*preprint*

**Inserted:** 17 nov 2021

**Last Updated:** 17 nov 2021

**Year:** 2021

**Abstract:**

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-Nikodym property.