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