Set functions and probability distributions of a finite random sets

This paper is the investigation of the probability distributions of a finite random set in which the set of random events are considered as a support of the finite random set. These probability distributions can be defined by six equivalent ways (distributions of the I-st–VI-th type). Each of these types of the probability distributions is the set function defined on the corresponding system of events. In this paper the sufficient conditions are formulated and proved. When these conditions are satisfied, then the set function determines the probability distributions of the finite random set of the II-nd and the V-th type. The found conditions supplement the known necessary conditions for the existence of the probability distributions of a finite random set of the II-nd and the V-th type.