## $$\perp$$-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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### References:

  Sugeno, M, Theory of fuzzy integrals and its applications, () · Zbl 0316.60005  Zadeh, L, Fuzzy sets, Inform. contr., 8, 338-353, (1965) · Zbl 0139.24606  Choquet, G, Theory of capacities, Ann. inst. Fourier (Grenoble), 5, 131-292, (1953/1954) · Zbl 0064.35101  Huber, P.J; Strassen, V, Minimax tests and the Neyman-Pearson lemma for capacities, Ann. statist., 1, 251-263, (1973) · Zbl 0259.62008  Menger, K, Statistical metrics, (), 535-537 · Zbl 0063.03886  Schweizer, B; Sklar, A, Espaces métriques aléatoires, C. R. acad. sci. Paris, 247, 2092-2094, (1958) · Zbl 0085.12503  Schweizer, B; Sklar, A, Associative functions and statistical triangle inequalities, Publ. math. debrecen, 8, 169-186, (1961) · Zbl 0107.12203  Ling, C.H, Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401  Birkhoff, G, Lattice theory, () · Zbl 0126.03801  Schweizer, B; Sklar, A, Associative functions and abstract semigroups, Publ. math. debrecen, 10, 69-81, (1963) · Zbl 0119.14001
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