Besicovitch covering theorem

Remarks

The proof chooses balls greedily, each time a ball of nearly maximal radius centred at a point not yet covered. A geometric argument in $\R^n$ shows that each chosen ball meets only a bounded number of earlier chosen balls of comparable or larger size, with a bound depending on $n$. The chosen balls are then sorted into $N_n$ disjoint subfamilies.

Unlike the Vitali covering lemma, no enlargement of the balls is needed and no measure enters the statement. So its consequences hold for every Radon measure on $\R^n$, doubling or not. A consequence concerns a locally finite Borel measure $\mu$ on $\R^n$ and a set $A$. If every point of $A$ is the centre of balls of $\mathcal{F}$ of arbitrarily small radius, then some countable disjoint subfamily of $\mathcal{F}$ covers $\mu$-almost all of $A$.

The fact that the balls are centred at points of $A$ is essential. The theorem does not extend to general metric spaces.

Going deeper

For a student

The theorem is the covering tool for measures that are not doubling. It gives the differentiation theorem $\lim_{r \to 0} \nu(B(x,r)) / \mu(B(x,r)) = \frac{d\nu}{d\mu}(x)$ for Radon measures $\mu, \nu$ on $\R^n$ with $\nu \ll \mu$, for $\mu$-almost every $x$. It is a basic tool of geometric measure theory.