TITLE: On The Convergence of Ergodic Measures
QUESTION [9 upvotes]: I would like to know an example (not using the Gibbs measure Theory) of a sequence of measures $\mu_n:\mathcal B\to[0,1]$ , where $\mathcal B$ is the $\sigma$-algebra of the borelians of a compact space $X$ such that :
1) $\mu_n$ is ergodic, with respect to a fixed continuous function $T:X\to X$, for all $n\in\mathbb N$;
2) $\mu_n\to \mu$ in the weak-$*$ topology and $\mu$ is not ergodic.

REPLY [4 votes]: consider the torus map $(x,y) \mapsto (x+y, y)$. For every $y$, this gives a rotation on the circle $S^1 \times y$. That is, Lebesgue measure on any "horizontal" circle is preserved. For $y$ rational it is not ergodic, but for $y$ irrational it is ergodic. Well, that's it. Lebesgue on a circle with rational rotation is weak* approximated by Lebesgue on circles with irrational rotation.