36 I. MEASURE PRESERVING AUTOMORPHISMS

unitary operators implies that if U(H) is the unitary group of a separable

infinite dimensional space H, then conjugacy in U(H) is Borel bireducible

to ∼. It is clear then that unitary (spectral) equivalence in Aut(X, µ) is

Borel reducible to measure equivalence ∼.

Theorem 5.7 (Kechris). (a) Measure equivalence is Borel bireducible to

unitary equivalence in MIX (and thus also in WMIX, ERG and Aut(X, µ)).

In particular, unitary equivalence in MIX cannot be classified by countable

structures.

(b) Measure equivalence is Borel reducible to conjugacy in MIX (and

thus also in WMIX, ERG and Aut(X, µ)). In particular, conjugacy in MIX

cannot be classified by countable structures.

Our proof will use the spectral theory of unitary operators and mea-

sure preserving transformations, for which some standard references are

Cornfeld-Fomin-Sinai [CFS], Glasner [Gl2], Goodson [Goo], Lema´ nczyk [Le],

Nadkarni [Na], Parry [Pa], Queff´ elec [Qu], and in particular the spectral the-

ory of the shift in Gaussian spaces (see Appendices C-E) for which we refer

the reader to Cornfeld-Fomin-Sinai [CFS]. Moreover, we will also make use

of some results in the harmonic analysis of measures on T, for which we refer

to Kechris-Louveau [KL].

Proof of 5.7. (a) For each finite (positive) Borel measure σ on T

consider ˆ σ : Z → C given by

ˆ(n) σ =

¯ndσ(z).

z

By Herglotz’s theorem (see, e.g., Parry [Pa]) σ ↔ ˆ σ is a 1-1 correspon-

dence between finite Borel measures on T and positive-definite functions on

Z. Note that ˆ σ is real iff σ is symmetric, i.e., invariant under conjugation

(σ(A) = σ(

¯),

A for every Borel set A, where

¯

A = {¯ z : z ∈ A}). To each sym-

metric σ then we can associate the real positive-definite function ϕ(σ) = ˆ σ

and then the Gaussian space

(RZ,µϕ(σ)),

as in Appendix C, and the corre-

sponding shift, which we will denote by Tσ, i.e., Tσ((xn)n∈Z) = (xn−1)n∈Z.

It is well-known (see, Cornfeld-Fomin-Sinai [CFS], p. 369) that Tσ is mixing

iff σ is a Rajchman measure, i.e., ˆ(n) σ → 0 as |n| → ∞.

Consider the Wiener chaos decomposition

L2(RZ,µϕ(σ))

=

L2(RZ,µϕ(σ),

C) =

n≥0

HC

:n:

discussed in Appendix D, and let UTσ = Uσ be the unitary operator on

L2(RZ,µϕ(σ))

induced by Tσ. Clearly each HC

:n:

is invariant under Uσ. Also

note that HC

:0:

= C is the subspace of constants and

L0(RZ,µϕ(σ)) 2

=

n≥1

HC

:n:.