Lattice separation, coseparation and regular measures (Q1922858)

Summary: Let \(X\) be an arbitrary non-empty set, and let \(\mathcal L\), \({\mathcal L}_1\), \({\mathcal L}_2\) be lattices of subsets of \(X\) containing \(\emptyset\) and \(X\). \({\mathcal A}({\mathcal L})\) denotes the algebra generated by \(\mathcal L\) and \(M({\mathcal L})\) the set of finite, non-trivial, non-negative finitely additive measures on \({\mathcal A}({\mathcal L})\). \(I({\mathcal L})\) denotes those elements of \(M({\mathcal L})\) which assume only the values zero and one. In terms of a \(\mu\in M({\mathcal L})\) or \(I({\mathcal L})\), various outer measures are introduced. Their properties are investigated. The interplay of measurability, smoothness of \(\mu\), regularity of \(\mu\) and lattice topological properties on these outer measures is also investigated. Finally, applications of these outer measures to separation type properties between pairs of lattices \({\mathcal L}_1\), \({\mathcal L}_2\), where \({\mathcal L}_1\subset{\mathcal L}_2\), are developed. In terms of measures from \(I({\mathcal L})\), necessary and sufficient conditions are established for \({\mathcal L}_1\) to semi-separate \({\mathcal L}_2\), for \({\mathcal L}_1\) to separate \({\mathcal L}_2\), and finally for \({\mathcal L}_1\) to coseparate \({\mathcal L}_2\).