Filter Convergence and Decompositions for Vector Lattice-Valued Measures

Abstract

Filter convergence of vector lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the σ-additive case is studied, without particular assumptions on the filter; later the finitely additive case is faced, first assuming uniform sboundedness (without restrictions on the filter), then relaxing this condition but imposing stronger properties on the filter. In order to obtain the last results, a Schur-type convergence theorem, obtained in [8], is used.2010 AMS Mathematics Subject Classification: 28B15, 28B05, 06A06, 54F05.

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

“…x ∈ X} (see also , Boccuto, Dimitriou and Papanastassiou (2012a-b), Candeloro and Sambucini (2015)). …”

Limit Theorems for k-Subadditive Lattice Group-Valued Capacities in The Filter Convergence Setting

“…Now, arbitrarily pick A ∈ Σ. Since Σ = T, there exists a (D)-sequence (a t,l ) t,l satisfying (14). Since R is super Dedekind complete and weakly σ-distributive, by Theorem 2.6, (2), in correspondence with (a t,l ) t,l , we find a sequence (ϕ n ) n in N N and an (O)-sequence (b n ) n such that, for each n ∈ N, ∞ t=1 a t,ϕ n (t) ≤ b n , and thus, there exist D *…”

“…A comprehensive survey can be found in [6]. There are also several versions of theorems of this kind for finitely or countably additive measures in the setting of filter convergence (for a related literature, see also [5][6][7][8][9], [14]). In [7], some Brooks-Jewett, Nikodým and Vitali-Hahn-Saks--type theorems are proved for positive and finitely additive lattice group-valued measures with respect to filter convergence, in which the pointwise convergence of the involved measures is required, not necessarily with respect to a single order sequence or regulator.…”

“…Moreover, for technical reasons, in our context we deal with (D)-convergence, because it is possible to use the Fremlin's lemma which allows to replace a series of (D)-sequences with a single regulator. In [5], [6], [8], [9], [14], some limit theorems are proved for finitely additive and not necessarily positive lattice group-valued measures, for diagonal filters which satisfy some additional properties. Finally, we pose some open problems.…”

Quantitative Approximation Properties for Iterates of Moment Operator

“…These operators have many applications to various fields of pure and applied mathematics: from Calculus of Variations (see [2,28]) to fractional calculus [6,20] and stochastic processes [14,18]. In the frame of approximation theory by sequences or nets of moment operators we quote the papers [3,13,27,29,30].…”