Outer measure analysis of topological lattice properties (Q1363343)

Summary: Let \(X\) be a set and \(\mathcal L\) a lattice of subsets of \(X\) such that \(\emptyset, X\in{\mathcal L}\). \({\mathcal A}({\mathcal L})\) is the algebra generated by \(\mathcal L\); \(M({\mathcal L})\) the set of nontrivial, finite, nonnegative, finitely additive measures on \({\mathcal A}({\mathcal L})\); and \(I({\mathcal L})\) those elements of \(M({\mathcal L})\) which just take the values zero and one. Various subsets of \(M({\mathcal L})\) and \(I({\mathcal L})\) are included which display smoothness and regularity properties. We consider several outer measures associated with elements of \(M({\mathcal L})\) and relate their behavior to smoothness and regularity conditions as well as to various lattice topological properties. In addition, their measurable sets are fully investigated. In the case of two lattices \({\mathcal L}_1\), \({\mathcal L}_2\) with \({\mathcal L}_1\subset{\mathcal L}_2\), we present consequences of separation properties between the pair of lattices in terms of these outer measures, and further demonstrate the extension of smoothness conditions on \({\mathcal L}_1\) to \({\mathcal L}_2\).