{
  "id": "1802.01878",
  "version": "v1",
  "published": "2018-02-06T10:33:25.000Z",
  "updated": "2018-02-06T10:33:25.000Z",
  "title": "Localizing Weak Convergence in $\\boldsymbol{ L_\\infty}$",
  "authors": [
    "J F Toland"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "For a general measure space $(X, \\sL, \\l)$ the pointwise nature of weak convergence in $\\Li$ is investigated using singular functionals analogous to $\\d$-functions in the theory of continuous functions on topological spaces. The implications for pointwise behaviour in $X$ of weakly convergent sequences in $\\Li$ are inferred and the composition mapping $u \\mapsto F(u)$ is shown to be sequentially weakly continuous on $\\Li$ when $F:\\RR \\to \\RR$ is continuous. When $\\sB$ is the Borel $\\sigma$-algebra of a locally compact Hausdorff topological space $(X,\\varrho)$ and $f \\in L_\\infty(X, \\sB, \\l)^*$ is arbitrary, let $\\nu$ be the finitely additive measure in the integral representation of $f$ on $L_\\infty(X, \\sB, \\l)$, and let $\\hat \\nu$ be the Borel measure in the integral representation of $f$ restricted to $C_0(X,\\varrho)$. From a minimax formula for $\\hat \\nu$ in terms $\\nu$ it emerges that when $(X,\\varrho)$ is not compact, $\\hat\\nu$ may be zero when $\\nu$ is not, and the set of $\\nu$ for which $\\hat \\nu$ has a singularity with respect to $\\l$ can be characterised. Throughout, the relation between $\\d$-functions and the analogous singular functionals on $\\Li$ is explored and weak convergence in $L_\\infty(X,\\sB,\\l)$ is localized about points of $(X_\\infty, \\varrho_\\infty)$, the one-point compactification of $(X,\\varrho)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-06T10:33:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "28C15",
      "46T99",
      "26A39",
      "28A25",
      "46B04"
    ],
    "keywords": [
      "localizing weak convergence",
      "integral representation",
      "general measure space",
      "locally compact hausdorff topological space",
      "analogous singular functionals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}