Fundamental triangular norm based tribes and measures
Let \(\{T_ s: 0< s< \infty\}\) be the family of Frank's \(t\)-norms. The author shows that for every \(T_ s\)-tribe \(\tau\) [as defined by \textit{D. Butnariu} and \textit{E. P. Klement}, J. Math. Anal. Appl. 162, No. 1, 111- 143 (1991; Zbl 0751.60003)] on a countable set \(X\) there exists a partition \((Y, Z)\) of \(X\) such that the set \(\tau| Z\) (of all restrictions of members of \(\tau\) to \(Z\)) is a \(\sigma\)-algebra of subsets of \(Z\) and \(\tau| Y\) consists of all measurable fuzzy sets on \(Y\) with respect to some \(\sigma\)-algebra of subsets of \(Y\). This result then leads to an integral representation of finite monotone \(T_ s\)- measures in the sense of \textit{Butnariu} and \textit{Klement} [op. cit.] on a countable set.
scientific article; zbMATH DE number 480467