CHAPTER I
Measure preserving automorphisms
1. The group Aut(X, µ)
(A) By a standard measure space (X, µ) we mean a standard Borel
space X with a non-atomic probability Borel measure µ. All such spaces are
isomorphic to ([0, 1],λ), where λ is Lebesgue measure on the Borel subsets
of [0,1]. Denote by MALGµ the measure algebra of µ, i.e., the algebra of
Borel subsets of X, modulo null sets. It is a Polish Boolean algebra under
the topology given by the complete metric
d(A, B) = dµ(A, B) = µ(A∆B),
where is symmetric difference.
Throughout we write
L2(X,
µ) =
L2(X,
µ, C) for the Hilbert space of
complex-valued square-integrable functions. When we want to refer explic-
itly to the space of real-valued functions, we will write it as
L2(X,
µ, R).
We denote by Aut(X, µ) the group of Borel automorphisms of X which
preserve the measure µ and in which we identify two such automorphisms
if they agree µ-a.e.
Convention. In the sequel, we will usually ignore null sets, unless there
is a danger of confusion.
There are two fundamental group topologies on Aut(X, µ): the weak and
the uniform topology, which we now proceed to describe.
(B) The weak topology on Aut(X, µ) is generated by the functions
T T(A), A MALGµ,
(i.e., it is the smallest topology in which these maps are continuous). With
this topology, denoted by w, (Aut(X, µ),w) is a Polish topological group. A
left-invariant compatible metric is given by
δw(S, T) =
2−nµ(S(An)∆T(An)),
where {An} is a dense set in MALGµ (e.g., an algebra generating the Borel
sets of X), and a complete compatible metric by
¯
δ w(S, T) = δw(S, T) +
δw(S−1,T −1).
We can also view Aut(X, µ) as the automorphism group of the measure
algebra (MALGµ,µ) (equipped with the pointwise convergence topology),
as well as the closed subgroup of the isometry group Iso(MALGµ,dµ) (again
1
http://dx.doi.org/10.1090/surv/160/01
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