On generalized limits and ultrafilters

TL;DR

This paper studies the structure of positive normalized linear functionals related to ideals on natural numbers, showing they can be expressed as ultrafilter limits, and explores their geometric properties and applications in sequence convergence.

Contribution

It demonstrates that all such functionals are ultrafilter limit averages, characterizes their diameter based on the ideal's maximality, and applies these results to sequence convergence theory.

Findings

Functionals are ultrafilter averages.

Diameter of the functional set is 2 iff the ideal is not maximal.

Identifies sequences where all functionals agree as $ extit{I}$-convergent sequences.

Abstract

Given an ideal $\mathcal{I}$ on $\omega$, 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.