Let $A \subseteq [a,b]$. Then $\mu^*(A)=0$ if and only if exists a sequence of open intervals $\{I_n\}_{n=1}^{\infty}$ that covers $A$, such that $\sum_{n=1}^{\infty}\mu^*(I_n) < \infty$ and every $x \in A$ belongs to an infinite number of $I_n$'s.
$\mu^*$ denotes the Lebesgue outer measure.
Can someone give me a proof or a help for this statement?
Thank you in advance!