On the extension of non-additive set functions

In this paper we prove theorems on the extension of non-additive set functions domain of definition of which, generally speaking, is not a ring, on the sigma-ring of sets. It is shown that continuous from the top at zero, the exhaustive compositional submersion of the first or second kind can be continued from the multiplicative class of sets to the sigma-ring of sets to a complete quasitriangular submerse complete at zero. Conditions are found under which the composition sub-measure of the first (second) kind extends to the composition sub-measure of the same kind. The continuation of the composite submerses obtained in the work is, in general, not unique. Some particular types of submeasures are considered, for which uniqueness of continuation takes place.