compactness of the set of invariant measures

Question

Suppose that we have a dynamical system on some compact space $X$ with discrete time space and transformation given by some $\phi : X \rightarrow X$. My question is when is the set $Prob(\phi)$ of all $\phi$-invariant probabilistic measures on $X$ compact? Compact in pretty much any sense will do (I just need the topology to be locally convex beacuse I want to use the Krein-Milman theorem). I've seen this: Space of $T$-invariant probability measures is compact. question, but this answer doesn't help me because it uses the assumption that $X$ is a metric space, while in my case I want to use it on a tore $T^d$. Can one get rid of this assumption?

Answer 1

I think what you need is Prohorov theorem, which is a probabilistic version (in fact simpler) of the general Banach-Alaoglu compacity theorem. It is extremely standard in probability, and works fine in Polish state spaces (metric, complete separable).

In essence, it tells you that a set of probability distribution is compact for the natural topology of convergence in law (the one of the central limit theorem), as soon as the latter is mainly weighting some compact uniformly (''tightness'').

In your case, if the state space is compact, you have nothing to prove, the whole set of probabilities is a compact. Oherwise, you need to ensure that the property of being an inviariant probability does imply tightness.

Note that the torus is obviously a metric space.