Vitali covering lemma

Remarks

The finite version is greedy: pick the largest ball, discard every ball that meets it, and repeat. A discarded ball meets a chosen ball of at least its radius, so it lies in the triple of that ball. The infinite version sorts the balls by radius into dyadic classes $R/2^{k+1} < r \le R/2^k$ and chooses a maximal disjoint subfamily in each class in turn, using Zorn's lemma. If $B$ meets $B'$ and the radius of $B$ is at most twice that of $B'$, then $B \subseteq 5B'$.

Example: in $\R^n$ the lemma gives $\lambda\big(\bigcup_i B_i\big) \le 3^n \sum_m \lambda(B_{j_m})$, which is the key step in the weak type estimate for the Hardy–Littlewood maximal function.

The bound on the radii is needed in part 2. The balls $\bar B(0, k)$ for $k = 1, 2, \dots$ in $\R$ pairwise meet, so a disjoint subfamily has one ball, and no fixed enlargement of it covers $\R$.

Going deeper

For a student

The lemma turns overlapping covers into disjoint ones at the price of a constant factor. It gives the weak $(1,1)$ bound for the maximal function and so the Lebesgue differentiation theorem. The Vitali covering theorem built on it says that a fine cover of a set by balls contains a disjoint subfamily that covers almost all of the set.