The Nikodym property and filters on ω

For a free filter F on ω, let NF=ω∪\, where pF∈ω, be equipped with the following topology: every element of ω is isolated whereas all open neighborhoods of pF are of the form A∪\ for A∈ F. The aim of this paper is to study spaces of the form NF in the context of the Nikodym property of Boolean algebras. By AN we denote the class of all those ideals I on ω such that for the dual filter I^* the space NI^* carries a sequence μₙ n∈ω of finitely supported signed measures such that \|μₙ\|→∞ and μₙ(A)→ 0 for every clopen subset A⊆ NI^*. We prove that I if and only if there exists a density submeasure φ on ω such that φ(ω)=∞ and I is contained in the exhaustive ideal Exh(φ). Consequently, we get that if I(φ) for some density submeasure φ on ω such that φ(ω)=∞ and NI^* is homeomorphic to a subspace of the Stone space St(A) of a given Boolean algebra A, then A does not have the Nikodym property. We observe that each I is Kat{e}tov below the asymptotic density zero ideal Z, and prove that the class AN has a subset of size d which is dominating with respect to the Kat{e}tov order ≤K, but AN has no ≤K-maximal element. We show that for a density ideal I it holds I if and only if I is totally bounded if and only if the Boolean algebra P(ω)/I contains a countable splitting family.