2. SOME BASIC FACTS ABOUT Aut(X, µ) 11

space of probability measures on T with the

weak∗-topology.

Then we have

T ∈ ERG ⇔ σT

0

({1}) = 0,

T ∈ WMIX ⇔ σT

0

is non-atomic,

T ∈ MMIX ⇔ σT

0

∈

D⊥,

T ∈ MIX ⇔ σT

0

∈ R,

where R is the class of Rajchman measures in P (T), i.e., those measures

µ ∈ P (T) such that ˆ(n) µ → 0 as |n| → ∞, D is the class of closed Dirichlet

sets in T, i.e., the class of closed E ⊆ T such that there is nk → ∞ with

znk

→ 1 uniformly for z ∈ E, and finally

D⊥

is the class of all µ ∈ P (T)

that annihilate all Dirichlet sets. All the preceding characterizations in

terms of UT , σT

0

can be found, for example, in Queff´ elec [Qu]. The reader

should note that in Queff´ elec [Qu], III.21 one finds the characterization:

T ∈ MMIX ⇔ σT

0

∈

D⊥

in the form T ∈ MMIX ⇔ σT

0

∈ LI, where LI is a

certain class of measures defined there. A proof that LI =

D⊥

can be found

in Proposition 3, p. 212 and Proposition 9, p. 243 of Host-M´ela-Parreau

[HMP]. For more about Dirichlet sets, see Lindahl–Poulsen [LP], Ch.1.

For each σ ∈ P (T) which is symmetric (i.e., invariant under conjuga-

tion), ˆ(n) σ =

z−ndσ(z)

is a positive-definite real function and let Tσ be

the Gaussian shift on

RZ

corresponding to ˆ σ (see Appendix C). It is known

(see Cornfeld-Fomin-Sinai [CFS], 14.3, Theorem 1) that the maximal spec-

tral type of UTσ

|C⊥

is σ∞ =

∑∞

n=1

1

2n

σ∗n,

where

σ∗n

= σ∗···∗σ is the n-fold

convolution of σ. We also have (see Cornfeld-Fomin-Sinai [CFS], 14.2) and

Queff´ elec [Qu], III. 21):

σ is non-atomic ⇔ Tσ ∈ ERG ⇔ Tσ ∈ WMIX,

σ ∈

D⊥

⇔ Tσ ∈ MMIX,

σ ∈ R ⇔ Tσ ∈ MIX.

For the equivalence σ ∈

D⊥

⇔ Tσ ∈ MMIX one needs to notice that

σ ∈

D⊥

⇒ σ∞ ∈

D⊥,

which follows from the fact that

D⊥

is closed un-

der convolution by any measure, since D is closed under translations (see

Lindahl–Poulsen [LP], p.4). We already noted that ERG, WMIX are Gδ in

(Aut(X, µ),w) and it is easy to see that MIX is Π3.

0

However we have the

following result.

Theorem 2.7 (Kaufman, Kechris). The set of mild mixing transfor-

mations in Aut(X, µ) is co-analytic but not Borel in the weak topology of

Aut(X, µ).

Proof. It is clear that MMIX is co-analytic. The map σ → Tσ ∈

Aut(X, µ) is Borel and

D⊥

restricted to symmetric measures is known to be

non-Borel (Kechris-Lyons [KLy], Host-Louveau-Parreau). ✷

Remark. Recall that a subset A of N \ {0} is an IP-set if there are

p1,p2, · · · ∈ N \ {0} such that A = {pi1 + · · · + pik : i1 · · · ik}. The filter