Capacities and Choquet averages of ultrafilters

Cerreia-Vioglio, Simone; Leonetti, Paolo; Maccheroni, Fabio; Marinacci, Massimo

doi:10.1090/proc/16642

Capacities and Choquet averages of ultrafilters
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by Simone Cerreia-Vioglio, Paolo Leonetti, Fabio Maccheroni and Massimo Marinacci

Proc. Amer. Math. Soc. 152 (2024), 1139-1151

DOI: https://doi.org/10.1090/proc/16642

Published electronically: January 5, 2024

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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.

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