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 $\Diamond$, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2021
- DOI:
- 10.48550/arXiv.2105.12467
- arXiv:
- arXiv:2105.12467
- Bibcode:
- 2021arXiv210512467S
- Keywords:
-
- Mathematics - Functional Analysis;
- Mathematics - General Topology;
- Mathematics - Logic;
- Primary: 06E15;
- 28A33;
- 03E75. Secondary: 28E15;
- 03E35
- E-Print:
- 23 pages, comments are welcome