Extension of quasi-Lipschitz set functions

In this article we consider so-called $\mathscr N$-triangular and quasi-Lipschitz set functions. In terms of $\mathscr N$ -semimeasures, we establish necessary and sufficient conditions for extending a quasi-Lipschitz set function which is continuous from above at zero from a ring of sets to the $\sigma$-ring generated by these sets, and also conditions for the uniqueness of the extension. As simple corollaries we obtain analogous results for vector-valued measures, continuous triangular measures, and real-valued finite $\mathscr N$ -triangular set functions which are continuous from above at zero.