I have been reading a paper on Ergodic Optimization ("Ergodic optimization in dynamical systems" by Jenkinson), where he claims that under nice conditions we can always pick an ergodic measure that is maximizing. I am not sure why...
Consider $T:X\to X$ and $f:X\to \mathbb{R}$ where $X$ is compact metric space and $T,f$ are continuous.
We say a measure $\gamma$ is maximizing for $f$ if $\int f d\gamma=\max_{\mu\in\mathcal{M}(X,T)}\int f d\mu$.
We know that $\mathcal{M}(X,T)$ is convex compact in weak* topology which means there is a maximizing measure (which justifies the use of max above). We also know that it's extreme points are the ergodic measures.
Now the set of maximizing measures $\mathcal{M}_{max}(f)$ is also convex and compact. Apparently it's extreme points are also ergodic. So it's extreme points are also extreme points of $\mathcal{M}(X,T)$. Why is that the case?
Thank you