The mean orbital pseudo-metric and the space of invariant measures

We study the mean orbital pseudo-metric for Polish dynamical systems and its connections with properties of the space of invariant measures. We give equivalent conditions for when the set of invariant measures generated by periodic points is dense in the set of ergodic measures and the space of invariant measures. We also introduce the concept of asymptotic orbital average shadowing property and show that it implies that every non-empty compact connected subset of the space of invariant measures has a generic point.