Let $T : X \to X$ be a measurable automorphism of a standard Borel measurable space $X$.
How can we tell if $T$ has an invariant probability measure?
For example, if $T$ is a rotation on the circle, then clearly an invariant probability measure exists. On the other hand, if $T$ is a translation on $\mathbb{R}$, then none exists.
Since I'm not assuming a given topology on $X$, the kind of answer that I'm looking for will not make reference to a topology on $X$. So the Krylov-Bogolyubov theorem is not really what I'm looking for. A good answer could for example be that no invariant probability measure exists if and only if a suitably defined cross product von Neumann algebra is properly infinite.