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.



On The Convergence of Ergodic Measures
Let $X=\{0,1\}^{\mathbb{N}}$ with the infinite product topology (which is metrisable). For each $n \geq 1$, define $x\_n$ to be the sequence given by $x\_i=0$ for $1 \leq i \leq n$, $x\_i=1$ for $n+1 \leq i \leq 2n$, and $x\_{2n+i}=x\_i$ for all $i$. Let $T \colon X \to X$ be the shift transformation $T[(x\_n)]= (x\_{n+1})$.
 We have $T^{2n}x\_n=x\_n$ for every $n \geq 1$, so the measure $\mu\_n$ defined by $\mu\_n:=(2n)^{-1}\sum\_{j=0}^{2n-1}\delta\_{T^jx\_n}$ is an ergodic invariant Borel probability measure for $T$. 
Let $\overline{0}$ denote the element of $X$ corresponding to an infinite sequence of zeroes, and similarly let $\overline{1}$ denote the infinite sequence of ones; we have $\lim\_{n \to \infty} \mu\_n = \frac{1}{2}(\delta\_{\overline{0}}+\delta\_{\overline{1}})$, and this limit is not ergodic (since the set containing only the point $\{\overline{0}\}$ has measure 1/2 but is invariant).


There is a nice paper by Parthasarathy - called, I think, "On the category of ergodic measures" - which shows that for this particular dynamical system and some of its generalisations, the set of all ergodic measures and the set of all non-ergodic measures are both weak-\* dense in the set of all invariant measures, so this phenomenon can actually happen quite a lot.


(Hmm, the definition of $X$ above is supposed to have curly set brackets in it, but I can't get them to appear for some reason. Anyway, it's supposed to be the set of all one-sided infinite sequences of zeroes and ones.) 


