Independent families in Boolean algebras with some separation properties

Koszmider, Piotr; Shelah, Saharon

doi:10.1007/s00012-013-0227-2

Cited by 16 publications

(13 citation statements)

References 11 publications

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“…The last two properties are the same and they imply the well known Vitaly-Hans-Saks property, which is stronger than the Nikodým property. Koszmider and Shelah have shown in [13] that if an infinite algebra A has the so-called Weak Subsequential Separation Property then the cardinal of A is greater than or equal to the continuum c. Since all algebras considered here have the Weak Subsequential Separation Property, it arises the natural question whether there exist algebras with the Nikodým property with cardinality less than c. This question has been solved positively by Sobota in [25]. On the other hand, in [14,Theorem 1] it was proved that the algebra J (K) of Jordan measurable subsets of the compact interval [27,Theorem 4].…”

Section: Preliminariesmentioning

confidence: 94%

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

Ferrando

López-Alfonso

Pellicer

2019

Filomat

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If R is a ring of subsets of a set Ω and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ∆ of R is called a Nikodým set for ba (R) if each set {µ α : α ∈ Λ} in ba (R) which is pointwise bounded on ∆ is norm-bounded in ba (R). If the whole ring R is a Nikodým set, R is said to have property (N), which means that R satisfies the Nikodým-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck's property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikodým sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N) ⇔ (wN) holds.

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Section: Preliminariesmentioning

confidence: 94%

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

Ferrando

López-Alfonso

Pellicer

2019

Filomat

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“…It is an easy fact that every σ-complete Boolean algebra contains an independent family of size c. Haydon [24] provided an argument (due to Argyros) proving that Boolean algebras with the Subsequential Completeness Property (SCP), being a weakening of the σ-completeness, always contain uncountable independent families (in fact, of size at least p). This result was later generalized by Koszmider and Shelah [30] who showed that Boolean algebras having a very weak form of completeness called the Weak Subsequential Separation Property (WSSP) must necessarily contain independent families of size c. Both the σ-completeness and Haydon's SCP imply the WSSP as well as the Grothendieck property and the Nikodym property, however note that there exists a Boolean algebra with the WSSP, but with neither the Grothendieck property nor the Nikodym property.…”

Section: Consequencesmentioning

confidence: 96%

Minimally generated Boolean algebras and the Nikodym property

Sobota¹,

Zdomskyy²

2021

Preprint

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A Boolean algebra A has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on A is uniformly bounded. Assuming the Diamond Principle ♦, we will construct an example of a minimally generated Boolean algebra A with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true-again under ♦ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The results have interesting measure-theoretic and topological consequences.

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“…if a sequence of non-negative measures on A is weakly* convergent, then it is weakly convergent. Such a variant of the Grothendieck property was introduced in Koszmider and Shelah [KS13], where it was called the positive Grothendieck property.…”

Section: By Taking Intersections We May Assume Thatmentioning

confidence: 99%

On sequences of homomorphisms into measure algebras and the Efimov Problem

Borodulin–Nadzieja¹,

Sobota²

2021

Preprint

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For given Boolean algebras A and B we endow the space H(A, B) of all Boolean homomorphisms from A to B with various topologies and study convergence properties of sequences in H(A, B). We are in particular interested in the situation when B is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on H(A, B) for a Boolean algebra B carrying a strictly positive measure and convergence properties of sequences of measures on A.

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Independent families in Boolean algebras with some separation properties

Cited by 16 publications

References 11 publications

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

Minimally generated Boolean algebras and the Nikodym property

On sequences of homomorphisms into measure algebras and the Efimov Problem

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