In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces.

Statement of the lemma

• Finite version: Let $B\_\{1\},...,B\_\{n\}$ be any collection of d-dimensional balls contained in d-dimensional Euclidean space $mathbb\{R\}^\{d\}$ (or, more generally, in an arbitrary metric space). Then there exists a subcollection $B\_\{j\_\{1\}\},B\_\{j\_\{2\}\},...,B\_\{j\_\{m\}\}$ of these balls which are disjoint and satisfy

$B\_\{1\}cup\; B\_\{2\}cupcdots\; cup\; B\_\{n\}subseteq\; 3B\_\{j\_\{1\}\}cup\; 3B\_\{j\_\{2\}\}cupcdots\; cup\; 3B\_\{j\_\{m\}\}$

where $3B\_\{j\_\{k\}\}$ denotes the ball with the same center as $B\_\{j\_\{k\}\}$ but with three times the radius.

• Infinite version: Let $\{B\_\{j\}:jin\; J\}$ be any collection (finite, countable, or uncountable) of d-dimensional balls in $mathbb\{R\}^\{d\}$ (or, more generally, in a metric space) such that

where $mathrm\{diam\}(B\_j)$ denotes the diameter of $B\_j$. Then there exists a subcollection $\{B\_j:jin\; J\text{'}\}$, $J\text{'}subset\; J$, of balls from our original collection which are disjoint and

$bigcup\_\{jin\; J\}\; B\_\{j\}subseteq\; bigcup\_\{jin\; J\text{'}\}\; 5,B\_\{j\}.$

Proof

The proof of the finite version is rather easy. With no loss of generality, we assume that the collection of balls is not empty; that is, $n>0$. Let $B\_\{j\_1\}$ be the ball of largest radius. Inductively, assume that $B\_\{j\_1\},dots,B\_\{j\_k\}$ have been chosen. If there is some ball in $B\_1,dots,B\_n$ that is disjoint from $B\_\{j\_1\}cup\; B\_\{j\_2\}cupcdotscup\; B\_\{j\_k\}$, let $B\_\{j\_\{k+1\}\}$ be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set $m:=k$ and terminate the inductive definition.

Now set $X:=bigcup\_\{k=1\}^m\; 3,B\_\{j\_k\}$. It remains to show that $B\_isubset\; X$ for every $i=1,2,dots,n$. This is clear if $iin\{j\_1,dots,j\_m\}$. Otherwise, there necessarily is some $kin\{1,dots,m\}$ such that $B\_i$ intersects $B\_\{j\_k\}$ and the radius of $B\_\{j\_k\}$ is at least as large as that of $B\_i$. The triangle inequality then easily implies that $B\_isubset\; 3,B\_\{j\_k\}subset\; X$, as needed. This completes the proof of the finite version.

We now prove the infinite version. Let $R$ be the supremum of the radii of balls in $\{B\_j\}$ and let $Z\_i$ be the collection of balls in $B\_j$ whose radius is in $(2^\{-i-1\},R,2^\{i\},R]$. We first take a maximal disjoint subcollection $Z\_0\text{'}$ of $Z\_0$, then take a maximal subcollection $Z\_\{1\}\text{'}$ of $Z\_\{1\}$ that is disjoint and disjoint from $bigcup\; Z\_\{0\}\text{'}$. Inductively, we take $Z\_\{k\}\text{'}$ to be a maximal disjoint and disjoint from $bigcup\_\{i=0\}^\{k-1\}bigcup\; Z\_\{i\}\text{'}$ subcollection of $Z\_k$. It is easy to check that the collection $bigcup\_\{k=0\}^infty\; Z\text{'}\_k$ satisfies the requirements.

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 $Esubseteqmathbb\{R\}^\{d\}$, which we know is contained in the union of a certain collection of balls $\{B\_\{j\}:jin\; J\}$, each of which has a measure we can more easily compute, or has a special property one would like to exploit. Hence, if we compute the measure of this union, we will have an upper bound on 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 subcollection $\{B\_\{j\}:jin\; J\text{'}\}$ which is disjoint and such that $bigcup\_\{jin\; J\text{'}\}5\; B\_jsupset\; bigcup\_\{jin\; J\}\; B\_jsupset\; E$. Therefore,

$m(E)leq\; mleft(bigcup\_\{jin\; J\}B\_\{j\}right)\; leq\; mleft(bigcup\_\{jin\; J\text{'}\}5B\_\{j\}right)leq\; sum\_\{jin\; J\text{'}\}\; m(5\; B\_\{j\}).$

Now, since increasing the radius of a d-dimensional ball by a factor of five increases its volume by a factor of $5^d$, we know that

$sum\_\{jin\; J\text{'}\}\; m(5B\_\{j\})=5^d\; sum\_\{jin\; J\text{'}\}\; m(B\_\{j\})$

and thus

$m(E)leq\; 5^\{d\}sum\_\{jin\; J\text{'}\}m(B\_\{j\}).$

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

Vitali covering theorem

For a set E ⊆ R^d, a Vitali class or Vitali covering $mathcal\{V\}$ for E is a collection of sets such that, for every x ∈ E and δ > 0, there is a set $Uinmathcal\{V\}$ such that x ∈ U and the diameter of U is non-zero and less than δ.

Theorem. Let H^s denote s-dimensional Hausdorff measure, let E ⊆ R^d be an 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

$H^\{s\}\; left(Ebackslash\; bigcup\_\{j\}U\_\{j\}\; right)=0\; mbox\{\; or\; \}sum\_\{j\}\; mathrm\{diam\}\; (U\_\{j\})^\{s\}=infty.$

Furthermore, if E has finite s-dimensional measure, then for any ε > 0, we may choose this subcollection {U_j} such that

$H^\{s\}(E)leq\; sum\_\{j\}\; mathrm\{diam\}\; (U\_\{j\})^\{s\}+varepsilon.$

Infinite-dimensional spaces

The Vitali covering theorem is not valid in infinite-dimensional settings. The first result in this direction was given by David Preiss in 1979: there exists a Gaussian measure γ on an (infinite-dimensional) separable Hilbert space H so that the Vitali covering theorem fails for (H, Borel(H), γ). This result was strengthened in 2003 by Jaroslav Tišer: the Vitali covering theorem in fact fails for every infinite-dimensional Gaussian measure on any (infinite-dimensional) separable Hilbert space.

References

• Falconer, Kenneth J. (1986). The geometry of fractal sets. Cambridge: Cambridge University Press.
• Preiss, David (1979). "Gaussian measures and covering theorems". Comment. Math. Univ. Carolin. 20 (1): 95–99.
• Stein, Elias M.; Shakarchi, Rami (2005). Real analysis. Princeton University Press.
• Tišer, Jaroslav (2003). "Vitali covering theorem in Hilbert space". Trans. Amer. Math. Soc. 355 3277–3289 (electronic).