On Decomposing Borel Functions - Beijing Normal …cst.bnu.edu.cn/conference/slides/Longyun...

On Decomposing Borel Functions2017 Chinese Mathematical Logic Conference

Beijing Normal University

Longyun Ding

School of Mathematical SciencesNankai University

20 May 2017

ω-decomposable functionsJayne-Rogers’ theorem

Generalizations

Outline

1 ω-decomposable functions

2 Jayne-Rogers’ theorem

3 Generalizations

L. Ding On Decomposing Borel Functions

Generalizations

Polish spaces

Definition

Polish space: a separable, completely metrizable topological space.

Example

1 Rn,Cn and I = [0, 1];

2 countable discrete spaces;

3 separable Banach spaces: c0, `p, C[0, 1], Lp[0, 1], · · · ;

4 products of countablely many Polish spaces:

(a) Hilbert cube Iω,(b) Cantor space {0, 1}ω,(c) Baire space ωω.

Generalizations

Polish spaces

Definition

Polish space: a separable, completely metrizable topological space.

Example

1 Rn,Cn and I = [0, 1];

2 countable discrete spaces;

3 separable Banach spaces: c0, `p, C[0, 1], Lp[0, 1], · · · ;

4 products of countablely many Polish spaces:

(a) Hilbert cube Iω,(b) Cantor space {0, 1}ω,(c) Baire space ωω.

Generalizations

Borel hierarchy

Definition

B(X): Borel sets of X is the σ-algebra generated by the open setsof X.

Σ01 = open, Π0

1 = closed;

Σ02 = Fσ, Π0

2 = Gδ;

for 1 ≤ α < ω1,

Σ0α = {

⋃n∈ω

An : An ∈ Π0αn, αn < α};

Π0α = the complements of Σ0

α sets;

∆0α = Σ0

α ∩Π0α.

Generalizations

Borel functions

Let f : X → Y .

Definition

Borel function: f−1(U) is Borel set in X for any U open in Y .

Definition

Σ0α-measurable function: f−1(U) is Σ0

α set in X for any U openin Y .

Continuous = Σ01-measurable.

Generalizations

Borel functions

Let f : X → Y .

Definition

Borel function: f−1(U) is Borel set in X for any U open in Y .

Definition

Σ0α-measurable function: f−1(U) is Σ0

α set in X for any U openin Y .

Continuous = Σ01-measurable.

Generalizations

Two examples

Example (Dirichlet function)

D(x) =

{1, x ∈ Q,0, x /∈ Q.

D(x) is Σ03-measurable but not continuous.

Both D � Q and D � (R \Q) are continuous.

Example (Riemann function)

R(x) =

{1/q, x = p/q with p, q coprime,0, x /∈ Q.

R is Σ02-measurable but not continuous.

R � (R \Q) is continuous;R � {p/q} is continuous for each p/q ∈ Q.

Generalizations

Two examples

D(x) =

{1, x ∈ Q,0, x /∈ Q.

R(x) =

Generalizations

Two examples

D(x) =

{1, x ∈ Q,0, x /∈ Q.

R(x) =

Generalizations

Two examples

D(x) =

{1, x ∈ Q,0, x /∈ Q.

R(x) =

Generalizations

ω-decomposable functions

Definition

A function f : X → Y is ω-decomposable if there exists partitionX =

⋃nXn such that each f � Xn is continuous.

Generalizations

Definition

Lusin, 1883–1950

Question: (Lusin) Is every Borel function ω-decomposable?

Generalizations

Definition

Lusin, 1883–1950

Question: (Lusin) Is every Borel function ω-decomposable?

Generalizations

Counterexamples

Keldys (1934), Kuratouski (1934), ...

Example (Pawlikowski function)

P : (ω ∪ {ω})ω → ωω with

P (x)(n) =

{x(n) + 1, x(n) < ω,0, x(n) = ω.

P is Σ02-measurable but not ω-decomposable.

Question: What kind of Borel functions is ω-decomposable?

Generalizations

Counterexamples

P (x)(n) =

{x(n) + 1, x(n) < ω,0, x(n) = ω.

Generalizations

Counterexamples

P (x)(n) =

{x(n) + 1, x(n) < ω,0, x(n) = ω.

Generalizations

Σ11 sets

Definition

Let X be a Polish space. A subset A ⊆ X is analytic (or Σ11) if

there is a Polish space Y and a closed subset C ⊆ X ×Y such that

x ∈ A ⇐⇒ ∃y ∈ Y ((x, y) ∈ C).

Theorem (Suslin)

Let A ⊆ X. Then A is Borel iff both A and X \A are Σ11.

Generalizations

Solecki’s dichotomy

Analytic space = Σ11 subspace of a Polish space.

Theorem (Solecki, 1998)

Let X,Y be separable metrizable spaces with X analytic. Letf : X → Y is Σ0

2-measurable. Then

f is NOT ω-decomposable ⇐⇒ P v f.

P v f : There exists Z ⊆ X such that the behavior of f � Zis same to P , i.e., there are homeomorphisms φ : Z → (ω ∪ {ω})ωand ψ : ωω → f(Z) with

f � Z = ψ ◦ P ◦ φ.

Generalizations

2-measurable. Then

f � Z = ψ ◦ P ◦ φ.

Generalizations

2-measurable. Then

f � Z = ψ ◦ P ◦ φ.

Generalizations

2-measurable. Then

f � Z = ψ ◦ P ◦ φ.

Generalizations

More dichotomy

Theorem (Motto Ros, 2013)

For n < ω, Solecki’s dichotomy is true for all Σ0n-measurable

functions.

— How about Σ0α-measurable functions for α ≥ ω?

— We don’t know.

Generalizations

More dichotomy

functions.

Generalizations

More dichotomy

functions.

Generalizations

Outline

3 Generalizations

Generalizations

dec(Σ0α) and dec(Σ0

α,∆0ξ)

Definition

dec(Σ0α): there exists a partition X =

⋃nXn such that each

f � Xn is Σ0α-measurable;

dec(Σ0α,∆

0ξ): there exists a partition X =

⋃nXn with Xn ∈∆0

such that each f � Xn is Σ0α-measurable.

ω-decomposable = dec(Σ01).

If f is Σ0β-measurable with α < β, then

f ∈ dec(Σ0α) ⇐⇒ f ∈ dec(Σ0

α,∆0β+1).

Generalizations

α,∆0ξ)

Definition

dec(Σ0α,∆

α,∆0β+1).

Generalizations

α,∆0ξ)

Definition

dec(Σ0α,∆

α,∆0β+1).

Generalizations

α,∆0ξ)

Definition

dec(Σ0α,∆

α,∆0β+1).

Generalizations

Some facts

The following are equivalent:

(i) f ∈ dec(Σ0γ ,∆

(ii) There exists a sequence (An) of Σ0δ subsets with X =

⋃nAn

such that every f � An is Σ0γ-measurable.

(iii) There exists a sequence (An) of Σ0δ subsets with X =

⋃nAn

such that every f � An ∈ dec(Σ0γ ,∆

Generalizations

Some facts

(i) f ∈ dec(Σ0γ ,∆

⋃nAn

Generalizations

Some facts

(i) f ∈ dec(Σ0γ ,∆

⋃nAn

Generalizations

Jayne-Rogers’s theorem

Definition

f−1Σ0α ⊆ Σ0

β: f−1(A) ∈ Σ0β for any A ∈ Σ0

f−1Σ01 ⊆ Σ0

β = Σ0β-measurable.

Theorem (Jayne-Rogers, 1982)

Let X,Y be separable metrizable spaces with X analytic. Then

f ∈ dec(Σ01,∆

02) ⇐⇒ f−1Σ0

2 ⊆ Σ02.

New proofs: Solecki (1998), Motto Ros-Semmes (2010).

Generalizations

Definition

f−1Σ0α ⊆ Σ0

f−1Σ01 ⊆ Σ0

f ∈ dec(Σ01,∆

02) ⇐⇒ f−1Σ0

2 ⊆ Σ02.

Generalizations

Definition

f−1Σ0α ⊆ Σ0

f−1Σ01 ⊆ Σ0

f ∈ dec(Σ01,∆

02) ⇐⇒ f−1Σ0

2 ⊆ Σ02.

Generalizations

Outline

3 Generalizations

Generalizations

The decomposition conjecture

Conjecture: Let X,Y be separable metrizable spaces with Xanalytic. Then for m ≤ n < ω,

f ∈ dec(Σ0n−m+1,∆

0n) ⇐⇒ f−1Σ0

m ⊆ Σ0n.

— How about α, β ≥ ω?

Example

P is Σ02-measurable, so P−1Σ0

n ⊆ Σ0n+1 for each n, and hence

P−1Σ0ω ⊆ Σ0

ω; but

P /∈ dec(Σ01,∆

0ω) ⊆ dec(Σ0

Generalizations

0n) ⇐⇒ f−1Σ0

m ⊆ Σ0n.

Example

P−1Σ0ω ⊆ Σ0

ω; but

P /∈ dec(Σ01,∆

0ω) ⊆ dec(Σ0

Generalizations

0n) ⇐⇒ f−1Σ0

m ⊆ Σ0n.

Example

P−1Σ0ω ⊆ Σ0

ω; but

P /∈ dec(Σ01,∆

0ω) ⊆ dec(Σ0

Generalizations

m ≤ n = 3

Theorem (Semmes, 2009)

The conjecture is true for m ≤ n = 3 within f : ωω → ωω.

Theorem (D.-Zhao, 2017)

The conjecture is true for m ≤ n = 3.

Generalizations

m ≤ n = 3

Theorem (Semmes, 2009)

The conjecture is true for m ≤ n = 3 within f : ωω → ωω.

Theorem (D.-Zhao, 2017)

The conjecture is true for m ≤ n = 3.

Generalizations

Σ0n−1 functions

By extending the Shore-Slaman Join Theorem onto thecontinuous-degree version, ...

Theorem (Gregoriades-Kihara-Ng, 2016)

Within Σ0n−1 functions, the conjecture is true for m ≤ n < ω.

Note: To proof the conjecture for m ≤ n < ω, we shouldconsider all Σ0

n functions.

Generalizations

Σ0n−1 functions

By extending the Shore-Slaman Join Theorem onto thecontinuous-degree version, ...

Theorem (Gregoriades-Kihara-Ng, 2016)

Within Σ0n−1 functions, the conjecture is true for m ≤ n < ω.

Note: To proof the conjecture for m ≤ n < ω, we shouldconsider all Σ0

n functions.

Generalizations

The end

Thank you!

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