Let $X={0,1}^{\mathbb{N}}$ 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.) 3 added 92 characters in body 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.reason. Anyway, it's supposed to be the set of all one-sided infinite sequences of zeroes and ones.) 2 added 372 characters in body; added 26 characters in body 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.)