The density function: some remarks, results, and open problems

The density function: some remarks, results, and open problems

This talk is about joint work with A. Andretta and C. Costantini. Most results are contained in the following papers:

[AC13] A. Andretta, R.C., The descriptive set theory of the Lebesgue density theorem, Advances in Mathematics 234 (2013)

[ACC] A. Andretta, C. Costantini, R.C., Lebesgue density and exceptional points, preprint

[AC] A. Andretta, R.C., The density function in the Cantor space, in preparation

. . . . . . .

The density function: some remarks, results, and open problems This talk is about joint work with A. Andretta and C. Costantini. Most

results are contained in the following papers:

[AC13] A. Andretta, R.C., The descriptive set theory of the Lebesgue density theorem, Advances in Mathematics 234 (2013)

[ACC] A. Andretta, C. Costantini, R.C., Lebesgue density and exceptional points, preprint

[AC] A. Andretta, R.C., The density function in the Cantor space, in preparation

. . . . . . .

The density function

Let (X , d , µ) be a metric measure space: a metric space endowed with a Borel measure.

Given a measurable A ⊆ X , define the density function: D A (x ) = lim

ε→0

µ(A ∩ B ε (x ))

µ(B ε (x )) . (1)

Let also

Φ(A) = {x ∈ X | D _A (x ) = 1} be the set of points with density 1 in A.

Note. To give always meaning to the defining equation (1) and to have

more control on the calculations, often some extra hypotheses on the

space are assumed.

The density function

Let (X , d , µ) be a metric measure space: a metric space endowed with a Borel measure.

Given a measurable A ⊆ X , define the density function:

D A (x ) = lim

ε→0

µ(A ∩ B ε (x ))

µ(B ε (x )) . (1)

Let also

Φ(A) = {x ∈ X | D _A (x ) = 1} be the set of points with density 1 in A.

Note. To give always meaning to the defining equation (1) and to have

more control on the calculations, often some extra hypotheses on the

space are assumed.

The density function

Let (X , d , µ) be a metric measure space: a metric space endowed with a Borel measure.

Given a measurable A ⊆ X , define the density function:

D A (x ) = lim

ε→0

µ(A ∩ B ε (x ))

µ(B ε (x )) . (1)

Let also

Φ(A) = {x ∈ X | D _A (x ) = 1}

be the set of points with density 1 in A.

Note. To give always meaning to the defining equation (1) and to have

more control on the calculations, often some extra hypotheses on the

space are assumed.

The density function

Let (X , d , µ) be a metric measure space: a metric space endowed with a Borel measure.

Given a measurable A ⊆ X , define the density function:

D A (x ) = lim

ε→0

µ(A ∩ B ε (x ))

µ(B ε (x )) . (1)

Let also

Φ(A) = {x ∈ X | D _A (x ) = 1}

be the set of points with density 1 in A.

Note. To give always meaning to the defining equation (1) and to have

more control on the calculations, often some extra hypotheses on the

space are assumed.

The density function

Most of the time, one works with metric measure spaces that are:

- fully supported (i.e., µ(U) > 0 for every non-empty open U)

- locally finite (i.e., every point has a neighbourhood of finite measure)

If A, B are measure equivalent, in symbols A ≡ B, then D A = D B , thus Φ(A) = Φ(B).

Thus Φ induces a function Φ : MALG (X ) → MEAS (X ) on the measure

algebra of X .

The density function

Most of the time, one works with metric measure spaces that are:

- fully supported (i.e., µ(U) > 0 for every non-empty open U)

- locally finite (i.e., every point has a neighbourhood of finite measure) If A, B are measure equivalent, in symbols A ≡ B, then D A = D B , thus Φ(A) = Φ(B).

Thus Φ induces a function Φ : MALG (X ) → MEAS (X ) on the measure

algebra of X .

The problem

What can be said, in general, about D and Φ?

Apparently not much. Very simple properties can be established, like:

I Φ(∅) = ∅, Φ(X ) = X

I A ⊆ B ⇒ Φ(A) ⊆ Φ(B)

I Φ(A ∩ B) = Φ(A) ∩ Φ(B)

I S

i ∈I Φ(A _i ) ⊆ Φ( S

i ∈I A _i ), if S

i ∈I A _i is measurable

I A ⊆ Φ(A) for open A

I Φ(¬A) ⊆ ¬Φ(A)

The problem

What can be said, in general, about D and Φ?

Apparently not much. Very simple properties can be established, like:

I Φ(∅) = ∅, Φ(X ) = X

I A ⊆ B ⇒ Φ(A) ⊆ Φ(B)

I Φ(A ∩ B) = Φ(A) ∩ Φ(B)

I S

i ∈I Φ(A _i ) ⊆ Φ( S

i ∈I A _i ), if S

i ∈I A _i is measurable

I A ⊆ Φ(A) for open A

I Φ(¬A) ⊆ ¬Φ(A)

The problem

What can be said, in general, about D and Φ?

Apparently not much. Very simple properties can be established, like:

I Φ(∅) = ∅, Φ(X ) = X

I A ⊆ B ⇒ Φ(A) ⊆ Φ(B)

I Φ(A ∩ B) = Φ(A) ∩ Φ(B)

I S

i ∈I Φ(A _i ) ⊆ Φ( S

i ∈I A _i ), if S

i ∈I A _i is measurable

I A ⊆ Φ(A) for open A

I Φ(¬A) ⊆ ¬Φ(A)

Lebesgue density theorem

There are Polish spaces such that, for all measurable subsets A, A ≡ Φ(A).

Other equivalent formulations:

I Φ : MALG → MALG is the identity

I Φ : MEAS → MEAS is a selector for the equivalence relation ≡

I for all measurable A, the map D A is defined almost everywhere in X and it coincides almost everywhere with the characteristic function of A

The last formulation says that with respect to the density function, all

interesting things happen on a set of measure 0.

Lebesgue density theorem

There are Polish spaces such that, for all measurable subsets A, A ≡ Φ(A).

Other equivalent formulations:

I Φ : MALG → MALG is the identity

I Φ : MEAS → MEAS is a selector for the equivalence relation ≡

I for all measurable A, the map D A is defined almost everywhere in X and it coincides almost everywhere with the characteristic function of A

The last formulation says that with respect to the density function, all

interesting things happen on a set of measure 0.

Lebesgue density theorem

There are Polish spaces such that, for all measurable subsets A, A ≡ Φ(A).

Other equivalent formulations:

I Φ : MALG → MALG is the identity

I Φ : MEAS → MEAS is a selector for the equivalence relation ≡

I for all measurable A, the map D A is defined almost everywhere in X and it coincides almost everywhere with the characteristic function of A

The last formulation says that with respect to the density function, all

interesting things happen on a set of measure 0.

Lebesgue density theorem

There are Polish spaces such that, for all measurable subsets A, A ≡ Φ(A).

Other equivalent formulations:

I Φ : MALG → MALG is the identity

I Φ : MEAS → MEAS is a selector for the equivalence relation ≡

I for all measurable A, the map D A is defined almost everywhere in X and it coincides almost everywhere with the characteristic function of A

The last formulation says that with respect to the density function, all

interesting things happen on a set of measure 0.

Examples of spaces satisfying LDT

I R ⁿ

I Polish ultrametric spaces

(B. Miller)

I Problem. Which Polish spaces satisfy Lebesgue density theorem?

I will focus on Cantor space 2 ^N with the usual distance and Lebesgue

(coin-tossing) measure

Examples of spaces satisfying LDT

I R ⁿ

I Polish ultrametric spaces (B. Miller)

I Problem. Which Polish spaces satisfy Lebesgue density theorem?

I will focus on Cantor space 2 ^N with the usual distance and Lebesgue

(coin-tossing) measure

Examples of spaces satisfying LDT

I R ⁿ

I Polish ultrametric spaces (B. Miller)

I Problem. Which Polish spaces satisfy Lebesgue density theorem?

I will focus on Cantor space 2 ^N with the usual distance and Lebesgue

(coin-tossing) measure

Examples of spaces satisfying LDT

I R ⁿ

I Polish ultrametric spaces (B. Miller)

I Problem. Which Polish spaces satisfy Lebesgue density theorem?

I will focus on Cantor space 2 ^N with the usual distance and Lebesgue

(coin-tossing) measure

The density function on 2 ^N

In 2 ^N the density function becomes D A (x ) = lim

n→∞

µ(A ∩ N _{x |}

) µ(N _{x |}

) = lim

n→∞ 2 ⁿ µ(A ∩ N _{x |}

)

and

x ∈ Φ(A) ⇔ ∀ε ∃n ∀m > n 2 ^m µ(A ∩ N _{x |}

) > 1 − ε, so that D A is a Borel (partial) function, and Φ(A) is Π ⁰ ₃ .

Sets of the form Φ(A) will be called regular. Since Φ ² = Φ, a set A is

regular iff Φ(A) = A.

The density function on 2 ^N

In 2 ^N the density function becomes D A (x ) = lim

n→∞

µ(A ∩ N _{x |}

) µ(N _{x |}

) = lim

n→∞ 2 ⁿ µ(A ∩ N _{x |}

) and

x ∈ Φ(A) ⇔ ∀ε ∃n ∀m > n 2 ^m µ(A ∩ N _{x |}

) > 1 − ε,

so that D A is a Borel (partial) function, and Φ(A) is Π ⁰ ₃ .

Sets of the form Φ(A) will be called regular. Since Φ ² = Φ, a set A is

regular iff Φ(A) = A.

The density function on 2 ^N

In 2 ^N the density function becomes D A (x ) = lim

n→∞

µ(A ∩ N _{x |}

) µ(N _{x |}

) = lim

n→∞ 2 ⁿ µ(A ∩ N _{x |}

) and

x ∈ Φ(A) ⇔ ∀ε ∃n ∀m > n 2 ^m µ(A ∩ N _{x |}

) > 1 − ε, so that D A is a Borel (partial) function, and Φ(A) is Π ⁰ ₃ .

Sets of the form Φ(A) will be called regular. Since Φ ² = Φ, a set A is

regular iff Φ(A) = A.

The density function on 2 ^N

In 2 ^N the density function becomes D A (x ) = lim

n→∞

µ(A ∩ N _{x |}

) µ(N _{x |}

) = lim

n→∞ 2 ⁿ µ(A ∩ N _{x |}

) and

x ∈ Φ(A) ⇔ ∀ε ∃n ∀m > n 2 ^m µ(A ∩ N _{x |}

) > 1 − ε, so that D A is a Borel (partial) function, and Φ(A) is Π ⁰ ₃ .

Sets of the form Φ(A) will be called regular. Since Φ ² = Φ, a set A is

regular iff Φ(A) = A.

Wadge hierarchy on Cantor space

For A, B ⊆ 2 ^N , set

A ≤ W B ⇔ there is a continuous f : 2 ^N → 2 ^N such that A = f ⁻¹ (B).

A ≡ W B ⇔ A ≤ W B ≤ W A.

A (equivalently, its Wadge degree [A] W ) is self dual iff A ≡ W ¬A. (Wadge; Martin) For all Borel A, B,

A ≤ W B ∨ ¬B ≤ W A;

moreover, ≤ _W is well founded on Borel sets.

Wadge hierarchy on Cantor space

For A, B ⊆ 2 ^N , set

A ≤ W B ⇔ there is a continuous f : 2 ^N → 2 ^N such that A = f ⁻¹ (B).

A ≡ W B ⇔ A ≤ W B ≤ W A.

A (equivalently, its Wadge degree [A] W ) is self dual iff A ≡ W ¬A. (Wadge; Martin) For all Borel A, B,

A ≤ W B ∨ ¬B ≤ W A;

moreover, ≤ _W is well founded on Borel sets.

Wadge hierarchy on Cantor space

For A, B ⊆ 2 ^N , set

A ≤ W B ⇔ there is a continuous f : 2 ^N → 2 ^N such that A = f ⁻¹ (B).

A ≡ W B ⇔ A ≤ W B ≤ W A.

A (equivalently, its Wadge degree [A] W ) is self dual iff A ≡ W ¬A.

(Wadge; Martin) For all Borel A, B,

A ≤ W B ∨ ¬B ≤ W A;

moreover, ≤ _W is well founded on Borel sets.

Wadge hierarchy on Cantor space

For A, B ⊆ 2 ^N , set

A ≤ W B ⇔ there is a continuous f : 2 ^N → 2 ^N such that A = f ⁻¹ (B).

A ≡ W B ⇔ A ≤ W B ≤ W A.

A (equivalently, its Wadge degree [A] W ) is self dual iff A ≡ W ¬A.

(Wadge; Martin) For all Borel A, B,

A ≤ W B ∨ ¬B ≤ W A;

moreover, ≤ _W is well founded on Borel sets.

How it looks like

{∅}

{2 ^N }

How it looks like

{∅}

∆ ⁰ ₁ \ {∅, 2 ^N }

{2 ^N }

How it looks like

{∅} Σ ⁰ ₁ \ Π ⁰ ₁

∆ ⁰ ₁ \ {∅, 2 ^N }

{2 ^N } Π ⁰ ₁ \ Σ ⁰ ₁

How it looks like

{∅} Σ ⁰ ₁ \ Π ⁰ ₁ • • •

∆ ⁰ ₁ \ {∅, 2 ^N } • • · · · •

{2 ^N } Π ⁰ ₁ \ Σ ⁰ ₁ • • •

· · ·

How it looks like

{∅} Σ ⁰ ₁ \ Π ⁰ ₁ • • •

∆ ⁰ ₁ \ {∅, 2 ^N } • • · · · •

{2 ^N } Π ⁰ ₁ \ Σ ⁰ ₁ • • •

· · ·

Self dual degrees and non-self dual pairs of degrees alternate. At all limit levels there is a non-self dual pair.

Each degree is assigned a rank, according to its position in the hierarchy (starting from 1).

The length of this hierarchy up to ∆ ⁰ ₂ is ω ₁ .

The length up to ∆ ⁰ ₃ is ω ₁ ^ω

.

How it looks like

{∅} Σ ⁰ ₁ \ Π ⁰ ₁ • • •

∆ ⁰ ₁ \ {∅, 2 ^N } • • · · · •

{2 ^N } Π ⁰ ₁ \ Σ ⁰ ₁ • • •

· · ·

Self dual degrees and non-self dual pairs of degrees alternate. At all limit levels there is a non-self dual pair.

Each degree is assigned a rank, according to its position in the hierarchy (starting from 1).

The length of this hierarchy up to ∆ ⁰ ₂ is ω ₁ .

The length up to ∆ ⁰ ₃ is ω ₁ ^ω

.

How it looks like

{∅} Σ ⁰ ₁ \ Π ⁰ ₁ • • •

∆ ⁰ ₁ \ {∅, 2 ^N } • • · · · •

{2 ^N } Π ⁰ ₁ \ Σ ⁰ ₁ • • •

· · ·

Self dual degrees and non-self dual pairs of degrees alternate. At all limit levels there is a non-self dual pair.

Each degree is assigned a rank, according to its position in the hierarchy (starting from 1).

The length of this hierarchy up to ∆ ⁰ ₂ is ω 1 .

The length up to ∆ ⁰ ₃ is ω ₁ ^ω

.

Question

Which of these Π ⁰ ₃ Wadgre degrees, arranged in ω ^ω ₁

+ 1 levels, contain some regular sets?

Remark. If A is clopen, then Φ(A) = A.

Question

Which of these Π ⁰ ₃ Wadgre degrees, arranged in ω ^ω ₁

+ 1 levels, contain some regular sets?

Remark. If A is clopen, then Φ(A) = A.

Climbing Wadge hierarchy of ∆ ⁰ ₃ sets

In order to prove that in every degree of Wadge rank < ω ^ω ₁

there is a regular set, it would be desirable to have operations on the degrees

◦ i (A ₀ , A ₁ , . . .) such that:

1. starting with ∆ ⁰ ₁ sets, they generate all ∆ ⁰ ₃ sets 2. Φ ◦ _i (A ₀ , A ₁ , . . .) = ◦ _i (Φ(A ₀ ), Φ(A ₁ ), . . .)

Mimicking some of the Wadge’s constructions on the Baire space, there

are candidate operations performing task 1.

Climbing Wadge hierarchy of ∆ ⁰ ₃ sets

In order to prove that in every degree of Wadge rank < ω ^ω ₁

there is a regular set, it would be desirable to have operations on the degrees

◦ i (A ₀ , A ₁ , . . .) such that:

1. starting with ∆ ⁰ ₁ sets, they generate all ∆ ⁰ ₃ sets

2. Φ ◦ _i (A ₀ , A ₁ , . . .) = ◦ _i (Φ(A ₀ ), Φ(A ₁ ), . . .)

Mimicking some of the Wadge’s constructions on the Baire space, there

are candidate operations performing task 1.

Climbing Wadge hierarchy of ∆ ⁰ ₃ sets

In order to prove that in every degree of Wadge rank < ω ^ω ₁

there is a regular set, it would be desirable to have operations on the degrees

◦ i (A ₀ , A ₁ , . . .) such that:

1. starting with ∆ ⁰ ₁ sets, they generate all ∆ ⁰ ₃ sets 2. Φ ◦ _i (A ₀ , A ₁ , . . .) = ◦ _i (Φ(A ₀ ), Φ(A ₁ ), . . .)

Mimicking some of the Wadge’s constructions on the Baire space, there

are candidate operations performing task 1.

Climbing Wadge hierarchy of ∆ ⁰ ₃ sets

In order to prove that in every degree of Wadge rank < ω ^ω ₁

there is a regular set, it would be desirable to have operations on the degrees

◦ i (A ₀ , A ₁ , . . .) such that:

1. starting with ∆ ⁰ ₁ sets, they generate all ∆ ⁰ ₃ sets 2. Φ ◦ _i (A ₀ , A ₁ , . . .) = ◦ _i (Φ(A ₀ ), Φ(A ₁ ), . . .)

Mimicking some of the Wadge’s constructions on the Baire space, there

are candidate operations performing task 1.

Operations generating ∆ ⁰ ₂ (2 ^N )

I A 7→ ¬A.

I (A, B) 7→ A ⊕ B.

If A is non-self dual, then A ⊕ ¬A is a self dual immediate successor of A.

I (A n ) 7→ (A n ) ^∇ , (A n ) 7→ (A n ) ^◦

If A is self-dual, then ( ~ A ^∇ , ~ A ^◦ ) is a pair of non-self dual sets immediate successors of A.

If ∀n A n < W A n+1 , then (A ^∇ _n , A ^◦ _n ) is the least non-self dual pair

immediately above all A n .

Operations generating ∆ ⁰ ₂ (2 ^N )

I A 7→ ¬A.

I (A, B) 7→ A ⊕ B.

If A is non-self dual, then A ⊕ ¬A is a self dual immediate successor of A.

I (A n ) 7→ (A n ) ^∇ , (A n ) 7→ (A n ) ^◦

If A is self-dual, then ( ~ A ^∇ , ~ A ^◦ ) is a pair of non-self dual sets immediate successors of A.

If ∀n A n < W A n+1 , then (A ^∇ _n , A ^◦ _n ) is the least non-self dual pair

immediately above all A n .

Operations generating ∆ ⁰ ₂ (2 ^N )

I A 7→ ¬A.

I (A, B) 7→ A ⊕ B.

If A is non-self dual, then A ⊕ ¬A is a self dual immediate successor of A.

I (A n ) 7→ (A n ) ^∇ , (A n ) 7→ (A n ) ^◦

If A is self-dual, then ( ~ A ^∇ , ~ A ^◦ ) is a pair of non-self dual sets immediate successors of A.

If ∀n A n < W A n+1 , then (A ^∇ _n , A ^◦ _n ) is the least non-self dual pair

immediately above all A n .

Operations generating ∆ ⁰ ₂ (2 ^N )

I A 7→ ¬A.

I (A, B) 7→ A ⊕ B.

If A is non-self dual, then A ⊕ ¬A is a self dual immediate successor of A.

I (A n ) 7→ (A n ) ^∇ , (A n ) 7→ (A n ) ^◦

If A is self-dual, then ( ~ A ^∇ , ~ A ^◦ ) is a pair of non-self dual sets immediate successors of A.

If ∀n A n < W A n+1 , then (A ^∇ _n , A ^◦ _n ) is the least non-self dual pair

immediately above all A n .

Operations generating ∆ ⁰ ₂ (2 ^N )

I A 7→ ¬A.

I (A, B) 7→ A ⊕ B.

If A is non-self dual, then A ⊕ ¬A is a self dual immediate successor of A.

I (A n ) 7→ (A n ) ^∇ , (A n ) 7→ (A n ) ^◦

If A is self-dual, then ( ~ A ^∇ , ~ A ^◦ ) is a pair of non-self dual sets immediate successors of A.

If ∀n A n < W A n+1 , then (A ^∇ _n , A ^◦ _n ) is the least non-self dual pair

immediately above all A n .

The definitions

• A ⊕ B = 0 ^a A ∪ 1 ^a B

• A ^∇ _n = S

n 0 ⁿ 1 ^a A n

• A ^◦ _n = A ^∇ _n ∪ {0 ^∞ }

The definitions

• A ⊕ B = 0 ^a A ∪ 1 ^a B

• A ^∇ _n = S

n 0 ⁿ 1 ^a A _n

• A ^◦ _n = A ^∇ _n ∪ {0 ^∞ }

Reaching all of ∆ ⁰ ₃ (2 ^N )

I (A, B) 7→ A + B.

If A is self-dual, then ||A + B|| _W = ||A|| _W + ||B|| _W .

I A 7→ A ^\ , A 7→ A ^[ .

If A is self-dual, then A ^\ , A ^[ are non-self dual, A ^\ ≡ W ¬A ^[ and their Wadge rank is ||A|| _W ω ₁ .

(Definitions omitted)

Reaching all of ∆ ⁰ ₃ (2 ^N )

I (A, B) 7→ A + B.

If A is self-dual, then ||A + B|| _W = ||A|| _W + ||B|| _W .

I A 7→ A ^\ , A 7→ A ^[ .

If A is self-dual, then A ^\ , A ^[ are non-self dual, A ^\ ≡ W ¬A ^[ and their Wadge rank is ||A|| _W ω ₁ .

(Definitions omitted)

Reaching all of ∆ ⁰ ₃ (2 ^N )

I (A, B) 7→ A + B.

If A is self-dual, then ||A + B|| _W = ||A|| _W + ||B|| _W .

I A 7→ A ^\ , A 7→ A ^[ .

If A is self-dual, then A ^\ , A ^[ are non-self dual, A ^\ ≡ W ¬A ^[ and their Wadge rank is ||A|| _W ω ₁ .

(Definitions omitted)

A serious obstacle

Except for ⊕, the above operations do not commute with Φ and they do not take regular sets to regular sets.

To fix this, they need to be replaced by more suitable ones.

The case of ∇ and ◦.

Let f : N → N \ {0} be such that lim ^n→∞ f (n) = +∞. Definition.

I Rake(f , A n ) = S

n 0 ⁿ 1 ^{f (n)a} A n

I Rake ⁺ (f , A _n ) = Rake(f , A _n ) ∪ {0 ^∞ } ∪ S

n,t {N ₀

t | lh(t) = f (n), t 6=

0 ^{f (n)} , 1 ^{f (n)} }

A serious obstacle

Except for ⊕, the above operations do not commute with Φ and they do not take regular sets to regular sets.

To fix this, they need to be replaced by more suitable ones.

The case of ∇ and ◦.

Let f : N → N \ {0} be such that lim ^n→∞ f (n) = +∞.

Definition.

I Rake(f , A n ) = S

n 0 ⁿ 1 ^{f (n)a} A n

I Rake ⁺ (f , A _n ) = Rake(f , A _n ) ∪ {0 ^∞ } ∪ S

n,t {N ₀

t | lh(t) = f (n), t 6=

0 ^{f (n)} , 1 ^{f (n)} }

A serious obstacle

Except for ⊕, the above operations do not commute with Φ and they do not take regular sets to regular sets.

To fix this, they need to be replaced by more suitable ones.

The case of ∇ and ◦.

Let f : N → N \ {0} be such that lim ^n→∞ f (n) = +∞.

Definition.

I Rake(f , A n ) = S

n 0 ⁿ 1 ^{f (n)a} A n

I Rake ⁺ (f , A _n ) = Rake(f , A _n ) ∪ {0 ^∞ } ∪ S

n,t {N ₀

t | lh(t) = f (n), t 6=

0 ^{f (n)} , 1 ^{f (n)} }

A serious obstacle

Except for ⊕, the above operations do not commute with Φ and they do not take regular sets to regular sets.

To fix this, they need to be replaced by more suitable ones.

The case of ∇ and ◦.

Let f : N → N \ {0} be such that lim ^n→∞ f (n) = +∞.

Definition.

I Rake(f , A n ) = S

n 0 ⁿ 1 ^{f (n)a} A n

I Rake ⁺ (f , A _n ) = Rake(f , A _n ) ∪ {0 ^∞ } ∪ S

n,t {N 0

t | lh(t) = f (n), t 6=

0 ^{f (n)} , 1 ^{f (n)} }

Theorem for Rake

I A ^∇ _n ≡ W Rake(f , A n ).

I Φ(Rake(f , A _n )) = Rake(f , Φ(A _n )). In particular, Rake(f , A _n ) is regular if all A _n are. In fact:

I If ∀n A n ∈ ran(Φ| _Π

1

) then Rake(f , A n ) ∈ ran(Φ| _Π

).

I If ∀n A _n ∈ ran(Φ| _Σ

1

) then Rake(f , A _n ) ∈ ran(Φ| _Σ

).

Theorem for Rake

I A ^∇ _n ≡ W Rake(f , A n ).

I Φ(Rake(f , A _n )) = Rake(f , Φ(A _n )).

In particular, Rake(f , A _n ) is regular if all A _n are. In fact:

I If ∀n A n ∈ ran(Φ| _Π

1

) then Rake(f , A n ) ∈ ran(Φ| _Π

).

I If ∀n A _n ∈ ran(Φ| _Σ

1

) then Rake(f , A _n ) ∈ ran(Φ| _Σ

).

Theorem for Rake

I A ^∇ _n ≡ W Rake(f , A n ).

I Φ(Rake(f , A _n )) = Rake(f , Φ(A _n )). In particular, Rake(f , A _n ) is regular if all A _n are.

In fact:

I If ∀n A n ∈ ran(Φ| _Π

1

) then Rake(f , A n ) ∈ ran(Φ| _Π

).

I If ∀n A _n ∈ ran(Φ| _Σ

1

) then Rake(f , A _n ) ∈ ran(Φ| _Σ

).

Theorem for Rake

I A ^∇ _n ≡ W Rake(f , A n ).

I Φ(Rake(f , A _n )) = Rake(f , Φ(A _n )). In particular, Rake(f , A _n ) is regular if all A _n are. In fact:

I If ∀n A n ∈ ran(Φ| _Π

1

) then Rake(f , A n ) ∈ ran(Φ| _Π

).

I If ∀n A _n ∈ ran(Φ| _Σ

1

) then Rake(f , A _n ) ∈ ran(Φ| _Σ

).

Theorem for Rake ⁺

Similarly,

I A ^◦ _n ≡ W Rake ⁺ (f , A n )

I Φ(Rake ⁺ (f , A _n )) = Rake ⁺ (f , Φ(A _n )). In particular, Rake ⁺ (f , A _n ) is regular if all A _n are. In fact:

I If ∀n A n ∈ ran(Φ| _Π

1

) then Rake ⁺ (f , A n ) ∈ ran(Φ| _Π

).

I If ∀n A n ∈ ran(Φ| _Σ

1

) then Rake ⁺ (f , A n ) ∈ ran(Φ| _Σ

).

Corollary. Given any Wadge degree d ⊆ ∆ ⁰ ₂ (2 ^N ) there is a regular set

A ∈ d such that A = Φ(C ) = Φ(U) for some C closed and U open.

Theorem for Rake ⁺

Similarly,

I A ^◦ _n ≡ W Rake ⁺ (f , A n )

I Φ(Rake ⁺ (f , A _n )) = Rake ⁺ (f , Φ(A _n )). In particular, Rake ⁺ (f , A _n ) is regular if all A _n are. In fact:

I If ∀n A n ∈ ran(Φ| _Π

1

) then Rake ⁺ (f , A n ) ∈ ran(Φ| _Π

).

I If ∀n A n ∈ ran(Φ| _Σ

1

) then Rake ⁺ (f , A n ) ∈ ran(Φ| _Σ

).

Corollary. Given any Wadge degree d ⊆ ∆ ⁰ ₂ (2 ^N ) there is a regular set

A ∈ d such that A = Φ(C ) = Φ(U) for some C closed and U open.

The other operations

It is possible to define operations Sum(A, B), Nat(A), Flat(A) such that:

I Sum(A, B) ≡ _W A + B, Nat(A) ≡ _W A ^\ , Flat(A) ≡ _W A ^[ .

I Sum(A, B), Nat(A), Flat(A) are regular when A, B are.

I If A, B are image under Φ of a closed set, the same holds for Sum(A, B), Nat(A), Flat(A).

I If A, B are image under Φ of an open set, the same holds for

Sum(A, B), Nat(A), Flat(A).

The other operations

It is possible to define operations Sum(A, B), Nat(A), Flat(A) such that:

I Sum(A, B) ≡ _W A + B, Nat(A) ≡ _W A ^\ , Flat(A) ≡ _W A ^[ .

I Sum(A, B), Nat(A), Flat(A) are regular when A, B are.

I If A, B are image under Φ of a closed set, the same holds for Sum(A, B), Nat(A), Flat(A).

I If A, B are image under Φ of an open set, the same holds for

Sum(A, B), Nat(A), Flat(A).

The other operations

It is possible to define operations Sum(A, B), Nat(A), Flat(A) such that:

I Sum(A, B) ≡ _W A + B, Nat(A) ≡ _W A ^\ , Flat(A) ≡ _W A ^[ .

I Sum(A, B), Nat(A), Flat(A) are regular when A, B are.

I If A, B are image under Φ of a closed set, the same holds for Sum(A, B), Nat(A), Flat(A).

I If A, B are image under Φ of an open set, the same holds for

Sum(A, B), Nat(A), Flat(A).

The result for ∆ ⁰ ₃ degrees

Every ∆ ⁰ ₃ Wedge degree contains a regular set. In fact, every ∆ ⁰ ₃ Wadge

degree contains a set A such that A = Φ(C ) = Φ(U) for some closed C

and open U.

Regular Π ⁰ ₃ -complete sets

Theorem

There are a closed set C and an open set U such that Φ(C ) = Φ(U) is Π ⁰ ₃ -complete.

In fact there are many regular Π ⁰ ₃ -complete sets:

Theorem. If a regular non-empty set has empty interior, then it is Π ⁰ ₃ -complete.

Corollary. If C is a closed set of positive measure with empty interior,

then Φ(C ) is Π ⁰ ₃ -complete.

Regular Π ⁰ ₃ -complete sets

Theorem

There are a closed set C and an open set U such that Φ(C ) = Φ(U) is Π ⁰ ₃ -complete.

In fact there are many regular Π ⁰ ₃ -complete sets:

Theorem. If a regular non-empty set has empty interior, then it is Π ⁰ ₃ -complete.

Corollary. If C is a closed set of positive measure with empty interior,

then Φ(C ) is Π ⁰ ₃ -complete.

Regular Π ⁰ ₃ -complete sets

Theorem

There are a closed set C and an open set U such that Φ(C ) = Φ(U) is Π ⁰ ₃ -complete.

In fact there are many regular Π ⁰ ₃ -complete sets:

Theorem. If a regular non-empty set has empty interior, then it is Π ⁰ ₃ -complete.

Corollary. If C is a closed set of positive measure with empty interior,

then Φ(C ) is Π ⁰ ₃ -complete.

Most regular sets are Π ⁰ ₃ -complete

Each element of the measure algebra MALG can be assigned as a colour the Wedge degree of the unique regular set belonging to it: let

W d = {[A] ∈ MALG | Φ(A) ∈ d }.

Theorem.

I Except for d = {∅}, {2 ^N }, all W d are topologically dense in MALG .

I W _Π

3

\∆

is comeagre in MALG .

I W _Π

3

\∆

the unique W _d that is dense in the sense of the forcing

(MALG , ≤).

Most regular sets are Π ⁰ ₃ -complete

Each element of the measure algebra MALG can be assigned as a colour the Wedge degree of the unique regular set belonging to it: let

W d = {[A] ∈ MALG | Φ(A) ∈ d }.

Theorem.

I Except for d = {∅}, {2 ^N }, all W d are topologically dense in MALG .

I W _Π

3

\∆

is comeagre in MALG .

I W _Π

3

\∆

the unique W _d that is dense in the sense of the forcing

(MALG , ≤).

Most regular sets are Π ⁰ ₃ -complete

Each element of the measure algebra MALG can be assigned as a colour the Wedge degree of the unique regular set belonging to it: let

W d = {[A] ∈ MALG | Φ(A) ∈ d }.

Theorem.

I Except for d = {∅}, {2 ^N }, all W d are topologically dense in MALG .

I W _Π

3

\∆

is comeagre in MALG .

I W _Π

3

\∆

the unique W _d that is dense in the sense of the forcing

(MALG , ≤).

Most regular sets are Π ⁰ ₃ -complete

Each element of the measure algebra MALG can be assigned as a colour the Wedge degree of the unique regular set belonging to it: let

W d = {[A] ∈ MALG | Φ(A) ∈ d }.

Theorem.

I Except for d = {∅}, {2 ^N }, all W d are topologically dense in MALG .

I W _Π

3

\∆

is comeagre in MALG .

I W _Π

3

\∆

the unique W _d that is dense in the sense of the forcing

(MALG , ≤).

A corollary of the construction

Fact. Every measurable set can be approximated by an F σ set from the inside and a G δ set from the outside. Hence each element of the measure algebra MALG contains both an F σ and a G δ set.

This cannot be improved:

Theorem. The elements of MALG that contain a ∆ ⁰ ₂ member form a

meagre subset of MALG .

A corollary of the construction

Fact. Every measurable set can be approximated by an F σ set from the inside and a G δ set from the outside. Hence each element of the measure algebra MALG contains both an F σ and a G δ set.

This cannot be improved:

Theorem. The elements of MALG that contain a ∆ ⁰ ₂ member form a

meagre subset of MALG .

Exceptional points

Definition

For A ⊆ 2 ^N measurable and x ∈ 2 ^N , let:

I density: D _A (x ) = lim _n→∞ ^µ(A∩N _µ(N

⁾

x |n

) I lower density: D _A ⁻ (x ) = lim inf n→∞

µ(A∩N

) µ(N

) I upper density: D _A ⁺ (x ) = lim sup _n→∞ ^µ(A∩N _µ(N

⁾

x |n

)

I oscillation: O A (x ) = D ⁺ _A (x ) − D ⁻ _A (x )

Exceptional points

Definition

For A ⊆ 2 ^N measurable and x ∈ 2 ^N , let:

I density: D _A (x ) = lim _n→∞ ^µ(A∩N _µ(N

⁾

x |n

)

I lower density: D _A ⁻ (x ) = lim inf n→∞

µ(A∩N

) µ(N

) I upper density: D _A ⁺ (x ) = lim sup _n→∞ ^µ(A∩N _µ(N

⁾

x |n

)

I oscillation: O A (x ) = D ⁺ _A (x ) − D ⁻ _A (x )

Exceptional points

Definition

For A ⊆ 2 ^N measurable and x ∈ 2 ^N , let:

I density: D _A (x ) = lim _n→∞ ^µ(A∩N _µ(N

⁾

x |n

) I lower density: D _A ⁻ (x ) = lim inf n→∞

µ(A∩N

) µ(N

) I upper density: D _A ⁺ (x ) = lim sup _n→∞ ^µ(A∩N _µ(N

⁾

x |n

)

I oscillation: O A (x ) = D ⁺ _A (x ) − D ⁻ _A (x )

Exceptional points

Definition

For A ⊆ 2 ^N measurable and x ∈ 2 ^N , let:

I density: D _A (x ) = lim _n→∞ ^µ(A∩N _µ(N

⁾

x |n

) I lower density: D _A ⁻ (x ) = lim inf n→∞

µ(A∩N

) µ(N

) I upper density: D _A ⁺ (x ) = lim sup _n→∞ ^µ(A∩N _µ(N

⁾

x |n

)

I oscillation: O A (x ) = D _A ⁺ (x ) − D ⁻ _A (x )

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise

, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0

A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

Definition

Point x is:

I regular for A if D A (x ) ∈ {0, 1}

I exceptional, otherwise, i.e., if it is

I

either sharp: if D

(x ) exists and it is neither 0 nor 1

I

or blurry: if D

(x ) does not exists, i.e., O

(x ) > 0 A measurable set A is:

I solid if D A is everywhere defined

I quasi-dualistic if D A takes value in {0, 1}

I dualistic if it is quasi-dualistic and solid

I spongy if it is quasi-dualistic and not solid

Exceptional points

By LDT, the set of exceptional points is null for every measurable set.

However:

Theorem

The generic [A] ∈ MALG is spongy, and has comeagerly many points with oscillation 1.

One can study classification problems arising from these meaure-theoretic notions both in MALG and in K (2 ^N ), the hyperspace of compact subsets of 2 ^N .

Notice however that in K (2 ^N ) all category arguments are void: the

generic compact set is null.

Exceptional points

By LDT, the set of exceptional points is null for every measurable set.

However:

Theorem

The generic [A] ∈ MALG is spongy, and has comeagerly many points with oscillation 1.

One can study classification problems arising from these meaure-theoretic notions both in MALG and in K (2 ^N ), the hyperspace of compact subsets of 2 ^N .

Notice however that in K (2 ^N ) all category arguments are void: the

generic compact set is null.

Exceptional points

By LDT, the set of exceptional points is null for every measurable set.

However:

Theorem

The generic [A] ∈ MALG is spongy, and has comeagerly many points with oscillation 1.

One can study classification problems arising from these meaure-theoretic notions both in MALG and in K (2 ^N ), the hyperspace of compact subsets of 2 ^N .

Notice however that in K (2 ^N ) all category arguments are void: the

generic compact set is null.

Exceptional points

By LDT, the set of exceptional points is null for every measurable set.

However:

Theorem

The generic [A] ∈ MALG is spongy, and has comeagerly many points with oscillation 1.

One can study classification problems arising from these meaure-theoretic notions both in MALG and in K (2 ^N ), the hyperspace of compact subsets of 2 ^N .

Notice however that in K (2 ^N ) all category arguments are void: the

generic compact set is null.

Some classification results

The main tool to obtain classification results is to construct several maps T 7→ K T that given a binary pruned tree T produce a suitable compact set K T , and mix them together.

Theorem

In K (2 ^N ):

I {K | K is solid} is Π ¹ ₁ -complete

I {K | K is (quasi-)dualistic} is Π ¹ ₁ -complete

I {K | K is spongy} is D 2 (Π ¹ ₁ )-complete

I if C is a non-empty, countable subset of ]0, 1[, then {K | rng (D K ) = {0, 1} ∪ C } is D 2 (Π ¹ ₁ )-complete

I {K | rng (D _K ) = [0, 1]} is Π ¹ ₂ -complete

Some classification results

The main tool to obtain classification results is to construct several maps T 7→ K T that given a binary pruned tree T produce a suitable compact set K T , and mix them together.

Theorem

In K (2 ^N ):

I {K | K is solid} is Π ¹ ₁ -complete

I {K | K is (quasi-)dualistic} is Π ¹ ₁ -complete

I {K | K is spongy} is D 2 (Π ¹ ₁ )-complete

I if C is a non-empty, countable subset of ]0, 1[, then {K | rng (D K ) = {0, 1} ∪ C } is D 2 (Π ¹ ₁ )-complete

I {K | rng (D _K ) = [0, 1]} is Π ¹ ₂ -complete

Some classification results

The main tool to obtain classification results is to construct several maps T 7→ K T that given a binary pruned tree T produce a suitable compact set K T , and mix them together.

Theorem

In K (2 ^N ):

I {K | K is solid} is Π ¹ ₁ -complete

I {K | K is (quasi-)dualistic} is Π ¹ ₁ -complete

I {K | K is spongy} is D 2 (Π ¹ ₁ )-complete

I if C is a non-empty, countable subset of ]0, 1[, then {K | rng (D K ) = {0, 1} ∪ C } is D 2 (Π ¹ ₁ )-complete

I {K | rng (D _K ) = [0, 1]} is Π ¹ ₂ -complete

Some classification results

The main tool to obtain classification results is to construct several maps T 7→ K T that given a binary pruned tree T produce a suitable compact set K T , and mix them together.

Theorem

In K (2 ^N ):

I {K | K is solid} is Π ¹ ₁ -complete

I {K | K is (quasi-)dualistic} is Π ¹ ₁ -complete

I {K | K is spongy} is D 2 (Π ¹ ₁ )-complete

I if C is a non-empty, countable subset of ]0, 1[, then {K | rng (D K ) = {0, 1} ∪ C } is D 2 (Π ¹ ₁ )-complete

I {K | rng (D _K ) = [0, 1]} is Π ¹ ₂ -complete

Some classification results

The main tool to obtain classification results is to construct several maps T 7→ K T that given a binary pruned tree T produce a suitable compact set K T , and mix them together.

Theorem

In K (2 ^N ):

I {K | K is solid} is Π ¹ ₁ -complete

I {K | K is (quasi-)dualistic} is Π ¹ ₁ -complete

I {K | K is spongy} is D 2 (Π ¹ ₁ )-complete

I if C is a non-empty, countable subset of ]0, 1[, then {K | rng (D K ) = {0, 1} ∪ C } is D 2 (Π ¹ ₁ )-complete

I {K | rng (D _K ) = [0, 1]} is Π ¹ ₂ -complete

A representation theorem for Σ ¹ ₁

Question.

What is the complexity of {K | rng (D _K ) = {0, 1} ∪ C } when C is an uncountable analytic subset of ]0, 1[?

Guess. It is Π ¹ ₂ -complete.

Theorem (Rep’n thm for analytic subsets of the interval)

The set

{(K , r ) ∈ K (2 ^N )×]0, 1[| ∃z ∈ 2 ^N D _K (z) = r }

is universal for analytic subsets of ]0, 1[.

A representation theorem for Σ ¹ ₁

Question. What is the complexity of {K | rng (D _K ) = {0, 1} ∪ C } when C is an uncountable analytic subset of ]0, 1[?

Guess. It is Π ¹ ₂ -complete.

Theorem (Rep’n thm for analytic subsets of the interval)

The set

{(K , r ) ∈ K (2 ^N )×]0, 1[| ∃z ∈ 2 ^N D _K (z) = r }

is universal for analytic subsets of ]0, 1[.

A representation theorem for Σ ¹ ₁

Question. What is the complexity of {K | rng (D _K ) = {0, 1} ∪ C } when C is an uncountable analytic subset of ]0, 1[?

Guess. It is Π ¹ ₂ -complete.

Theorem (Rep’n thm for analytic subsets of the interval)

The set

{(K , r ) ∈ K (2 ^N )×]0, 1[| ∃z ∈ 2 ^N D _K (z) = r }

is universal for analytic subsets of ]0, 1[.

A representation theorem for Σ ¹ ₁

Question. What is the complexity of {K | rng (D _K ) = {0, 1} ∪ C } when C is an uncountable analytic subset of ]0, 1[?

Guess. It is Π ¹ ₂ -complete.

Theorem (Rep’n thm for analytic subsets of the interval)

The set

{(K , r ) ∈ K (2 ^N )×]0, 1[| ∃z ∈ 2 ^N D _K (z) = r }

is universal for analytic subsets of ]0, 1[.

A representation theorem for Σ ¹ ₁

Question. What is the complexity of {K | rng (D _K ) = {0, 1} ∪ C } when C is an uncountable analytic subset of ]0, 1[?

Guess. It is Π ¹ ₂ -complete.

Theorem (Rep’n thm for analytic subsets of the interval)

The set

{(K , r ) ∈ K (2 ^N )×]0, 1[| ∃z ∈ 2 ^N D _K (z) = r }

is universal for analytic subsets of ]0, 1[.

Representing Borel sets injectively

Corollary (of the proof)

For every Borel B ⊆]0, 1[ there is K ∈ K (2 ^N ) such that rng (D _K ) = {0, 1} ∪ B and

∀r ∈ B ∃!z ∈ 2 ^N D _K (z) = r

Representing Borel sets injectively

Corollary (of the proof)

For every Borel B ⊆]0, 1[ there is K ∈ K (2 ^N ) such that rng (D _K ) = {0, 1} ∪ B and

∀r ∈ B ∃!z ∈ 2 ^N D _K (z) = r

Higher in the projective hierarchy of K (2 ^N )

Using the representation theorem, rng (D _K ) is Borel if and only if there is H such that

rng (D _K )∩]0, 1[ and rng (D _H )∩]0, 1[ partition ]0, 1[. So a direct calculation shows that

R ∆ = {K | rng (D K ) ∈ ∆ ¹ ₁ } is Σ ¹ ₃ hence

R Σ = {K | rng (D K ) ∈ Σ ¹ ₁ \ ∆ ¹ ₁ } is Π ¹ ₃ On the other hand

R _Σ ^∗ = {K | rng (D K ) is Σ ¹ ₁ -complete} is Σ ¹ ₃

Higher in the projective hierarchy of K (2 ^N )

Using the representation theorem, rng (D _K ) is Borel if and only if there is H such that

rng (D _K )∩]0, 1[ and rng (D _H )∩]0, 1[

partition ]0, 1[.

So a direct calculation shows that R ∆ = {K | rng (D K ) ∈ ∆ ¹ ₁ } is Σ ¹ ₃ hence

R Σ = {K | rng (D K ) ∈ Σ ¹ ₁ \ ∆ ¹ ₁ } is Π ¹ ₃ On the other hand

R _Σ ^∗ = {K | rng (D K ) is Σ ¹ ₁ -complete} is Σ ¹ ₃

Higher in the projective hierarchy of K (2 ^N )

Using the representation theorem, rng (D _K ) is Borel if and only if there is H such that

rng (D _K )∩]0, 1[ and rng (D _H )∩]0, 1[

partition ]0, 1[. So a direct calculation shows that R ∆ = {K | rng (D K ) ∈ ∆ ¹ ₁ } is Σ ¹ ₃

hence

R Σ = {K | rng (D K ) ∈ Σ ¹ ₁ \ ∆ ¹ ₁ } is Π ¹ ₃ On the other hand

R _Σ ^∗ = {K | rng (D K ) is Σ ¹ ₁ -complete} is Σ ¹ ₃

Higher in the projective hierarchy of K (2 ^N )

Using the representation theorem, rng (D _K ) is Borel if and only if there is H such that

rng (D _K )∩]0, 1[ and rng (D _H )∩]0, 1[

partition ]0, 1[. So a direct calculation shows that R ∆ = {K | rng (D K ) ∈ ∆ ¹ ₁ } is Σ ¹ ₃ hence

R Σ = {K | rng (D K ) ∈ Σ ¹ ₁ \ ∆ ¹ ₁ } is Π ¹ ₃

On the other hand

R _Σ ^∗ = {K | rng (D K ) is Σ ¹ ₁ -complete} is Σ ¹ ₃

Higher in the projective hierarchy of K (2 ^N )

Using the representation theorem, rng (D _K ) is Borel if and only if there is H such that

rng (D _K )∩]0, 1[ and rng (D _H )∩]0, 1[

partition ]0, 1[. So a direct calculation shows that R ∆ = {K | rng (D K ) ∈ ∆ ¹ ₁ } is Σ ¹ ₃ hence

R Σ = {K | rng (D K ) ∈ Σ ¹ ₁ \ ∆ ¹ ₁ } is Π ¹ ₃ On the other hand

R _Σ ^∗ = {K | rng (D K ) is Σ ¹ ₁ -complete} is Σ ¹ ₃

Higher in the projective hierarchy of K (2 ^N )

So under Σ ¹ ₁ -determinacy, R ∆ , R Σ are ∆ ¹ ₃ .

Question.

I What is the complexity of R _∆ , R _Σ , R _Σ ^∗ ?

I How it depends on additional axioms?

Higher in the projective hierarchy of K (2 ^N )

So under Σ ¹ ₁ -determinacy, R ∆ , R Σ are ∆ ¹ ₃ . Question.

I What is the complexity of R _∆ , R _Σ , R _Σ ^∗ ?

I How it depends on additional axioms?