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
There are two basic versions of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space R d {\displaystyle \mathbb {R} ^{d}} . In both theorems we will use the following notation: if B = B ( x , r ) {\textstyle B=B(x,r)} is a ball and c ≥ 0 {\displaystyle c\geq 0} , we will write c B {\displaystyle cB} for the ball B ( x , c r ) {\textstyle B(x,cr)} .
Finite version
Infinite version The following proof is based on (Evans & Gariepy 1992, section 1.5.1). Remarks
In the infinite version, the initial collection of balls can be countable or uncountable. In a separable metric space, any pairwise disjoint collection of balls must be countable. In a non-separable space, the same argument shows a pairwise disjoint subfamily exists, but that family need not be countable. The result may fail if the radii are not bounded: consider the family of all balls centered at 0 in Rd; any disjoint subfamily consists of only one ball B, and 5 B does not contain all the balls in this family. The constant 5 is not optimal. If the scale c−n, c > 1, is used instead of 2−n for defining Fn, the final value is 1 + 2c instead of 5. Any constant larger than 3 gives a correct statement of the lemma, but not 3. Using a finer analysis, when the original collection F is a Vitali covering of a subset E of Rd, one shows that the subcollection G, defined in the above proof, covers E up to a Lebesgue-negligible set.
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 d-dimensional Lebesgue measure, λ d {\displaystyle \lambda _{d}} , of a set E ⊂ Rd, which we know is contained in the union of a certain collection of balls { B j : j ∈ J } {\displaystyle \{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 ∈ J ′ } {\displaystyle \left\{B_{j}:j\in J'\right\}} which is disjoint and such that ⋃ j ∈ J ′ 5 B j ⊃ ⋃ j ∈ J B j ⊃ E {\textstyle \bigcup _{j\in J'}5B_{j}\supset \bigcup _{j\in J}B_{j}\supset E} . Therefore,
λ d ( E ) ≤ λ d ( ⋃ j ∈ J B j ) ≤ λ d ( ⋃ j ∈ J ′ 5 B j ) ≤ ∑ j ∈ J ′ λ d ( 5 B j ) . {\displaystyle \lambda _{d}(E)\leq \lambda _{d}{\biggl (}\bigcup _{j\in J}B_{j}{\biggr )}\leq \lambda _{d}{\biggl (}\bigcup _{j\in J'}5B_{j}{\biggr )}\leq \sum _{j\in J'}\lambda _{d}(5B_{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
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