Ascoli-type theorems and ideal (α)-convergence

Abstract

We investigate fundamental properties of I-exhaustiveness and I-convergence of real-valued function sequences, giving some characterizations. Furthermore, we establish new versions of Ascoli and Helly theorems, giving also applications to measure theory. Finally, we pose an open problem.

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

“…As an application of filter (α)-convergence and the related notions of filter exhaustiveness, we give an abstract Ascoli-type theorem, where the involved concepts of closedness and compactness are formulated in terms of suitable convergences of nets, without assuming necessarily the existence of a structure of Hausdorff topology. In this framework, we extend some results proved in Athanassiadou, Boccuto, Dimitriou and Papanastassiou (2012), , Gregoriades and Papanastassiou (2008) and Kelley (1955). Furthermore, we consider filter (weak) exhaustiveness and filter (α)-convergence with respect to a triple of filters, in relation with the problem of finding necessary/sufficient conditions for filter continuity of the limit of a function net.…”

“…As an application, we give a result, which extends some classical versions of Ascoli theorems related to exhaustiveness (see also Athanassiadou, Boccuto, Dimitriou and Papanastassiou (2012), Theorem 3.7, Caserta, Di Maio and Kočinac (2011), Theorem 4.7, Gregoriades and Papanastassiou (2008, Theorem 3.1.1) and some other Ascoli-type theorems, where the topological space on which the involved functions are defined is not necessarily metrizable (see for instance Gregoriades andPapanastassiou (2008), Theorems 3.2.19, 3.2.20, andKelley (1955), Theorems 7.6, 7.18 and7.21). The approach which we use in investigating the concepts of filter (α)-convergence and exhaustiveness with respect to a pair of filters, and consequently (strong uniform) continuity, is very natural to formulate some concepts of "filter closedness" and "filter compactness" in connection with convergences, which are not generated by any Hausdorff topology.…”

“…Moreover we give some comparison results between filter (α)-convergence with respect to a pair of filters and usual (α)-convergence with respect to one filter and between filter exhaustiveness with respect to a pair of filters and usual exhaustiveness with respect to a single filter, and illustrate some differences between strong filter (α)-convergence and filter (α)-convergence, and between (strong) exhaustiveness and (strong) weak exhaustiveness. Some related results have been proved in Caserta, Di Maio and Holá (2010), Gregoriades and Papanastassiou (2008) for the classical case, in Caserta, Di Maio and Kočinac (2011) for the statistical convergence and in Athanassiadou, Boccuto, Dimitriou and Papanastassiou (2012), Athanassiadou, Dimitriou, Papachristodoulos and Papanastassiou (2012), Dimitriou (2011, 2014), Boccuto, Dimitriou, Papanastassiou and Wilczyński (2011) in connection with filter/ideal continuous convergence for function sequences, where exhaustiveness is intended with respect to one filter. In this setting, we investigate also some property of the filter F st of all subsets of N having asymptotic density one.…”

Asymmetric Ascoli-type theorems and filter exhaustiveness

“…For example, in [16], some Ascoli-type theorems are proved, in connection with various kinds of convergence and exhaustiveness of function nets. In [1] and [4] these convergences, together with the concept of exhaustiveness, are considered in the context of filter/ideal convergence, and in this setting some Ascoli-type theorems in the metric space context are extended. In [15] some Ascoli-type theorem is proved, when the involved distance function is not required to be necessarily symmetric.…”

Ideal convergence and divergence of nets in (ℓ)-groups

“…In [10] we introduced the concept of ideal convergence in (ℓ)-groups, we dealt with the main basic properties and in [8], [10] we proved some versions of basic matrix theorems. Moreover, in [9] we obtained some limit theorems for ideal pointwise convergent measures taking values in an (ℓ)-group R. Some other applications of I-convergence in the real case (that is, when R = R) can be found in [1], [7], concerning weak compactness, Ascoli-type theorems, ideal exhaustiveness and uniform convergence on compact sets. In [7], [11] some results contained in [21] were extended to the ideal setting.…”