  We prove a version of the Lebesgue Differentiation Theorem for mappings that
are defined on a measure space and take values into a metric space, with
respect to the differentiation basis induced by a von Neumann lifting. As a
consequence, we obtain a lifting theorem for the space of sections of a
measurable Banach bundle and a disintegration theorem for vector measures whose
target is a Banach space with the Radon-Nikod\'{y}m property.
