Schur-type theorems for k-triangular lattice group-valued set functions with respect to filter convergence

Abstract

We prove some Schur and limit theorems for lattice group-valued -triangular set functions with respect to filter convergence, by means of sliding hump-type techniques. As consequences, we deduce some Vitali-Hahn-Saks and Nikodýmtype theorems. Furthermore, we pose some open problems.

Dieudonné-type theorems for lattice group-valued k-triangular set functions

“…a t,l + k w j2 ≤ 3 k w j2 ≤ (k j 2 + 1)w j2 , which contradicts (20). Let j 3 > n 2 be an integer such that {m n (H j2 ) : n ∈ N} ≤ w j3 .…”

“…which contradicts (20). Proceeding by induction, it is possible to construct two strictly increasing sequences (j h ) h , (n h ) h , such that n h > j h ≥ h for every h ∈ N, and…”

“…Some examples of non-monotone 1-triangular set functions are the Saeki measuroids (see [43]). In [17,19,20,21,22,23] some limit theorems were proved for lattice group-valued k-subadditive capacities and k-triangular set functions.…”

Equivalence between limit theorems for lattice group-valued k-triangular set functions

“…In this paper we deal with k-triangular lattice group-valued set functions. We continue the investigation started in [7,10,11], where some limit theorems were proved for k-subadditive, positive and monotone set functions. In particular we treat (s)-boundedness and continuity from above at ∅.…”