Convergence of measures after adding a real

TL;DR

This paper proves that adding certain types of reals via forcing destroys the Nikodym and Grothendieck properties of infinite Boolean algebras in the generic extension, highlighting the impact of forcing on measure-theoretic properties.

Contribution

It establishes that adding Cohen, unsplit, or random reals, as well as dominating reals, via forcing eliminates the Nikodym and Grothendieck properties in Boolean algebras.

Findings

Adding Cohen, unsplit, or random reals destroys these properties.

Adding a dominating real also destroys the Nikodym property.

The results hold in any generic extension after forcing.

Abstract

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.