3. FULL GROUPS OF EQUIVALENCE RELATIONS 13 3. Full groups of equivalence relations (A) Let (X, µ) be a standard measure space and E a Borel equivalence relation on X. We say that E is a countable equivalence relation if every E-class is countable. Feldman-Moore [FM] have shown that E is countable iff there is a countable (discrete) group Γ acting in a Borel way on X such that if (γ, x) γ · x denotes the action and we let X = {(x, y) : ∃γ Γ(γ · x = y)} be the induced equivalence relation, then E = EX. Γ We say that E is a measure preserving equivalence relation if the action of Γ is measure pre- serving, i.e., x γ · x is measure preserving for each γ Γ. This condition is independent of Γ and the action that induces E and is equivalent to the condition that every Borel automorphism T of X for which T(x)Ex, ∀x, is measure preserving. For any Borel E we define its full group or inner automorphism group, [E], by [E] = {T Aut(X, µ) : T(x)Ex, µ-a.e.(x)}. It is easy to check that this is a closed subgroup of Aut(X, µ) in the uniform topology. Convention. Below we will assume that equivalence relations E are countable and measure preserving, unless otherwise explicitly stated. Recall that E is an ergodic equivalence relation if every E-invariant Borel set is null or co-null. We call E a finite equivalence relation or a periodic equivalence relation if (almost) all its classes are finite and an aperiodic equivalence relation if (almost) all its classes are infinite. Proposition 3.1. E is ergodic iff [E] is dense in (Aut(X, µ),w). Proof. Assume E is ergodic. Then if A, B are Borel sets of the same measure, there is T [E] with T(A) = B. Now let S Aut(X, µ). A nbhd basis for S is given by the sets of the form {T : ∀i n(dµ(T(Ai),S(Ai)) )}, where 0, and A1,...,An is a Borel partition of X. Find T1,...,Tn [E] such that Ti(Ai) = S(Ai). Then T = n i=1 Ti|Ai is in [E] and clearly T(Ai) = S(Ai), i = 1,...,n. If E is not ergodic, there is A MALGµ with 0 µ(A) 1 which is [E]-invariant. If [E] was weakly dense in Aut(X, µ), A would be T-invariant for any T Aut(X, µ), which is absurd. T.-J. Wei [We] has shown in fact that [E] is closed in w iff E is a finite equivalence relation. Otherwise, it is Π0-complete 3 in w and its closure in the weak topology is equal to [F ], where F is the (not necessarily countable) equivalence relation corresponding to the ergodic decomposition of E. Also clearly [E] is a meager subset of (Aut(X, µ),w), since it is a Borel non-open subgroup of (Aut(X, µ),w). Note now the following basic fact about the uniform topology of [E].
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