Properties of discrete uncertainty spaces. Refinement of the basic coding theorem (Q1842439)

The notion of discrete uncertainty space (DUS) [\textit{N. N. Diduk}, Dokl. Akad. Nauk Ukr. SSR, Ser. A 1983, No. 1, 63-65 (1983; Zbl 0511.94013); Cybernetics 20, 258-264 (1984); translation from Kibernetika, No. 2, 69- 73 (1984; Zbl 0562.94006)] is a mathematical abstraction with two important features. First, some results of classical information theory can be extended to these spaces; second, DUS can be used to formalize various uncertain situations (both purely probabilistic and purely ``probability-less'' situations, as well as mixed situations). Yet some essential questions relating to DUS properties still remain without an answer. In particular, they include the following questions: (1) Does a DUS always have an entropy? (2) Are there upper and lower bounds on the entropy of an arbitrary DUS? (3) Do improper DUS exist? The absence of unambiguous and clear answers to these questions is a serious obstacle to further development of uncertainty theory and to solutions of various applied problems. The present article attempts to answer these questions. It also refines the statement of the basic coding theory and provides a new proof, since it has been recently established that the original theory can be substantially strengthened.