Fuzzy Integrals

Abstract

In this chapter, we assume that (X, ℱ) is a measurable space, where X ∈ ℱ, μ : ℱ → [0, ∞] is a fuzzy measure (or a nonnegative monotone set function for Section 7.6), and that F is the class of all finite nonnegative measurable functions defined on (X, ℱ). For any given f ∈ F, we write F _α = {x|f(x) ≥ α}, F _{α +} = {x f(x) > α}, where α ∈ [0, ∞]. Let the sets F _α and F _α+ be called an α-cut and a strict α-cut of f, respectively.