Canonical Behaviour of Borel Functions on Polish Planes

Otmar Spinas

A subset of the Baire space^{ω}ω is called superperfect if it is closed and homeomorphic to^{ω}ω. A rectangle in (^{ω}ω)^{2} is called superperfect if both sides are superperfect. I will present the canonization of all Borel measurable functions from (^{ω}ω)^{2} to ℝ modulo restriction to superperfect rectangles. By Hurewicz’ Theorem this yields, for every Polish space X which is not σ-compact, the canonization of all Borel functions f : X^{2} → ℝ modulo restriction to rectangles with both sides closed and not σ-compact. It is known that an analogous result is impossible in dimension 3 and higher. Below I explain how the canonization looks like. This is a continuation of the work in [Sp1] and is presented in [Sp2].

Definition 1. We say that (x, y) ∈ (^{ω}ω)^{2} oscillates infinitely often if there exists a strictly increasing sequence ⟨k_{i} : i < ø⟩ ∈ ^{ω}ω such that k_{0} = 0 and the following hold for all n < ω:

1. k_{2n} = min{i < ω : y(i) > max{k_{2n+1}, x(j) : j < k_{2n+1}}},

2. k_{2n+1} = min{i < ω : x(i) > max{k_{2n+2}, y(j) : j < k_{2n+2}}}.

As ⟨k_{i} : i < ω⟩ is uniquely determined, we let μ_{0}(x, y) = ∅, μ_{n+1}(x, y) = x↾k_{2n+1}, ν_{n}(x, y) = y↾k_{2n}, for all n < ω. Let IO denote the set of all (x, y) ∈ (^{ω}ω)^{2} which oscillate infinitely often. Let ETP (the set of even type-pairs) denote the set of all pairs of the form (μ_{n}(x, y), ν_{n}(x, y)), for some (x, y) ∈ IO and n < ω. Similarly, OTP denotes the set of all odd type-pairs, i.e. those of the form (μ_{n+1}(x, y), ν_{n}(x, y)), for some (x, y) ∈ IO and n < ω.

Definition 2. Let c :ETP→ 2, d :OTP→ 2 be arbitrary. Define functions f_{c} and f_{d} with domain IO as follows: Fix (x, y) ∈ IO and let (μ_{n}, ν_{n}) = (μ_{n}(x, y), ν_{n}(x, y)). If ⟨(μ_{n_{k}}, ν_{n_{k}}) : k < N ≤ ω⟩ is the subsequence of those pairs (μ_{n}, ν_{n}) with c(μ_{n}, ν_{n}) = 1, then let f_{c}(x, y) = ⟨μ_{n_{k}+1} : k < N⟩. Similarly, if ⟨(μ_{n_{k}+1}, ν_{n_{k}}) : k < N ≤ ω⟩ is the subsequence of those (μ_{n+1}, ν_{n}) with d(μ_{n+1}, ν_{n}) = 1, let f_{d}(x, y) = ⟨ν_{n_{k}+1} : k < N⟩.

Theorem. Suppose that f : (^{ω}ω)^{2} → ℝ is Borel measurable. There exists a superperfect rectangle S_{0} ×S_{1}⊆ IO such that precisely one of the following alternatives holds:

1. ∀(x, y), (x′, y′) ∈ S_{0} × S_{1} f(x, y) = f(x′, y′).

2. ∀(x, y), (x′, y′) ∈ S_{0} × S_{1} f(x, y) = f(x′, y′) ⇔ x = x′.

3. ∀(x, y), (x′, y′) ∈ S_{0} × S_{1} f(x, y) = f(x′, y′) ⇔ y = y′.

4. There exists c :ETP→ 2 such that for all (x, y), (x′, y′) ∈ S_{0}×S_{1} we have c(μ_{n}(x, y), ν_{n}(x, y)) = 1 for infinitely many n and f(x, y) = f(x′, y′) ⇔ f_{c}(x, y) = f_{c}(x′, y′).

5. There exists d :OTP→ 2 such that for all (x, y), (x′, y′) ∈ S_{0} × S_{1} we have c(μ_{n+1}(x, y), ν_{n}(x, y)) = 1 for infinitely many n and f(x, y) = f(x′, y′) ⇔ f_{d}(x, y) = f_{d}(x′, y′).

6. ∀(x, y), (x′, y′) ∈ S_{0} × S_{1} f(x, y) = f(x′, y′) ⇔ (x, y) = (x′, y′).

[Sp1] O. Spinas, Ramsey and freeness properties of Polish planes, Proc. London Math. Soc., to appear.

[Sp2] O. Spinas, Canonical behaviour of Borel functions on superperfect rectangles, preprint.

Address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Strasse 4, 24098 Kiel, Germany

E-mail: spinas@math.uni-kiel.de