Probability theory in fuzzy sample spaces.

*(English)*Zbl 1083.60004Summary: This paper tries to develop a neat and comprehensive probability theory for sample spaces where the events are fuzzy subsets of \(\mathbb R^k\). The investigations are focussed on the discussion how to equip those sample spaces with suitable \(\sigma\)-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.

##### MSC:

60A05 | Axioms; other general questions in probability |

03E72 | Theory of fuzzy sets, etc. |

60D05 | Geometric probability and stochastic geometry |