Measures on Bells space Piotr BorodulinNadzieja Finite and infinite - - PowerPoint PPT Presentation

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Measures on Bells space Piotr BorodulinNadzieja Finite and infinite - - PowerPoint PPT Presentation

Motivation Construction Measures on Bells space Piotr BorodulinNadzieja Finite and infinite sets, 2011, Budapest joint work with Mirna D zamonja Piotr BorodulinNadzieja Measures on Bells space Motivation Construction Measures


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Motivation Construction

Measures on Bell’s space

Piotr Borodulin–Nadzieja

Finite and infinite sets, 2011, Budapest

joint work with Mirna Dˇ zamonja

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Measures on Boolean algebras

We consider finitely–additive measures on Boolean algebras; A measure µ is strictly positive on A if µ(A) > 0 for each A ∈ A+. In this case we say that A supports µ; Every (finitely–additive) measure on A can be uniquely extended to a (σ–additive) Radon measure on Stone(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Measures on Boolean algebras

We consider finitely–additive measures on Boolean algebras; A measure µ is strictly positive on A if µ(A) > 0 for each A ∈ A+. In this case we say that A supports µ; Every (finitely–additive) measure on A can be uniquely extended to a (σ–additive) Radon measure on Stone(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Measures on Boolean algebras

We consider finitely–additive measures on Boolean algebras; A measure µ is strictly positive on A if µ(A) > 0 for each A ∈ A+. In this case we say that A supports µ; Every (finitely–additive) measure on A can be uniquely extended to a (σ–additive) Radon measure on Stone(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separable measures

Definition A measure µ on a Boolean algebra A is separable if there is a countable B ⊆ A such that inf{µ(A △ B): B ∈ B} = 0 for each A ∈ A

  • Equivalently. . .

A measure µ on A is separable iff the (pseudo–)metric space (A, dµ) is separable, dµ(A, B) = µ(A △ B) the space L1(µ) is separable.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separable measures

Definition A measure µ on a Boolean algebra A is separable if there is a countable B ⊆ A such that inf{µ(A △ B): B ∈ B} = 0 for each A ∈ A

  • Equivalently. . .

A measure µ on A is separable iff the (pseudo–)metric space (A, dµ) is separable, dµ(A, B) = µ(A △ B) the space L1(µ) is separable.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separable measures

Definition A measure µ on a Boolean algebra A is separable if there is a countable B ⊆ A such that inf{µ(A △ B): B ∈ B} = 0 for each A ∈ A

  • Equivalently. . .

A measure µ on A is separable iff the (pseudo–)metric space (A, dµ) is separable, dµ(A, B) = µ(A △ B) the space L1(µ) is separable.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Spaces with small measures

Problem How to characterize Boolean algebras carrying only separable measures? Theorem (Fremlin) Under MA(ω1) a Boolean algebra A carries a non–separable measure if and only if A contains an uncountable independent sequence. In ZFC: ? (This is one of the problems connected to the programme of the classification of finitely–additive measures.)

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Spaces with small measures

Problem How to characterize Boolean algebras carrying only separable measures? Theorem (Fremlin) Under MA(ω1) a Boolean algebra A carries a non–separable measure if and only if A contains an uncountable independent sequence. In ZFC: ? (This is one of the problems connected to the programme of the classification of finitely–additive measures.)

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Spaces with small measures

Problem How to characterize Boolean algebras carrying only separable measures? Theorem (Fremlin) Under MA(ω1) a Boolean algebra A carries a non–separable measure if and only if A contains an uncountable independent sequence. In ZFC: ? (This is one of the problems connected to the programme of the classification of finitely–additive measures.)

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Uniformly regular measures

Definition A measure µ on A is uniformly regular if there is a countable family D ⊆ A such that inf{µ(A \ D): D ∈ D, D ⊆ A} = 0 for every A ∈ A.

  • Equivalently. . .

A measure µ on A is uniformly regular if and only if µ is a Gδ point in the space of probability Radon measures on Stone(A) with weak∗ topology (Pol, 1982).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Uniformly regular measures

Definition A measure µ on A is uniformly regular if there is a countable family D ⊆ A such that inf{µ(A \ D): D ∈ D, D ⊆ A} = 0 for every A ∈ A.

  • Equivalently. . .

A measure µ on A is uniformly regular if and only if µ is a Gδ point in the space of probability Radon measures on Stone(A) with weak∗ topology (Pol, 1982).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Characterization of uniform regularity

Theorem (Dˇ zamonja, Pbn) If a Boolean algebra supports a non–atomic uniformly regular measure, then it is isomorphic to a subalgebra of the Jordan algebra containing a dense Cantor subalgebra.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability versus uniform regularity

The obvious connection: Remark Every uniformly regular measure is separable. Less obvious connection: Theorem (Plebanek, PBN) All Boolean algebras without a non–separable measure carry a uniformly regular measure.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability versus uniform regularity

The obvious connection: Remark Every uniformly regular measure is separable. Less obvious connection: Theorem (Plebanek, PBN) All Boolean algebras without a non–separable measure carry a uniformly regular measure.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability versus uniform regularity

The obvious connection: Remark Every uniformly regular measure is separable. Less obvious connection: Theorem (Plebanek, PBN) All Boolean algebras without a non–separable measure carry a uniformly regular measure.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability versus uniform regularity

The obvious connection: Remark Every uniformly regular measure is separable. Less obvious connection: Theorem (Plebanek, PBN) All Boolean algebras without a non–separable measure carry a uniformly regular measure.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

What about strictly positive measures?

Theorem (Plebanek, PBN) All Boolean algebras without a non–separable measure carry a uniformly regular measure. Question Can we prove an analogous theorem for strictly positive measures? I.e. is it true that all Boolean algebras supporting a measure either supports a uniformly regular measure or a non–separable one?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

What about strictly positive measures?

Theorem (Plebanek, PBN) All Boolean algebras without a non–separable measure carry a uniformly regular measure. Question Can we prove an analogous theorem for strictly positive measures? I.e. is it true that all Boolean algebras supporting a measure either supports a uniformly regular measure or a non–separable one?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Bell’s space

Theorem (Bell) There is a zero–dimensional compact separable space K without a countable π–base and which cannot be mapped continuously onto [0, 1]ω1. K is compact zerodimensional, so A = Clopen(K) is a Boolean algebra; K is separable, so A supports a measure; K has no countable π–base, so it does not support a uniformly regular measure; is every measure on K separable?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Bell’s space

Theorem (Bell) There is a zero–dimensional compact separable space K without a countable π–base and which cannot be mapped continuously onto [0, 1]ω1. K is compact zerodimensional, so A = Clopen(K) is a Boolean algebra; K is separable, so A supports a measure; K has no countable π–base, so it does not support a uniformly regular measure; is every measure on K separable?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Bell’s space

Theorem (Bell) There is a zero–dimensional compact separable space K without a countable π–base and which cannot be mapped continuously onto [0, 1]ω1. K is compact zerodimensional, so A = Clopen(K) is a Boolean algebra; K is separable, so A supports a measure; K has no countable π–base, so it does not support a uniformly regular measure; is every measure on K separable?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Bell’s space

Theorem (Bell) There is a zero–dimensional compact separable space K without a countable π–base and which cannot be mapped continuously onto [0, 1]ω1. K is compact zerodimensional, so A = Clopen(K) is a Boolean algebra; K is separable, so A supports a measure; K has no countable π–base, so it does not support a uniformly regular measure; is every measure on K separable?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Bell’s space

Theorem (Bell) There is a zero–dimensional compact separable space K without a countable π–base and which cannot be mapped continuously onto [0, 1]ω1. K is compact zerodimensional, so A = Clopen(K) is a Boolean algebra; K is separable, so A supports a measure; K has no countable π–base, so it does not support a uniformly regular measure; is every measure on K separable?

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Construction

For A ⊆ ω let A0 = {x ∈ 2ω : x(n) = 0 for each n ∈ A} For A ⊆ P(ω) let A0 = {A0 : A ∈ A}. Let K(A) be K(A) = Stone(algebra generated by A0) ⊆ P(2ω). Example: K(Fin) = 2ω.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Construction

For A ⊆ ω let A0 = {x ∈ 2ω : x(n) = 0 for each n ∈ A} For A ⊆ P(ω) let A0 = {A0 : A ∈ A}. Let K(A) be K(A) = Stone(algebra generated by A0) ⊆ P(2ω). Example: K(Fin) = 2ω.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Construction

For A ⊆ ω let A0 = {x ∈ 2ω : x(n) = 0 for each n ∈ A} For A ⊆ P(ω) let A0 = {A0 : A ∈ A}. Let K(A) be K(A) = Stone(algebra generated by A0) ⊆ P(2ω). Example: K(Fin) = 2ω.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Construction

For A ⊆ ω let A0 = {x ∈ 2ω : x(n) = 0 for each n ∈ A} For A ⊆ P(ω) let A0 = {A0 : A ∈ A}. Let K(A) be K(A) = Stone(algebra generated by A0) ⊆ P(2ω). Example: K(Fin) = 2ω.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability

Fact K(A) is separable for each A ⊆ P(ω). consider g ∈ 2ω; let Fg = {B ∈ alg(A0): g ∈ B}; it is a filter on alg(A0); let xg ∈ K(A) be any ultrafilter extending Fg; the set {xg : supp(g) is finite}. is dense in K(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability

Fact K(A) is separable for each A ⊆ P(ω). consider g ∈ 2ω; let Fg = {B ∈ alg(A0): g ∈ B}; it is a filter on alg(A0); let xg ∈ K(A) be any ultrafilter extending Fg; the set {xg : supp(g) is finite}. is dense in K(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability

Fact K(A) is separable for each A ⊆ P(ω). consider g ∈ 2ω; let Fg = {B ∈ alg(A0): g ∈ B}; it is a filter on alg(A0); let xg ∈ K(A) be any ultrafilter extending Fg; the set {xg : supp(g) is finite}. is dense in K(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability

Fact K(A) is separable for each A ⊆ P(ω). consider g ∈ 2ω; let Fg = {B ∈ alg(A0): g ∈ B}; it is a filter on alg(A0); let xg ∈ K(A) be any ultrafilter extending Fg; the set {xg : supp(g) is finite}. is dense in K(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability

Fact K(A) is separable for each A ⊆ P(ω). consider g ∈ 2ω; let Fg = {B ∈ alg(A0): g ∈ B}; it is a filter on alg(A0); let xg ∈ K(A) be any ultrafilter extending Fg; the set {xg : supp(g) is finite}. is dense in K(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Separability

Fact K(A) is separable for each A ⊆ P(ω). consider g ∈ 2ω; let Fg = {B ∈ alg(A0): g ∈ B}; it is a filter on alg(A0); let xg ∈ K(A) be any ultrafilter extending Fg; the set {xg : supp(g) is finite}. is dense in K(A).

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Cofinality and π-bases

Fact If A does not contain a countable cofinite subfamily, then K(A) does not have a countable π–base.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Independence

Remark If A contains an uncountable almost disjoint family, then alg(A0) contains an uncountable independent sequence. let {Aα : α < ω1} ⊆ A be almost disjoint (Aα ∩ Aβ is finite for each α = β); then {A0

α : α < ω1} is an uncountable independent sequence.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Independence

Remark If A contains an uncountable almost disjoint family, then alg(A0) contains an uncountable independent sequence. let {Aα : α < ω1} ⊆ A be almost disjoint (Aα ∩ Aβ is finite for each α = β); then {A0

α : α < ω1} is an uncountable independent sequence.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

Independence

Remark If A contains an uncountable almost disjoint family, then alg(A0) contains an uncountable independent sequence. let {Aα : α < ω1} ⊆ A be almost disjoint (Aα ∩ Aβ is finite for each α = β); then {A0

α : α < ω1} is an uncountable independent sequence.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

The space

Theorem (Dˇ zamonja, PBN) Let {Tα : α < ω1} ⊆ P(ω) be such that for each α < β < ω1 T0 = ∅, Tβ \ Tα is infinite, Tα \ Tβ is finite. Let T = {T : T =∗ Tα for some α < ω1}. Then K(T ) supports only separable measures. Consequently, the Boolean algebra alg(T 0) supports only separable measures but not a uniformly regular one.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

The space

Theorem (Dˇ zamonja, PBN) Let {Tα : α < ω1} ⊆ P(ω) be such that for each α < β < ω1 T0 = ∅, Tβ \ Tα is infinite, Tα \ Tβ is finite. Let T = {T : T =∗ Tα for some α < ω1}. Then K(T ) supports only separable measures. Consequently, the Boolean algebra alg(T 0) supports only separable measures but not a uniformly regular one.

Piotr Borodulin–Nadzieja Measures on Bell’s space

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Motivation Construction

The end

Thank you for your attention! This research was supported by the ESF Research Networking Programme INFTY.

Piotr Borodulin–Nadzieja Measures on Bell’s space