Atoms of monotone set-valued measures and integrals

This paper first introduces the concept of a monotone set-valued measure and then focuses on the atoms and pseudo-atoms of the monotone set-valued measure space. Moreover, the paper proposes an integral, which is the integral of real-valued function with respect to the monotone set-valued measure, and shows some properties of this integral. Particularly, the paper gives the representations of integrals defined on atoms and pseudo-atoms. Some classical results are extended to the case of the monotone set-valued measure.