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Strong Law of Large Numbers for Fuzzy Random Variables in Fuzzy Metric Space
International Journal of Fuzzy Logic and Intelligent Systems 2020;20(4):278-289
Published online December 25, 2020
© 2020 Korean Institute of Intelligent Systems.

Reza Ghasemi, Mohammad Reza Rabiei, and Ahmad Nezakati

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
Correspondence to: Mohammad Reza Rabiei (
Received October 21, 2019; Revised February 26, 2020; Accepted March 17, 2020.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Despite uncertainty in fuzzy random variables, crisp metrics have always been used. In this study, we attempt to introduce the concept of fuzzy metric space and fuzzy normed space for fuzzy sets and some of its properties, and we investigate the strong law of large numbers. The embedded theorem for fuzzy compact sets in fuzzy normed space and the generalized Hukuhara difference are the most important tools used to prove this theorem. In addition, as a result and application, we use the strong law of large numbers for fuzzy random variables in the fuzzy metric space for the bootstrap mean.
Keywords : Fuzzy metric space, Limit theorems, Random set, Fuzzy random variable, Bootstrap mean