Regular fuzzy measure and representation of comonotonically additive functional
Two real-valued functions \(f,g: X\to\mathbb{R}\) are called comonotonic if \(f(x)< f(x')\) implies \(g(x)\leq g(x')\) for all \(x,x'\in X\). A functional \(I: K\to\mathbb{R}\) on a space of real-valued functions is called comonotonically additive if \(I(f+g)= I(f)+ I(g)\) for any pair of comonotonic functions of \(K\). It is proved that a functional \(I:K\to \mathbb{R}\) on the space \(K\) of all continuous functions on a compact Hausdorff space is monotone and comonotonically additive iff it can be represented as a Choquet integral \(I(f)= \int^\infty_0 \mu(\{x: f(x)\geq r\}) dr\) with respect to a regular fuzzy measure \(\mu\) on the \(\sigma\)-algebra of Borel subsets of \(X\). Here, a fuzzy measure \(\mu\) is a real-valued monotone set function with \(\mu(\emptyset)= 0\). It is also given a representation of monotone and monotonically additive functionals on the space of all continuous functions with compact support defined on a locally compact Hausdorff space.
scientific article; zbMATH DE number 1462784