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

In this paper, we study the relationship between the pseudo-metric and sigma-decomposable measures with respect to a triangular conorm (sigma-perpendicular to-decomposable measures, for short). We first construct a pseudo-metric on the measurable sets of a given sigma-perpendicular to-decomposable measure, and then discuss several properties such as completeness and continuity of the constructed pseudometric space. Finally, we show that the mu-separabilityand nonatom of the sigma-perpendicular to-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 sigma-perpendicular to-decomposable measure. (C) 2015 Elsevier B.V. All rights reserved.