Basic Ergodic Theory pp 87-97 | Cite as

# The Glimm-Effros Theorem

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## 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.

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## Bibliography

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