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 $\mathcal{AN}$ 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 $\langle\mu_n\colon n\in\omega\rangle$ of finitely supported signed measures such that $\|\mu_n\|\rightarrow\infty$ and $\mu_n(A)\rightarrow 0$ for every clopen subset $A\subseteq N_{\mathcal{I}^*}$. We prove that $\mathcal{I}\in\mathcal{AN}$ 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 $\mbox{Exh}(\varphi)$. Consequently, we get that if $\mathcal{I}\subseteq\mbox{Exh}(\varphi)$ 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 $\mathcal{I}\in\mathcal{AN}$ is Kat\v{e}tov below the asymptotic density zero ideal $\mathcal{Z}$, and prove that the class $\mathcal{AN}$ has a subset of size $\mathfrak{d}$ which is dominating with respect to the Kat\v{e}tov order $\leq_K$, but $\mathcal{AN}$ has no $\leq_K$-maximal element. We show that for a density ideal $\mathcal{I}$ it holds $\mathcal{I}\not\in\mathcal{AN}$ 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.