Let $E\subseteq \mathbb{R}$ be a compact set and positive Lebesgue measure. Does there exits $a>0$ such that $$\cap_{0\leq x\leq a}E+x$$ is a set of positive Lebesgue measure?
I proved that above is true if $E$ is an interval.
Edit: I changed the question and add the condition that $E$ is compact.
Best Answer
Take $E$ to be any compact set with positive Lebesgue measure and empty interior, for example the fat Cantor set. Then for any $a>0$, $$\bigcap_{0\leq x \leq a} E +x = \emptyset.$$ (In particular its Lebesgue measure is zero). Indeed, assume by contradiction that there exists $a >0$ such that $\bigcap_{0\leq x \leq a} E+x$ is not empty. Then, there exists $y$ such that $y-x \in E$ for every $0\leq x \leq a$. In other words $[y-a,y] \subset E$. This contradicts the empty interior assumption.