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 ∞,