existence of a ball in a measurable set Let $\lambda$ be a Lebesgue-measure on $\mathbb{R}^n$ and let $A\subset \mathbb{R}^n$ be Lebesgue-measurable set with $\lambda(A)>0$. I know that the difference set $A - A$ contains an open ball $B_r(x)$. My question is if the set A also contains an open ball with $B_r(x) \subset A$ where $r>0$?

No, not necessarily. One interesting counterexample is a "fat" Cantor set.
