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\(\perp\)-decomposable measures and integrals for Archimedean t-conorms \(\perp\). (English) Zbl 0614.28019

Let \((\Omega, \mathcal B)\) be a measurable space, \(m\) a mapping from \(\mathcal B\) into \([0,1]\) such that \(m(\emptyset)=0\) and \(m(\Omega)=1,\) and \(\perp\) a t-conorm. The author calls \(m\) a \(\perp\)-decomposable measure if \(m(A\cup B)=m(A)\perp m(B)\) whenever \(A\cap B=\emptyset\), i.e., if \(m\) is \(\perp\)-additive, and a \(\sigma\)-\(\perp\)-decomposable measure if \(m\) is \(\perp\text{-}\sigma\)-additive. The author studies such measures for strict and Archimedean t-conorms, with emphasis on the nonstrict Archimedean case. He defines an integral of a measurable functions with respect to any such \(\perp\)-decomposable measure and compares it to similar integrals defined by Choquet and Sugeno. In the process, he develops some interesting properties of t-norms and t-conorms. Thus, for any nonstrict Archimedean t-conorm \(\perp\) (t-norm \(T)\), with additive generator \(g(f)\), he defines the complementary t-norm \(\perp'\) of \(\perp\) (t-conorm \(T'\) of \(T)\) as the t-norm (t-conorm) additively generated by \(g(1)-g(x)(f(0)-f(x))\). The functions so related possess a number of pleasant duality properties.

MSC:

28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
28A25 Integration with respect to measures and other set functions
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