The Nikodym property and filters on \(\omega \)

Abstract

For a free filter F on \(\omega \), let \(N_F=\omega \cup \{p_F\}\), where \(p_F\not \in \omega \), be equipped with the following topology: every element of \(\omega \) is isolated whereas all open neighborhoods of \(p_F\) are of the form \(A\cup \{p_F\}\) for \(A\in F\). The aim of this paper is to study spaces of the form \(N_F\) in the context of the Nikodym property of Boolean algebras. By \(\mathcal{A}\mathcal{N}\) we denote the class of all those ideals \(\mathcal {I}\) on \(\omega \) such that for the dual filter \(\mathcal {I}^*\) the space \(N_{\mathcal {I}^*}\) carries a sequence \(\langle \mu _n:n\in \omega \rangle \) of finitely supported signed measures such that \(\Vert \mu _n\Vert \rightarrow \infty \) and \(\mu _n(A)\rightarrow 0\) for every clopen subset \(A\subseteq N_{\mathcal {I}^*}\). We prove that \(\mathcal {I}\in \mathcal{A}\mathcal{N}\) if and only if there exists a density submeasure \(\varphi \) on \(\omega \) such that \(\varphi (\omega )=\infty \) and \(\mathcal {I}\) is contained in the exhaustive ideal \(\text{ Exh }(\varphi )\). Consequently, we get that if \(\mathcal {I}\subseteq \text{ Exh }(\varphi )\) for some density submeasure \(\varphi \) on \(\omega \) such that \(\varphi (\omega )=\infty \) and \(N_{\mathcal {I}^*}\) is homeomorphic to a subspace of the Stone space \(St(\mathcal {A})\) of a given Boolean algebra \(\mathcal {A}\), then \(\mathcal {A}\) does not have the Nikodym property. We observe that each \(\mathcal {I}\in \mathcal{A}\mathcal{N}\) is Katětov below the asymptotic density zero ideal \(\mathcal {Z}\), and prove that the class \(\mathcal{A}\mathcal{N}\) has a subset of size \(\mathfrak {d}\) which is dominating with respect to the Katětov order \(\le _K\), but \(\mathcal{A}\mathcal{N}\) has no \(\le _K\)-maximal element. We show that, when \(\mathcal {I}\) is a density ideal, \(\mathcal {I}\not \in \mathcal{A}\mathcal{N}\) holds if and only if \(\mathcal {I}\) is totally bounded if and only if the Boolean algebra \(\mathcal {P}(\omega )/\mathcal {I}\) contains a countable splitting family. Our results shed some new light on differences between the Nikodym property and the Grothendieck property of Boolean algebras.