Hubal O. Additive and non-additive measures on ultrametric spaces.

The thesis is devoted to investigation of the categorical properties of an ultrametric on the set of probability measures defined on ultrametric spaces, on the set of idempotent measures defined on ultrametric spaces and on the set of capacities defined on ultrametric spaces. It is shown that the functor of probability measures with compact supports forms a monad in the category of ultrametric spaces and nonexpanding maps. It is also proven that the G-symmetric power functor has an extension onto the Kleisli category of the probability measure monad. One can find counterparts of the mentioned results for the category complete ultrametric spaces and nonexpanding maps .An ultrametric шs also introduced on the set of idempotent probability measures (Maslov measures) defined on an ultrametric space. This construction determines a functor on the category of ultrametric spaces and nonexpanding maps. This functor defines a monad on the category . The G-symmetric power functor has an extension onto the Kleisli category of the idempotent measure monad.Analogously an ultrametric was introduced on the set of capacities with compact supports on an ultrametric space. This ultrametric proves to be functorial in the category of ultrametric spaces and nonexpanding maps. There is, however, a substantial difference between the case of capacities and that of probability (idempotent) measures. The capacity functor does not preserve completeness of the space. A construction used to define monads in the category of ultrametric spaces and non-expanding maps generated by functors of idempotent and probability measures does not work for the case of the capacity functor.