Continuous Ergodic Capacities

Abstract

The objective of this paper is to characterize the structure of the set $Θ$ for a continuous ergodic upper probability $\mathbb{V}=\sup_{P\inΘ}P$ (Theorem \ref {main result}): . $Θ$ contains a finite number of ergodic probabilities; . Any invariant probability in $Θ$ is a convex combination of those ergodic ones in $Θ$; . Any probability in $Θ$ coincides with an invariant one in $Θ$ on the invariant $σ$-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 $ξ$ is bounded by the upper expectation $\sup_{P\inΘ}E_P[ξ]$, instead of the Choquet integral. Generally, the former is strictly smaller.