I have an assignment asking me to prove that any closed box denoted $B = \{ x : x_i \in [a_i, b_i]\}$ is measurable. I believe I have worked through it, but please tell me if my idea for the solution needs more work.
Outline of my attempt:
I considered $(a, \infty)^n$ to be an interval in $n$ dimensions (if there is a better way to denote/describe this, let me know please).
Then I took some $A \subset \mathbb{R}^n$ and actually showed that $(a, \infty)^n$ is measurable by establishing $m(A) \geq m(A \cap (a, \infty)^n) + m(A \cap (-\infty, a)^n)$ where $m(\cdot)$ denotes the Lebesgue measure.
Then by basically the same proof, $(-\infty, a)^n$ is measurable.
Thus, because the intersection of a countable number of measurable sets is measurable, I concluded that $(-\infty, b)^n \cap (a, \infty)^n = (a,b)^n$ is measurable and hence, all open boxes are measurable.
And therefore, because the complement of a measurable set is measurable, any closed box is measurable as well.
Does this outline look alright?
Additionally, would it then suffice to say that the Lebesgue measure of a closed box is equivalent to the outer measure of a closed box since we found closed boxes to be Lebesgue measurable sets?