Capacities and Choquet Averages of Ultrafilters

TL;DR

This paper characterizes capacities invariant under ideals on natural numbers as Choquet averages of ultrafilter-based measures, linking measure theory with Riesz space structures.

Contribution

It provides a novel representation of invariant capacities as Choquet averages of ultrafilter measures, extending the theory to Archimedean Riesz spaces.

Findings

Capacities invariant under ideals are characterized as Choquet averages.

Representation links ultrafilter measures with Riesz space theory.

Abstract results generalize classical measure invariance concepts.

Abstract

We show that a normalized capacity $\nu: \mathcal{P}(\mathbf{N})\to \mathbf{R}$ is invariant with respect to an ideal $\mathcal{I}$ on $\mathbf{N}$ 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 $\mathcal{I}$. This is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces.