Jonathan M. Keith

Completeness and additive property for submeasures

Given an extended real-valued submeasure $ν$ defined on a field of subsets $Σ$ of a given set, we provide necessary and sufficient conditions for which the pseudometric $d_ν$ defined by $d_ν(A,B):=\min\{1,ν(A\bigtriangleup B)\}$ for all $A,B \in Σ$ is complete. As an application,...