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.
Categories: math.FA math.GN