• Besicovitch covering theorem
• covering limits
• covering properties
• differentiation of measures
• directionally limited
• Federer density
• measure
• metric spaces

The present thesis deals with the differentiation of measures in metric spaces. We present both the cases where we have a Vitali relation, and the case where we have no special assumptions on the metric space.
Among the applications we provide a chacterization of measurable functions in terms of approximate continuity. We give important sufficient conditions to recognize adequate families of closed subsets of metric spaces with respect to a given Borel measure that is finite on bounded sets.
We establish a very weak assumption to obtain the Besicovitch Covering Theorem in metric spaces. This assumption is given by the directionally limited metric spaces. The latter family of metric spaces includes all real finite dimensional normed spaces and Riemannian manifolds. As a consequence of the Besicovitch Covering Theorem we show that in a separable directionally limited metric space we can reconstruct a Borel measure that is finite on bounded sets from its values on the closed metric balls. Using the Jordan Decomposition Theorem we extend this result to the case of signed measures and of vector-valued measures. We also investigate the relations among Besicovitch type covering properties, giving concrete examples of metric spaces where these properties are not equivalent among them, and illustrating some classes of metric spaces where, instead, two of them are equivalent to each other. In particular, we focus on the so called Weak Besicovitch Covering Property, showing its strong dependence on the metric and illustrating its behaviour under natural operations on metric spaces (restriction to a metric subspace, product of metric spaces, and submetry between metric spaces). Moreover, we prove that every metric space satisfying the so called Besicovitch Covering Property has finite topological dimension. This result tells us that the above mentioned covering properties are conditions that require a certain finite-dimensionality of the metric space. We study the equivalence between the Lebesgue Differentiation Property and some ``finite dimensionality'' of the metric spaces. We provide the complete proof of one implication, relying on the works of Assouad and de Gromard. Finally, we give some examples of metric spaces where the Besicovitch Covering Theorem does not hold. In this case, Magnani introduced the so-called Federer density and established a measure-theoretic area formula, which however differs from an effective differentiation theorem. On the other hand, it is still useful to compute the spherical measure of submanifolds in classes of nilpotent Lie groups.