Vitali Cover Review

We will now review some of the recent material regarding Vitali covers.

• On the Vitali Covers of a Set page that if $E$ is a set then a Vitali Cover of $E$ is a collection of intervals $\mathcal I$ such that for all $\epsilon > 0$ we have that for all $x \in E$ there exists an interval $I \in \mathcal I$ such that $x \in I$ and $l(I) < \epsilon$ and we say that $\mathcal I$ Covers $E$ in the Sense of Vitali.

• We then looked at a very important result called the Vitali Covering Lemma which we proved in two parts on the The Vitali Covering Lemma Part 1 and the The Vitali Covering Lemma Part 2. It states that if $E \subseteq \mathbb{R}$, $m^*(E) < \infty$, and $\mathcal I$ is a Vitali cover of $E$ then for all $\epsilon > 0$ there exists a finite collection of mutually disjoint intervals $\{ I_1, I_2, ..., I_N \}$ in $\mathcal I$ such that:

(1)

\begin{align} \quad m^* \left ( E \setminus \bigcup_{n=1}^{N} I_n \right ) < \epsilon \end{align}

• In other words, if $E$ is a subset of $\mathbb{R}$ with finite Lebesgue outer measure, then for every Vitali cover of $E$ there exists a finite collection of mutually disjoint intervals $\{ I_1, I_2, ..., I_N \}$ in the Vitali cover such that the Lebesgue outer measure of $E$ removing the points from the union of the $I_n$s is less than $\epsilon$.