Vitali Covering: Proving Finite Interval Covering for m*(E)<∞

[joypav]

Problem:
Let $E$ be a subset of $R$ with $m^∗(E) < ∞$, and let $K$ be a collection of compact intervals $I$ covering $E$. Show that there exists a positive constant $β$ and a finite number of disjoint intervals $I_1, . . . , I_N$ in $K$ such that

$$\sum_{n=1}^N |I_n| ≥ β \cdot m^∗(E)$$Could someone help me out here? I'm guessing this is an application of the Vitali Covering theorem? I'm not sure how to apply it. However, I will keep working and edit if I get somewhere.

 

[Jhoneine]

Solution:This is an application of the Vitali Covering Theorem. By Vitali Covering Theorem, there exists a disjoint collection of intervals $\{I_1, I_2, \dots , I_N\}$ such that $E \subset \bigcup_{n=1}^NI_n$ and $\sum_{n=1}^N|I_n| \geq \beta m^*(E)$ for some positive constant $\beta$. Hence $\sum_{n=1}^N|I_n| \geq \beta m^*(E)$.

 

[math771]

Hi there, yes, this problem can be solved using the Vitali Covering theorem. Here's how you can approach it:

First, note that since $K$ covers $E$, we have $\bigcup_{I\in K} I \supseteq E$. Therefore, by the Vitali Covering lemma, there exists a countable disjoint subcollection $K'=\{I_n\}_{n\in \mathbb{N}}$ of $K$ such that
$$
\bigcup_{n\in\mathbb{N}} I_n \supseteq E\quad \text{and}\quad \sum_{n\in\mathbb{N}} |I_n| \leq 2\cdot m^*(E).
$$
Now, since $m^*(E)<\infty$, we can choose $N$ large enough so that
$$
\sum_{n=1}^N |I_n| \geq \frac{1}{2} \sum_{n\in\mathbb{N}} |I_n| \geq \frac{1}{2} \cdot m^*(E).
$$
Finally, let $\beta = \frac{1}{2}$ and we have
$$
\sum_{n=1}^N |I_n| \geq \beta \cdot m^*(E).
$$
Hope this helps! Let me know if you have any further questions.