3. FULL GROUPS OF EQUIVALENCE RELATIONS 13

3. Full groups of equivalence relations

(A) Let (X, µ) be a standard measure space and E a Borel equivalence

relation on X. We say that E is a countable equivalence relation if every

E-class is countable. Feldman-Moore [FM] have shown that E is countable

iff there is a countable (discrete) group Γ acting in a Borel way on X such

that if (γ, x) → γ · x denotes the action and we let

EΓ

X

= {(x, y) : ∃γ ∈ Γ(γ · x = y)}

be the induced equivalence relation, then E = EΓ

X

. We say that E is a

measure preserving equivalence relation if the action of Γ is measure pre-

serving, i.e., x → γ · x is measure preserving for each γ ∈ Γ. This condition

is independent of Γ and the action that induces E and is equivalent to the

condition that every Borel automorphism T of X for which T(x)Ex, ∀x, is

measure preserving.

For any Borel E we define its full group or inner automorphism group,

[E], by

[E] = {T ∈ Aut(X, µ) : T(x)Ex, µ-a.e.(x)}.

It is easy to check that this is a closed subgroup of Aut(X, µ) in the uniform

topology.

Convention. Below we will assume that equivalence relations E are

countable and measure preserving, unless otherwise explicitly stated.

Recall that E is an ergodic equivalence relation if every E-invariant Borel

set is null or co-null. We call E a finite equivalence relation or a periodic

equivalence relation if (almost) all its classes are finite and an aperiodic

equivalence relation if (almost) all its classes are infinite.

Proposition 3.1. E is ergodic iff [E] is dense in (Aut(X, µ),w).

Proof. Assume E is ergodic. Then if A, B are Borel sets of the same

measure, there is T ∈ [E] with T(A) = B.

Now let S ∈ Aut(X, µ). A nbhd basis for S is given by the sets of the

form {T : ∀i ≤ n(dµ(T(Ai),S(Ai)) )}, where 0, and A1,...,An is a

Borel partition of X. Find T1,...,Tn ∈ [E] such that Ti(Ai) = S(Ai). Then

T =

n

i=1

Ti|Ai is in [E] and clearly T(Ai) = S(Ai), i = 1,...,n.

If E is not ergodic, there is A ∈ MALGµ with 0 µ(A) 1 which is

[E]-invariant. If [E] was weakly dense in Aut(X, µ), A would be T-invariant

for any T ∈ Aut(X, µ), which is absurd. ✷

T.-J. Wei [We] has shown in fact that [E] is closed in w iff E is a finite

equivalence relation. Otherwise, it is Π3-complete

0

in w and its closure in

the weak topology is equal to [F ], where F is the (not necessarily countable)

equivalence relation corresponding to the ergodic decomposition of E. Also

clearly [E] is a meager subset of (Aut(X, µ),w), since it is a Borel non-open

subgroup of (Aut(X, µ),w).

Note now the following basic fact about the uniform topology of [E].