Uniform (s)-boundedness and regularity for (l)-group-valued measures

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

“…A historical comprehensive survey can be found in [16]. Among the most important developments existing in the literature about these subjects, see for instance [2,3,30,31,32,33,38,45], and in particular, concerning the setting of lattice group-valued measures, we quote [6,9,10,12,13]. In [14,24] some Dieudonnétype theorems were proved for lattice group-valued finitely additive regular measures in the context of filter convergence, while some versions of uniform boundedness theorems in this setting are proved in [11,25].…”

“…(seeBoccuto and Dimitriou [22, Lemma 3.4]) Let L ⊂ P(G) be an algebra, G and H be two sublattices of L, such that the complement of every element of H belongs to G, m j : L → R, j ∈ N, be a sequence of k-triangular and G-uniformly (s)-bounded set functions. Fix W ∈ H and a decreasing sequence (H n ) n in G, with W ⊂ H n for each n ∈ m j (A) = 0 for every j ∈ N(13) with respect to a single (D)-sequence (a t,l ) t,l respect to (a t,l ) t,l . The next step is to prove a Dieudonné-type theorem for k-triangular lattice groupvalued set functions, which extends[10, Lemma 3.2].Theorem 3.3.…”