ERGODIC MEASURES ON COMPACT METRIC SPACES FOR ISOMETRIC ACTIONS BY INDUCTIVELY COMPACT GROUPS

We obtain a partial converse of Vershik's description of ergodic probability measures on a compact metric space with respect to an isometric action by an inductively compact group. This allows us to identify, in this setting, the set of ergodic probability measures with the set of weak limit points of orbital measures. We also show that for a general action of an inductively compact group, the weak limit of orbital measures can fail to be ergodic.