Lemma of covering of Vitali

The lemma of covering of Vitali is a result Combinatoire of Théorie of integration of the Euclidean spaces. It is largely used in demonstrations in real Analyze.

The basic idea of the lemma is the following one: let us suppose that one has a collection of Cercle S in the plan, authorized to superimpose itself. Then it is possible to extract from it a under-collection whose circles are not intersected, and if one multiplies by 3 their radii, these circles recover the initial collection.

Statement

Version finished: Is $B_ {1},…, B_ {N}$ a collection of balls of Dimension D in a space euclien of dimension D $\ mathbb {R} ^ {D}$ . Then, there exists a under-collection disjoined $B_ {j_ {1}}, B_ {j_ {2}},…, B_ {j_ {m}}$ of these balls satisfying

$B_ {1} \ cup B_ {2} \ cup \ cdots \ cup B_ {N} \ subseteq 3B_ {j_ {1}} \ cup 3B_ {j_ {2}} \ cup \ cdots \ cup 3B_ {j_ {m}}$

where $3B_ {j_ {K}}$ indicating the of the same ball centers than $B_ {j_ {K}}$ but having 3 times its ray.

Version inifnie: Is $\ {B_ {J} \}$ a collection (finished, countable, or indénombrable) of balls of Dimension D in a space euclien of dimension D $\ mathbb {R} ^ {D}$ . Then, there exists a countable under-collection of disjoined balls $\ {B_ {j_ {K}} \} _ {k=1} ^ {\ infty}$ of the initial collection collection with

$\ bigcup_ {J} B_ {J} \ subseteq \ bigcup_ {k=1} ^ {\ infty} 3B_ {j_ {K}}.$

Applications

A direct application of the lemma of covering of Vitali makes it possible to prove the maximum inequality of Hardy-Littlewood. As in this proof, the lemma of Vitali is frequently used when, for example, one studies the Mesure of Lebesgue, $m$ , of a Ensemble $E \ subseteq \ mathbb {R} ^ {D}$ , that one can be contained in the union of a certain collection of balls $\ {B_ {J} \}$ , each one of them having a measurement being able to be calculated easily, or having a particular property which one wishes to exploit. Therefore, if the measurement of this union is calculated, one will have an upper limit of the measurement of $E$ . However, it is difficult to calculate the measurement of the union of these balls if they are superimposed. With the theorem of Vitali, one can choose a under-collection $\ {B_ {j_ {K}} \}$ disjoined. Then, by tripling their ray, this transformed subcollection will contain the volume occupied by the original collection of balls, and thus will cover $E$ . One thus has,

$m (E) \ Leq m \ left (\ bigcup_ {J} B_ {J} \ right) \ Leq m \ left (\ bigcup_ {K} 3B_ {j_ {K}} \ right) \ Leq \ sum_k m (3B_ {j_ {K}})$

As one triples the ray of a ball of dimension D amounts multiplying his volume by a factor of $3^d$ , one a:

$\ sum_k m (3B_ {j_ {K}}) =3^d \ sum_ {K} m (B_ {j_ {K}})$

and thus:

$m (E) \ Leq 3^ {D} \ sum_ {K} m (B_ {j_ {K}}).$

One can use this approach by considering the Dimension of Hausdorff in the place of the Mesure of Lebesgue. In this case, one obtains the following theorem.

Theorem of covering of Vitali

Definition. For a unit $E \ subseteq \ mathbb {R} ^ {D}$ , one defines the class of Vitali $\ mathcal {V}$ for $E$ as being a collection of whole such as for all $x \ in E$ and $\ delta>0$ there exists a unit $U \ in \ mathcal {V}$ such as $x \ in U$ and the Diamètre de $U$ is smaller than $\ delta.$

Theorem. Is $E \ subseteq \ mathbb {R} ^ {D}$ a unit $H^ {S}$ -mesurable and $\ mathcal {V}$ a class of Vitali pour $E$ . Then there exists a disjoined collection, countable $\ {U_ {J} \} \ subseteq \ mathcal {V}$ such as is

$H^ {S} (E \ backslash \ bigcup_ {J} U_ {J}) =0 \ mbox {or} \ sum_ {J} D (U_ {J}) ^ {S} = \ infty.$

Moreover, if $E$ with a measurement of Hausdorff finished, then for all $\ epsilon>0$ , one can choose this under-collection $\ {U_ {J} \}$ such as

$H^ {S} (E) \ Leq \ sum_ {J} D (U_ {J}) ^ {S} + \ epsilon.$

Sources

Measure theory and inegration , Michael E. Taylor, American Mathematical Society.
K.J. Falconer, The Geometry off Fractal Sets , Cambridge University Near, 1985.
Rami Shakarchi & Elias Stein, Princeton Readings in Analysis III: Real Analysis , Princeton University Near, 2005.

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