This paper discusses the properties of Skorokhod metric on normal and upper semi-continuous fuzzy sets on metric space. All fuzzy sets mentioned below refer to this type of fuzzy sets. We confirm that the Skorokhod metric and the enhanced-type Skorokhod metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the enhanced-type Skorokhod metric need not be equivalent on $L_p$-integrable fuzzy sets, which include compact fuzzy sets. We point out that the $L_p$-type $d_p$ metric, $p\ge 1$, is well-defined in common cases but the $d_p$ metric is not always well-defined on all fuzzy sets. We introduce the $d_p^{⁎}$ metric which is an expansion of the $d_p$ metric, and write $d_p^{⁎}$ as $d_p$ in the sequel. Then, we investigate the relationship between these two Skorokhod-type metrics and the $d_p$ metric. We show that the relationship can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the $d_p$ metric. On $L_p$-integrable fuzzy sets, the Skorokhod metric is not necessarily stronger than the $d_p$ metric, but the enhanced-type Skorokhod metric is still stronger than the $d_p$ metric. On all fuzzy sets, even the enhanced-type Skorokhod metric is not necessarily stronger than the $d_p$ metric. We also show that the Skorokhod metric is stronger than the sendograph metric. At last, we give a simple example to answer some recent questions involved the Skorokhod metric.

本文讨论了度量空间上正态和上半连续模糊集上Skorokhod度量的性质。下文提到的所有模糊集均指这种类型的模糊集。我们确认，紧致模糊集上的Skorokhod度量和增强型Skorokhod度量是等效的。但是，Skorokhod指标和增强型Skorokhod指标不需要等效于${\mathrm{大号}}_p$-可积分模糊集，其中包括紧凑型模糊集。我们指出${\mathrm{大号}}_p$-类型 $d_p$ 指标， $p\ge \mathrm{1个}$在常见情况下定义明确，但 $d_p$度量标准并非总是在所有模糊集上都得到很好的定义。我们介绍$d_p^{⁎}$ 指标，是 $d_p$ 度量并编写 $d_p^{⁎}$ 作为 $d_p$在续集中。然后，我们调查这两个Skorokhod类型指标与$d_p$指标。我们证明这种关系可以分为三种情况。在紧致模糊集上，Skorokhod度量比$d_p$指标。上${\mathrm{大号}}_p$-可积分模糊集，则Skorokhod指标不一定强于 $d_p$ 指标，但增强型Skorokhod指标仍然比 $d_p$指标。在所有模糊集上，即使增强型Skorokhod指标也不一定比$d_p$指标。我们还显示Skorokhod指标比发送器指标更强。最后，我们给出一个简单的示例来回答一些最近的有关Skorokhod指标的问题。