It is well known that the fuzzy sets theory can be successfully used in quantum models ([5, 26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations - multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.
A variant of Alexandrov theorem is proved stating that a compact, subadditive $D$-poset valued mapping is continuous. Then the measure extension theorem is proved for MV-algebra valued measures.
In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space-valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among them the fact that our integral contains under suitable hypothesis the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.
Calculus for observables in a space of functions from an abstract set to the unit interval is developed and then the individual ergodic theorem is proved.
This paper generalizes the results of papers which deal with the Kurzweil-Henstock construction of an integral in ordered spaces. The definition is given and some limit theorems for the integral of ordered group valued functions defined on a Hausdorff compact topological space $T$ with respect to an ordered group valued measure are proved in this paper.
A definition of “Šipoš integral” is given, similarly to [3],[5],[10], for real-valued functions and with respect to Dedekind complete Riesz-space-valued “capacities”. A comparison of Choquet and Šipoš-type integrals is given, and some fundamental properties and some convergence theorems for the Šipoš integral are proved.