\(\perp\)-decomposable measures and integrals for Archimedean t-conorms \(\perp\)
type: Item
instance of: scholarly article
title: \(\perp\)-decomposable measures and integrals for Archimedean t-conorms \(\perp\)
zbMATH Open document ID: 0614.28019
DOI: 10.1016/0022-247X(84)90061-1
author: S. Weber
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: Let \((\Omega, \mathcal B)\) be a measurable space, \(m\) a mapping from \(\mathcal B\) into \([0,1]\) such that \(m(\emptyset)=0\) and \(m(\Omega)=1,\) and \(\perp\) a t-conorm. The author calls \(m\) a \(\perp\)-decomposable measure if \(m(A\cup B)=m(A)\perp m(B)\) whenever \(A\cap B=\emptyset\), i.e., if \(m\) is \(\perp\)-additive, and a \(\sigma\)-\(\perp\)-decomposable measure if \(m\) is \(\perp\text{-}\sigma\)-additive. The author studies such measures for strict and Archimedean t-conorms, with emphasis on the nonstrict Archimedean case. He defines an integral of a measurable functions with respect to any such \(\perp\)-decomposable measure and compares it to similar integrals defined by Choquet and Sugeno. In the process, he develops some interesting properties of t-norms and t-conorms. Thus, for any nonstrict Archimedean t-conorm \(\perp\) (t-norm \(T)\), with additive generator \(g(f)\), he defines the complementary t-norm \(\perp'\) of \(\perp\) (t-conorm \(T'\) of \(T)\) as the t-norm (t-conorm) additively generated by \(g(1)-g(x)(f(0)-f(x))\). The functions so related possess a number of pleasant duality properties.
Mathematics Subject Classification ID: 28E10
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 28A25
zbMATH DE Number: 3993913
zbMATH Keywords: fuzzy sets
zbMATH Keywords: Choquet's integral
zbMATH Keywords: Sugeno's integral
zbMATH Keywords: associative functions
zbMATH Keywords: fuzzy integral
zbMATH Keywords: decomposable measure
zbMATH Keywords: Archimedean t-conorms
zbMATH Keywords: integral of a measurable functions
zbMATH Keywords: t-norms
MaRDI profile type: Publication
cites work: Q4077278
cites work: Fuzzy sets
cites work: Theory of capacities
cites work: Minimax tests and the Neyman-Pearson lemma for capacities
cites work: Statistical Metrics
cites work: Q3254667
cites work: Q3293719
cites work: Q5511448
cites work: Q5331549
cites work: Q5730522
description: scientific article; zbMATH DE number 3993913