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