Today I was wondering about the relation between the measure of $ E\setminus (E+x)$ where $x\in \mathbb{R}^+ $ and $E+x=\{e+x:e\in E\} $ (off course $E$ is measurable). First I took $E=[a,b] $ for which is clear that $\mu( E\setminus (E+x))=x$. Then I realized that if $E=\mathbb{Q} $ then $0=\mu( E\setminus (E+x))<x$. This examples leaded me to the following conclusion:
$$\mu( E\setminus (E+x)) \leq x$$
I don't know how to prove it so i came here but it looks like a fact to me. I was wondering too what conditions do we need on $ E$ to have an equality (conectedness perhaps since in the intervals it works (?)).
It is false: $E=(0,1)\cup(2,3)$ and $x=\frac{1}{2}$ satisfy
$\mu(E\setminus(E+x))=\mu((0,\frac{1}{2})\cup(2,2+\frac{1}{2}))=1>\frac{1}{2}$.