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

Motivated by the recent results on topology determined by the Sugeno and Choquet integrals we study the topology on the space of measurable functions for a non-additive measure, which is determined by the integral IS based on a semicopula S. We define a family of mappings ρS on the set of measurable functions parameterized by a semicopula S and study their properties. We show that for a semicopula S without zero divisors the convergence in ρS is equivalent to convergence in (non-additive) measure m as well as other two types of convergences of measurable functions. For each mapping ρS we construct a family TS of subsets of measurable functions and we provide sufficient conditions for TS 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.