34 I. MEASURE PRESERVING AUTOMORPHISMS
be the equivalence relation generated by E and ϕ. Clearly F is measure
preserving and ergodic. Since every F -class consists of at most 2 E-classes,
F is hyperfinite.
Assume now that the result is true up to n and say E = {e1,...,en,en+1}.
By induction hypothesis we can find a hyperfinite measure preserving ergodic
equivalence relation En on Xe1 · · · Xen extending E|(Xe1 · · · Xen ).
Then if Fn = E∪En,Fn is hyperfinite, measure preserving and has 2 ergodic
invariant measures, so by the n = 2 case, we can find F Fn, F hyperfinite,
ergodic and measure preserving.
Case 2. E is countably infinite.
Say E = {e1,e2,... ). By Case 1, there is hyperfinite, measure preserving,
ergodic En on Xe1 ∪· · ·∪Xen with E|Xe1 ∪· · ·∪E|Xen En and En En+1.
Let F =
n
En. This clearly works.
Case 3. E is uncountable and ν is non-atomic.
Let then T Aut(E,ν) be ergodic. For each e E, let ϕe : Xe
XT
(e)
be a Borel bijection such that ϕ(e, x) = ϕe(x) is Borel and ϕe is an
isomorphism of (E|Xe,e) with (E|XT
(e)
, T(e)) modulo null sets. Let then
ϕ(x) = ϕπ(x)(x) (so if x Xe,ϕ(x) = ϕe(x)). Note first that ϕ Aut(X, µ).
To see this fix a Borel set A X. Then
µ(ϕ−1(A))
=
e(ϕ−1(A)
Xe)dν(e)
=
T(e)(ϕe(ϕ−1(A)
Xe))dν(e)
= T(e)(A XT (e))dν(e)
= e(A Xe)dν(e)
= µ(A),
as ν is T-invariant.
Let F be the equivalence relation induced by E and ϕ, so that E F
and F is measure preserving, ergodic. Finally, E is hyperfinite (µ-a.e.) as
every F class can be (Borel uniformly) ordered in order type
ζ2
(where ζ is
the order type of Z) (see Jackson-Kechris-Louveau [JKL], Section 2).
Case 4. E is uncountable but ν has atoms.
Let then {e1,e2,... } be the atoms of ν. Let X0 =
i
Xei , X1 = X \ X0.
By applying the preceding cases to (E|Xi, (µ|Xi)/µ(Xi)),i = 0, 1, we see
that we can find measure preserving, ergodic, hyperfinite Fi on Xi such that
E|Xi Fi. If F = F1 F2, then F is measure preserving, hyperfinite and
E F . Then, by Case 1 again, we can find measure preserving, ergodic,
hyperfinite F F and the proof is complete.
Similar arguments also show the following.
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