Q:

Sets of measure zero on the real line

Is there anything wrong with what I have written here?
Let $D$ be a non-empty subset of the real numbers, with Lebesgue measure $0$.
Say $x$  is in $D$. Since $D$ has measure $0$, it cannot contain any intervals. Therefore,
for any $d > 0$, we know $(x - d,x + d)$ is not contained in $D$. Hence, for any $d > 0$, there is a real number $z$ in $(x - d,x + d)$ so that $z$ is not in $D$.

A:

Yes, the statement is correct: a set $D$ of measure $0$ (a null set) in the reals (or any so-called strictly positive measure space, where $\mu(U) >0$ for any $U$ that is open and non-empty) has empty interior, or $D^\complement$ is dense.

