## $$\perp$$-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:

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