then µ(Bn) 0 (as Sn 1 in N[E]). If x Bn, then
x An T
= T
= Sn(T
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
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
] closed in N[EΓ
but Cµ(EΓ
) = 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
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,
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