Nonstandard entropy and energy measures of a fuzzy set
\textit{A. De Luca} and \textit{S. Termini} [Inf. Control 20, 301-312 (1972; Zbl 0239.94028)] have defined the entropy for a fuzzy set in Zadeh's sense and they have introduced the idea of energy. Successively, A. De Luca and S. Termini, E. Trillas and T. Riera, J. Knopfmacher and other authors have given several interesting examples of energy and entropy. Unfortunately, when X is infinite, these examples have several indesirable features. For examples, they are only defined for measurable fuzzy sets, and many properties which are true when X is finite are only ''almost everywhere'' true when X is infinite. Moreover, the contribution to entropy and energy of a finite subset of points is nonregistered. The aim of this paper is to show that, if we replace the real number field by a nonstandard model of analysis, it is possible to avoid such inconveniences. In fact I give some examples of nonstandard entropies and energies which are natural extensions to the infinite case of the known energies and entropies defined when X is finite [see also \textit{A. Di Nola} and \textit{G. Gerla}, Fuzzy Sets Syst. 18, 173-181 (1986)].
scientific article; zbMATH DE number 3963742