On set-valued measures

Abstract

As a rule, a measure is a mapping from a σ-field of sets into the set of reals, or more generally, into some Banach space. A concept of set-valued measure (SV-measure) is introduced in the paper being a specific mapping from a σ-field of sets into a power set of a set. Properties of SV-measures are analyzed and illustrated on examples. Close relationship between SV-measures and a new nonstandard approach in artificial intelligence (AI) is explained. Then, the construction of factorization of the measures is mentioned, a special class of σ-quasiatomic SV-measures is defined and corresponding characterization theorem is proved. This class involves SV-measures ranging in a countable set which were used in modelling uncertainty in AI. It enables to answer one question arising in connection with this application.