I'm struggle with proof about statement as follow. I tried to prove. But I think that it doesn't work..
PROPERTY:
Assume that $m^{*}(A)<\infty$. Then, $\forall \varepsilon>0, \exists \mathcal{I}_{\varepsilon}=\left(I_{k}^{\varepsilon}\right)$ countable collection of open intervals with $\bigcup_{k=1}^{\infty} I_{k}^{\varepsilon} \supset A$ such that $\sum_{k=1}^{\infty} l\left(I_{k}^{\varepsilon}\right)<m^{*}(A)+\varepsilon$.