In this paper, a characterization of normed barrelled spaces is given, as well as a similar characterization of rings of sets with the Nikodym Property. Finally, a condition that is equivalent to a Banach space having a separable quotient is discussed.

Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems for scalar-valued measures on rings of sets that are in general not σ-rings are presented. As a consequence, the rings of subsets of N with density zero and uniform density zero are shown to have the Nikodym property. In addition, vector measure generalizations of the Vitali–Hahn–Saks theorem are given.

In this note, we give a gliding hump characterization of the dense barrelled solid subspaces of $\ell^1$ (recall that a sequence is solid if $\ell^\infty\cdot E=E$). Also, we present a sufficient condition for a dense subspace of $\ell^1$ to be barrelled, without assuming solidness, and generalizations to dense subspaces of arbitrary Banach spaces.