Convergence of measures in forcing extensions

Abstract

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 situation in which there exists an Efimov space.