Computational Statistics & Data Analysis 51 (2006) 109 – 114 www.elsevier.com/locate/csda Tools for fuzzy random variables: Embeddings and measurabilities Miguel López-Díaz a , ∗ , Dan A. Ralescu b a Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, E-33071 Oviedo, Spain b Department of Mathematical Sciences, University of Cincinnati, OH 45221-0025, USA Available online 11 May 2006 Abstract The concept of fuzzy random variable has been shown to be as a valuable model for handling fuzzy data in statistical problems. The theory of fuzzy-valued random elements provides a suitable formalization for the management of fuzzy data in the probabilistic setting. A concise overview of fuzzy random variables, focussed on the crucial aspects for data analysis, is presented. © 2006 Elsevier B.V.All rights reserved. Keywords: Embedding; Fuzzy random variable; Measurability of fuzzy set-valued random elements 1. Introduction An overview on some mathematical aspects of the concept of fuzzy random variable, also referred to in the literature as random fuzzy set or random upper semicontinuous function, is given. Statistical analyses involve observations taken on a sample of individuals with the aim of making inferences about the general population from which the sample is drawn. In many real-life problems the available observations are not real/vectorial-valued but rather imprecisely valued. For instance, when subjective perceptions are involved. Fuzzy sets have been shown to be a valuable tool to model this type of observations leading to the so-called fuzzy data. From a theoretical point of view fuzzy data could be considered as a particular case of functional data, although they have many distinguishing features especially those derived from the arithmetic of fuzzy data. Due to these differences, the problem of modeling and handling fuzzy data deserves a separate analysis, as will be commented later. The concept of fuzzy random variable has been introduced to formalize fuzzy data associated with outcomes from a random experiment. Historically, the concept was presented for the first time by Kwakernaak (1978). Later, Kruse and Meyer (1987) have reformulated the model in a probabilistic setting. The model by Kwakernaak et al., have assumed the existence of an underlying original real-valued random variable, the fuzzy random variable being a fuzzy perception from it. Puri and Ralescu (1986) have stated a more general model in a probabilistic setting. This model assumes that fuzzy random variables are fuzzy-valued random elements. The current theory of fuzzy random variables, which concerns probabilistic and statistical studies, is based on Puri and Ralescu’s definition. Developments in probability theory have shown that the fundamentals of this theory are soundly and rigourously supported. These developments also justify the need for a separate discussion of the matter. ∗ Corresponding author. Tel.: +34 985103360; fax: +34 985103354. E-mail address: mld@uniovi.es (M. López-Díaz). 0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.04.017