For any subset $S \subset \mathbb{R}$ of the reals, I know that it is possible to define a 'length' of $S$ as $$\mu(S) = \inf \{\lambda(U) \mid S \subset U\text{, } U \text{ is an open subset of } \mathbb{R} \},$$ where $\lambda(U)$ is the length of the open set $U$. $\lambda$ is well defined given the fact that open subsets of the reals can be represented as the union of (at most countably many) disjoint open intervals.
Then I thought, well then if we define some $$\overline{\mu}(S) = \sup \{\lambda(U) \mid U \subset S\text{, } U \text{ is an open subset of } \mathbb{R} \},$$ would this be an equivalent definition? In other words, would $\mu = \overline{\mu}$ hold?
I am assuming yes, since the statements are seeminly symmetric; but I have no clue on how to prove/disprove this proposition. I would appreciate any help.
If $S=[0,1]\backslash \mathbb{Q}$, then $\mu(S)=1$ but $\overline{\mu}(S)=0$.