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THE RECONSTRUCTION THEOREM 21 induced transformation. 3, S = T (T )−1 = [U, V ] = U1 V1 , for some U, V, U1 , V1 ∈ [E], U1 , V1 involutions. Also supp(T ) = A and µ(A) ≤ 12 µ(supp(T )) = 12 , as A ∩ T (A) = ∅, so δu (T , 1) ≤ 12 . ✷ Fix now T ∈ [E]. All transformations below are in [E]. 4, T = S0 T1 , where S0 is a commutator and δu (T1 , 1) = µ(supp(T1 )) ≤ 12 . We concentrate on T1 . Split X = X1 ∪ X2 ∪ . . , with µ(Xi ) = 2−i , supp(T1 ) ⊆ X1 . 4 again, T1 = S1 T2 , where S1 is a commutator of elements with support contained in X1 , T2 has support also contained in X1 , and δu (T2 , 1) ≤ 14 .

Fix a free Borel action (n, x) → n · x of Z on (X, µ) which generates E. To show that F is smooth, it is enough to show that if (Y, ν) is one of 4. THE RECONSTRUCTION THEOREM 29 the ergodic components of the action of Γ, with ν non-atomic, then F |Y is smooth. Consider the cocycle α from Γ × Y into Z : α(γ, x) = (the unique n such that γ · x = n · x). Then there is Borel f : Y → Z such that α(γ, x) = f (γ · x) − f (x). Fix a set A of positive ν-measure, thus meeting every F -class in Y , on which f is constant.

The following simple result was originally proved for residually ﬁnite groups. Ben Miller then noticed that the argument really shows the following stronger fact. 13. Let E be an ergodic, hyperﬁnite equivalence relation. Given a countable group Γ, if for every γ ∈ Γ \ {1} there is a homomorphism π : Γ → [E] such that γ ∈ ker(π), then Γ embeds into [E]. In particular every residually amenable Γ embeds into [E] and for every countable group Γ there is unique normal subgroup N such that Γ/N embeds into [E] and every homomorphism from N into [E] is trivial.