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. 
Yanqi Qiu
math.DS math.PR