On pseudo-metric spaces induced by σ-⊥-decomposable measures

In this paper, we study the relationship between the pseudo-metric and σ-decomposable measures with respect to a triangular conorm (σ-⊥-decomposable measures, for short). We first construct a pseudo-metric on the measurable sets of a given σ-⊥-decomposable measure, and then discuss several properties such as completeness and continuity of the constructed pseudo-metric space. Finally, we show that the μ-separability and nonatom of the σ-⊥-decomposable measure can be characterized in the constructed pseudo-metric space. Our results suggest that the standard approach for obtaining a metric from a given probability measure can be generalized to the setting of σ-⊥-decomposable measure.