Abstract
This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view.
Highlights
A Polish space is a separable, completely metrizable topological space and a standard Borel space is a Polish space equipped with the σ-algebra of its Borel sets
Denote by Ea the equivalence relation on X generated by a: xEay ⇐⇒ ∃γ ∈ Γ(γ · x = y), whose classes are the orbits of the action
We will consider locally countable, Borel graphs on standard Borel spaces X. These are Borel subsets of X2 that are symmetric, avoid the diagonal and have countable sections. For any such graph G denote by G∗ the equivalence relation generated by G, whose classes are the connected components of G
Summary
A Polish space is a separable, completely metrizable topological space and a standard Borel space is a Polish space equipped with the σ-algebra of its Borel sets. If (X, μ) is a standard probability space and a is a Borel action of Γ on X, we say that a is measure preserving if for every γ ∈ Γ and every Borel set A ⊆ X, μ(γ · A) = μ(A). Two measure preserving actions Γ a X and ∆ a Y , on standard probability spaces (X, μ), (Y, ν), are orbit equivalent if there are invariant (under the corresponding actions) Borel sets A ⊆ X, B ⊆ Y with μ(A) = ν(B) = 1 and a Borel isomorphism f : A → B that sends μ to ν and satisfies xEay ⇐⇒ f (x)Ebf (y). These are Borel subsets of X2 that are symmetric, avoid the diagonal and have countable sections For any such graph G denote by G∗ the equivalence relation generated by G, whose classes are the connected components of G. This work in progress is contained in the preprint [K3] and the goal of the present paper is to present a survey of the current state of affairs in this subject and its open problems, referring to [K3] for a more complete development, including detailed proofs of all the results discussed here
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