On outer measures and semi-separation of lattices (Q1345782)

Summary: This present paper is concerned with set functions related to \(\{0, 1\}\) two-valued measures. These set functions are either outer measures or have many of the same characteristics. We investigate their properties and look at relations among them. We note in particular their association with the semi-separation of lattices. To be more specific, we define three set functions \(\mu'\), \(\mu''\), and \(\widetilde \mu\) related to \(\mu\in I(\mathbb{L})\), the \(\{0, 1\}\) two-valued set functions defined on the algebra generated by the lattice of sets \(\mathbb{L}\) such that \(\mu\) is a finitely additive monotone set function for which \(\mu(\emptyset)= 0\). We note relations among them and properties they possess. In particular, necessary and sufficient conditions are given for the semi-separation of lattices in terms of equality of set functions over a lattice of subsets. Finally, the notion of \(I\)-lattice is defined, we look at some of its properties and certain other side conditions assumed and end with an application involving semi-separation and \(I\)-lattices.