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If $E $ can be covered by a sequence of intervals $(I_n)$ such that $\sum _{n=1}^\infty m^*(I_n)<\infty $ and each $x\in E$ is in infinitely many $I_n$,show that $m^*(E)=0$.

In order to show that $m^*(E)=0$ we have to show that there exists a sequence of intervals $(I_n)$ such that $E\subset \cup I_n;$ with $\sum l(I_n)<\epsilon$ for all $\epsilon>0$.

Suppose for every sequence of intervals $(I_n)$ with $E\subset \cup I_n$ we have $\sum l(I_n)>r$ for some real number $r>0$. Then I think I have to some how exploit the condition that each $x\in E$ is in infinitely many $I_n$,

I am stuck here.Please help.

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    This is the Borel-Cantelli lemma – 2017-01-04
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    You may find this useful: http://math.stackexchange.com/questions/2070972/show-sum-n-ma-n-infty-implies-mb-0 – 2017-01-04

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