GENERAL ENTROPY OF GENERAL MEASURES

Abstract

The concept of entropy is an important part of the theory of additive measures. In this paper, a definition of entropy is introduced for general (not necessarily additive) measures as the infinum of the Shannon entropies of "subordinate" additive measures. Several properties of the general entropy are discussed and proved. Some of the properties require that the measure belongs to the class of so-called "equientropic" general measures introduced and studied in this paper. The definition of general entropy is extended to the countable case for which a sufficient condition of convergence is proved. We introduce a method of "conditional combination" of general measures and prove that in that case the general entropy possesses the "subset independence" property.