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