Measures on coallocation and normal lattices (Q1205610)

Summary: Let \({\mathcal L}_ 1\) and \({\mathcal L}_ 2\) be lattices of subsets of a nonempty set \(X\). Suppose \({\mathcal L}_ 2\) coallocates \({\mathcal L}_ 1\) and \({\mathcal L}_ 1\) is a subset of \({\mathcal L}_ 2\). We show that any \({\mathcal L}_ 1\)-regular finitely additive measure on the algebra generated by \({\mathcal L}_ 1\) can be uniquely extended to an \({\mathcal L}_ 2\)-regular measure on the algebra generated by \({\mathcal L}_ 2\). The case when \({\mathcal L}_ 1\) is not necessarily contained in \({\mathcal L}_ 2\), as well as the measure enlargement problem are considered. Furthermore, some discussions on normal lattices and separation of lattices are also given.