Generalization of a theorem of Alexandroff (Q1842059)

The author shows that under a certain regularity condition a finitely additive set function from a ring of sets \(\mathcal R\) into an Abelian Hausdorff topological group \(G\) is countably additive. He also studies the question whether regularity of an exhausting additive set function \(\mu: {\mathcal R}\to G\) with range in a sequentially complete subset of \(G\) is inherited by its unique extension to the \(\sigma\)-ring generated by \(\mathcal R\).