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.
be any finite collection of balls contained in an arbitrary metric space.
Proof: Without loss of generality, we assume that the collection of balls is not empty; that is, n > 0.
be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set m := k and terminate the inductive definition.
be an arbitrary collection of balls in a separable metric space such that
Proof: Consider the partition of F into subcollections Fn, n ≥ 0, defined by
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 of the theorem: G is a disjoint collection, and is thus countable since the given metric space is separable.
In any case, B intersects a ball C that belongs to the union of G0, ..., Gn.
Since the radius of B is less than or equal to 2−nR, we can conclude by the triangle inequality that B ⊂ 5 C, as claimed.
[2] 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,
, each of which has a measure we can more easily compute, or has a special property one would like to exploit.
of measurable subsets of Rd is a regular family (in the sense of Lebesgue) if there exists a constant C such that for every set V in the collection
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
The original result of Vitali (1908) is a special case of this theorem, in which d = 1 and
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.
The main differences between the Besicovitch covering theorem and the Vitali covering lemma 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.
[6] 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 (finite or countably infinite) disjoint subcollection
Furthermore, if E has finite s-dimensional Hausdorff measure, then for any ε > 0, we may choose this subcollection {Uj} such that This theorem implies the result of Lebesgue given above.
Proof: Without loss of generality, one can assume that all balls in F are nondegenerate and have radius less than or equal to 1.
By the infinite form of the covering lemma, there exists a countable disjoint subcollection
are contained in B(r+2), and these balls are disjoint we see Therefore, the term on the right side of the above inequality converges to 0 as N goes to infinity, which shows that Z is negligible as needed.
[7] The Vitali covering theorem is not valid in infinite-dimensional settings.
The first result in this direction was given by David Preiss in 1979:[8] 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.
