Sukhorukova K. Non-additive measures and their application in equilibrium theory

The dissertation is dedicated to the study of classes of non-additive measures generated by triangular norms *-measures). Such measures are defined as functionals on spaces of continuous functions with values in the unit interval. The spaces of *-measures are equipped with the weak* topology and form compact Hausdorff spaces in this topology. The functoriality of the construction of *-measure spaces in the category of compact Hausdorff spaces and their continuous mappings is demonstrated. For *-measure spaces, an analogue of the Milyutin mapping is constructed, which is known for probability measures and idempotent measures, allowing the reduction of the general case to the zero-dimensional one. Idempotent mathematics is a part of mathematics where one of the usual arithmetic operations in R is replaced by an idempotent operation (such as maximum). The results and methods of idempotent mathematics find numerous applications in various branches of mathematics, as well as in computer science and other disciplines. The purpose of the dissertation work is to investigate triangular norms * in the category of compact Hausdorff spaces, to study *-measure spaces with compact supports on ultrametric (non-Archimedean) spaces, to explore the structure of the monad generated by the functor of *-measure spaces with compact supports in the category of ultrametric spaces and non-expanding mappings, and to establish certain fundamental properties of such monads. Furthermore, the dissertation defines games in *-valued strategies and proves the continuity of payoff functions for these games for applications in game theory and equilibrium. It also covers the structure of the monad generated by the *-measure functor and its application to equilibrium in games with *-valued strategies. The research in the dissertation employs methods from functor theory in topological categories, general topology, idempotent mathematics, category theory, and game theory. In the section “Spaces of *-measure on Compact Hausdorff Spaces”, for each triangular norm *, we introduce the concept of *-measure as a functional on the space of continuous functions C(X, I). The set of all -measures on a compact Hausdorff space is equipped with the weak* topology. It is shown that the resulting *-measure space is a compact Hausdorff space. This construction defines a covariant functor in the category of compact Hausdorff spaces and continuous mappings. Moreover, an analog of the Milyutin mapping, first defined for probability measures, is constructed for *-measures. Additionally, a description of *-measures as closed subsets of the product space with a unit segment and certain properties is provided. It is proved that the set of ∗-measures with finite supports is everywhere dense in the space of all ∗- measures. One of the main results of this section is the description of *-measure spaces as hyperspaces of sets with certain properties. This allows for the comparison of *-measure spaces for different triangular norms *. In the section "Spaces of *-measure with Compact Supports on Ultrametric Spaces", we consider the case of ultrametric spaces (recall that a metric is called ultrametric if it satisfies the strong triangle inequality) and construct the ultrametricization of spaces of *-measures with compact supports on ultrametric spaces. It is shown that this construction defines a covariant functor in the category of ultrametric spaces and non-expansive mappings. It serves as an analogue for *-measures of functors defined for probability measures, idempotent measures, and upper semicontinuous capacities with compact supports. The dissertation proves that this functor is locally non-expansive. Additionally, it is demonstrated that the space of *-measures with compact supports on a complete ultrametric space is itself a complete ultrametric space. One of the main results of this section is the preservation of the class of complete ultrametric spaces by the functor of *-measures. This section, "Monads generated by functors of *-measure" is dedicated to the structure of a monad defined by *-measure functors on the category of Ultr, which consists of ultrametric spaces and non-expansive mappings. Several fundamental properties of such monads are established. Examples of non-isomorphic monads for different triangular norms * are provided. In particular, this structure allows for defining the tensor product of *-measures in the category Ultr. For this purpose, the maximal ultrametric on the product of ultrametric spaces is considered. Furthermore, we define games in *-valued strategies on ultrametric spaces and prove the continuity of payoff functions for these games. Finally, it is proven that any equilibrium for games in *-valued strategies can be approximated by almost equilibria composed of *-measures with finite supports.