if the Lebesgue outer measure of a set A which is a subset of R is positive does the set A contain for certain an interior point? I was looking at this  exercise from measure theory.
If the set $A \subset \mathbb{R}$ contains at least one interior point show that $m^*(A)>0$.
I have done the exercise but I am wondering whether the inverse logical statement is true, meaning if
$m^*(A)>0$, does that imply that $A$ contains at least one interior point? I haven't figured anything out. Could someone help me? I feel that the statement is true because for $A$ to have a positive outer measure it must contain an interval and therefore an interior point but I am not certain.

No, it is not true. Take a fat Cantor set, which is measurable with Lebesgue measure greater than $0$ (and therefore its outer measure is greater than $0$). But its interior is empty.
