Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic?
I know that ergodic measures are the extreme points of the convex set of invariant measures, but the proof of this fact requires the Radon-Nikodym Theorem. I'm just wondering if the above case has an elementary proof. So far, I don't see it.