Real functions and the extension of generalized probability measures (Q2843895)

The paper discusses the topological aspects of the extensions of generalized probability measures (states) taking advantage of the theory of D-posets. The author works with D-posets of fuzzy sets which are represented by real functions and which are objects of the category ID (\textit{R. Frič} and \textit{M. Papčo} [Internat. J. Theoret. Phys. 50, 3778--3786 (2011; Zbl 1254.60009)]). The advantage of this category is that fields of sets and classical probabilistic measures are objects and morphisms of this category, respectively.NEWLINENEWLINEThe main aim of the paper is to build an appropriate classification of the extensions of generalized probability measures (probability measures and integrals with respect to probability measures, states) from a suitable class of generalized random events to a larger class having some additional (algebraic and topological) properties. These properties are divided into internal ones (expressed by using algebraic and topological properties of the system of all functions with values into the closed unit interval [0,1]) and external ones (expressed by using the extension of states).NEWLINENEWLINEThe proposed classification allows smooth and accurate characterization of the extensions of ID-objects. This paper provides an innovative view on the issue of the extensions of objects.