Applications of outer measures to separation properties of lattices and regular or \(\sigma\)-smooth measures (Q1912318)

Summary: Associated with a 0-1 measure \(\mu\in I({\mathcal L})\), where \(\mathcal L\) is a lattice of subsets of \(X\), are outer measures \(\mu'\) and \(\widetilde\mu\); associated with a \(\sigma\)-smooth 0-1 measure \(\mu\in I_\sigma({\mathcal L})\) is an outer measure \(\mu''\) or with \(\mu\in I_\sigma({\mathcal L}')\), \({\mathcal L}'\) being the complementary lattice, another outer measure \(\overset\approx \mu\). These outer measures and their associated measurable sets are used to establish separation properties on \(\mathcal L\) and regularity and \(\sigma\)-smoothness of \(\mu\). Separation properties between two lattices \({\mathcal L}_1\) and \({\mathcal L}_2\), \({\mathcal L}_1\subseteq {\mathcal L}_2\), are similarly investigated. Notions of strongly \(\sigma\)-smooth and slightly regular measures are also used.