1. THE GROUP Aut(X, µ) 3
for A1,...,An MALGµ, 0,S Aut(X, µ), form an open basis for
w. The easiest way to see this, as pointed out by O. Ageev, is to use the
identification of (Aut(X, µ),w) with a closed subgroup of
µ)) with
the weak topology, which is generated by the functions U U(f),g =
U(f)gdµ, for f, g
µ). Simply notice that if we denote by χA the
characteristic function of a set A, then the linear combinations
of characteristic functions of Borel sets form a dense set in the Hilbert space
µ) and UT (χA),χB = (χA T
= µ(T(A) B).
It is clear that we can restrict in VS,A1...An, or WS,A1...An, the sets
A1,...,An to any countable dense set in MALGµ and to rationals and
still obtain bases. Also it is easy to check that we can restrict above
the A1,...,An to belong to any countable dense (Boolean) subalgebra of
MALGµ and moreover assume that A1,...,An form a partition of X (by
looking, for each A1,...,An, at the atoms of the Boolean algebra generated
by A1,...,An).
(C) We now come to the uniform topology, u. This is defined by the
δu(S, T) = sup
This is a 2-sided invariant complete metric on Aut(X, µ) but it is not sepa-
rable (e.g., if denotes the rotation by α T of the group T, then {Sα} is
discrete). Clearly w u.
An equivalent to δu metric is defined by
δu(S, T) = µ({x : S(x) = T(x)}).
In fact,
δu δu δu.
The metric δu is also 2-sided invariant. For T : X X let
supp(T) = {x : T(x) = x}.
Then δu(T, 1) = µ(supp(T)), where 1 = id (the identity function on X).
We note that each closed δu-ball is closed in w and therefore each open
δu-ball is in w. To see this, note that for each T0 Aut(X, µ), 0, the
{T : δu(T0,T) }
coincides with the set of T satisfying
∀A MALGµ(dµ(T(A),T0(A)) ).
Similarly, each closed δu-ball is closed in w. To see this, it is enough to
show, for each 0, that the set {T : δu(T, 1) } is open in w. Fix T with
δu(T, 1) . Then we can find pairwise disjoint Borel sets A1,...,An with

µ(Ai) and T(Ai) Ai = ∅, ∀i n. Fix δ

µ(Ai) ).
Then if µ(S(Ai)∆T(Ai)) δ, ∀i n, we have
µ(S(Ai)∆Ai) µ(T(Ai)∆Ai) µ(S(Ai)∆T(Ai)) 2µ(Ai) δ,
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