# Convergence of measures in forcing extensions

@article{Sobota2019ConvergenceOM,
title={Convergence of measures in forcing extensions},
author={Damian Sobota and Lyubomyr Zdomskyy},
journal={Israel Journal of Mathematics},
year={2019},
pages={1-29}
}
• Published 27 May 2019
• Mathematics
• Israel Journal of Mathematics
We prove that if A is a σ-complete Boolean algebra in a model V of set theory and ℙ ∈ V is a proper forcing with the Laver property preserving the ground model reals non-meager, then every pointwise convergent sequence of measures on A is weakly convergent, i.e., A has the Vitali- Hahn-Saks property. This yields a consistent example of a whole class of infinite Boolean algebras with this property and of cardinality strictly smaller than the dominating number ∂. We also obtain a new consistent…
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## References

SHOWING 1-10 OF 34 REFERENCES
A non-reflexive Grothendieck space that does not containl∞
A compact spaceS is constructed such that, in the dual Banach spaceC(S)*, every weak* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol∞. The construction
On the density of Banach spaces C(K) with the Grothendieck property
Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the
Maximal almost disjoint families of functions
We study maximal almost disjoint (MAD) families of functions in ω that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of
Relaciones entre propiedades de supremo y propiedades de interpolación en álgebras de Boole
In this paper the relations between some properties of interpolation and some properties of supremum, on Boolean algebras with the countable chain condition, are studied. It is also proved that a
On sequences without weak* convergent convex block subsequences
• Mathematics
• 1987
Let X be a Banach space such that X* contains a bounded sequence without a weak* convergent convex block subsequence. Then, subject to Martin's Axiom and the negation of the Continuum Hypothesis, X
Un Nouveau ℓ (K) Qui Possede la Propriete de Grothendieck
Using the continuum hypothesis, we construct a compact spaceK such that ℓ(K) possesses the Grothendieck property, but such that the unit ball of ℓ(K)′ does not containβN, and hence, in particular,
COMPACT SETS WITHOUT CONVERGING SEQUENCES IN THE RANDOM REAL MODEL
• Mathematics
• 2000
It is shown that in the model obtained by adding any number of random reals to a model of CH, there is a compact Hausdor space of weight !1 which contains no non-trivial converging sequences. It is