Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of  $A$ minus its interior is positive ?
When studying weak border conditions (in Sobolev Spaces), the usual motivation for the weak meaning of inequalities is that the boundary of most open sets in $\mathbb{R}^n$ has zero Lebesgue measure. But is there any open set $U\subseteq\mathbb{R}^n$ such that it's boundary has positive Lebesgue measure? I really can't think of anything like that.
This is a question from Stromberg related to Steinhaus' Theorem:     If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.   I can't quite see how to modify the Steinhaus proof though.