Characterizations of outer measures associated with lattice measures (Q1586746)

Summary: Let \(\nu\) be a finite countably subadditive outer measure defined on all subsets of a set \(X\) and take a collection \(\mathbb{C}\) of subsets of \(X\) containing \(X\) and \(\emptyset\). We derive an outer measure \(\rho\) using \(\nu\) on sets in \(\mathbb{C}\). By applying this general framework on two special cases in which \(\nu= \mu''\), one where \(\mu\in M_\sigma({\mathcal L})\) and the other where \(\mu\in M_\sigma({\mathcal L}_1)\), \({\mathcal L}_1\subseteq{\mathcal L}_2\) being lattices on a set \(X\), we obtain new characterizations of the outer measure \(\mu''\). These yield useful relationships between various set functions including \(\mu_i\), \(\mu_j\), \(\mu''\), and \(\mu'\).