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

الملخص EN

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.