Denote the Lebesgue outer measure $m^*(A) = inf \{\sum_{n=1}^{\infty} \ell (I_n): A \subseteq \cup_{n=1}^{\infty} I_n$ where $I_n$’s open intervals$\}$.
Consider the “measure” (which is, as it turns out, not actually a measure) $m^{**}$ given by $m^{**}(A) = inf \{\sum_{n=1}^{N} \ell (I_n): A \subseteq \cup_{n=1}^{N} I_n$ where $I_n$’s open intervals$\}$. So, basically, a finite version of the Lebesgue measure.
i) Show that $m^*(A) = m^{**}(A)$ for compact sets $A \subseteq \mathbb{R}$.
ii) Show that $m^{**}(A) = m^{**}(\bar{A})$ for bounded sets $A \subseteq \mathbb{R}$.
I am not really sure how to proceed. Obviously, for the first one, I want to try and use the definition of compactness which says that any open cover of A has a finite sub cover. But then how do I “translate” it to open intervals?
I am not sure exactly how to proceed for the second, but I am hoping to get a better idea on how to work with these functions from the first. Also, obviously I have that $(\bar{A})$ is compact for bounded A.