Some remarks on the double asymptotic null-additivity of monotonic measures and related applications

Abstract

In this paper, we discuss the inheriting of convergence of the monotonic measures under the following operations: addition, multiplication, max and min and the uniqueness of convergence of a monotonic measure. Moreover, we also point out that autocontinuity from above cannot imply double asymptotic null-additivity for monotonic measures using a counterexample contrary to the case of fuzzy measures proved by Ha et al. [Fundamental convergence of sequences of measurable functions on fuzzy measure space, Fuzzy Sets and Systems 95 (1998) 77–81]. Finally, we show that for monotonic measure convergence, double asymptotic null-additivity is better than autocontinuity from above.