Convergence of measures in forcing extensions

Sobota, Damian; Zdomskyy, Lyubomyr

doi:10.1007/s11856-019-1872-8

Cited by 14 publications

(14 citation statements)

References 33 publications

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“…Note that several consistent examples of Boolean algebras with the Nikodym or Grothendieck property of size strictly less than c have been obtained, e.g. by Brech [4], Sobota [41], Sobota and Zdomskyy [42]; those algebras cannot of course contain independent families of size c and thus are far from being σ-complete.…”

Section: Consequencesmentioning

confidence: 99%

“…The property may look at first sight a bit strange, but by the virtue of the Riesz representation theorem for the duals of Banach C(K)-spaces it is closely related to the well-known Uniform Boundedness Principle (or, the Banach-Steinhaus theorem) of Banach spaces; it may also be expressed in terms of convergence of measures on totally disconnected compact spaces, see [42,Prop. 2.4].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Consequencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimally generated Boolean algebras and the Nikodym property

Sobota¹,

Zdomskyy²

2021

Preprint

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“…The Stone spaces of such Boolean algebras of 'old' subsets of ω were considered e.g. in [SZ19] and in the context of random forcing in [DF07]. The measures on P(ω) were studied e.g.…”

Section: Homomorphisms and Names For Ultrafiltersmentioning

confidence: 99%

On measures induced by forcing names for ultrafilters

Borodulin–Nadzieja¹,

Cegiełka²

2021

Preprint

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We study the interplay between properties of measures on a Boolean algebra A and forcing names for ultrafilters on A. We show that several well known measure theoretic properties of Boolean algebras (such as supporting a strictly positive measure or carrying only separable measures) have quite natural characterizations in the forcing language. We show some applications of this approach. In particular, we reprove a theorem of Kunen saying that in the classical random model there are no towers of height ω2.

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“…Let us note that consistently there exist Efimov spaces with the Grothendieck property and hence without the fsJNP, see e.g. Talagrand [79], Brech [19], or Sobota and Zdomskyy [77].…”

Section: Systems Of Simple Extensions and The Fsjnpmentioning

confidence: 99%

“…In fact, Cembranos [20] proved that a space C(K) is Grothendieck if and only if it does not contain any complemented copy of c 0 . For more information on Grothendieck C(K)-spaces we refer the reader to the papers of Haydon [44], Koszmider [52], or Sobota and Zdomskyy [77].…”

Section: Introductionmentioning

confidence: 99%

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

Ka̧kol¹,

Sobota²,

Zdomskyy³

2020

Preprint

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The celebrated Josefson-Nissenzweig theorem implies that for a Banach space C(K) of continuous real-valued functions on an infinite compact space K there exists a sequence of Radon measures µ n : n ∈ ω on K which is weakly* convergent to the zero measure on K and such that µ n = 1 for every n ∈ ω. We call such a sequence of measures a Josefson-Nissenzweig sequence. In this paper we study the situation when the space K admits a Josefson-Nissenzweig sequence of measures such that its every element has finite support. We prove among the others that K admits such a Josefson-Nissenzweig sequence if and only if C(K) does not have the Grothendieck property restricted to functionals from the space ℓ 1 (K). We also investigate miscellaneous analytic and topological properties of finitely supported Josefson-Nissenzweig sequences on general Tychonoff spaces.We prove that various properties of compact spaces guarantee the existence of finitely supported Josefson-Nissenzweig sequences. One such property is, e.g., that a compact space can be represented as the limit of an inverse system of compact spaces based on simple extensions. An immediate consequence of this result is that many classical consistent examples of Efimov spaces, i.e. spaces being counterexamples to the famous Efimov problem, admit such sequences of measures.Similarly, we show that if K and L are infinite compact spaces, then their product K × L always admits a finitely supported Josefson-Nissenzweig sequence. As a corollary we obtain a constructive proof that the space C p (K × L) contains a complemented copy of the space c 0 endowed with the pointwise topology-this generalizes results of Cembranos and Freniche.Finally, we provide a direct proof of the Josefson-Nissenzweig theorem for the case of Banach spaces C(K).

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Convergence of measures in forcing extensions

Cited by 14 publications

References 33 publications

Minimally generated Boolean algebras and the Nikodym property

Minimally generated Boolean algebras and the Nikodym property

On measures induced by forcing names for ultrafilters

The Josefson--Nissenzweig theorem, Grothendieck property, and finitely supported measures on compact spaces

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