Cardinal invariants of the Haar null ideal M ark Po or E otv os - - PowerPoint PPT Presentation

Introduction Uniformity and covering number Cofinality Questions

Cardinal invariants of the Haar null ideal

M´ ark Po´

• r

E¨

• tv¨
• s Lor´

and University, Budapest

Descriptive Set Theory in Turin September 2017 joint work with M´ arton Elekes

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Introduction

The Haar null ideal HN(G).

Definition

Let G be a Polish group, N ⊆ G be a set. Then N is Haar null, N ∈ HN(G) if there exists a Borel set B ⊇ N, and a probability Borel measure µ on G such that ∀g, h ∈ G µ(hBg) = 0.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Introduction

The Haar null ideal HN(G).

Definition

Let G be a Polish group, N ⊆ G be a set. Then N is Haar null, N ∈ HN(G) if there exists a Borel set B ⊇ N, and a probability Borel measure µ on G such that ∀g, h ∈ G µ(hBg) = 0. The original notion HN UM(G), in the sense of Christensen:

Definition

Let G be a Polish group, N ⊆ G be a set. Then N is generalized Haar null, N ∈ HN UM(G) if there exists a universally measurable set U ⊇ N, and a probability Borel measure µ on G such that ∀g, h ∈ G µ(hUg) = 0.

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

It is a generalisation of the null ideal:

Theorem

If G is a locally compact Polish group, then HN(G) = HN UM(G), moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

It is a generalisation of the null ideal:

Theorem

If G is a locally compact Polish group, then HN(G) = HN UM(G), moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. However, for non-locally compact Polish groups there is no Haar measure.

Theorem

If G is a Polish group then HN(G) and HN UM(G) form σ-ideals.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

It is a generalisation of the null ideal:

Theorem

If G is a locally compact Polish group, then HN(G) = HN UM(G), moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. However, for non-locally compact Polish groups there is no Haar measure.

Theorem

If G is a Polish group then HN(G) and HN UM(G) form σ-ideals.

Theorem (M. Elekes, Z. Vidny´ anszky)

Suppose that G is a non-locally compact Polish group that admits an invariant

• metric. Then there is a set C ∈ HN UM(G) ∩ Π1

1 such that

C / ∈ HN(G).

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

It is a generalisation of the null ideal:

Theorem

If G is a locally compact Polish group, then HN(G) = HN UM(G), moreover it is the set of null sets w.r.t. the (complete) left or right Haar measure. However, for non-locally compact Polish groups there is no Haar measure.

Theorem

If G is a Polish group then HN(G) and HN UM(G) form σ-ideals.

Theorem (M. Elekes, Z. Vidny´ anszky)

Suppose that G is a non-locally compact Polish group that admits an invariant

• metric. Then there is a set C ∈ HN UM(G) ∩ Π1

1 such that

C / ∈ HN(G). i.e. ∃µ probability Borel measure s.t. µ∗(gCh) = 0 (∀g, h), ∄ ν, B ∈ ∆1

1 such that C ⊆ B, and ν(gBh) = 0 (∀g, h).

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

A sufficient condition for non-Haar nullness

Definition

In a topological group G a set S is called compact-catcher, if for every compact set C ⊆ G there exist elements g, h such that gCh ⊆ S ( ⇐ ⇒ C ⊆ g −1Sh−1)

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

A sufficient condition for non-Haar nullness

Definition

In a topological group G a set S is called compact-catcher, if for every compact set C ⊆ G there exist elements g, h such that gCh ⊆ S ( ⇐ ⇒ C ⊆ g −1Sh−1) A compact-catcher (Borel set) B cannot be Haar null. If µ is an arbitrary probability Borel measure (on a Polish group G), then (by regularity) there is a compact set C with µ(C) > 0.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

A sufficient condition for non-Haar nullness

Definition

In a topological group G a set S is called compact-catcher, if for every compact set C ⊆ G there exist elements g, h such that gCh ⊆ S ( ⇐ ⇒ C ⊆ g −1Sh−1) A compact-catcher (Borel set) B cannot be Haar null. If µ is an arbitrary probability Borel measure (on a Polish group G), then (by regularity) there is a compact set C with µ(C) > 0. Then translating B so that it covers C C ⊆ gBh ⇒ 0 < µ(C) ≤ µ(gBh).

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

A Haar null set B ∈ ∆1

1 does not necessarily have a Gδ hull, in fact the

following holds.

Theorem (M. Elekes, Z. Vidny´ anszky)

Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω1. Then there exists a Haar null Borel set Bα ∈ HN(G) such that there is no B′ ∈ HN(G) with Bα ⊆ B′ and B′ ∈ Π0

α.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

A Haar null set B ∈ ∆1

1 does not necessarily have a Gδ hull, in fact the

following holds.

Theorem (M. Elekes, Z. Vidny´ anszky)

Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω1. Then there exists a Haar null Borel set Bα ∈ HN(G) such that there is no B′ ∈ HN(G) with Bα ⊆ B′ and B′ ∈ Π0

α.

Corollary

If G is non-locally compact with an invariant metric then add(HN(G)) = ω1.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

A Haar null set B ∈ ∆1

1 does not necessarily have a Gδ hull, in fact the

following holds.

Theorem (M. Elekes, Z. Vidny´ anszky)

Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω1. Then there exists a Haar null Borel set Bα ∈ HN(G) such that there is no B′ ∈ HN(G) with Bα ⊆ B′ and B′ ∈ Π0

α.

Corollary

If G is non-locally compact with an invariant metric then add(HN(G)) = ω1.

• Proof. Let Bα-s (α < ω1) be given by the theorem.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Facts

A Haar null set B ∈ ∆1

1 does not necessarily have a Gδ hull, in fact the

following holds.

Theorem (M. Elekes, Z. Vidny´ anszky)

Suppose that G is a non-locally compact Polish group with an invariant metric, and let α < ω1. Then there exists a Haar null Borel set Bα ∈ HN(G) such that there is no B′ ∈ HN(G) with Bα ⊆ B′ and B′ ∈ Π0

α.

Corollary

If G is non-locally compact with an invariant metric then add(HN(G)) = ω1.

• Proof. Let Bα-s (α < ω1) be given by the theorem. Now if B ⊇

α<ω1 Bα is a

Borel Haar null set, and B ∈ Π0

β for some countable β, Bβ ⊆ B is a

contradiction.

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Uniformity and covering number

Theorem (T. Banakh)

cov(HN UM(Zω)) = min(b, cov(N)) non(HN UM(Zω)) = max(d, non(N))

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Uniformity and covering number

Theorem (T. Banakh)

cov(HN UM(Zω)) = min(b, cov(N)) non(HN UM(Zω)) = max(d, non(N)) The theorem also holds in a bit more general setting, as does the following, which is obtained by a minor modification of Banakh’s proof.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Uniformity and covering number

Theorem (T. Banakh)

cov(HN UM(Zω)) = min(b, cov(N)) non(HN UM(Zω)) = max(d, non(N)) The theorem also holds in a bit more general setting, as does the following, which is obtained by a minor modification of Banakh’s proof.

Theorem (M. Elekes, M.P.)

cov(HN(Zω)) = min(b, cov(N)) non(HN(Zω)) = max(d, non(N))

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Theorem (T. Banakh)

Let G be a non-locally compact Polish group admitting an invariant metric. Then cof(HN UM(G)) > min(d, non(N)).

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Theorem (T. Banakh)

Let G be a non-locally compact Polish group admitting an invariant metric. Then cof(HN UM(G)) > min(d, non(N)).

Corollary

(MA) Let G be a non-locally compact Polish group admitting an invariant

• metric. Then

cof(HN UM(G)) > c.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Theorem (M. Elekes, M.P.)

Let G be a non-locally compact Polish group with an invariant metric. Then cof(HN(G)) = c.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Theorem (M. Elekes, M.P.)

Let G be a non-locally compact Polish group with an invariant metric. Then cof(HN(G)) = c. The proof consists of the following parts. Recall that for a Polish space X the Effros Borel space F(X) is the standard Borel space of the closed subsets of X. The essence of the proof lies in the following technical statement.

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Theorem (M. Elekes, M.P.)

Let G be a non-locally compact Polish group with an invariant metric. Then cof(HN(G)) = c. The proof consists of the following parts. Recall that for a Polish space X the Effros Borel space F(X) is the standard Borel space of the closed subsets of X. The essence of the proof lies in the following technical statement.

Proposition

Let G be a non-locally compact Polish group with an invariant metric. Then there exists a family of closed Haar null sets {Nx : x ∈ 2ω} ⊆ HN(G) ∩ F(G) such that the mapping x → Nx is an injective Borel mapping from 2ω to F(G), and for each nonempty perfect set P ⊆ 2ω

• x∈P

Nx is compact-catcher.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Proof.(Theorem) For cof(HN(G)) = c it is enough to show that for each Borel set B ∈ HN(G) |{x ∈ 2ω : Nx ⊆ B}| ≤ ω1.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Proof.(Theorem) For cof(HN(G)) = c it is enough to show that for each Borel set B ∈ HN(G) |{x ∈ 2ω : Nx ⊆ B}| ≤ ω1. Then, since {(g, F) ∈ G × F(G) : g ∈ F} ⊆ G × F(G) is Borel, {F ∈ F(G) : F ⊆ B} = {F : ∀g g ∈ F → g ∈ B} is Π1

1,

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Proof.(Theorem) For cof(HN(G)) = c it is enough to show that for each Borel set B ∈ HN(G) |{x ∈ 2ω : Nx ⊆ B}| ≤ ω1. Then, since {(g, F) ∈ G × F(G) : g ∈ F} ⊆ G × F(G) is Borel, {F ∈ F(G) : F ⊆ B} = {F : ∀g g ∈ F → g ∈ B} is Π1

1, so is its intersection with {Nx : x ∈ 2ω}:

{Nx ∈ F(G) : Nx ⊆ B} ∈ Π1

1

({Nx : x ∈ 2ω} is Borel, since x → Nx was injective and Borel).

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Proof.(Theorem) For cof(HN(G)) = c it is enough to show that for each Borel set B ∈ HN(G) |{x ∈ 2ω : Nx ⊆ B}| ≤ ω1. Then, since {(g, F) ∈ G × F(G) : g ∈ F} ⊆ G × F(G) is Borel, {F ∈ F(G) : F ⊆ B} = {F : ∀g g ∈ F → g ∈ B} is Π1

1, so is its intersection with {Nx : x ∈ 2ω}:

{Nx ∈ F(G) : Nx ⊆ B} ∈ Π1

1

({Nx : x ∈ 2ω} is Borel, since x → Nx was injective and Borel). Now taking its preimage under the mapping x → Nx {x ∈ 2ω : Nx ⊆ B} is also Π1

1, thus it is the union of ω1-many Borel sets.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Proof.(Theorem) For cof(HN(G)) = c it is enough to show that for each Borel set B ∈ HN(G) |{x ∈ 2ω : Nx ⊆ B}| ≤ ω1. Then, since {(g, F) ∈ G × F(G) : g ∈ F} ⊆ G × F(G) is Borel, {F ∈ F(G) : F ⊆ B} = {F : ∀g g ∈ F → g ∈ B} is Π1

1, so is its intersection with {Nx : x ∈ 2ω}:

{Nx ∈ F(G) : Nx ⊆ B} ∈ Π1

1

({Nx : x ∈ 2ω} is Borel, since x → Nx was injective and Borel). Now taking its preimage under the mapping x → Nx {x ∈ 2ω : Nx ⊆ B} is also Π1

1, thus it is the union of ω1-many Borel sets. Hence if

|{x ∈ 2ω : Nx ⊆ B}| was larger than ω1, than there would be an uncountable Borel set D ⊆ {x ∈ 2ω : Nx ⊆ B} ⊆ 2ω, which contains a nonempty perfect set P ⊆ D.

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Cofinality

Proof.(Theorem) For cof(HN(G)) = c it is enough to show that for each Borel set B ∈ HN(G) |{x ∈ 2ω : Nx ⊆ B}| ≤ ω1. Then, since {(g, F) ∈ G × F(G) : g ∈ F} ⊆ G × F(G) is Borel, {F ∈ F(G) : F ⊆ B} = {F : ∀g g ∈ F → g ∈ B} is Π1

1, so is its intersection with {Nx : x ∈ 2ω}:

{Nx ∈ F(G) : Nx ⊆ B} ∈ Π1

1

({Nx : x ∈ 2ω} is Borel, since x → Nx was injective and Borel). Now taking its preimage under the mapping x → Nx {x ∈ 2ω : Nx ⊆ B} is also Π1

1, thus it is the union of ω1-many Borel sets. Hence if

|{x ∈ 2ω : Nx ⊆ B}| was larger than ω1, than there would be an uncountable Borel set D ⊆ {x ∈ 2ω : Nx ⊆ B} ⊆ 2ω, which contains a nonempty perfect set P ⊆ D. But

• x∈P

Nx is compact-catcher,

• x∈P

Nx ⊆ B is Haar null, a contradiction.

M´ ark Po´

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Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Question

Question

Can the results cov(HN(Zω)) = min(b, cov(N)) and non(HN(Zω)) = max(d, non(N)) be generalised to non-locally compact Polish groups with invariant metric? which are Abelian?

• r at least for Banach spaces?

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal

Introduction Uniformity and covering number Cofinality Questions

Thank you for your attention!

M´ ark Po´

• r

Cardinal invariants of the Haar-null ideal