The topology on the space of measurable functions that is compatible with convergence in nonadditive measure

Abstract

In this paper, given a nonadditive measure μ, a topology, which is compatible with convergence in μ-measure, is defined on the real linear space of all (equivalence classes of) measurable functions by using the distance introduced by Dunford and Schwartz. Some fundamental properties of the topology, such as the open sphere condition, the Hausdorff separation axiom, separability, approximation by continuous functions, linearlization, and metrizability, are related to the characteristics of a nonadditive measure.