42 I. MEASURE PRESERVING AUTOMORPHISMS

iM
µ(Ai) . Then for some N and all n N,
µ(Tn(Ai)∆Ai)
(M + 1)
, ∀i M,
TnγiTn
−1(x)
= γi(x), ∀i M,
for all x An,
i
where µ(An)
i
(M+1)
, i M. Thus off a set of measure
2 , we have that if i M and x Ai, then Tn
−1(x)
Ai and
TTn
−1(x)
= γi(Tn
−1(x))
= Tn
−1(γi(x))
= Tn
−1T
(x).
Thus µ({x : TTn
−1(x)
= Tn
−1T
(x)}) 3 , if n N, so δu(TTn
−1,Tn −1T
) =
δu(TnTTn
−1,T
) 0.
Thus the following is also a complete metric for N[E]:
¯
δ N[E](T1,T2) =
¯w(T1,T2)
δ +
n
2−nδu(T1γnT1 −1,T2γnT2 −1),
where Γ = {γn}.
Remark. It is clear that in the preceding one can equivalently use δu
instead of δu.
(B) There is also another way to understand the topology of N[E].
Consider the measure M on E defined by
M(A) = card(Ax)dµ(x),
where Ax = {y : (x, y) A}, for any Borel set A E. This is a σ-finite
Borel measure on E.
Given now T N[E] consider the map T × T on E defined by
T × T(x, y) = (T(x),T(y)).
Then it is easy to see that T ×T preserves the measure M, therefore induces
a unitary operator on the Hilbert space
L2(E,
M)
denoted by UT
×T
:
UT
×T
(f)(x, y) = f(T
−1(x),T −1(y)).
The map T UT
×T
is a group isomorphism between N[E] and a sub-
group of
U(L2(E,
M)). (To see that it is 1-1, assume that UT
×T
= 1, i.e.,
f(T
−1(x),T −1(y))
= f(x, y), M-a.e., for all f
L2(E,
M). Then for any
g
L2(X,
µ), if
f(x, y) =
g(x), if x = y,
0, otherwise,
then f
L2(E,
M) and g(T
−1(x))
= f(T
−1(x),T −1(x))
= f(x, x) = g(x), µ-
a.e., so T = 1.)
Now notice that UTn×Tn 1 in
U(L2(E,
M)) Tn 1 in N[E]. This
is because for any T N[E],S [E], if f is the characteristic function of
Previous Page Next Page