On the basic solutions to the generalized fuzzy integral equation
Fuzzy measures and fuzzy integrals were introduced and examined by M. Sugeno [cf. \textit{M. M. Gupta, G. N. Saridis} and \textit{B. R. Gaines} (eds.), Fuzzy automata and decision processes, North-Holland, Amsterdam, 89-102 (1977; Zbl 0378.68035)]. Generalized fuzzy integrals based on generalized triangular norms were introduced by \textit{C. Wu, S. Wang}, and \textit{M. Ma} [Fuzzy Sets Syst. 57, No. 2, 219-226 (1993; Zbl 0786.28016)]. The authors of the present paper consider a condition for a function \(f\) with a constant fuzzy integral (e.g. all probability distribution functions fulfil a similar condition with the Lebesgue-Stieltjes integral). A condition of the form \(\sup_{\alpha>0} \min(\alpha,\mu(\{x\in A: \min(f(x),h(x)) > \alpha\})) = \beta\) with given constant \(\beta\), measure \(\mu\), measurable set \(A\), and measurable function \(h\) is rather not connected with the theory of integral equations. Constant and step functions \(f\) fulfilling such conditions are described.
scientific article; zbMATH DE number 1305162