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?

Entropy in Dynamical Systemsby Tomasz Downarowicz? I think his Theorem 6.8.1 is what you are after. $\endgroup$