Let $A \subseteq \Bbb R$, the intervals $I_n$'s in the definition $$m^*(A)=\inf\{\sum_{n=1}^\infty \ell(I_n)~:~A \subseteq \cup_{n=1}^\infty I_n\},$$ of Lebesgue outer measure are often assumed to be open. Is it necessary? Can we take closed intervals instead? Or half open half closed?
Also, what about the $\sup\{ \sum_{n=1}^\infty\ell(I_n)~:~\cup_{n=1}^\infty I_n \subseteq A \}$? Can we equate the supremum with $m^*(A)$?
Best Answer
Note that one of the properties of the Lebesgue outer measure is that any singleton has measure $0$ (easy exercise - to do this can just construct arbitrarily small intervals around the singleton). You can then prove that the union of countable null sets is null. Hence we can add countably many points to our open unions and it won’t change the value we get for our Lebesgue outer measure. In particular, we can make the intervals closed or half-open half-closed.
One of the benefits of the Lebesgue integral over the Riemann/Darboux integral is that we need not take an inner measure/lower bound. Your definition is not correct as the intervals are not disjoint, however even if they were it is not necessary. We basically say sets are nice (Lebesgue measurable) if they don’t cause any problems with countable additivity: if $A_n$ are disjoint then $$m^*(\bigcup\limits_{n=1}^\infty A_n)= \sum\limits_{n=1}^\infty m^*(A_n)$$ These sets turn out to have all the nice properties you could possibly want (the real definition is slightly more complicated but not worth going into here). Note that it is not obvious how to construct a set of sets for which countable additivity fails.