20 I. MEASURE PRESERVING AUTOMORPHISMS
4. The Reconstruction Theorem
(A) We will prove here another beautiful result of Dye that asserts that
any ergodic E is completely determined by [E] as an abstract group. Given
two countable Borel measure preserving equivalence relations E, F we say
that E, F are isomorphic, in symbols
if there is T ∈ Aut(X, µ) such that except on a null set
xEy ⇔ T(x)FT(y),
or, equivalently, T[E]T
= [F ], i.e., [E], [F ] are conjugate in Aut(X, µ).
Theorem 4.1 (Dye). Let E, F be ergodic equivalence relations. Then the
following are equivalent:
= F .
(ii) [E], [F ] are conjugate in Aut(X, µ).
(iii) ([E],u), ([F ],u) are isomorphic as topological groups.
(iv) [E], [F ] are isomorphic as abstract groups.
Moreover, for any algebraic isomorphism f : [E] → [F ], there is unique
ϕ ∈ Aut(X, µ) with f(T) =
∀T ∈ [E].
(B) In preparation for the proof, we prove some results that are also
useful for other purposes.
Lemma 4.2. Let E be ergodic. Then every element of [E] is a product of 5
commutators in [E].
Proof. We will need two sublemmas.
Sublemma 4.3. Let E be ergodic and let T ∈ [E] be periodic. Then T is a
commutator in [E] and the product of two involutions in [E].
Proof. We can assume that for some n ≥ 2 all T-orbits have exactly
n elements. Let A be a Borel selector for the T-orbits. Split A = A1 ∪ A2,
where µ(A1) = µ(A2) =
µ(A). Let Q ∈ [E] be an involution that sends
to B2 =
and conjugates T|B1 to T
Let T1 = T|B1 ∪ id|B2,T2 = id|B1 ∪ T|B2. Then QT2
= T1 and T =
T1T2 = QT2
= [Q, T2
Sublemma 4.4. Let E be ergodic. If T ∈ [E], then we can write T =
ST , where S ∈ [E],S = [U, V ] = U1V1 with U, V, U1,V1 ∈ [E],δu(T , 1) ≤
δu(T, 1),U1,V1 involutions and
supp(U) ∪ supp(V ) ∪ supp(U1) ∪ supp(V1) ∪ supp(T ) ⊆ supp(T).
Proof. We can assume that T is aperiodic (since on the periodic part
of T we can apply 4.3 and take T = 1). We can then find a Borel complete
section A for the T-orbits such that in the usual Z-order of each T-orbit the
distance between successive elements of A is at least 2, and A is unbounded
in each direction in every orbit (see, e.g., [KM], 6.7). Let T = TA be the