Triangular norms and measures of fuzzy sets

Abstract The classical measure and probability theory is based on the notion of σ-algebra of subsets of a set. Butnariu and Klement [8] generalized it to fuzzy sets by considering collections of fuzzy sets called T-tribes (where T denotes a fixed triangular norm). Their concept of T-measure is fundamental in the fuzzification of classical measure theory. However, it has been successfully applied elsewhere, too (e.g., in finding solutions to games with fuzzy coalitions, see [9]). Here we summarize results about characterization of measures on tribes. More generally, we study signed measures (called charges). Unlike preceding papers, we put emphasis on σ-order continuous charges which preserve limits of increasing as well as decreasing sequences of fuzzy sets. We argue that this notion could be considered as a promising alternative to the original notion of Butnariu and Klement.

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