Good day, I have been working on a problem and one part of it just seems to get over my head.
We are given that $\lambda^n$ is the Lebesgue measure on $\Bbb R^n$ and $E$ is a Lebesgue Measurable set
Show that a dense subset of $[0, 1]^n$ might have Lebesgue measure 0
I know that a subset $A \subset [0, 1]^n $ is dense iff $\forall x \in [0, 1]^n , \forall \delta>0 \ \exists b \in A$ such that
$\vert b-x \vert < \delta$
And I have found some other question on the site that talked about considering the intersection of $\Bbb Q^n \cap [0, 1]^n$ with the Cantor set. But I did not quite understand what those were hinting at.
Your specific question should be answered by considering the set $\mathbb{Q}^n\cap [0,1]^n$, this is countable, and hence both measurable and has Lebesgue measure $0$. In the general case second countability of $\mathbb{R}^n$ should hand you the solution, however this might prove tricky, since measurable sets can look quite ugly (for instance $\mathbb{R}\setminus \mathbb{Q}$).