Linear $σ$-additivity and some applications

We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily sigma-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, Miller, Scheepers and Szeptycki [COC2], Gruenhage and Szeptycki [FUfin], Tsaban and Zdomskyy [SFT], and Tsaban [o-bdd, OPiT], and construct topological groups with very strong combinatorial properties.