Base topologies and convergence in nonadditive measure

Abstract

The existing literature on convergence in nonadditive measure has exclusively focused on the construction of weak base topologies whereby different authors have used different notions of balls in their respective constructions. Whenever the nonadditive measure fails to satisfy specific structural properties, some notions of balls might fail to be open sets. Building on existing results, we show that these weak base topologies are equivalent for finite nonadditive measures regardless of whether the different notions of balls are open sets or not. As our main contribution we complement the existing literature through the construction of base topologies which are finer than the corresponding weak base topologies. In contrast to weak base topologies, a decision theoretic modeler can directly ensure through a base topology that his/her preferred notions of balls are always open sets for arbitrary nonadditive measures.