It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words,
if
$$M(X,T) := \{ \mu\,\, \text{probability measure} : \mu= T\_\*\mu \} $$
is the set of invariant measures for $T$, then


$$h\_\text{top}(
T)= \sup\_{\mu \in M(X,T)} h\_{\mu}(T) $$


where $h\_\text{top}(T)$ is the topological entropy and $h\_{\mu}(T)$ is the metric entropy relative to $\mu$.


I have seen somewhere that, if we denote by $E(X,T) \subset M(X,T)$ the set of invariant ergodic measures for $T$, then


$$h\_\text{top}(T)= \sup\_{\mu \in E(X,T)} h\_{\mu}(T) $$


My questions are: is this true? If it is true, how is it proven?



Metric entropy and topological entropy
It's true for a very simple reason: the entropy of a dynamical system with respect to a (not necessarily ergodic) invariant measure is the average of the entropies of its ergodic components.


