# Prove the set of invariant measures of a continous action is w* closed

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 $$\int fd\mu_i \to \int f d\mu \ \ \forall f\in C(\Omega)$$

• You're a bit vague on what you mean by probability measure. If you consider arbitrary measures, the natural weak-$^*$ topology would be that of $\mathcal{L}^\infty(X)^*$ (the space of bounded measurable functions), and then I don't see a reason why the action of $\mathrm{Homeo}(X)$ on the set of measures would be continuous (even for a fixed self-homeomorphism). But in this context it's more common and natural to consider the space of Radon measures, and the weak-$^*$ would be that of $C(X)^*$. Then the action is continuous and hence the set of fixed points is closed. – YCor Nov 8 at 15:17
• You are right! I am going to make the clarification – HFKy Nov 8 at 16:11
• Ive added the clarifications! Thanks for the advice – HFKy Nov 8 at 16:24