Limit Theorems in (l)-Groups with Respect to (D)-Convergence

Abstract

Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for \((l)\)-group-valued measures are given in the context of \((D)\)-convergence. We consider both the \(\sigma\)-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measures is intended relatively to a common \((D)\)-sequence. Among the tools, we use the Fremlin lemma, which allows us to replace a countable family of \((D)\)-sequence with one regulator, and the Maeda-Ogasawara-Vulikh representation theorem for Archimedean lattice groups.