doiSerbia

Ascoli-type theorems and ideal (α)-convergence

Athanassiadou E. (Department of Mathematics, University of Athens, Panepistimiopolis, Athens, Greece)
Boccuto A. (Department of Mathematics and Computer Sciences, University of Perugia, Perugia, Italy)
Dimitriou X. (Department of Mathematics and Computer Sciences, University of Perugia, Perugia, Italy)
Papanastassiou N. (Department of Mathematics and Computer Sciences, University of Perugia, Perugia, Italy)

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.

Keywords: Admissible ideal, ideal convergence, ideal (α)-convergence, (ideal) compactness, (ideal) exhaustiveness, uniform (ideal)exhaustiveness, weak (ideal) exhaustiveness, equicontinuity, Ascoli theorem, Heine theorem, Helly theorem, limit theorems, stronguniform convergence (on finite sets)