Interior of a non measurable set

Question

Reading some exercises about Lebesgue measure I found the following True/False statement:

"Every non Lebesgue measurable set in $\Bbb{R}^N$ has empty interior"

We have only really grasped the concept of non measurable sets in class, but it looks like it should be true from the examples I have found online.

Any solid ideas about the veracity of this statement?

Answer 1

Let $N$ be a non-measurable subset of $(2,3)$ and define $M = N \cup (0,1)$.

Then the interior of $M$ is clearly non-empty (contains $(0,1)$) but is non-measurable because otherwise $N = M\setminus (0,1)$ would be measurable, quod non.