Outer measures, measurability, and lattice regular measures (Q1912332)

Summary: Let \(X\) be an arbitrary non-empty set and \(\mathcal L\) a lattice of subsets of \(X\) such that \(\emptyset, X\in {\mathcal L}\). \({\mathcal A}({\mathcal L})\) denotes the algebra generated by \(\mathcal L\) and \(I({\mathcal L})\) those zero-one valued, non-trivial, finitely additive measures on \({\mathcal A}({\mathcal L})\). \(I_\sigma({\mathcal L})\) denotes those elements of \(I({\mathcal L})\) that are \(\sigma\)-smooth on \(\mathcal L\), and \(I_R({\mathcal L})\) denotes those elements of \(I({\mathcal L})\) that are \({\mathcal L}\)-regular while \(I^\sigma_R({\mathcal L})= I_R({\mathcal L})\cap I_\sigma({\mathcal L})\). In terms of those and other subsets of \(I({\mathcal L})\), various outer measures are introduced, and their properties are investigated. Also, the interplay between the measurable sets associated with these outer measures, regularity properties of the measures, smoothness properties of the measures, and lattice topological properties are thoroughly investigated -- yielding new results for regularity or weak regularity of these measures, as well as domination on a lattice of a suitably given measure by a regular one. Finally, elements of \(I_\sigma({\mathcal L})\) are fully characterized in terms of induced measures on a certain generalized Wallman space.