Characterizations of endograph metric and $Γ$-convergence on fuzzy sets

Abstract: This paper is devoted to the relationships and properties of the endograph metric and the $\Gamma$-convergence. The main contents can be divided into three closely related parts. Firstly, on the class of upper semi-continuous fuzzy sets with bounded $\alpha$-cuts, we find that an endograph metric convergent sequence is exactly a $\Gamma$-convergent sequence satisfying the condition that the union of $\alpha$-cuts of all its elements is a bounded set in $\mathbb{R}^m$ for each $\alpha > 0$. Secondly, based on investigations of level characterizations of fuzzy sets themselves, we present level characterizations (level decomposition properties) of the endograph metric and the $\Gamma$-convergence. It is worth mentioning that, using the condition and the level characterizations given above, we discover the fact: the endograph metric and the $\Gamma$-convergence are compatible on a large class of general fuzzy sets which do not have any assumptions of normality, convexity or star-shapedness. Its subsets include common particular fuzzy sets such as fuzzy numbers (compact and noncompact), fuzzy star-shaped numbers (compact and noncompact), and general fuzzy star-shaped numbers (compact and noncompact). Thirdly, on the basis of the conclusions presented above, we study various subspaces of the space of upper semi-continuous fuzzy sets with bounded $\alpha$-cuts equipped with the endograph metric. We present characterizations of total boundedness, relative compactness and compactness in these fuzzy set spaces and clarify relationships among these fuzzy set spaces. It is pointed out that the fuzzy set spaces of noncompact type are exactly the completions of their compact counterparts under the endograph metric.