Differentiation of measures on complete Riemannian manifolds

In this note we give a new proof of a version of the Besicovitch covering theorem, given in \cite{EG1992}, \cite{Bogachev2007} and extended in \cite{Federer1969}, for locally finite Borel measures on finite dimensional complete Riemannian manifolds $(M,g)$. As a consequence, we prove a differentiation theorem for Borel measures on $(M,g)$, which gives a formula for the Radon-Nikodym density of two nonnegative locally finite Borel measures $\nu_1, \nu_2$ on $(M, g)$ such that $\nu_1 \ll \nu_2$, extending the known case when $(M, g)$ is a standard Euclidean space.