Minimally generated Boolean algebras and the Nikodym property

TL;DR

Under the Diamond Principle, the paper constructs minimally generated Boolean algebras with the Nikodym property and explores their measure-theoretic and topological implications, including the relationship with Efimov spaces.

Contribution

It provides the first known constructions of minimally generated Boolean algebras with and without the Nikodym property under $ riangle$, highlighting their topological and measure-theoretic characteristics.

Findings

Existence of minimally generated Boolean algebras with the Nikodym property under $ riangle$.

Existence of minimally generated Boolean algebras with Efimov Stone spaces lacking the Nikodym property.

Implications for measure theory and topology related to these algebraic structures.

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 $\Diamond$, 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 $\Diamond$ 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.