Some properties of Skorokhod metric on fuzzy sets

Abstract

In this paper, we have our discussions on normal and upper semi-continuous fuzzy sets on metric spaces. The Skorokhod-type metric is stronger than the Skorokhod metric. It is found that the Skorokhod metric and the Skorokhod-type metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the Skorokhod-type metric need not be equivalent on $L_p$-integrable fuzzy sets. Based on this, we investigate relations between these two metrics and the $L_p$-type $d_p$ metric. It is found that the relations 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, which take compact fuzzy sets as special cases, the Skorokhod metric is not necessarily stronger than the $d_p$ metric, but the Skorokhod-type metric is still stronger than the $d_p$ metric. On general fuzzy sets, even the Skorokhod-type metric is not necessarily stronger than the $d_p$ metric. We also show that the Skorokhod metric is stronger than the sendograph metric.