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

X
= {(x, y) : ∃γ Γ(γ · x = y)}
be the induced equivalence relation, then E =
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].
Previous Page Next Page