Nonstandard entropy and energy measures of a fuzzy set
type: Item
instance of: scholarly article
title: Nonstandard entropy and energy measures of a fuzzy set
zbMATH Open document ID: 0597.94031
DOI: 10.1016/0022-247X(84)90035-0
author: Giangiacomo Gerla
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: \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)].
Mathematics Subject Classification ID: 94D05
Mathematics Subject Classification ID: 94A17
Mathematics Subject Classification ID: 26E35
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 3963742
MaRDI profile type: Publication
cites work: A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory
cites work: On the Nonstandard Representation of Measures
cites work: On measures of fuzziness
cites work: Q4104164
cites work: Entropies in finite fuzzy sets
cites work: Fuzzy sets
cites work: Probability measures of fuzzy events
Recommended article: Nonprobabilistic entropies and indetermination measures in the setting of fuzzy sets theory
Recommended article: Entropy and information energy for fuzzy sets
Recommended article: ON ENTROPY OF FUZZY SETS
Recommended article: Entropies in finite fuzzy sets
Recommended article: Q4680255
Recommended article: Entropy of Fuzzy Measure
Recommended article: Entropy of discrete fuzzy measures
Recommended article: The entropy of hyperfuzzy sets
Recommended article: Q5291628
Recommended article: Higher order fuzzy entropy and hybrid entropy of a set
description: scientific article; zbMATH DE number 3963742