Ergodic measures on compact metric spaces for isometric actions by inductively compact groups

TL;DR

This paper explores the relationship between ergodic measures and orbital measures in compact metric spaces under isometric actions by inductively compact groups, providing a partial converse to a known description.

Contribution

It establishes a partial converse to Vershik's description, linking ergodic measures to weak limits of orbital measures in this setting.

Findings

Identifies ergodic measures with weak limit points of orbital measures.

Provides a partial converse to Vershik's description.

Enhances understanding of measure dynamics under group actions.

Abstract

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.