Monotone Set-Valued Function Defined by Set-Valued Choquet Integral

Structural characteristics and absolute continuities of monotone set-valued function defined by set-valued Choquet integral are discussed. Similar to the single-valued monotone set function,several important structural characteristics of set-valued function are defined and have been proven the same as those in the original set functions,such as null-additivity,weakly null-additivity,order continuity,strong order continuity and property(S). A counterexample shows that order continuity and strong order continuity of the original set functions are no longer kept in a monotone set-valued function when Choquet integrably bounded assumption is abandoned. Four kinds of absolute continuities are defined for set-valued function,and all been proven valid with respect to the original set functions.