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

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.