Weber, Siegfried \(\perp\)-decomposable measures and integrals for Archimedean t-conorms \(\perp\). (English) Zbl 0614.28019 J. Math. Anal. Appl. 101, 114-138 (1984). 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. Reviewer: Siegfried Weber (Mainz) Cited in 7 ReviewsCited in 142 Documents MSC: 28E10 Fuzzy measure theory 03E72 Theory of fuzzy sets, etc. 28A25 Integration with respect to measures and other set functions Keywords:fuzzy sets; Choquet’s integral; Sugeno’s integral; associative functions; fuzzy integral; decomposable measure; Archimedean t-conorms; integral of a measurable functions; t-norms × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Sugeno, M., Theory of Fuzzy Integrals and Its Applications, (Doctoral dissertation (1974), Tokyo Institute of Technology) · Zbl 0316.60005 [2] Zadeh, L., Fuzzy sets, Inform. Contr., 8, 338-353 (1965) · Zbl 0139.24606 [3] Choquet, G., Theory of capacities, Ann. Inst. Fourier (Grenoble), 5, 131-292 (1953/1954) · Zbl 0064.35101 [4] Huber, P. J.; Strassen, V., Minimax tests and the Neyman-Pearson lemma for capacities, Ann. Statist., 1, 251-263 (1973) · Zbl 0259.62008 [5] Menger, K., Statistical metrics, (Proc. Nat. Acad. Sci. USA, 28 (1942)), 535-537 · Zbl 0063.03886 [6] Schweizer, B.; Sklar, A., Espaces métriques aléatoires, C. R. Acad. Sci. Paris, 247, 2092-2094 (1958) · Zbl 0085.12503 [7] Schweizer, B.; Sklar, A., Associative functions and statistical triangle inequalities, Publ. Math. Debrecen, 8, 169-186 (1961) · Zbl 0107.12203 [8] Ling, C. H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401 [9] Birkhoff, G., Lattice Theory, (Amer. Math. Soc. Colloq. Publications, Vol. XXV (1960), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0126.03801 [10] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.