We have defined the outer measure (in $\Bbb{R}^n$) in class as:
$$\mu (A) = \inf \left\{ \sum_{i=1}^\infty v(I_i)\quad \{I_i\}_{i=1}^\infty \text{ collection of open cubes with } A \subset \bigcup_{i=1}^\infty I_i \right\}.$$
I have proved, as our exercises suggested, that the definition holds if we use $I_i$ as open and bounded cubes and also if $I_i$ are compact cubes.
However, since the exercise doesn't suggest to do the same with closed (not necessarily bounded) cubes, I suppose it's false.
Is this true? Would you know any counterexample?