The Absolute Continuity And Bound Variation Of Closed-convex-set-valued Function And Its Aumann Integral Representations

In this paper, we convert the set-valued function to a real-valued function by using the support function of the set , and then we discuss the set-valued function by means of the classical real analysis.This paper deals with the characters of the support function, Aumann integral, bound variation and absolute continuity. Firstly, we obtain some characters of the support function of closed-convex set. Secondly, using the support function, the Aumann integral of set-valued function can be characterized by the integrability of real-valued function. Thirdly, the relations between the total variations of bound variation function and it's Aumann integrability are discussed, i.e the integral representation of Aumann integration for the total variation of the bound variation function. Finally, the necessary and sufficient conditions for the Newton-leibniz formula in the sense of Aumann integral is given.