Vitali covering lemma

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

tatement 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

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

: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

::

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 . 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 . Inductively, we take $Z\_\{k\}\text{'}$ to be a maximal disjoint and disjoint from subcollection of $Z\_k$. It is easy to check that the collection 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 . Therefore,

:

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

:

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

* cite book
last = Falconer
first = Kenneth J.
title = The geometry of fractal sets
series = Cambridge Tracts in Mathematics 85
publisher = Cambridge University Press
location = Cambridge
year = 1986
pages = xiv+162
isbn = 0-521-25694-1 MathSciNet|id=867284
* cite journal
last = Preiss
first = David
title = Gaussian measures and covering theorems
journal = Comment. Math. Univ. Carolin.
volume = 20
year = 1979
issue = 1
pages = 95–99
issn = 0010-2628 MathSciNet|id=526149
* cite book
last = Stein
first = Elias M.
coauthors = Shakarchi, Rami
title = Real analysis
series = Princeton Lectures in Analysis, III
publisher = Princeton University Press
address = Princeton, NJ
year = 2005
pages = xx+402
isbn = 0-691-11386-6 MathSciNet|id=2129625
* cite journal
last = Tišer
first = Jaroslav
title = Vitali covering theorem in Hilbert space
journal = Trans. Amer. Math. Soc.
volume = 355
year = 2003
number = 8
pages = 3277–3289 (electronic)
issn = 0002-9947
doi = 10.1090/S0002-9947-03-03296-3 MathSciNet|id=1974687