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 $\sum_{n=1}^N|I_n| \geq \beta m^*(E)$.