{
  "id": "2403.07484",
  "version": "v1",
  "published": "2024-03-12T10:20:33.000Z",
  "updated": "2024-03-12T10:20:33.000Z",
  "title": "The Nikodym property and filters on $ω$",
  "authors": [
    "Tomasz Żuchowski"
  ],
  "comment": "21 pages",
  "categories": [
    "math.LO",
    "math.FA",
    "math.GN"
  ],
  "abstract": "For a free filter $F$ on $\\omega$, let $N_F=\\omega\\cup\\{p_F\\}$, where $p_F\\not\\in\\omega$, be equipped with the following topology: every element of $\\omega$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\\cup\\{p_F\\}$ for $A\\in F$. The aim of this paper is to study spaces of the form $N_F$ in the context of the Nikodym property of Boolean algebras. By $\\mathcal{AN}$ we denote the class of all those ideals $\\mathcal{I}$ on $\\omega$ such that for the dual filter $\\mathcal{I}^*$ the space $N_{\\mathcal{I}^*}$ carries a sequence $\\langle\\mu_n\\colon n\\in\\omega\\rangle$ of finitely supported signed measures such that $\\|\\mu_n\\|\\rightarrow\\infty$ and $\\mu_n(A)\\rightarrow 0$ for every clopen subset $A\\subseteq N_{\\mathcal{I}^*}$. We prove that $\\mathcal{I}\\in\\mathcal{AN}$ if and only if there exists a density submeasure $\\varphi$ on $\\omega$ such that $\\varphi(\\omega)=\\infty$ and $\\mathcal{I}$ is contained in the exhaustive ideal $\\mbox{Exh}(\\varphi)$. Consequently, we get that if $\\mathcal{I}\\subseteq\\mbox{Exh}(\\varphi)$ for some density submeasure $\\varphi$ on $\\omega$ such that $\\varphi(\\omega)=\\infty$ and $N_{\\mathcal{I}^*}$ is homeomorphic to a subspace of the Stone space $St(\\mathcal{A})$ of a given Boolean algebra $\\mathcal{A}$, then $\\mathcal{A}$ does not have the Nikodym property. We observe that each $\\mathcal{I}\\in\\mathcal{AN}$ is Kat\\v{e}tov below the asymptotic density zero ideal $\\mathcal{Z}$, and prove that the class $\\mathcal{AN}$ has a subset of size $\\mathfrak{d}$ which is dominating with respect to the Kat\\v{e}tov order $\\leq_K$, but $\\mathcal{AN}$ has no $\\leq_K$-maximal element. We show that for a density ideal $\\mathcal{I}$ it holds $\\mathcal{I}\\not\\in\\mathcal{AN}$ if and only if $\\mathcal{I}$ is totally bounded if and only if the Boolean algebra $\\mathcal{P}(\\omega)/\\mathcal{I}$ contains a countable splitting family.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-12T10:20:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E75",
      "28A33",
      "03E05",
      "54A20",
      "28A60",
      "06E15"
    ],
    "keywords": [
      "nikodym property",
      "boolean algebra",
      "density submeasure",
      "asymptotic density zero ideal",
      "stone space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}