Continuous ergodic capacities.

The objective of this paper is to characterize the structure of the set \Theta for a continuous ergodic upper probability \mathbb {V}=\sup _{P\in \Theta }P \Theta contains a finite number of ergodic probabilities; Any invariant probability in \Theta is a convex combination of those ergodic ones in \Theta; Any probability in \Theta coincides with an invariant one in \Theta on the invariant \sigma-algebra. The last property has already been obtained in Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396], which first studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result, which improves the result of Cerreia-Vioglio, Maccheroni, and Marinacci [Proc. Amer. Math. Soc. 144 (2016), pp. 3381–3396] in the sense that the limit of the time means of \xi is bounded by the upper expectation \sup _{P\in \Theta }E_P[\xi ], instead of the Choquet integral. Generally, the former is strictly smaller. [ABSTRACT FROM AUTHOR]