Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

Abstract

We prove some equivalence results between limit theorems for sequences of ( )-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same (O)-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems.

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.…”

Asymmetric Ascoli-type theorems and filter exhaustiveness

“…Asymmetric distance has different applications in several branches of Mathematics (see for instance [15] and the bibliography therein), and is connected also with the study of various semicontinuity properties and related topics (see also, for example, [2]). Filter convergence and filter exhaustiveness have many developments in the very recent literature, for instance in limit and decomposition theorems for measures (see also [5,6,7,8,9,10,11,13,14]). A comprehensive survey about these topics can be found in [12].…”

Addendum to: “Some New Types of Filter Limit Theorems for Topological Group-Valued Measures”

“…To prove equivalence results between filter limit theorems, we apply some results about existence of suitable σ-additive restrictions of (s)-bounded measures, like in [41,42], and without considering the Stone Isomorphism technique, though it is possible to get Stone-type extensions also for (s)-bounded topological group-valued measures (see also [49,50]). In the lattice group setting (see [15]) it is dealt with the Stone extensions, since the nature of the convergence in such groups is not necessarily topological, and hence it is not advisable to argue with σ-additive restrictions. However, the Drewnowski-type technique here used is in general easier to handle than the Stone Isomorphism technique.…”