Continuous ergodic capacities

Abstract:

The objective of this paper is to characterize the structure of the set $\Theta$ for a continuous ergodic upper probability $\mathbb {V}=\sup _{P\in \Theta }P$

1. $\Theta$ contains a finite number of ergodic probabilities;
2. Any invariant probability in $\Theta$ is a convex combination of those ergodic ones in $\Theta$;
3. 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.