On the Dobrakov submeasure on fuzzy sets

The well-known fact that any nonatomic measure has the Darboux property has been generalized in many directions. Particularly by Dobrakov (Dissertationes Math. 112 (1974) 1) and recently by Klimkin and Svistula (Mat. Sb. 192 (2001) 41). In this paper the fuzzy sets are taken instead of sets as elements of domain of considered mappings.