Ideal Limit Theorems and Their Equivalence in $(ell)$-Group Setting

We prove some equivalence results between limit theorems for sequences of $(ell)$-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same $(O)$-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems.