For a subset $E subset mathbb{R}^d$ define the outer measure
$$m_*(E)=inf {Sigma vert I_j vert}$$
Where the infimum is taken over all countable coverings of $E$ by closed intervals. So $E subset bigcup_{j=1}^infty I_j$.
Show $m_*((0,1))=1.$
How do I go about this rigorously? What would my $I_j$ be?
