Convergence of measures after adding a real

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.