The smallest semicopula-based universal integrals III: Topology determined by the integral

Motivated by the recent results on topology determined by the prominent Sugeno and Choquet integrals, we study the topology on the space of measurable functions for a non-additive measure, which is determined by the universal integral based on a semicopula. We define a family of (pseudo-)metrics on the set of measurable functions parameterized by a semicopula and study their properties. We show that for a semicopula without zero divisors the convergence in the constructed (pseudo-)metric is equivalent to convergence in (non-additive) measure as well as other two types of convergences of measurable functions. For each such a (pseudo-)metric we construct a family of subsets of measurable functions and we provide sufficient conditions for this family to be a topology on the set of measurable functions. Moreover, we show that for some natural class of semicopulas the corresponding topological spaces are all equivalent, being equivalent to the topology determined by the Choquet integral.