Information on the thesis

he thesis focuses on elaborating methods for approximation of non-additive regular measures (also called capacities by Choquet) on  metric spaces, with non-additive measures that are of “simpler nature” or more convenient for calculations.

In the thesis 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 . Therefore most of the approximations have been obtained for capacities on metric compacta. In this case the considered classes are compact w.r.t. the topology induced from the space of capacities, which ensures that the required optimal  approximation exists. It  has been proved that there exist continuous almost optimal  approximations.  The construction uses properties of idempotent semimodules.  In the thesis a “finite representation” of an arbitrary subnormalized capacity  on 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 foundation of the capacity is described with the fractal dimensions, which are analogous to Hausdorff dimension and lower/upper box dimensions. They have been compared with the respective dimensions for supports and additive measures. Methods for calculation and estimation of dimensions for self-similar capacities, based on similar dimensions for inclusion hyperspaces (also introduced in the thesis), have been obtained.

Keywords: non-additive measure, capacity, Sugeno integral, Prokhorov metric, metric, optimal approximation, continuous approximation, idempotent semimodule, idempotent convex combination, Hausdorff dimension, box dimension, self-similarity, iterated function system.