Filter exhaustiveness and filter limit theorems for $k$-triangular lattice group-valued set functions

Abstract

We give some limit theorems for sequences of lattice group-valued $k$-triangular set functions, in the setting of filter convergence, and some results about their equivalence. We use the tool of filter exhaustiveness to get uniform $(s)$-boundedness, uniform continuity and uniform regularity of a suitable subsequence of the given sequence, whose indexes belong to the involved filter. Furthermore we pose some open problems.