The Glimm-Effros Theorem

Abstract

Let (X, Ɓ) be a standard Borel space and let G be a countable group of Borel automorphisms acting freely on X. (Acting freely means that for every x ∈ X, gx = x only when g = e the identity of the group.) If μ is a probability measure supported on an orbit of G, then clearly the G-action is ergodic with respect to μ. Thus there always exists, in a trivial sense, a probability measure with respect to which the G-action is ergodic. But the μ above is discrete and supported on an orbit. A continuous probability measure μ (necessarily giving mass zero to each orbit which is countable) ergodic with respect to the G-action need not always exist. An obvious necessary condition for the existence of a continuous measure for which the G-action is ergodic is that the orbit space of G should not admit a Borel cross-section. We will prove in this chapter that the converse of this holds. (See Shelah and Weiss [7].) The result will follow as a consequence of the Ramsay-Mackey theorem of Chapter 8 and the following theorem due to Glimm [3] and Effros [2]. We will also discuss some related results and a recent theorem of R. Dougherty, S. Jackson and A. Kechris [1] which in some sense is a complement of the Glimm-Effros theorem for the case of a single automorphism.