The title is basically my question: How do I find the Lebesgue measure of an arbitrary closed box $B$ in $\mathbb{R}^n$?
My idea was to prove that closed boxes are Lebesgue measurable then note that on Lebesgue measurable sets, the Lebesgue measure and the outer measure are equal so that I would have $m(B) = m^*(B) = \prod\limits_{i = 1}^n (b_i - a_i)$ pretty painlessly. But (referencing my previous question) I don't know that measurability = Lebesgue measurability in this case, so I don't know that this is a valid argument here.
Is there better/easier/standard method for finding the Lebesgue measure of a closed box?