TITLE: Closed sets with empty interior measure zero
QUESTION [9 upvotes]: Is the Lebesgue measure of a closed set with empty interior in $\mathbb{R}^{n}$ always zero?
Trying to understand something in the math notes that I don't understand, and if the above is true, it would make more sense. Not sure if true though!

REPLY [8 votes]: You can just take
$$
U =\bigcup_n B_{1/2^n}(x_n),
$$
where $(x_n)_n$ is an enumeration of $\Bbb{Q}$. 
Then $U$ is open an of finite measure, so that $A=U^c$ is closed and of positive (infinite) measure. 
But the interior of $A$ is void, because it contains no point of $\Bbb{Q}$. 
This example avoids having to understand the construction of the Cantor set.