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.