Negligible sets and differentiable measures in Banach spaces

We introduce the notion of a negligible set in a separable Banach space and then prove that every Gaussian null set is negligible. From this and an analogous theorem of R. Phelps (Pacif. J. Math., 1978, 77, № 2, 523–531; MR 80m: 46040) we deduce that a locally Lipschitz map from a separable Banach space into a Banach space with the Radon–Nikodym property is Gateaux differentiable outside the negligible set. A criterion of differentiability of a measure along a direction is given.