First Alexandroff decomposition theorem for topological lattice group valued measures (Q1583846)

Let \((X,{\mathcal F})\) be an Alexandroff space, i.e., a system of subsets of a set \(X\) with respect to finite unions and countable intersections with \(\emptyset,X\in{\mathcal F}\). Alexandroff proved that any inner regular bounded measure \(\mu: a({\mathcal F})\to\mathbb{R}\) defined on the algebra generated by \({\mathcal F}\) has a decomposition \(\mu= \xi+\eta\) where \(\xi\) is an \({\mathcal F}\)-smooth measure and any ``compact'' set of \(a({\mathcal F})\) is an \(\eta\)-null set. The authors generalize this result for measures with values in a Dedekind complete lattice group endowed with an order continuous locally solid group topology having a neighbourhood base consisting of sublattices.