Vitali covering lemma


In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E  of Rd by a disjoint family extracted from a Vitali covering of E.

Vitali covering lemma

Statement of the lemma

Comments.

Finite version

Without loss of generality, we assume that the collection of balls is not empty; that is, n > 0. Let be the ball of largest radius. Inductively, assume that have been chosen. If there is some ball in that is disjoint from, let be such ball with maximal radius, otherwise, we set m := k and terminate the inductive definition.
Now set. It remains to show that for every. This is clear if. Otherwise, there necessarily is some such that Bi intersects and the radius of is at least as large as that of Bi. The triangle inequality then easily implies that, as needed. This completes the proof of the finite version.

Infinite version

Let F denote the collection of all balls Bj, j ∈ J, that are given in the statement of the covering lemma. The following result provides a certain disjoint subcollection G of F. If this subcollection G is described as, the property of G, stated below, readily proves that
Precise form of the covering lemma. Let  F be a collection of balls in a metric space, with bounded radii. There exists a disjoint subcollection  G of  F with the following property:


Let R  be the supremum of the radii of balls in F. Consider the partition of F into subcollections Fn, n ≥ 0, consisting of balls B  whose radius is in (2−n−1R, 2−nR]. A sequence Gn, with Gn ⊂ Fn, is defined inductively as follows. First, set H0 = F0 and let G0 be a maximal disjoint subcollection of H0. Assuming that G0,...,Gn have been selected, let
and let Gn+1 be a maximal disjoint subcollection of Hn+1. The subcollection
of F satisfies the requirements: G is a disjoint collection, and every ball B ∈ F intersects a ball C ∈ G such that B ⊂ 5 C.

Indeed, let n  be such that B  belongs to Fn. Either B  does not belong to Hn, which implies n > 0 and means that B  intersects a ball from the union of G0,...,Gn−1, or B ∈ Hn and by maximality of Gn, B  intersects a ball in Gn. In any case, B  intersects a ball C  that belongs to the union of G0,...,Gn. Such a ball C  has radius > 2−n−1R. Since the radius of B  is ≤ 2−nR, it is less than twice that of C  and the conclusion B ⊂ 5 C  follows from the triangle inequality as in the finite version.

Remarks

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 d-dimensional Lebesgue measure,, of a set E ⊂ Rd, which we know is contained in the union of a certain collection of balls, 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 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 5d, we know that
and thus

Vitali covering theorem

In the covering theorem, the aim is to cover, up to  a "negligible set", a given set E ⊆ Rd by a disjoint subcollection extracted from a Vitali covering for E : a Vitali class or Vitali covering for E  is a collection of sets such that, for every x ∈ E  and δ > 0, there is a set U  in the collection such that x ∈ U  and the diameter of U  is non-zero and less than δ.
In the classical setting of Vitali, the negligible set is a Lebesgue negligible set, but measures other than the Lebesgue measure, and spaces other than Rd have also been considered, as is shown in the relevant section below.
The following observation is useful: if is a Vitali covering for E  and if E  is contained in an open set Ω ⊆ Rd, then the subcollection of sets U  in that are contained in Ω  is also a Vitali covering for E.

Vitali's covering theorem for the Lebesgue measure

The next covering theorem for the Lebesgue measure λd is due to. A collection of measurable subsets of Rd is a regular family if there exists a constant C  such that
for every set V  in the collection.

The family of cubes is an example of regular family, as is the family of rectangles in R2 such that the ratio of sides stays between m−1 and m, for some fixed m ≥ 1. If an arbitrary norm is given on Rd, the family of balls for the metric associated to the norm is another example. To the contrary, the family of all  rectangles in R2 is not  regular.
Theorem. Let E ⊆ Rd be a measurable set with finite Lebesgue measure, and let be a regular family of closed subsets of Rd that is a Vitali covering for E. Then there exists a finite or countably infinite disjoint subcollection such that
The original result of is a special case of this theorem, in which d = 1 and is a collection of intervals that is a Vitali covering for a measurable subset E  of the real line having finite measure.


The theorem above remains true without assuming that E  has finite measure. This is obtained by applying the covering result in the finite measure case, for every integer n ≥ 0, to the portion of E  contained in the open annulus Ωn of points x such that n < |x| < n+1.
A somewhat related covering theorem is the Besicovitch covering theorem. To each point a of a subset A ⊆ Rd, a Euclidean ball B with center a and positive radius ra is assigned. Then, as in the Vitali theorem, a subcollection of these balls is selected in order to cover A in a specific way. The main differences with the Vitali covering theorem are that on one hand, the disjointness requirement of Vitali is relaxed to the fact that the number Nx of the selected balls containing an arbitrary point x ∈ Rd is bounded by a constant Bd  depending only upon the dimension d; on the other hand, the selected balls do cover the set A of all the given centers.

Vitali's covering theorem for the Hausdorff measure

One may have a similar objective when considering Hausdorff measure instead of Lebesgue measure. The following theorem applies in that case.
Theorem. Let Hs denote s-dimensional Hausdorff measure, let E ⊆ Rd be an Hs-measurable set and a Vitali class
of closed sets for E. Then there exists a disjoint subcollection such that either
Furthermore, if E  has finite s-dimensional Hausdorff measure, then for any ε > 0, we may choose this subcollection such that
This theorem implies the result of Lebesgue given above. Indeed, when s = d, the Hausdorff measure Hs on Rd coincides with a multiple of the d-dimensional Lebesgue measure. If a disjoint collection is regular and contained in a measurable region B  with finite Lebesgue measure, then
which excludes the second possibility in the first assertion of the previous theorem. It follows that E  is covered, up to a Lebesgue-negligible set, by the selected disjoint subcollection.

From the covering lemma to the covering theorem

The covering lemma can be used as intermediate step in the proof of the following basic form of the Vitali covering theorem. Actually, a little more is needed, namely the precised form of the covering lemma obtained in the "proof of the infinite version".
Without loss of generality, one can assume that all balls in F are nondegenerate and have radius ≤ 1. By the precised form of the covering lemma, there exists a disjoint subcollection G of F such that every ball B ∈ F intersects a ball C ∈ G for which B ⊂ 5 C. Let r > 0 be given, and let Z  denote the set of points z ∈ E  that are not contained in any ball from G and belong to the open ball B of radius r, centered at 0. It is enough to show that Z  is Lebesgue-negligible, for every given r.
Let G  denote the subcollection of those balls in G that meet B. Consider the partition of G  into sets Gn, n ≥ 0, consisting of balls that have radius in is contained in B. It follows from the disjointness property of G that
This implies that Gn is a finite set for every n. Given
ε > 0, we may select N  such that
Let z ∈ Z  be fixed. By definition of Z, this point z does not belong to the closed set K  equal to the union of balls in Gk, k ≤ N. By the Vitali cover property, one can find a ball B ∈ F containing z, contained in B and disjoint from K. By the property of G, the ball B  meets C  and is included in 5 C  for some ball C ∈ G. One sees that C ∈ G  because C  intersects B, but C  does not belong to any family Gk, k ≤ N, since B  meets C  but is disjoint from K. This proves that every point z ∈ Z  is contained in the union of 5 C, when C  varies in Gn, n > N, hence
and
Since ε > 0 is arbitrary, this shows that Z  is negligible.

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 separable Hilbert space H so that the Vitali covering theorem fails for. This result was strengthened in 2003 by Jaroslav Tišer: the Vitali covering theorem in fact fails for every infinite-dimensional Gaussian measure on any separable Hilbert space.