The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals

TL;DR

This paper investigates the Nikodym and Grothendieck properties in Boolean algebras related to ideals, establishing conditions for their equivalence and constructing diverse examples with specific properties.

Contribution

It provides new insights into the relationship between Nikodym and Grothendieck properties, constructs numerous examples of Boolean algebras with these properties, and extends existing results on P-ideals.

Findings

If the Boolean algebra generated by an ideal lacks the Nikodym property, it also lacks the Grothendieck property.

Constructs of many non-isomorphic Boolean algebras with Nikodym but not Grothendieck properties.

Characterizes when an analytic P-ideal is totally bounded and relates it to other properties.

Abstract

For an ideal $\mathcal{I}$ in a $\sigma$-complete Boolean algebra $\mathcal{A}$, we show that if the Boolean algebra $\mathcal{A}\langle\mathcal{I}\rangle$ generated by $\mathcal{I}$ does not have the Nikodym property, then it does not have the Grothendieck property either. The converse however does not hold -- we construct a family of $\mathfrak{c}$ many pairwise non-isomorphic Boolean subalgebras of the power set $\wp(\omega)$ of the form $\wp(\omega)\langle\mathcal{I}\rangle$ which, when thought of as subsets of the Cantor space $2^\omega$, belong to the Borel class $\mathbb{F}_{\sigma\delta}$ and have the Nikodym property but not the Grothendieck property, and a family of $2^\mathfrak{c}$ many pairwise non-isomorphic non-analytic Boolean algebras of the form $\wp(\omega)\langle\mathcal{I}\rangle$ with the Nikodym property but without the Grothendieck property. Extending a result of Hern\'{a}ndez-Hern\'{a}ndez and Hru\v{s}\'{a}k, we show that for an analytic P-ideal $\mathcal{I}$ on $\omega$ the following are equivalent: 1) $\mathcal{I}$ is totally bounded, 2) $\mathcal{I}$ has the Local-to-Global Boundedness Property for submeasures, 3) $\wp(\omega)/\mathcal{I}$ contains a countable splitting family, 4) $\mbox{conv}\le_K\mathcal{I}$. Moreover, proving a conjecture of Drewnowski, Florencio, and Pa\'ul, we present examples of analytic P-ideals on $\omega$ with the Nikodym property but without the Local-to-Global Boundedness Property for submeasures (and so not totally bounded). Exploiting a construction of Alon, Drewnowski, and {\L}uczak, we also describe a family of $\mathfrak{c}$ many pairwise non-isomorphic ideals on $\omega$, induced by sequences of Kneser hypergraphs, which all have the Nikodym property but not the Nested Partition Property -- this answers a question of Stuart. Finally, Tukey reducibility of a class of ideals without the Nikodym property is studied.