4. THE RECONSTRUCTION THEOREM 25
has positive measure. Then we can find C ⊆ B of positive measure such
∩ C = ∅,
∩ f(T)(C) = ∅,
f(T)(C) ∩ C = ∅.
Let D = C ∪ f(T)(C). Then D is f(T)-invariant but not
Write f(T) = U1U2, where U1 = f(T) on D, = id on X \ D and U2 = f(T)
on X \ D, id on D, so that U1,U2 ∈ [F ] are involutions. Then
T = f
= supp(U1) and similarly the
has support equal to supp(U2), thus
leaves D invariant, a contradiction. ✷
Finally, we verify that there is unique ϕ such that f(S) =
that we use the following lemmas which, for further reference, we state in
more generality than we need here.
Lemma 4.10. For any aperiodic E (not necessarily ergodic) and Borel set
A, there is T ∈ [E] with supp(T) = A.
Proof. Since E is aperiodic, measure preserving, we can assume that
E|A is aperiodic. Thus, by 3.5, there is aperiodic T0 : A → A with T0(x)Ex.
Let T = T0 ∪ id|(X \ A). ✷
Corollary 4.11. For any aperiodic E (not necessarily ergodic) the central-
izer C[E] of [E] in Aut(X, µ) is trivial.
Proof. If S ∈ C[E] and A ∈ MALGµ, then by 4.10 there is T ∈ [E] with
supp(T) = A. Then A = supp(T) =
= S(A), so S = 1. ✷
The uniqueness now follows immediately from 4.11 and the proof of 4.1
Remark. Note that the argument following the statement of 4.9 shows
that if ϕ is a non-singular Borel automorphism of X, E is ergodic and
⊆ Aut(X, µ), then ϕ ∈ Aut(X, µ).
(D) Dye’s Reconstruction Theorem suggests the problem of distinguish-
ing, up to isomorphism, ergodic equivalence relations by finding algebraic
or perhaps topological group distinctions of their corresponding full groups
(it is understood here that full groups are equipped with the uniform topol-
ogy). For example, 3.17 provides such a distinction between hyperfinite and
One might explore the following possibility. Recall that an action of Γ
on (X, µ) is called a free action if ∀γ = 1(γ · x = x, µ-a.e.). We know by
Gaboriau [Ga1] that if Em is given by a free, measure preserving action of