Triangular set functions and the Nikodym, Brooks-Jewett, and Vitali-Hahn-Saks theorems on convergent sequences of measures

We consider triangular set functions which have no escaping load and are defined on classes of sets which are closed with respect to the union of at most countably many disjoint sets in the given class. We refine and generalize the Nikodym, Brooks-Jewett, and Vitali-Hahn-Saks theorems to these classes of set functions in a nontrivial manner; these include quasi-Lipschitz and finitely additive set functions, vector-valued measures, semimeasures, and outer measures. As immediate consequences of the theorems proved here we obtain a series of statements concerning the extension of properties of set functions, such as convergence, compactness, uniform absolute continuity and absolute continuity, from a ring of sets to the $\sigma$-ring generated by the ring.
Bibliography: 18 titles.