Capacities and Choquet averages of ultrafilters

We show that a normalized capacity ν : P(N) → R is invariant with respect to an ideal I on N if and only if it can be represented as a Choquet average of {0, 1}-valued finitely additive probability measures corresponding to the ultrafilters containing the dual filter of I. This is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces.