Nonlinearity of the fuzzy integral
type: Item
instance of: scholarly article
title: Nonlinearity of the fuzzy integral
zbMATH Open document ID: 0573.28003
DOI: 10.1016/S0165-0114(83)80089-X
published in: Fuzzy Sets and Systems
publication date: 1983-01-01T00:00:00Z
review text: The authors introduce the notion of a fuzzy measure \(\mu\) as a real- valued set function on a \(\sigma\)-algebra of subsets of a set X, which is isotone, continuous from above and below, and satisfies \(\mu (\emptyset)=0\) and \(\mu (X)=1\). For any real-valued, measurable functions f with values in the unit interval the fuzzy integral \(\int fd\mu\) of f is defined by sup\(\{\) min\(\{\) \(\alpha\),\(\mu\) (\(\{\) \(f\geq \alpha \})\}:\alpha \in [0,1]\}\). It is shown that the fuzzy integral conicides with the Lebesgue integral of \(\mu\) if and only if \(\mu\) is \(\{\) 0,1\(\}\)- valued. This last condition is also necessary and sufficient for a linear extension of the fuzzy integral to the set of all bounded, measurable functions.
Mathematics Subject Classification ID: 28A10
Mathematics Subject Classification ID: 28A99
Mathematics Subject Classification ID: 28A25
zbMATH DE Number: 3914627
zbMATH Keywords: fuzzy expected value
zbMATH Keywords: fuzzy measure
zbMATH Keywords: fuzzy integral
zbMATH Keywords: linear extension of the fuzzy integral to the set of all bounded, measurable functions
author: Erich Peter Klement
author: Dan A. Ralescu
MaRDI profile type: Publication
full work available at URL: s0165-0114(83)80089-x
OpenAlex ID: W4243711248
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description: scientific article; zbMATH DE number 3914627