Continuous Ergodic Capacities

The objective of this paper is to characterize the structure of the set $\Theta$ for a continuous ergodic upper probability $\mathbb{V}=\sup_{P\in\Theta}P$ (Theorem \ref {main result}): . $\Theta$ 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 \textsl{Cerreia-Vioglio, Maccheroni, and Marinacci} \cite{ergodictheorem}, which firstly studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result (Theorem \ref {improve}), which improves the result in \cite{ergodictheorem} in the sense that the limit of the time mean 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.