ABSTRACT. Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for ( )-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.

In this paper we introduce the notion of exhaustiveness which applies for both families and nets of functions. This new notion is close to equicontinuity and describes the relation between pointwise convergence for functions and α-convergence (continuous convergence). Using these results we obtain some Ascoli-type theorems dealing with exhaustiveness instead of equicontinuity. Also we deal with the corresponding notions of separate exhaustiveness and separate α-convergence. Finally we give conditions under which the pointwise limit of a sequence of arbitrary functions is a continuous function.

ABSTRACT. Some aspects of the theory of order and (D)-convergence in ( )-groups with respect to ideals are investigated. Moreover some new Basic Matrix Theorems are proved.

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.

Some limit and Dieudonné-type theorems in the setting of (ℓ)-groups with respect to filter convergence are proved, extending earlier results.