Convergence of measures in forcing extensions

TL;DR

This paper proves that under certain forcing conditions, pointwise convergent measures on Boolean algebras in generic extensions are weakly convergent, establishing the Vitali--Hahn--Saks property and constructing related examples.

Contribution

It demonstrates that forcing with proper Laver property preserves measure convergence properties on Boolean algebras, leading to new examples of algebras with the Vitali--Hahn--Saks property.

Findings

Pointwise convergence implies weak convergence in generic extensions.

Existence of Boolean algebras with the Vitali--Hahn--Saks property smaller than the dominating number.

New consistent example of an Efimov space.

Abstract

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 $\mathcal{A}$ in a $\mathbb{P}$-generic extension $V[G]$ is weakly convergent, i.e. $\mathcal{A}$ has the Vitali--Hahn--Saks property in $V[G]$. 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 $\mathfrak{d}$. We also obtain a new consistent situation in which there exists an Efimov space.