Minimally generated Boolean algebras and the Nikodym property

Abstract

A Boolean algebra $\mathcal{A}$ has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on $\mathcal{A}$ is uniformly bounded. Assuming the Diamond Principle $◊$, we will construct an example of a minimally generated Boolean algebra $\mathcal{A}$ with the Nikodym property. The Stone space of such an algebra must necessarily be an Efimov space. The converse is, however, not true - again under $◊$ we will provide an example of a minimally generated Boolean algebra whose Stone space is Efimov but which does not have the Nikodym property. The results have interesting measure-theoretic and topological consequences.

23 pages, comments are welcome