4. THE RECONSTRUCTION THEOREM 25
has positive measure. Then we can find C B of positive measure such
that
ϕTϕ−1(C)
C = ∅,
ϕTϕ−1(C)
f(T)(C) = ∅,
f(T)(C) C = ∅.
Let D = C f(T)(C). Then D is f(T)-invariant but not
ϕTϕ−1
invariant.
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
−1(U1)f −1(U2),
so
ϕTϕ−1
= (ϕf
−1(U1)ϕ−1)(ϕf −1(U2)ϕ−1).
Now ϕf
−1(U1)ϕ−1
has sup-
port ϕ(supp(f
−1(U1)))
= supp(ff
−1(U1))
= supp(U1) and similarly the
automorphism ϕf
−1(U2)ϕ−1
has support equal to supp(U2), thus
ϕTϕ−1
leaves D invariant, a contradiction.
Finally, we verify that there is unique ϕ such that f(S) =
ϕSϕ−1.
For
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) =
supp(STS−1)
= S(A), so S = 1.
The uniqueness now follows immediately from 4.11 and the proof of 4.1
is complete.
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
ϕ[E]ϕ−1
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
non-hyperfinite ones.
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
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