8 I. MEASURE PRESERVING AUTOMORPHISMS

where

∀∗T

means “for comeager many T.” Fix a dense set {ξm} ⊆

L2(X,

µ).

Then

m

{T : fT (χA) − ξm 1/20} = Aut(X, µ). So there is an open

nonempty set W ⊆ Aut(X, µ) and some m so that, putting ξm = ξ, we have

∀∗T

∈ W fT (χA) − ξ

1

20

.

We can then find S, T ∈ W and n such that

US

n(χA)

− χA

1

20

, UT

n(χA)

− χA

1

3

and fSUSfS

−1

= fT UT fT

−1

= UT0 ,

fS(χA) − ξ

1

20

, fT (χA) − ξ

1

20

.

Then if fS(χA) = ξ1,fT (χA) = ξ2, we have

UT0

n

(ξ1) − ξ1

1

20

, UT0

n

(ξ2) − ξ2

1

3

,

and ξ1 − ξ , ξ2 − ξ

1

20

, so

UT0

n

(ξ) − ξ ≤ UT0

n

(ξ) − UT0

n

(ξ1) + UT0

n

(ξ1) − ξ1 + ξ1 − ξ ,

which is smaller than

3

20

, while also

UT0

n

(ξ) − ξ ≥ UT0

n

(ξ2) − ξ2 − UT0

n

(ξ2) − UT0

n

(ξ) − ξ − ξ2 ,

which is bigger than

7

30

, a contradiction. ✷

Remark. One can employ a similar strategy to give another proof of

the well-known fact (see Nadkarni [Na], Ch. 8) that in the unitary group

U(H) of a separable infinite dimensional Hilbert space H (with the weak,

equivalently strong, topology) every conjugacy class is meager.

Fix a unit vector v ∈ H. As in the proof of 2.5, it is enough to show

that for any positive , δ 1 the open set

W = {(S, T) ∈

U(H)2

: ∃n(

Sn(v)

− v and T

n(v)

− v δ)}

is dense in

U(H)2.

Fixing a basis {bi}i=1

∞

for H, with v = b1, we can view

the unitary group U(m) of

Cm

as the unitary group of the subspace bi

m

i=1

and it is easy to see that

m

U(m) is dense in U(H). So it is enough to show

that given any m and S0,T0 ∈ U(m), there are S, T ∈ U(m) as close we want

to S0,T0 such that for some n,

Sn

−I and T

n

+I 2−δ, where I is

the identity of U(m) and · refers here to the norm of operators on

Cn.

For

this it is enough to show again that given any U ∈ U(m) and ρ 0, there is

N such that for n ≥ N, there are V, W ∈ U(m) with U − V , U − W ρ

and V

n

= I, W

n

= −I.

Since any U0 ∈ U(m) is conjugate in U(m) to some U1 ∈ U(m) which

is diagonal for an orthonormal basis of

Cm,

i.e., for such a basis {e1,...em}

we have U1(ei) = λiei, where λi ∈ T, we can assume that U is already of

that form. Given ρ 0, there is σ 0, such that if P ∈ U(m) satisfies

P (ei) = µiei, where |µi − λi| σ, 1 ≤ i ≤ m, then U − P ρ. Now we