Some topologies on the set of lattice regular measures

Abstract

We consider the general setting of A.D. Alexandroff, namely, an arbitrary set X and an arbitrary lattice of subsets of X, ℒ. 𝒜(ℒ) denotes the algebra of subsets of X generated by ℒ and MR(ℒ) the set of all lattice regular, (finitely additive) measures on 𝒜(ℒ).First, we investigate various topologies on MR(ℒ) and on various important subsets of MR(ℒ), compare those topologies, and consider questions of measure repleteness whenever it is appropriate.Then, we consider the weak topology on MR(ℒ), mainly when ℒ is δ and normal, which is the usual Alexandroff framework. This more general setting enables us to extend various results related to the special case of Tychonoff spaces, lattices of zero sets, and Baire measures, and to develop a systematic procedure for obtaining various topological measure theory results on specific subsets of MR(ℒ) in the weak topology with ℒ a particular topological lattice.