The Brooks–Jevett theorem on uniform dimentricularity on a non-sigma-full class of sets

For a sequence of exhaustive composition-triangular set functions defined on a non-sigma-complete class of sets, more general than the ring of sets, the Brooks–Jewett theorem on uniform exhaustibility is proved. As a corollary, we have obtained analogue of the Brooks–Jewett theorem for functions defined on a sigma-summable class of sets. It is shown that if, in addition to the property compositional triangularity, the set functions have the composite semi-additivity property and are continuous from above at zero, then an analog of Nikodym's theorem on equicontinuous weak continuity is valid for them. The corresponding results are obtained for a family of quasi-Lipschitz set functions.