Measurability and Convergence for some Functionals of Fuzzy Random Variables

Abstract

In practical applications of fuzzy set theory, a fuzzy set is produced by some process and very often its information is summarized in the form of one or several numbers (e.g., if a clear-cut decision needs to be taken which depends on a single value). While those fuzzy sets are typically considered to be deterministic, a more realistic modelling would take into account the random variations in the system producing the fuzzy sets. In this paper, we consider some functionals mapping fuzzy numbers to real numbers and study two questions which are important in order to use them in a random environment. First, whether applying the functional to a fuzzy random variable yields an ordinary random variable (measurability). Second, to what extent the functional preserves convergence of sequences of fuzzy random variables (continuity). This abstract continuity property allows its general application to statistical consistency of estimators, and we use the convergence of the sample mean to the population mean of a fuzzy random variable to provide a concrete example that functionals of an estimator consistently estimate the functional evaluated at the true population value.