  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.
