Vitali covering lemma

In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. The basic intuition of plane geometry behind the result is that, if you have an arbitrary collection of circles, allowed to overlap, then one may choose a subcollection of these circles that do not touch, and such that if you increase their radii by a factor of three, they contain the area covered by the original circles.

Statement of the lemma

• Finite version: Let $B_{1},...,B_{n}$ be any collection of d-dimensional balls contained in d-dimensional Euclidean space $\mathbb {R} ^{d}$. Then there exists a subcollection $B_{j_{1}},B_{j_{2}},...,B_{j_{m}}$ of these balls which are disjoint and satisfy

$B_{1}\cup B_{2}\cup \cdots \cup B_{n}\subseteq 3B_{j_{1}}\cup 3B_{j_{2}}\cup \cdots \cup 3B_{j_{m}}$

where $3B_{j_{k}}$ denotes the ball with the same center as $B_{j_{k}}$ but with three times the radius.

• Infinite version: Let $\{B_{j}\}$ be any collection (finite, countable, or uncountable) collection of d-dimensional balls in $\mathbb {R} ^{d}$. Then there exists a countable subcollection $\{B_{j_{k}}\}_{k=1}^{\infty }$ of balls from our original collection which are disjoint and

$\bigcup _{j}B_{j}\subseteq \bigcup _{k=1}^{\infty }3B_{j_{k}}.$

Applications and method of use

An application of the Vitali lemma is in proving the Hardy-Littlewood maximal inequality. As in this proof, the Vitali lemma is frequently used when we are, for instance, considering the Lebesgue measure, $m$, of a set $E\subseteq \mathbb {R} ^{d}$, which we know is contained under the union of a certain collection of balls $\{B_{j}\}$, each of which has a measure we can more easily compute or has a special property we'd like to exploit. Hence, if we compute the measure of this union, we will have an upper bound to the measure of $E$. However, it is difficult to compute the measure of the union of all these balls if they overlap. By the Vitali lemma, we may choose a disjoint subcollection $\{B_{j_{k}}\}$ of this collection which is disjoint and, by tripling their radii, will contain the area covered by the original collection of balls, and hence will cover $E$. Hence,

$m(E)\leq m\left(\bigcup _{j}B_{j}\right)\leq m\left(\bigcup _{k}3B_{j_{k}}\right)\leq \sum _{j=k}m(3B_{j_{k}})$

Now, since increasing the radius of a d-dimensional ball by a factor of three increases it's volume by a factor of $3^{d}$, we know that

$\sum _{j=k}m(3B_{j_{k}})=3^{d}\sum _{k}m(B_{j_{k}})$

and thus

$m(E)\leq 3^{d}\sum _{k}m(B_{j_{k}}).$

One may also have a similar objective when considering Hausdorff measure instead of Lebesgue measure. In this case, we have the theorem below.

Vitali covering theorem

Definition. For a set $E\subseteq \mathbb {R} ^{d}$, define a Vitali Class ${\mathcal {V}}$ for $E$ to be a collection of sets such that for every $x\in E$ and $\delta >0$ there is a set $U\in {\mathcal {V}}$ such that $x\in U$ and the diameter of $U$ is less than $\delta .$

Theorem. Let $E\subseteq \mathbb {R} ^{d}$ be a $H^{s}$-measurable set and ${\mathcal {V}}$ a Vitali class for $E$. Then there exists a (finite or countably infinite) disjoint subcollection $\{U_{j}\}\subseteq {\mathcal {V}}$ such that either

$H^{s}(E\backslash \bigcup _{j}U_{j})=0{\mbox{ or }}\sum _{j}d(U_{j})^{s}=\infty .$

Furthermore, if $E$ has finite s-dimensional measure, then for any $\epsilon >0$, we may choose this subcollection $\{U_{j}\}$ such that

$H^{s}(E)\leq \sum _{j}d(U_{j})^{s}+\epsilon .$

References

• K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985.
• Rami Shakarchi & Elias Stein, Princeton Lectures in Analysis III: Real Analysis, Princeton University Press, 2005.