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On generalized limits and ultrafilters

Paolo Leonetti, Cihan Orhan

TL;DR

The paper addresses the structure of $S_{\\mathcal{I}}$-limit functionals on $\\ell_\\infty$ by representing $\\mathrm{SL}(\\mathcal{I})$ as both a Choquet integral over $\\mathrm{Ult}(\\mathcal{I})$ and as the weak$^*$ closed convex hull of ultrafilter limit functionals. It proves a sharp diameter result, $\\mathrm{diam}(\\mathrm{SL}(\\mathcal{I}))=2$ if and only if $\\mathcal{I}$ is not maximal, with a stronger version for meager ideals, and provides a complete decomposition description of elements in $\\mathrm{SL}(\\mathcal{I})$ via $f=g-h$ under explicit constraints. As an application, it recovers Freedman’s identification $\\mathscr{V}(\\mathcal{I})=c(\\mathcal{I})\\cap \\ell_\\infty$ of the bounded $\\mathcal{I}$-convergent sequences and clarifies the relation between $\\mathcal{I}$-cluster, $\\mathcal{I}$-limit, and ultrafilter convergence. The work combines Choquet theory, finitely additive measures, and ultrafilter techniques to illuminate the geometry of $\\mathrm{SL}(\\mathcal{I})$ and the $\\mathcal{I}$-convergence landscape, and ends with an open problem on characterizing ideals with the strong diameter property.

Abstract

Given an ideal $\mathcal{I}$ on $ω$, we denote by $\mathrm{SL}(\mathcal{I})$ the family of positive normalized linear functionals on $\ell_\infty$ which assign value $0$ to all characteristic sequences of sets in $\mathcal{I}$. We show that every element of $\mathrm{SL}(\mathcal{I})$ is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of $\mathrm{SL}(\mathcal{I})$ is $2$ if and only if $\mathcal{I}$ is not maximal, and that the latter claim can be considerably strengthened if $\mathcal{I}$ is meager. Lastly, we provide several applications: for instance, recovering a result of Freedman in [Bull. Lond. Math. Soc. 13 (1981), 224--228], we show that the family of bounded sequences for which all functionals in $\mathrm{SL}(\mathcal{I})$ assign the same value coincides with the closed vector space of bounded $\mathcal{I}$-convergent sequences.

On generalized limits and ultrafilters

TL;DR

The paper addresses the structure of -limit functionals on by representing as both a Choquet integral over and as the weak closed convex hull of ultrafilter limit functionals. It proves a sharp diameter result, if and only if is not maximal, with a stronger version for meager ideals, and provides a complete decomposition description of elements in via under explicit constraints. As an application, it recovers Freedman’s identification of the bounded -convergent sequences and clarifies the relation between -cluster, -limit, and ultrafilter convergence. The work combines Choquet theory, finitely additive measures, and ultrafilter techniques to illuminate the geometry of and the -convergence landscape, and ends with an open problem on characterizing ideals with the strong diameter property.

Abstract

Given an ideal on , we denote by the family of positive normalized linear functionals on which assign value to all characteristic sequences of sets in . We show that every element of is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of is if and only if is not maximal, and that the latter claim can be considerably strengthened if is meager. Lastly, we provide several applications: for instance, recovering a result of Freedman in [Bull. Lond. Math. Soc. 13 (1981), 224--228], we show that the family of bounded sequences for which all functionals in assign the same value coincides with the closed vector space of bounded -convergent sequences.

Paper Structure

This paper contains 6 sections, 17 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Proofs of Main results
  5. Further results
  6. Concluding Remarks

Key Result

Theorem 1.3

Let $\mathcal{I}$ be an ideal on $\omega$. Then $\mathscr{V}(\mathcal{I})=\overline{c(\mathcal{I}^\star) \cap \ell_\infty}=c(\mathcal{I}) \cap \ell_\infty$. If, in addition, $\mathcal{I}$ is a $P$-ideal, then also $\mathscr{V}(\mathcal{I})=c+(c_{00}(\mathcal{I})\cap \ell_\infty)$.

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • ...and 29 more