52 I. MEASURE PRESERVING AUTOMORPHISMS

then µ(Bn) → 0 (as Sn → 1 in N[E]). If x ∈ Bn, then

x ∈ An ⇒ T

−1Sn(x)

= T

−1(x)

= Sn(T

−1(x))

⇒ T

−1(x)

∈ An ⇒ x ∈ T(An),

so An\T(An) ⊆ Bn and therefore µ(An\T(An)) → 0. Thus µ(An∆T(An)) =

2µ(An \ T(An)) → 0. It follows that E is not E0-ergodic, a contradiction.

Thus δu(Sn, 1) → 1. Let Un ∈ [E] be an involution with support An and

put

Tn(x) =

Sn(x), if x ∈ An,

Un(x), if x ∈ An.

Then Tn ∈ [E],δu(Sn,Tn) ≤ µ(An) → 0 and δu(Tn, 1) = 1. Also clearly

Tn → 1 in N[E].

Remark. The converse of 8.1 is not true. By 9, (A) and 9.1 below there

are groups Γ such that for any measure preserving, free, ergodic action of Γ,

we have [EΓ

X

] closed in N[EΓ

X]

but Cµ(EΓ

X

) = 1. For example, one can take

any ICC (infinite conjugacy classes) group with property (T), fixed price

and cost equal to 1, for instance SL3(Z).

(B) The preceding result 8.1 can be used to give a partial answer to a

question of Jones-Schmidt [JS], p. 113 and Schmidt [Sc2], 4.6. They ask for

a characterization of when [E] is closed in N[E]. The following corollary

resolves this when E is treeable, i.e., there is a Borel acyclic graph whose

connected components are the E-classes.

Corollary 8.6. Let E be an ergodic equivalence relation. Then if E is

treeable, the following are equivalent:

(i) [E] is closed in N[E].

(ii) E is not hyperfinite.

Proof. We have already seen that if E is hyperfinite, [E] is dense in (and

not equal to) N[E]. Conversely, if [E] is not closed, then by 8.1, Cµ(E) = 1,

so, as E is treeable, E is hyperfinite (see Kechris-Miller [KM], 22.2 and

27.10). ✷

It is interesting to note here that although the statement of this result

does not involve costs, the proof given here makes use of this concept.

For further results concerning the question of when [E] is closed in N[E],

see the next section and Section 29, (C).

9. Inner amenability

(A) Recall that a countable group Γ is inner amenable if there is a mean

on Γ \{1} invariant under the conjugacy action of Γ or equivalently there is

a finitely additive probability measure (f.a.m.) on Γ\{1} invariant under the

conjugacy action of Γ. See Effros [Ef1], where this notion was introduced,

and the survey Bedos-de la Harpe [BdlH]. Examples of such groups are:

(i) amenable groups,