Completeness and Separability of the Space of a Class of Integrable Fuzzy Valued Functions Based on the tK-Integral Norm Metric

摘要

When a class of fuzzy value functions constitute a metric space, the completeness and separability is an important problem that must be considered to discuss the approximation of fuzzy systems. In this paper, Firstly, a new tK-integral norm is defined by introducing two induced operators, and prove that the class of tK-integrable fuzzy value functions is a metric space. And then, the integral transformation theorems and tK-integrable Borel-Cantelli Lemma are applied to study the completeness of the space, furthermore, its separability is discussed by means of the approximation of fuzzy valued simple functions and fuzzy valued Bernstein polynomials. The results show that the space of the tK-integrable fuzzy valued functions constitutes a complete separable metric space in the sense of the tK-integral norm.