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Journal of Function Spaces
Volume 2016 (2016), Article ID 4237423, 10 pages
http://dx.doi.org/10.1155/2016/4237423
Research Article

Abstract Theorems on Exchange of Limits and Preservation of (Semi)continuity of Functions and Measures in the Filter Convergence Setting

1Dipartimento di Matematica e Informatica, Via Vanvitelli 1, 06123 Perugia, Italy
2Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece

Received 28 April 2016; Accepted 7 June 2016

Academic Editor: Ioan Rasa

Copyright © 2016 Antonio Boccuto and Xenofon Dimitriou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give necessary and sufficient conditions for exchange of limits of double-indexed families, taking values in sets endowed with an abstract structure of convergence, and for preservation of continuity or semicontinuity of the limit family, with respect to filter convergence. As a consequence, we give some filter limit theorems and some characterization of continuity and semicontinuity of the limit of a pointwise convergent family of set functions. Furthermore, we pose some open problems.