On generalized limits and ultrafilters

Given an ideal $I$ on ω, we denote by $\mathrm{SL}\left(I\right)$ the family of positive normalized linear functionals on ${\ell }_{\infty }$ which assign value 0 to all characteristic sequences of sets in $I$. We show that every element of $\mathrm{SL}\left(I\right)$ is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of $\mathrm{SL}\left(I\right)$ is 2 if and only if $I$ is not maximal, and that the latter claim can be considerably strengthened if $I$ is meager. Lastly, we provide several applications: for instance, recovering a result of Freedman (1981) [19], we show that the family of bounded sequences for which all functionals in $\mathrm{SL}\left(I\right)$ assign the same value coincides with the closed vector space of bounded $I$-convergent sequences.