On Filter $(\alpha)$-convergence and Exhaustiveness of Function Nets in Lattice Groups and Applications

Abstract

We consider (strong uniform) continuity of the limit of a pointwise convergent net of lattice group-valued functions, (strong weak) exhaustiveness and (strong) (α)-convergence with respect to a pair of filters, which in the setting of nets are more natural than the corresponding notions formulated with respect to a single filter. Some comparison results are given between such concepts, in connection with suitable properties of filters. Moreover, some modes of filter (strong uniform) continuity for lattice group-valued functions are investigated, giving some characterization. As an application, we get some Ascoli-type theorem in an abstract setting, extending earlier results to the context of filter (α)-convergence. Furthermore, we pose some open problems.

Abstract Theorems on Exchange of Limits and Preservation of (Semi)continuity of Functions and Measures in the Filter Convergence Setting

“…for each ∈ * . From (6), (7), and (H1) we get that for every ∈ Π there is ∈ S such that for each ∈ there exists * ∈ F with ( , , , ) ∈ 2 * + whenever ∈ * . Thus the family ( , ) , is weakly (UF)-backward exhaustive at .…”

“…A widely investigated problem in convergence theory and topology is to find necessary and/or sufficient conditions for continuity and/or semicontinuity of the limit of a pointwise convergent net of functions or measures. There have been many recent related studies in abstract structures, like topological spaces, lattice groups, metric semigroups, and cone metric spaces, with respect to usual, statistical, or filter/ideal convergence and associated with the notions of equicontinuity, filter exhaustiveness, and filter continuous convergence (see also [1][2][3][4][5][6][7][8][9]). The study of semicontinuous functions is associated with quasimetric spaces, that is, spaces endowed with an asymmetric distance function (for a related literature, see, e.g., [3][4][5][10][11][12][13]).…”