The standard definition for outer measure of a set of real numbers $A$ is:
$$ m*(A) = inf {\Large \{} \sum_{k=1}^{\infty} \ell(I_k) \; {\Large |} \; A \subseteq \bigcup_{k=1}^{\infty} I_k {\Large \}} $$
(as, for example, in Royden, Real Analysis, 4th ed., p. 31) where the $I_k$ are required to be nonempty, open, bounded intervals.
My question is: Would it amount to the same thing if we instead required that the $I_k$ are nonempty, closed, bounded intervals? It feels like there should be some reason why inner measure uses closed sets and outer measure uses open sets, but I can't think of an example set $A$ where $m*(A)$ would differ from $m**(A)$ defined exactly as above except that each $I_k$ is required to be closed.