Vitali covering lemma

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

Contents

[edit] Statement of the lemma

  • Finite version: Let B1,...,Bn 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}\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}:j\in 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
\sup_j \mathrm{diam}(B_j)<\infty
where diam(Bj) denotes the diameter of Bj. Then there exists a subcollection \{B_j:j\in J'\}, J'\subset J, of balls from our original collection which are disjoint and
\bigcup_{j\in J} B_{j}\subseteq \bigcup_{j\in J'} 5\,B_{j}.

[edit] 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}\cup\cdots\cup 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_i\subset X for every i=1,2,\dots,n. This is clear if i\in\{j_1,\dots,j_m\}. Otherwise, there necessarily is some k\in\{1,\dots,m\} such that Bi intersects B_{j_k} and the radius of B_{j_k} is at least as large as that of Bi. The triangle inequality then easily implies that B_i\subset 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 {Bj} and let Zi be the collection of balls in Bj whose radius is in (2^{-i-1}\,R,2^{i}\,R]. We first take a maximal disjoint subcollection Z0' of Z0, then take a maximal subcollection Z1' of Z1 that is disjoint and disjoint from \bigcup Z_{0}'. Inductively, we take Zk' to be a maximal disjoint and disjoint from \bigcup_{i=0}^{k-1}\bigcup Z_{i}' subcollection of Zk. It is easy to check that the collection \bigcup_{k=0}^\infty Z'_k satisfies the requirements.

[edit] 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 in the union of a certain collection of balls \{B_{j}:j\in 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}:j\in J'\} which is disjoint and such that \bigcup_{j\in J'}5 B_j\supset \bigcup_{j\in J} B_j\supset E. Therefore,

m(E)\leq m\left(\bigcup_{j\in J}B_{j}\right) \leq m\left(\bigcup_{j\in J'}5B_{j}\right)\leq \sum_{j\in J'} 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 5d, we know that

\sum_{j\in J'} m(5B_{j})=5^d \sum_{j\in J'} m(B_{j})

and thus

m(E)\leq 5^{d}\sum_{j\in J'}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.

[edit] Vitali covering theorem

For a set E ⊆ Rd, 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 U\in\mathcal{V} such that x ∈ U and the diameter of U is less than δ.

Theorem. Let Hs denote s-dimensional Hausdorff measure, let E ⊆ Rd be an Hs-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( E\backslash \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 {Uj} such that

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

[edit] Infinite-dimensional spaces

Unfortunately, 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.

[edit] References

  • Falconer, Kenneth J. (1986). The geometry of fractal sets, Cambridge Tracts in Mathematics 85. Cambridge: Cambridge University Press, xiv+162. ISBN 0-521-25694-1.  MR867284
  • Preiss, David (1979). "Gaussian measures and covering theorems". Comment. Math. Univ. Carolin. 20 (1): 95–99. ISSN 0010-2628.  MR526149
  • Stein, Elias M.; Shakarchi, Rami (2005). Real analysis, Princeton Lectures in Analysis, III. Princeton University Press, xx+402. ISBN 0-691-11386-6.  MR2129625
  • Tišer, Jaroslav (2003). "Vitali covering theorem in Hilbert space". Trans. Amer. Math. Soc. 355: 3277–3289 (electronic). doi:10.1090/S0002-9947-03-03296-3. ISSN 0002-9947.  MR1974687
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