My task is the following:
Construct an open set $E \subset \mathbb{R}^n$ such that $$\lim_{r \to 0^+} \frac{|E \cap B_r(0)|}{|B_r(0)|}$$ does not exist
Here, $| \cdot |$ is the Lebesgue measure on $\mathbb{R}^n$ and $B_r(0)$ is a ball of radius $r$ centered at the origin. I'm a bit unsure of how to go about this particular problem. We previously used the Lebesgue points Theorem to show that for any Lebesgue measurable set $E \subset \mathbb{R}^n$, $$1 = \lim_{r \to 0^+} \frac{|E \cap B_r(x)|}{|B_r(x)|}$$ held for a.e. $x \in E$. Any help would be appreciated!