Subequivalence Relations and Positive - Definite Functions

Consider a standard probability space (X,μ), i.e., a space isomorphic to the unit interval with Lebesgue measure. We denote by Aut(X,μ) the automorphism group of (X,μ), i.e., the group of all Borel automorphisms of X which preserve μ (where two such automorphisms are identified if they are equal μ-a.e.). A Borel equivalence relation E ⊆ X is called countable if every E-class [x]E is countable and measure preserving if every Borel automorphism T of X for which T (x)Ex is measure preserving. Equivalently, E is countable, measure preserving iff it is induced by a measure preserving action of a countable (discrete) group on (X,μ) (see Feldman-Moore [FM]). To each countable, measure preserving equivalence relation E one can assign the positive-definite function φE(S) on Aut(X,μ) given by φE(S) = μ({x : S(x)Ex}); see Section 1. Intuitively, φE(S) measures the amount by which S is “captured” by E. This positive-definite function completely determines E. We use this function to measure the proximity of a pair E ⊆ F of countable, measure preserving equivalence relations. In Section 2, we show, among other things, the next result, where we use the following notation: If a countable group Γ acts on X, we also write γ for the automorphism x 7→ γ · x; if A ⊆ X and E is an equivalence relation on X, then E|A = E ∩ A is the restriction of E to A; if E ⊆ F are equivalence relations, then [F : E] = m means that every F -class contains exactly m classes; if F is a countable, measure preserving equivalence relation on (X,μ), then [F ] is the full group of F , i.e., [F ] = {T ∈ Aut(X,μ) : T (x)Fx, μ−a.e.(x)}.