I am learning measure theory, and here is a question I would like to ask.
Definition 1. Let $E \subset \mathbb{R}$ be a measurable set and $x \in E$. The point $x$ is called a density point if $$ \lim _{h \rightarrow 0} \frac{m(E \cap[x-h, x+h])}{m([x-h, x+h])}=1 \, . $$
Definition 2. $\operatorname{dist}(x,E)=\inf\{|x-e|:e\in E\}$
Is it true that every density point of $E$ has $\operatorname{dist}(x,E)=0$?
I think this should be true because, if $\operatorname{dist}(x,E)\neq0$, then $\lim _{h \rightarrow 0} {m(E \cap[x-h, x+h])}=0$,and so the limit above will be $0$, which is a contradiction.