8. COSTS AND THE OUTER AUTOMORPHISM GROUP 49
If x Ci,x B, there is |n| N with
Sn(x)
A and Ti(x) =
S−nTiSn(x).
Thus {S, Ti|(A B Ci)} forms an L-graphing of E
S,Ti
and
if Ai = A B Ci, then µ(Ai) δ.
It follows that we can find i0 i1 · · · ik ik+1 . . . such that
i ik ∃Bi(µ(Bi)
2k+1
and {S, Ti|Bi} is an L-graphing of E
S,Ti
).
Let ik0 ik1 . . . be a subsequence of {ik}. Then ik i , so we can find
D with µ(D )
2 +1
and {S, Tik |D } is an L-graphing of E
S,Tik
, so that
{S, Tik |D } is an L-graphing of E
S,Tik
and therefore Cµ(E
S,Tik
)
1 +
∑∞
=0
2 +1
= 1 + .
Lemma 8.4. If [E] is not closed in N[E] and there are aperiodic transfor-
mations S1,...,Sk [E] with E = E
S1,...,Sk
, then Cµ(E) = 1.
Proof. Fix 0. Repeatedly applying the preceding lemma, we can
find a good sequence {Tn} such that
Cµ(E
Si,Tn
n
) 1 +
k
, ∀i k.
Since, by Lemma 8.2, E
Tn
n
is aperiodic, it follows (see Kechris-Miller [KM],
23.5) that
Cµ(E) 1 = Cµ(
k
i=1
E
Si,Tn
n
) 1

k
i=1
[Cµ(E
Si,Tn
n
) 1] .
Since is arbitrary, Cµ(E) = 1.
The next fact is true in the pure Borel category (where we interpret [E]
as consisting of all Borel automorphisms T with T(x)Ex, for all x).
Lemma 8.5. Let E be a countable aperiodic Borel equivalence relation and
S [E]. Then there is an aperiodic T [E] with ES ET .
Proof. We can assume that every orbit of S is finite. Let then A be a
Borel transversal for ES. Then E|A is aperiodic, so (by the proof of 3.5, (i))
let F E|A be aperiodic hyperfinite. Then F ES is aperiodic, hyperfinite
and so of the form ET for some aperiodic T [E].
Now, to complete the proof, assume E is as in the statement of the
theorem. By 3.5 there is U0 [E] which is ergodic (and thus aperiodic).
By 8.3, there is a good sequence {Tn} such that Cµ(E
U0,Tn
n
) 2. Let
E = E
U1,U2,...
, Ui [E]. Then if Em = E
U0,...,Um,Tn
n
, E1 E2 . . . and
each Em is ergodic. Also
Cµ(Em) m + Cµ(E
U0,Tn
n
) m + 2 ∞,
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