Non-Linearity of the Seminormed Fuzzy Integral

준노름 퍼지적분의 비 선형성

  • Published : 2002.06.01

Abstract

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.

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