2. SOME BASIC FACTS ABOUT Aut(X, µ) 9

can find large enough N, such that if n ≥ N, there are nth roots of 1, say

λ1,...,λm, with |λi − λi| σ, and nth roots of −1, say λ1,...,λm, with

|λi − λi| σ. Then if V (ei) = λiei,W (ei) = λi ei,V,W clearly work.

Addendum. Rosendal [Ro] has recently found the following simple

proof that all unitary equivalence classes in Aut(X, µ) are meager in the

weak topology.

For each infinite set I ⊆ N, let A(I) = {T ∈ Aut(X, µ) : ∃i ∈ I(T

i

= 1)}.

By 2.2 and 2.3, A(I) is dense in (Aut(X, µ),w). Let V0 ⊇ V1 ⊇ . . . be a

basis of open nbhds of 1 and consider the set B(I, k) = {T ∈ Aut(X, µ) :

∃i ∈ I(i k & T

i

∈ Vk}. It contains A(I \ {0,...,k}), so it is open dense.

Thus

C(I) =

k

B(I, k) = {T : ∃{in} ⊆ I(T

in

→ 1)}

is comeager and clearly invariant under unitary equivalence. If a unitary

equivalence class C is non-meager, it is contained in all C(I),I ⊆ N infinite.

It follows that if T ∈ C, then T

n

→ 1, so T = 1, a contradiction.

Notice that this proof shows that in any Polish group G in which the

sets A(I) = {g ∈ G : ∃i ∈

I(gi

= 1)} are dense for all infinite I ⊆ N, every

conjugacy class is meager. Another group that has this property is U(H)

(see the preceding remark) and this gives a simple proof that conjugacy

classes in U(H) are meager.

Denote by ERG the set of ergodic T ∈ Aut(X, µ).

Theorem 2.6 (Halmos [Ha]). ERG is dense Gδ in (Aut(X, µ),w).

Proof. Take X =

2N.

For T ∈ Aut(X, µ), define fT : X →

XZ

by

fT (x)n = T

−n(x).

Then if S is the (left) shift action on

XZ,

S((xn)) = (xn−1),

clearly fT (T

n(x))

=

Sn(fT

(x)). So if νT = (fT )∗µ, νT is shift-invariant. We

will verify that T → νT is continuous from (Aut(X, µ),w) to the space of

shift-invariant probability measures on

XZ,

which is a convex compact set

in the usual

weak∗-topology

of probability measures on the compact space

XZ.

Then

T ∈ ERG ⇔ νT is an ergodic, shift-invariant measure.

But the ergodic shift-invariant measures are exactly the extreme points of

the convex, compact set of shift-invariant measures, so they form a Gδ set

and thus ERG is Gδ.

To check the continuity of T → νT we need to verify that for some

uniformly dense D ⊆

C(XZ),

and any f ∈ D,T → fdνT is continuous. By

Stone-Weierstrass we can take D to be the complex algebra generated by

the functions (xi)i∈Z → χA(xn), for n ∈ Z,A ⊆ X clopen. Thus it is enough