On set-valued measures

Article type: Research Article

Authors: Lachout, Petr | Studený, Milan | Šindelář, Jan

Affiliations: Institute of Information Theory and Automation, Academy of Sciences Czech Republic, Pod vodárenskou věží 4, 18208 Prague 8, Czech Republic

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.

Keywords: set-valued measure, factorization, quasiatom of set-valued measure

DOI: 10.3233/INF-1993-41-202

Journal: Informatica, vol. 4, no. 1-2, pp. 21-44, 1993