Some properties of Skorokhod metric on fuzzy sets

Abstract

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 are not necessarily equivalent on Lp-integrable fuzzy sets, which include compact fuzzy sets. We point out that the Lp-type dp metric, p≥1, is well-defined in common cases but the dp metric is not always well-defined on all fuzzy sets. We introduce the dp⁎ metric which is an expansion of the dp metric, and write dp⁎ as dp in the sequel. Then, we investigate the relationship between these two Skorokhod-type metrics and the dp metric. We show that the relationship can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the dp metric. On Lp-integrable fuzzy sets, the Skorokhod metric is not necessarily stronger than the dp metric, but the enhanced-type Skorokhod metric is still stronger than the dp metric. On all fuzzy sets, even the enhanced-type Skorokhod metric is not necessarily stronger than the dp 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.