Integrals of random fuzzy sets
type: Item
instance of: scholarly article
title: Integrals of random fuzzy sets
zbMATH Open document ID: 1137.28002
DOI: 10.1007/BF02607061
author: Volker Krätschmer
published in: Test
publication date: 2007-05-25T00:00:00Z
review text: The paper presents a unified and systematic investigation of integration of random fuzzy sets. The mostly accepted approach is the so called Aumann-integral introduced by \textit{M. L. Puri} and \textit{D. A. Ralescu} [J. Math. Anal. Appl. 114, 409--422 (1986; Zbl 0592.60004)]. Another approach has been initialized by \textit{P. Diamond} and \textit{P. Kloeden} [Fuzzy Sets Syst. 35, No. 2, 241--249 (1990; Zbl 0704.54006)] which allows an embedding of random fuzzy sets into the general probability theory in Banach spaces. Therefore, as an alternative to Aumann-integral, an adaption of Bochner- and Pettis-integration can be considered. The author discusses mutual relationships of these competing concepts. He completes and improves former results from literature and presents, as a by product, strong laws of large numbers and central limit theorems for random fuzzy sets which are based on weaker assumptions than previous versions from literature.
reviewed by: Wolfgang Näther
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 26E50
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 5157113
zbMATH Keywords: random fuzzy sets
zbMATH Keywords: Aumann-integral
zbMATH Keywords: Pettis-integral
zbMATH Keywords: Bochner-integral
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: central limit theorem
MaRDI profile type: Publication
cites work: Integrals of set-valued functions
cites work: A generalized strong law of large numbers
cites work: Metric spaces of fuzzy sets
cites work: Q3994935
cites work: An elementary proof of the strong law of large numbers
cites work: Integrals, conditional expectations, and martingales of multivalued functions
cites work: The law of large numbers and the central limit theorem in Banach spaces
cites work: Limit theorems for fuzzy random variables
cites work: On the variance of fuzzy random variables
cites work: A unified approach to fuzzy random variables
cites work: Limit theorems for fuzzy-random variables
cites work: Some complete metrics on spaces of fuzzy subsets
cites work: Probability theory in fuzzy sample spaces
cites work: Constructive definitions of fuzzy random variables
cites work: Approximating integrably bounded fuzzy random variables in terms of the ``generalized'' Hausdorff metric
cites work: Q4273174
cites work: Q4884357
cites work: On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data.
cites work: Fuzzy random variables
cites work: Fuzzy random variables, expectation, and martingales
cites work: Pettis integral and measure theory
cites work: A strong law of large numbers for random upper semicontinuous functions under exchangeability conditions.
cites work: Q4389738
cites work: \(L_p\) metrics for compact convex sets
cites work: The concept of a linguistic variable and its application to approximate reasoning. I
description: scientific article; zbMATH DE number 5157113