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# Vitali covering lemma

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 $B_\left\{1\right\},...,B_\left\{n\right\}$ be any collection of d-dimensional balls contained in d-dimensional Euclidean space $mathbb\left\{R\right\}^\left\{d\right\}$ (or, more generally, in an arbitrary metric space). Then there exists a subcollection $B_\left\{j_\left\{1\right\}\right\},B_\left\{j_\left\{2\right\}\right\},...,B_\left\{j_\left\{m\right\}\right\}$ of these balls which are disjoint and satisfy

$B_\left\{1\right\}cup B_\left\{2\right\}cupcdots cup B_\left\{n\right\}subseteq 3B_\left\{j_\left\{1\right\}\right\}cup 3B_\left\{j_\left\{2\right\}\right\}cupcdots cup 3B_\left\{j_\left\{m\right\}\right\}$

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

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

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

$bigcup_\left\{jin J\right\} B_\left\{j\right\}subseteq bigcup_\left\{jin J\text{'}\right\} 5,B_\left\{j\right\}.$

## 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 $B_\left\{j_1\right\}$ be the ball of largest radius. Inductively, assume that $B_\left\{j_1\right\},dots,B_\left\{j_k\right\}$ have been chosen. If there is some ball in $B_1,dots,B_n$ that is disjoint from $B_\left\{j_1\right\}cup B_\left\{j_2\right\}cupcdotscup B_\left\{j_k\right\}$, let $B_\left\{j_\left\{k+1\right\}\right\}$ be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set $m:=k$ and terminate the inductive definition.

Now set $X:=bigcup_\left\{k=1\right\}^m 3,B_\left\{j_k\right\}$. It remains to show that $B_isubset X$ for every $i=1,2,dots,n$. This is clear if $iin\left\{j_1,dots,j_m\right\}$. Otherwise, there necessarily is some $kin\left\{1,dots,m\right\}$ such that $B_i$ intersects $B_\left\{j_k\right\}$ and the radius of $B_\left\{j_k\right\}$ is at least as large as that of $B_i$. The triangle inequality then easily implies that $B_isubset 3,B_\left\{j_k\right\}subset X$, 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 $\left\{B_j\right\}$ and let $Z_i$ be the collection of balls in $B_j$ whose radius is in $\left(2^\left\{-i-1\right\},R,2^\left\{i\right\},R\right]$. We first take a maximal disjoint subcollection $Z_0\text{'}$ of $Z_0$, then take a maximal subcollection $Z_\left\{1\right\}\text{'}$ of $Z_\left\{1\right\}$ that is disjoint and disjoint from $bigcup Z_\left\{0\right\}\text{'}$. Inductively, we take $Z_\left\{k\right\}\text{'}$ to be a maximal disjoint and disjoint from $bigcup_\left\{i=0\right\}^\left\{k-1\right\}bigcup Z_\left\{i\right\}\text{'}$ subcollection of $Z_k$. It is easy to check that the collection $bigcup_\left\{k=0\right\}^infty Z\text{'}_k$ 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 $Esubseteqmathbb\left\{R\right\}^\left\{d\right\}$, which we know is contained in the union of a certain collection of balls $\left\{B_\left\{j\right\}:jin J\right\}$, 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 $\left\{B_\left\{j\right\}:jin J\text{'}\right\}$ which is disjoint and such that $bigcup_\left\{jin J\text{'}\right\}5 B_jsupset bigcup_\left\{jin J\right\} B_jsupset E$. Therefore,

$m\left(E\right)leq mleft\left(bigcup_\left\{jin J\right\}B_\left\{j\right\}right\right) leq mleft\left(bigcup_\left\{jin J\text{'}\right\}5B_\left\{j\right\}right\right)leq sum_\left\{jin J\text{'}\right\} m\left(5 B_\left\{j\right\}\right).$

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

$sum_\left\{jin J\text{'}\right\} m\left(5B_\left\{j\right\}\right)=5^d sum_\left\{jin J\text{'}\right\} m\left(B_\left\{j\right\}\right)$

and thus

$m\left(E\right)leq 5^\left\{d\right\}sum_\left\{jin J\text{'}\right\}m\left(B_\left\{j\right\}\right).$

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

Theorem. Let Hs denote s-dimensional Hausdorff measure, let E ⊆ Rd be an Hs-measurable set and $mathcal\left\{V\right\}$ a Vitali class for E. Then there exists a (finite or countably infinite) disjoint subcollection $\left\{U_\left\{j\right\}\right\}subseteq mathcal\left\{V\right\}$ such that either

$H^\left\{s\right\} left\left(Ebackslash bigcup_\left\{j\right\}U_\left\{j\right\} right\right)=0 mbox\left\{ or \right\}sum_\left\{j\right\} mathrm\left\{diam\right\} \left(U_\left\{j\right\}\right)^\left\{s\right\}=infty.$

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

$H^\left\{s\right\}\left(E\right)leq sum_\left\{j\right\} mathrm\left\{diam\right\} \left(U_\left\{j\right\}\right)^\left\{s\right\}+varepsilon.$

## 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).

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