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

TL;DR

This paper extends the Lebesgue Differentiation Theorem to metric-valued functions in measure spaces, providing new tools for analysis in Banach bundles and vector measures with the Radon-Nikodým property.

Contribution

It introduces a metric-valued differentiation theorem using von Neumann liftings, enabling new results in Banach bundle sections and vector measure disintegration.

Findings

Established a Lebesgue Differentiation Theorem for metric-valued functions.

Derived a lifting theorem for sections of measurable Banach bundles.

Proved a disintegration theorem for vector measures into Banach spaces with Radon-Nikodým property.

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-Nikod\'{y}m property.