About exterior measure

Question

Given sets $E_n \subset [0,1]$ with $\sum m^*E_n =\infty$, determine whether

$$m^*(\limsup E_n)>0$$

(we define $\limsup E_n=\cap_{k=1}^{\infty}\cup_{n=k}^{\infty}E_n$).

My attempt: I think $m^*(\limsup E_n)>0$. I want to use contradiction: if $m^*(\limsup E_n)=0$, then $\limsup E_n$ doesn’t contain any open interval ... but I don’t know what to do next .

Answer 1

What about $E_n=[0,1/n]$ ($n\geq1$)?

We have $m^* E_n = 1/n$ so that $\sum m^* E_n$ diverges, and $\limsup E_n = \{0\}$.