Non-Linearity of the Seminormed Fuzzy Integral

준노름 퍼지적분의 비 선형성

Let (X, F, g) be a fuzzy measure space. Then for any h$\in$ $L^{0}$ (X) , a$\in$[0 , 1] , and $A\in$F ∫$_{A}$aㆍh($\chi$)┬g＝aㆍ∫$_{A}$h($\chi$)┬g with the t-seminorm ┬(x, y)= xy. And we prove that the Seminormed fuzzy integral has some linearity properties only for ｛0,1｝-classes of fuzzy measure as follow, For any f, h$\in$ $L^{0}$ ($\chi$), any a, b$\in$R+: af＋bh$\in$ $L^{0}$ ($\chi$)⇒ ∫$_{A}$(af＋bh)┬g＝a∫$_{A}$f┬g＋b∫$_{A}$h┬g; if and only if g is a probability measure fulfilling g(A) $\in$｛0, 1｝ for all $A\in$F.n$F.