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.
Categories: math.LO math.FA math.GN