Non-atomicity for fuzzy and non-fuzzy multivalued set functions

In this paper, we extend the concept of set-valued fuzzy measures (introduced and studied by Guo and Zhang [2004. On set-valued fuzzy measures. Information Sciences 160 (2004) 13–25]) to the class of multivalued set functions taking values in the family of non-empty closed subsets of a real normed space. Also, we introduce the notions of atom and non-atomicity of a multivalued set function and present some of their properties. We also establish an extension result by preserving non-atomicity for a fuzzy multisubmeasure/multimeasure.