Sugeno integral

In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. Let ( X , Ω ) {displaystyle (X,Omega )} be a measurable space and let h : X → [ 0 , 1 ] {displaystyle h:X o } be an Ω {displaystyle Omega } -measurable function. The Sugeno integral over the crisp set A ⊆ X {displaystyle Asubseteq X} of the function h {displaystyle h} with respect to the fuzzy measure g {displaystyle g} is defined by: where F α = { x | h ( x ) ≥ α } {displaystyle F_{alpha }=left{x|h(x)geq alpha ight}} . The Sugeno integral over the fuzzy set A ~ {displaystyle { ilde {A}}} of the function h {displaystyle h} with respect to the fuzzy measure g {displaystyle g} is defined by: where h A ( x ) {displaystyle h_{A}(x)} is the membership function of the fuzzy set A ~ {displaystyle { ilde {A}}} .