10 I. MEASURE PRESERVING AUTOMORPHISMS
to check that for each finite {n1,...,nk} Z and clopen sets {A1,...,Ak}
the function
T
k
i=1
χAi (xni )dνT ((xn))
=
k
i=1
χAi (T
−ni
(x))dµ(x)
=
k
i=1
χT
ni
(Ai)(x)dµ(x)
= χ
k
i=1
T
ni
(Ai)

= µ(
k
i=1
T
ni
(Ai))
is continuous, which is clear.
Concerning stronger notions of ergodicity, the set of weak mixing trans-
formations, WMIX, is also dense (Halmos [Ha]; see 12.1 for a more
general statement) but the set of mild mixing transformations, MMIX, and
the set of (strong) mixing transformations, MIX, are meager (Rokhlin for
MIX; see Katok-Thouvenot [KTh], 5.48 and Nadkarni [Na], Chapter 8).
(B) Recall that if T Aut(X, µ) and UT
U(L2(X,
µ)) is the cor-
responding unitary operator, then ERG, MMIX, WMIX, MIX are charac-
terized as follows in terms of UT , where we put below L0(X,
2
µ) = {f
L2(X,
µ) : fdµ = 0} =
C⊥
(the orthogonal of the constant functions):
T ERG UT (f) = f, ∀f L0(X,
2
µ) \ {0},
T WMIX UT (f) = λf, ∀λ T∀f L0(X,
2
µ) \ {0},
T MMIX UT
nk
(f) f, ∀nk ∞, ∀f L0(X,
2
µ) \ {0},
T MIX U
n(f),f
0, ∀f L0(X,
2
µ).
We have MIX MMIX WMIX ERG and these are proper inclusions.
(Note that if for any T Aut(X, µ) we denote by κT
0
the Koopman repre-
sentation of Z on L0(X,
2
µ) induced by UT , then T is in ERG, etc., iff κT
0
is
ergodic, etc., according to the definition in Appendix H.)
Also denote by σT
0
the maximal spectral type of UT |L0(X,
2
µ) (thus σT
0
is
uniquely defined up to measure equivalence); see Appendix F (where we take
= Z,H = L0(X,
2
µ),π(n) = UT
n|L0(X, 2
µ)). Note that σT
0
can be chosen
so that the map T σT
0
is continuous from (Aut(X, µ),w) into P (T), the
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