigma \\to Z^*$ be a countably additive $\\mu$-continuous vector measure. Let $X \\subseteq Z^*$ be a norm-closed subspace which is norming for $Z$. Write $\\sigma(Z,X)$ (resp. $\\mu(X,Z)$) to denote the weak (resp. Mackey) topology on $Z$ (resp. $X$) associated to the dual pair $\\langle X,Z\\rangle$. Suppose that, either $(Z,\\sigma(Z,X))$ has the Mazur property, or $(B_{X^*},w^*)$ is convex block compact and $(X,\\mu(X,Z))$ is complete. We prove that the range of $\\nu$ is contained in $X$ if, for each $A\\in \\Sigma$ with $\\mu(A)>0$, the $w^*$-closed convex hull of $\\{\\frac{\\nu(B)}{\\mu(B)}: \\, B\\in \\Sigma, \\, B \\subseteq A, \\, \\mu(B)>0\\}$ intersects $X$. This extends results obtained by Freniche [Proc. Amer. Math. Soc. 107 (1989), no. 1, 119--124] when $Z=X^*$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-31T15:54:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A50",
      "46G10"
    ],
    "keywords": [
      "finite measure space",
      "convex block compact",
      "banach space",
      "continuous vector measure",
      "extends results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}