Set Valued Measure And Set Valued Bartle Integral

It is the purpose of the first part in this paper to develop a new Bartle integral for a strong additive and weakσ-additive vector measure on a certain field.Then we get some properties as the traditional Bartle integral which defines on theΣ-field forσ-additive vector measures does.In the second part we define the semivariation of a set-valued measure and examine the essential properties of the semevariation.Then generalize the Vitali-Hahn-Saks-Nikodym theorem about finitely additive set valued measure by using the properties which are equivalent to uniformly strongly bounded set valued measures.In the last part,we explore the integral properties for non-negatively bounded function with respect to a weakly compact and convex set valued measure and establishes a limit theorem for the set valued Bartle integral.