Let $E $ be compact.For $\delta > 0$ let $E (\delta)$ be the set of points which have distance less than $\delta $ to at least one of the points in $E$. Show that $E (\delta)$ is measurable.
How can I proceed? Please help me.
Thank you in advance.
EDIT $:$
It is clear that $E (\delta) \subset B (x;\delta)$ for some $x \in E$. Can it help in proving this fact?