Probability theory in fuzzy sample spaces

Abstract

This paper tries to develop a neat and comprehensive probability theory for sample spaces where the events are fuzzy subsets of ℝk. The investigations are focussed on the discussion how to equip those sample spaces with suitable σ-algebras and metrics. In the end we can point out a unified concept of random elements in the sample spaces under consideration which is linked with compatible metrics to express random errors. The result is supported by presenting a strong law of large numbers, a central limit theorem and a Glivenko-Cantelli theorem for these kinds of random elements, formulated simultaneously w.r.t. the selected metrics. As a by-product the line of reasoning, which is followed within the paper, enables us to generalize as well as to bring together already known results and concepts from literature.