Let $\Gamma\curvearrowright (X,\mu)$ be a probability measure preserving action of a countable group $\Gamma$ on a standard probability space $(X,\mu)$.
It is well known that $X$ can be equipped with a compact topology so that the action is by homeomorphisms. My question is: Can $X$ be equipped with a metric so that the action is by isometries?
Clearly, in general one cannot wish for a compact metric space since there are groups (minimally almost periodic) that have no non-trivial actions by isometries on compact metric spaces. But perhaps one could get some non-compact metric?