Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. It is known that $$\mathcal{M}_{\sigma}(X)= \{ \mu \in \mathcal{M}(X) \mid \sigma_* \mu=\mu \} \subset \mathcal{M}(X)= \{ \text{Borelian probability measure} \} \subset C(X,\mathbb{R})^*$$ where $\mathcal{M}_{\sigma}(X)$ and $\mathcal{M}(X)$ are both non-empty compact convex subspaces. Moreover, $\mu \in \mathcal{M}(X)$ is ergodic iff it is an extremal point of $\mathcal{M}_{\sigma}(X)$.
Is there an elementary way to show that such extremal points are dense in $\mathcal{M}_{\sigma}(X)$?