Monotone set-valued functions defined by set-valued Choquet integrals

In this paper, some properties of the monotone set-valued function defined by the set-valued Choquet integral are discussed. It is shown that several important structural characteristics of the original set function, such as null-additivity, strong order continuity, property(S) and pseudometric generating property, etc., are preserved by the new set-valued function. It is also shown that integrable assumption is inevitable for the preservation of strong order continuous and pseudometric generating property. Several kind of absolute continuity of set-valued function with respect to set function are also discussed.