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

U(L2(X,

µ)) with

the weak topology, which is generated by the functions U → U(f),g =

U(f)gdµ, for f, g ∈

L2(X,

µ). Simply notice that if we denote by χA the

characteristic function of a set A, then the linear combinations

∑n

i=1

αiχAi

of characteristic functions of Borel sets form a dense set in the Hilbert space

L2(X,

µ) and UT (χA),χB = (χA ◦ T

−1)χBdµ

= 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

metric

δu(S, T) = sup

A∈MALGµ

µ(S(A)∆T(A)).

This is a 2-sided invariant complete metric on Aut(X, µ) but it is not sepa-

rable (e.g., if Sα 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,

2

3

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 Fσ in w. To see this, note that for each T0 ∈ Aut(X, µ), 0, the

ball

{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

∑

n

i=1

µ(Ai) and T(Ai) ∩ Ai = ∅, ∀i ≤ n. Fix δ

1

n

(

∑

n

i=1

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,