Let $X$ be a topological space and $T: X \to X$ an invertible continuous map.
Recall that $T$ is ergodic with respect to a $T$-invariant probability measure $\mu$ if and only if any $T$-invariant function in the space $L^2(X, \mu)$ of square-integrable functions with respect to $\mu$ is constant almost everywhere.
Is there any simple property of this kind that characterizes when a $T$-invariant probability measure $\mu$ is a finite convex combination of ergodic probability measures?
A reference to any other characterizations of finite convex combinations of ergodic probability measures would also be appreciated.