Differentiation of measures in metric spaces

TL;DR

This paper reviews classical and recent results on the differentiation of measures in metric spaces, emphasizing covering properties like the weak Besicovitch covering property and their role in measure differentiation theorems.

Contribution

It recalls fundamental differentiation theorems in metric spaces and analyzes the weak Besicovitch covering property, highlighting its significance in measure differentiation.

Findings

Classical differentiation theorems rely on covering properties.

The weak Besicovitch covering property characterizes spaces where differentiation holds.

Recent results determine when this covering property is valid or not.

Abstract

The theory of differentiation of measures originates from works of Besicovitch in the 1940's. His pioneering works, as well as subsequent developments of the theory, rely as fundamental tools on suitable covering properties. The first aim of these notes is to recall nowadays classical results about differentiation of measures in the metric setting together with the covering properties on which they are based. We will then focus on one of these covering properties, called in the present notes the weak Besicovitch covering property, which plays a central role in the characterization of (complete separable) metric spaces where the differentiation theorem holds for every (locally finite Borel regular) measure. We review in the last part of these notes recent results about the validity or non validity of this covering property.