We prove that if ▫$\mathcal{A}$▫ is a ▫$\sigma$▫-complete Boolean algebra in a model ▫$V$▫ of set theory and ▫$\mathbb{P} \in V$▫ is a proper forcing with the Laver property preserving the ground ...model reals non-meager, then every pointwise convergent sequence of measures on ▫$\mathcal{A}$▫ is weakly convergent, i.e., ▫$\mathcal{A}$▫ 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 ▫$\partial$▫. We also obtain a new consistent situation in which there exists an Efimov space.
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