Maximal linked systems and ultrafilters: main representations and topological properties

Questions connected with representation of the ultrafilter (UF) set for widely understood measurable space are investigated; this set is considered as a subspace of bitopological space of maximal linked systems (MLS) under equipment with topologies of Wallman and Stone types (measurable structure is defined as a π -system with “zero” and “unit”). Analogous representations connected with generalized variant of cohesion is considered also; in this variant, for corresponding set family, it is postulated the nonemptyness of intersection for finite subfamilies with power not exceeding given. Conditions of identification of UF and MLS (in the above-mentioned generalized sense) are investigated. Constructions reducing to bitopological spaces with points in the form of MLS and 𝑛-supercompactness property generalizing the “usual” supercompactness are considered. Finally, some characteristic properties of MLS and their corollaries connected with the MLS contraction to a smaller π -system are being studied. The case of algebras of sets is selected separately.

Phillips Lemma on effect algebras of sets

We prove the classical Phillips Lemma in the setting of measures defined on effect algebras of sets. This leads to several Vitali-Hahn-Saks-type results for these measures.

On Lower and Semi-Lower Density Operators

This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ-algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.

On sets not belonging to algebras

Let be a finite sequence of algebras of sets given on a set with more than pairwise disjoint sets not belonging to It was shown in [4] and [5] that in this case Let us consider, instead a finite sequence of algebras It turns out that if for each natural i ≤ l there exist no less than pairwise disjoint sets not belonging to then But if l ≥ 195 and if for each natural i ≤ l there exist no less than pairwise disjoint sets not belonging to then After consideration of finite sequences of algebras, it is natural to consider countable sequences of algebras. We obtained two essentially important theorems on a countable sequence of almost σ-algebras (the concept of almost σ-algebra was introduced in [4]).