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.
Categories: math.LO math.FA math.GN