On generalized limits and ultrafilters

Given an ideal I on ω, we denote by SL(I) the family of positive normalized linear functionals on l∞ which assign value 0 to all characteristic sequences of sets in I. We show that every element of SL(I) is a Choquet average of certain ultrafilter limit functionals. Also, we prove that the diameter of SL(I) 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 SL(I) assign the same value coincides with the closed vector space of bounded I-convergent sequences