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 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 = 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:.
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