Remarks on \(\mu''\)-measurable sets: regularity, \(\sigma\)-smoothness, and measurability (Q1301932)

Summary: Let \({\mathbf X}\) be an arbitrary nonempty set and \({\mathcal L}\) a lattice of subsets of \({\mathbf X}\) such that \(\phi, {\mathbf X}\in{\mathcal L}\). \({\mathcal A}({\mathcal L})\) is the algebra generated by \({\mathcal L}\) and \({\mathcal M}({\mathcal L})\) denotes those nonnegative, finite, finitely additive measures \(\mu\) on \({\mathcal A}({\mathcal L})\). \(I({\mathcal L})\) denotes the subset of \({\mathcal M}({\mathcal L})\) of nontrivial zero-one valued measures. Associated with \(\mu\in I({\mathcal L})\) (or \(I_\sigma({\mathcal L})\)) are the outer measures \(\mu'\) and \(\mu''\) considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for \({\mathcal S}_{\mu''}\), the set of \(\mu''\)-measurable sets. Notions of strongly \(\sigma\)-smooth and vaguely regular measures are also discussed. Relationships between regularity, \(\sigma\)-smoothness and measurability are investigated for zero-one valued measures and certain results are extended to the case of a pair of lattices \({\mathcal L}_1\), \({\mathcal L}_2\) where \({\mathcal L}_1\subset{\mathcal L}_2\).