Some versions of limit and Dieudonné-type theorems with respect to filter convergence for (ℓ)-group-valued measures

Abstract

Some limit and Dieudonné-type theorems in the setting of (ℓ)-groups with respect to filter convergence are proved, extending earlier results.

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

“…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.…”

Ideal Exhaustiveness, Weak Convergence and Weak Compactness in Banach Spaces

“…in the literature particularly in convergence of functions (see [4,8,13,28,31]) and convergence of measures and integrals (see [7,8,9,10,11,12]). Note that, in general, ideal convergence is strictly weaker than ordinary convergence (see [30,31]).…”

“…Under suitable hypotheses it is possible to prove some versions of limit theorems even if we require the simple ideal setwise convergence of the involved measure sequence (see e.g. [9,10,11,12]). Within this framework new results are established and some classical results in the literature are reproved under strictly weaker hypotheses.…”

“…The main tools are the notions of ideal (α)-convergence and ideal exhaustiveness previously introduced and studied for function sequences in [13] (see also [27,29]). Recently, some other versions of limit theorems for measures were proved in [3,9,10,11,12] with respect to this kind of convergence.…”