Hlushak I. Approximations of non-additive measures

The thesis focuses on elaborating methods for approximation of non-additive regular measures (also called capacities by Choquet) on infinite metric spaces, with non-additive measures that are of “simpler nature” or more convenient for calculations. In the thesis possible ways of metrization of the sets of capacities are considered. It is proved that Prokhorov metric on the set of the capacities on a metric compactum is analogous to Uspenskii metric, when Choquet integral, which is used in the definition of the latter, is replaced with Sugeno integral, which is more adequate for non-additive measures. It has been shown that, to use Prokhorov metric on noncompact metric spaces, one has to restrict the class of capacities that are regular w.r.t. the topology, to the class of capacities that are regular w.r.t. the metric. The main goal of the thesis was to find, for an arbitrary capacity on a metric space, a capacity in a certain class, that is the closest to the given capacity w.r.t. Prokhorov metric. Approximation problems have been solved for the following classes: of the capacities that are Lipschitz w.r.t. Hausdorff metric; of the additive measure on a finite subspace; of the necessity measures; of the possibility measures; of the normalized capacities with supports in a closed subspaces. Such an approximation may not be unique, hence the conditions have been obtained to describe the set of the optimal approximations in the chosen class for an arbitrary capacity. The question on existence of a continuous selection of the obtained multivalued mapping was studied, and answered in negative in general case. It has been proved that there exist continuous almost optimal approximations. The construction uses properties of idempotent semimodules, relying on the fact that the space of subnormalized capacities on a metric compactum is a compact Lawson I-semimodule, and all the considered classes are I-convex compacta in it. In the thesis a “finite representation” of an arbitrary subnormalized capacity an infinite metric compactum has been obtained, as an approximation with a capacity, which is determined with a finite set of the values of the original capacity on all unions of elements of a finite family, of subsets of the space, called a foundation of the capacity. The least (in terms of cardinality or “total smallness”) foundation of the capacity is described with the introduced fractal dimensions, which are analogous to Hausdorff dimension and lower/upper box dimensions. They have been compared with the respective dimensions for sets and additive measures. Methods for calculation and estimation of dimensions for selfsimilar capacities, based on similar dimensions for inclusion hyperspaces, have been obtained.