Nonequivalence of Various Definitions of Differentiability Directions for Vector Measures

It is proved that the definitions of differentiability directions for vector measures in various topologies, namely, the topology of convergence on a system measurable sets, the topology of convergence with respect to semivariation, and the topology of convergence in variation, are generally pairwise nonequivalent. It is also proved that, for measures with values in a Banach space with the Radon–Nikodym property, these definitions are equivalent.