Convergence of measures after adding a real

Damian Sobota, Lyubomyr Zdomskyy

Published 2021-10-09, updated 2022-09-08Version 2

We prove that if $\mathcal{A}$ is an infinite Boolean algebra in the ground model $V$ and $\mathbb{P}$ is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any $\mathbb{P}$-generic extension $V[G]$, $\mathcal{A}$ has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.