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

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications