I'm trying to reason through the following problem. (Note: $m$ denotes the Lebesgue measure on $\mathbb{R}$)
For a subset $E\subset[0,1]$, put $U_{\varepsilon}(E)=\{x~:~|x-y|<\varepsilon~\mathrm{for~some}~y\in E\}$. Is it true that for every Borel set $E\subset [0,1]$, $\lim_{\varepsilon\to 0}m(U_{\varepsilon}(E))=m(E)$? Does this limit hold for every closed subset of $[0,1]$?
I'm really not sure how to determine if the statement is true or false (and, moreover, how to prove my answer). I was thinking that as $\varepsilon\to 0$, we may have some sort of decreasing sequence of sets and thus be able to use the continuity of measure. Is this the direction I should take this, or am I completely off track? Thanks in advance for any help -- it would be greatly appreciated!