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Jincheng Yang

The University of Texas at Austin

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Research Blog

Covering Lemma

2020, April 11

Vitali Covering Lemma

Statement

Let $E$ be a measurable set in $\mathbb R ^n$, and let $\{ B _\alpha \} _{\alpha \in \Lambda}$ be a collection of balls that covers $E$ with uniformly bounded radii. Then there exists a pairwise disjoint subcollection $\{ B _{\alpha _j} \}$ such that $\bigcup _j 5 B _{\alpha _j}$ covers $E$.

Algorithm

In $j$-th iteration, we

  • Find the largest ball $B _{\alpha _j}$
  • Remove every $B _\beta$ that intersects with $B _{\alpha _j}$

Besicovitch (Безико́вич) Covering Lemma

Statement

Let $E$ be a measurable set in $\mathbb R ^n$, and Let $\{ Q _\alpha \} _{\alpha \in \Lambda}$ be a collection of cubes with uniformly bounded radii, such that for every $x \in E$ there exists at least one $Q _\alpha$ centering at $x$. Then there exists a finitely overlapping subcover of $E$.

Algorithm

In $j$-th iteration, we

  • Find the largest cube $Q _{\alpha _j}$
  • Remove every $Q _\beta$ that centers within $Q _{\alpha _j}$

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