Why is it enough to prove the sentence?

Question

I am looking at the proof of the theorem that for any rectangle the outer measure is equal to the volume.

At the beginning of the proof there is the following sentence:

It is enough to look at the case where the rectangle R is closed and bounded.

Why does it stand?

Answer 1

Let $M^*$ the outer measure, $M$ the measure, defined on open sets.

1. If the rectangle is not bounded, then both measures are $\infty$.
2. Now assume that the rectangle is bounded. Consider a non-closed rectangle, that is a measurable set $R$ such as $$ \prod_{i=1}^d (a_i,b_i) \subset R \subset \prod_{i=1}^d [a_i,b_i] $$ Then it is known that $R$ has the same volume as $$\prod_{i=1}^d [a_i,b_i] $$ And if $O$ is an open set such as $\prod_{i=1}^d [a_i,b_i] \subset O$ then $ R\subset O $ as well, and so $M^*(R) \le M^*(\prod_{i=1}^d [a_i,b_i])$
3. Now assume that $R\subset O$, with $O$ an open set. Let $r>0$ be a small number. Then $$ \prod_{i=1}^d [a_i,b_i] \subset O \cup \prod (a_i -r,b_i+r) $$ and $$ M\left( O \cup \prod (a_i -r,b_i+r) \right) \downarrow_{r\downarrow 0} M(O) $$ so $M^*(R) = \inf_{R\subset O \text{ open}} M(O) \ge M^*(\prod_{i=1}^d [a_i,b_i])$