I am stuck on the following problem, which is a previous step for proving existence of invariant measures, and also is a previous step for proving the existence of a Ergodic System.
Let $G$ be an abelian group, $\Omega$ a Hausdorff compact space, and $\alpha:G\times\Omega \to \Omega$ a continuous action on $\Omega$. A probability measure $\mu \in M(\Omega)$ is called $\alpha$-invariant if $\mu(\alpha_t(E)) = \mu(E) \ \forall E$ Borel set, and for all $t \in G$. Let $M_{\alpha}$ be the set of $\alpha$-invariant measures. Prove that $M_\alpha$ is convex and w$*$-closed.
I've already proved that $M(\Omega)$ is w$*$-compact, using Banach-Alaoglu and some tricks. Hence if I prove that $M_\alpha$ is w$*$-closed I finish. To prove this I take a convergence net $\mu_i\to \mu$ and I've tried to prove that $\mu$ belongs to $M_{\alpha}$. The best I can do was that:
Since $\mu_i$ is invariant, it can be proved that $$\int_{\Omega}fd\mu_i = \int_{\Omega}f\alpha_t^{-1}d\mu_i,$$ now taking limits the left side converges to $\int_{\Omega}fd\mu$ and the right to $\int_{\Omega}f\alpha_t^{-1}d\mu$. The right member converges to this because $f\alpha_t^{-1} \in C(\Omega)$.
I would appreciate any ideas.
EDIT 1: By probability measure I mean regular borel measures on $\Omega$. In this case, we have that $C(\Omega)^*\cong M(\Omega) $ where $M(\Omega)$ are the sets of all complex regular borel measures on $\Omega$. We consider the set of probability measures meaning the subset of $M(\Omega)$ which are probability measures. We have a natural topology here that is the weak star topology of $M(\Omega)$ restricted to the sets of probability measures. Here a net $\mu_i$ of probability measures converges to a probability measures $\mu$ if and only if $$\int fd\mu_i \to \int f d\mu \ \ \forall f\in C(\Omega)$$