Nykyforchyn O. Non-additive measures: categories and topological properties

The aim of the thesis is to develop a categorical and topological theory of non-additive measures. We propose topologizations of the set of regular capacities on a compact Hausdorff topological space: with weak* topologies with respect to the Sugeno integrals and other fuzzy integrals; with the Lawson topology that is determined with the lattice structure on the set of capacities; and with an embedding of the subgraphs of the capacities into a hyperspace with the Vietoris topology. It has been proved that all the mentioned methods, and in the metric case an analogue of the Prokhorov metric, determine the vague topology, which is equivalent to the weak* topology with respect to the Choquet integrals.