Minimally generated Boolean algebras and the Nikodym property

Abstract

A Boolean algebra A has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on A is uniformly bounded. Assuming the Diamond Principle ♦, we will construct an example of a minimally generated Boolean algebra 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.