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
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