Capacities and Choquet averages of ultrafilters

Abstract

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