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

We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions. The starting point for writing this paper was the desire to improve the results of V.K. Maslyuchenko et al. [MMMS], [MS], [KM], [KMM] who generalized a classical theorem of W.Rudin [Ru]...