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Characterizations of endograph metric and Γ-convergence on fuzzy sets

机译：模糊集的内向度量和Γ收敛性的刻画

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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 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. (C) 2018 Elsevier B.V. All rights reserved.
机译：本文致力于内窥镜度量标准与伽玛收敛的关系和性质。主要内容可以分为三个紧密相关的部分。首先，在有界α割的上半连续模糊集的类别上，我们发现一个内向度量度量收敛序列恰好是满足所有元素的α割的并集是有界的条件的伽马收敛序列。设置每个Rm> 0的Rm。其次，基于对模糊集本身的级别表征的研究，我们介绍了内窥镜度量标准和Gamma收敛的级别表征（级别分解属性）。值得一提的是，使用上面给出的条件和级别表征，我们发现一个事实：内窥镜度量标准和Gamma收敛在大类通用模糊集上兼容，这些模糊集不具有任何正态性，凸性或星形。它的子集包括常见的特定模糊集，例如模糊数（紧凑和非紧凑），模糊星形数（紧凑和非紧凑）和常规模糊星形数（紧凑和非紧凑）。第三，基于以上得出的结论，我们研究了带有内窥镜度量的有界alpha切口的上半连续模糊集空间的各个子空间。我们介绍了这些模糊集空间中总有界度，相对紧性和紧致性的表征，并阐明了这些模糊集空间之间的关系。要指出的是，非紧缩类型的模糊集空间恰好是内窥镜度量下其紧缩对应物的补全。 （C）2018 Elsevier B.V.保留所有权利。

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