On the g-entropy and its Hudetz correction

The fuzzy entropy h(T) of a dynamical system has been introduced in [5] (see also [1, 3, 8, 10]). Generalizing the notion of a fuzzy partition Mesiar and Rybarik have studied the p-entropy (see [7, 10, 11]) based on the Pap y-calculus ([9]). The notion is based on an increasing bijective function g : [0,oo] —> [0, oo], such that #(0) = 0 and #(1) = 1. The choice g(x) = x leads to the fuzzy entropy. The corresponding theorem states that to any ^-decomposable measure there exists a fuzzy measure such that the p-entropy can be expressed by the fuzzy entropy. Of course, the fuzzy entropy depends on a family T of fuzzy sets. If ^contains all constant functions, then the fuzzy entropy equals infinity. This defect has been corrected by Hudetz ([4]) by introducing a correcting member in the definition of the entropy of a fuzzy partition. The aim of this paper is a study of an analogous correction in the case of ^-entropy. Similarly as Mesiar and Rybarik in [7] we prove the corresponding representation theorem. We construct also an example demonstrating that the Hudetz modification of ^-entropy can be used although the usual g entropy is not available.