The descriptive set theory of the Lebesgue density theorem joint with A. Andretta

### The density function

Let (X , d , µ) be a Polish metric space endowed with a Borel probability measure giving positive measures to open sets.

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

ε→0^{+}

µ(A ∩ B_{ε}(x ))
µ(Bε(x )) .

Let also

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

### The density function

Let (X , d , µ) be a Polish metric space endowed with a Borel probability measure giving positive measures to open sets.

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

DA(x ) = lim

ε→0^{+}

µ(A ∩ B_{ε}(x ))
µ(Bε(x )) .

Let also

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

### The density function

Let (X , d , µ) be a Polish metric space endowed with a Borel probability measure giving positive measures to open sets.

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

DA(x ) = lim

ε→0^{+}

µ(A ∩ B_{ε}(x ))
µ(Bε(x )) .

Let also

Φ(A) = {x ∈ X | DA(x ) = 1}

be the set of points with density 1 in A.

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

Thus Φ induces a function Φ : MALG (X ) → MALG (X ) on the measure algebra of X .

### The problem

What can be said, in general, about 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 ∈IΦ(A_{i}) ⊆ Φ(S

i ∈IA_{i})

I Φ(¬A) ⊆ ¬Φ(A)

I A ⊆ Φ(A) for open A

Thus {A ∈ MEAS (X ) | A ⊆ Φ(A)} is a topology (the density topology) extending the topology of X .

### The problem

What can be said, in general, about 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 ∈IΦ(A_{i}) ⊆ Φ(S

i ∈IA_{i})

I Φ(¬A) ⊆ ¬Φ(A)

I A ⊆ Φ(A) for open A

Thus {A ∈ MEAS (X ) | A ⊆ Φ(A)} is a topology (the density topology) extending the topology of X .

### The problem

What can be said, in general, about 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 ∈IΦ(A_{i}) ⊆ Φ(S

i ∈IA_{i})

I Φ(¬A) ⊆ ¬Φ(A)

I A ⊆ Φ(A) for open A

Thus {A ∈ MEAS (X ) | A ⊆ Φ(A)} is a topology (the density topology) extending the topology of X .

### Lebesgue density theorem

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

This means that map Φ : MALG → MALG is the identity

and Φ : MEAS → Bor is a selector for the equivalence relation ≡. Remark. At least in these cases the inclusion Φ(¬A) ⊆ ¬Φ(A) cannot be replaced by equality. Otherwise Φ : MEAS → Bor would be a

homomorphism of Bolean algebras such that ∀A Φ(A) ≡ A. The existence of such Borel liftings is independent of ZFC (Shelah).

### Lebesgue density theorem

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

This means that map Φ : MALG → MALG is the identity

and Φ : MEAS → Bor is a selector for the equivalence relation ≡. Remark. At least in these cases the inclusion Φ(¬A) ⊆ ¬Φ(A) cannot be replaced by equality. Otherwise Φ : MEAS → Bor would be a

homomorphism of Bolean algebras such that ∀A Φ(A) ≡ A. The existence of such Borel liftings is independent of ZFC (Shelah).

### Lebesgue density theorem

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

This means that map Φ : MALG → MALG is the identity

and Φ : MEAS → Bor is a selector for the equivalence relation ≡.

Remark. At least in these cases the inclusion Φ(¬A) ⊆ ¬Φ(A) cannot be replaced by equality. Otherwise Φ : MEAS → Bor would be a

homomorphism of Bolean algebras such that ∀A Φ(A) ≡ A. The existence of such Borel liftings is independent of ZFC (Shelah).

### Lebesgue density theorem

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

This means that map Φ : MALG → MALG is the identity

and Φ : MEAS → Bor is a selector for the equivalence relation ≡.

Remark. At least in these cases the inclusion Φ(¬A) ⊆ ¬Φ(A) cannot be replaced by equality. Otherwise Φ : MEAS → Bor would be a

### Examples of spaces satisfying LDT

I R^{n}

I Polish ultrametric spaces (B. Miller)

I Problem. Which Polish spaces satisfy Lebesgue density theorem?
Here 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?

Here 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?

Here I will focus on Cantor space 2^{N}with the usual distance and
Lebesgue (coin-tossing) measure.

In 2^{N}the density function becomes
DA(x ) = lim

n→∞

µ(A ∩ N_{x |}_{n})
µ(N_{x |}_{n}) = lim

n→∞2^{n}µ(A ∩ N_{x |}_{n})
and

x ∈ Φ(A) ⇔ ∀ε ∃n ∀m > n 2^{m}µ(A ∩ N_{x |}_{m}) > 1 − ε,
so that 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.

In 2^{N}the density function becomes
DA(x ) = lim

n→∞

µ(A ∩ N_{x |}_{n})
µ(N_{x |}_{n}) = lim

n→∞2^{n}µ(A ∩ N_{x |}_{n})
and

x ∈ Φ(A) ⇔ ∀ε ∃n ∀m > n 2^{m}µ(A ∩ N_{x |}_{m}) > 1 − ε,
so that 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

∅
2^{N}

### How it looks

∅

∆^{0}_{1}\ {∅, 2^{N}}
2^{N}

### How it looks

∅ Σ^{0}_{1}\ Π^{0}_{1}

∆^{0}_{1}\ {∅, 2^{N}}

2^{N} Π^{0}_{1}\ Σ^{0}_{1}

### How it looks

∅ Σ^{0}_{1}\ Π^{0}_{1} • • •

∆^{0}_{1}\ {∅, 2^{N}} • • · · · •

2^{N} Π^{0}_{1}\ Σ^{0}_{1} • • •

· · ·

### How it looks

∅ 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 ∆^{2}_{0}is ω_{1}.
The length up to ∆^{0}_{3}is ω_{1}^{ω}^{1}.

### How it looks

∅ 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 ∆^{2}_{0}is ω1.

The length up to ∆^{0}_{3}is ω_{1}^{ω}^{1}.

### How it looks

∅ 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 ∆^{2}_{0}is ω1.
The length up to ∆^{0}_{3}is ω_{1}^{ω}^{1}.

### Question

Which of these Π^{0}_{3} Wadgre degrees, arranged in ω^{ω}_{1}^{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}+ 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}^{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}^{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}^{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

^{ω}_{1}^{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}), . . .)

### 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 (An) 7→ (An)^{∇}, (An) 7→ (An)^{◦}

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

If ∀n An<W An+1, then (A^{∇}_{n}, A^{◦}_{n}) is the least non-self dual pair
immediately above all An.

### 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 (An) 7→ (An)^{∇}, (An) 7→ (An)^{◦}

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

If ∀n An<W An+1, then (A^{∇}_{n}, A^{◦}_{n}) is the least non-self dual pair
immediately above all An.

### 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 (An) 7→ (An)^{∇}, (An) 7→ (An)^{◦}

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

If ∀n An<W An+1, then (A^{∇}_{n}, A^{◦}_{n}) is the least non-self dual pair
immediately above all An.

### The definitions

• A ⊕ B = 0^{a}A ∪ 1^{a}B

• A^{∇}_{n} =S

n0^{n}1^{a}An

• A^{◦}_{n}= A^{∇}_{n} ∪ {0^{∞}}

### The definitions

• A ⊕ B = 0^{a}A ∪ 1^{a}B

• A^{∇}_{n} =S

n0^{n}1^{a}An

• 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 (An) 7→ (An)^{\}, (An) 7→ (An)^{[}.

If A is self-dual, then A^{\}, A^{[}are non-self dual, A^{\}≡W ¬A^{[}and their
Wadge rank is ||A||Wω1.

### 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 (An) 7→ (An)^{\}, (An) 7→ (An)^{[}.

If A is self-dual, then A^{\}, A^{[}are non-self dual, A^{\}≡W ¬A^{[}and their
Wadge rank is ||A||Wω1.

### The definition

For s = (s0, s1, . . .) ∈ 2^{≤ω}, let ¯s = (s0, s0, s1, s1, . . .). Also, ¯A = {¯a}a∈A.

• A + B = {¯sta | s ∈ 2^{<ω}, t ∈ {01, 10}, a ∈ A} ∪ ¯B.

• A^{\}= { ¯s1t1. . . ¯sntna | s¯ i ∈ 2^{<ω}, ti ∈ {01, 10}, a ∈ A}

• A^{[}= A^{\}∪ {x ∈ 2^{N}| ∃^{∞}n x (2n) 6= x (2n + 1)}
Warning:

Something suspicious here: sets of the form ¯A have measure 0.

### The definition

For s = (s0, s1, . . .) ∈ 2^{≤ω}, let ¯s = (s0, s0, s1, s1, . . .). Also, ¯A = {¯a}a∈A.

• A + B = {¯sta | s ∈ 2^{<ω}, t ∈ {01, 10}, a ∈ A} ∪ ¯B.

• A^{\}= { ¯s1t1. . . ¯sntna | s¯ i ∈ 2^{<ω}, ti ∈ {01, 10}, a ∈ A}

• A^{[}= A^{\}∪ {x ∈ 2^{N}| ∃^{∞}n x (2n) 6= x (2n + 1)}

Warning:

Something suspicious here: sets of the form ¯A have measure 0.

### The definition

For s = (s0, s1, . . .) ∈ 2^{≤ω}, let ¯s = (s0, s0, s1, s1, . . .). Also, ¯A = {¯a}a∈A.

• A + B = {¯sta | s ∈ 2^{<ω}, t ∈ {01, 10}, a ∈ A} ∪ ¯B.

• A^{\}= { ¯s1t1. . . ¯sntna | s¯ i ∈ 2^{<ω}, ti ∈ {01, 10}, a ∈ A}

• A^{[}= A^{\}∪ {x ∈ 2^{N}| ∃^{∞}n x (2n) 6= x (2n + 1)}

Warning:

Something suspicious here: sets of the form ¯A have measure 0.

### 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 , An) =S

n0^{n}1^{f (n)a}An

I Rake^{+}(f , A_{n}) = Rake(f , A_{n}) ∪ {0^{∞}} ∪S

n,t{N_{0}nt | 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 , An) =S

n0^{n}1^{f (n)a}An

I Rake^{+}(f , A_{n}) = Rake(f , A_{n}) ∪ {0^{∞}} ∪S

n,t{N_{0}nt | 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 , An) =S

n0^{n}1^{f (n)a}An

I Rake^{+}(f , A_{n}) = Rake(f , A_{n}) ∪ {0^{∞}} ∪S

n,t{N0^{n}t | lh(t) = f (n), t 6=

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

### Theorem for Rake

I A^{∇}_{n} ≡W Rake(f , An).

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 An∈ ran(Φ|_{Π}0

1) then Rake(f , An) ∈ ran(Φ|_{Π}0
1).

I If ∀n A_{n}∈ ran(Φ|_{Σ}0

1) then Rake(f , A_{n}) ∈ ran(Φ|_{Σ}0
1).

### Theorem for Rake

^{+}

Similarly,

I A^{◦}_{n}≡_{W} Rake^{+}(f , A_{n})

I Φ(Rake^{+}(f , An)) = Rake^{+}(f , Φ(An)). In particular, Rake^{+}(f , An) is
regular if all An are. In fact:

I If ∀n An∈ ran(Φ|_{Π}0

1) then Rake^{+}(f , An) ∈ ran(Φ|_{Π}0
1).

I If ∀n A_{n}∈ ran(Φ|_{Σ}^{0}

1) then Rake^{+}(f , A_{n}) ∈ ran(Φ|_{Σ}0
1).

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 , An)) = Rake^{+}(f , Φ(An)). In particular, Rake^{+}(f , An) is
regular if all An are. In fact:

I If ∀n An∈ ran(Φ|_{Π}0

1) then Rake^{+}(f , An) ∈ ran(Φ|_{Π}0
1).

I If ∀n A_{n}∈ ran(Φ|_{Σ}^{0}

1) then Rake^{+}(f , A_{n}) ∈ ran(Φ|_{Σ}0
1).

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).

### Did you ask for a definition?

Here you go for Sum(A, B):

Fix an increasing sequence rn of positive real numbers converging to 1.
Sum(A, B) = ¯B ∪S{¯st^{a}O(2^{length(s)}rlength(s)µ(A ∩ Ns)) | s ∈ 2^{<ω}, t ∈
{01, 10}} ∪S

e∈E(B)e^{a}B,
where

O(r ) = 2^{N}\ N_{0}min{h>0|r ≤1−2−h }1

E(B) = {¯st0min{h>0|2^{length(s)}r_{length(s)}µ(B∩Ns)≤1−2^{−h}}1 | s ∈ 2^{<ω}, t ∈ {01, 10}}

### Did you ask for a definition?

Here you go for Sum(A, B):

Fix an increasing sequence rn of positive real numbers converging to 1.

Sum(A, B) = ¯B ∪S{¯st^{a}O(2^{length(s)}r_{length(s)}µ(A ∩ Ns)) | s ∈ 2^{<ω}, t ∈
{01, 10}} ∪S

e∈E(B)e^{a}B,

where

O(r ) = 2^{N}\ N_{0}min{h>0|r ≤1−2−h }1

E(B) = {¯st0min{h>0|2^{length(s)}r_{length(s)}µ(B∩Ns)≤1−2^{−h}}1 | s ∈ 2^{<ω}, t ∈ {01, 10}}

### Did you ask for a definition?

Here you go for Sum(A, B):

Fix an increasing sequence rn of positive real numbers converging to 1.

Sum(A, B) = ¯B ∪S{¯st^{a}O(2^{length(s)}r_{length(s)}µ(A ∩ Ns)) | s ∈ 2^{<ω}, t ∈
{01, 10}} ∪S

e∈E(B)e^{a}B,
where

O(r ) = 2^{N}\ N_{0}min{h>0|r ≤1−2−h }1

E(B) = {¯st0min{h>0|2^{length(s)}r_{length(s)}µ(B∩Ns)≤1−2^{−h}}1 | s ∈ 2^{<ω}, t ∈ {01, 10}}

### 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

Where everything began: 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

Where everything began: 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

Where everything began: 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

Wd = {[A] ∈ MALG | Φ(A) ∈ d }.

Theorem.

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

I W_{Π}0

3\∆^{0}_{3} is comeagre in MALG .

I W_{Π}0

3\∆^{0}_{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

Wd = {[A] ∈ MALG | Φ(A) ∈ d }.

Theorem.

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

I W_{Π}0

3\∆^{0}_{3} is comeagre in MALG .

I W_{Π}0

3\∆^{0}_{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 Fσ set from the inside and a Gδ set from the outside. Hence each element of the measure algebra MALG contains both an Fσ and a Gδ 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 Fσ set from the inside and a Gδ set from the outside. Hence each element of the measure algebra MALG contains both an Fσ and a Gδ set.

This cannot be improved:

Theorem. The elements of MALG that contain a ∆^{0}_{2} member form a
meagre subset of MALG .