I want to show that if is a Lebesgue measurable set where , then for some , where .

MY approach is this. I take , a box in with equal side lengths such that . Setting , take such that . Then and . Since Lebesgue measure is translation invariant, it follows that , and so . If it were empty, then , thus , a contradiction.

Then , and so . Thus the box centered at 0 is contained in .

Is this valid? Thank you.