On the continuity of fuzzy entropies

Abstract

The entropy as a measure of diversity has been used by ecologists to characterize a community by its stability process. After the introduction of the concept of a fuzzy subset by Zadeh (1965), many definitions of entropies were given emphasizing the subjectivity in evaluations. A pioneering work which relates the classical meaning of entropy (Shannon index) with the modern fuzzy theory was due to De Luca and Termini (1972); Knopfmacher (1975) formulated a generalization of the axiomatics given by De Luca and Termini; Batle and Trillas (1979) obtained a result which is essentially analogous to Knopfmacher’s by considering a finite fuzzy measure space and the Sugeno’s integrals; and Trillas and Riera (1978) introduced the concept of fuzzy algebraic entropics. Analyses the continuity properties for these fuzzy entropies and establishes conditions which guarantee the convergence E(fn) → E(f), where (fn) is a sequence of fuzzy sets and E is an entropy.