Volume of an $\epsilon$-neighborhood of a compact zero measure set

Question

Let $E$ be a compact subset of $\mathbb{R}^n$, s.t. $\lambda(E)=0$, where $\lambda$ is the Lebesgue volume. For $\epsilon>0$, let $E_\epsilon:=\{x:d(x,E)<\epsilon\}$ be the $\epsilon$-neighborhood of $E$. My question is, does $\lambda(E_\epsilon)$ go to 0 when $\epsilon$ goes to 0? And if so, in which rate?

Edit: I’ll rephrase the part I wished to focus on. What can we say about the rate of $\lim\limits_{\epsilon\rightarrow 0^+}\lambda(E_\epsilon)=0$? Can we say there is a $\delta>0$ s.t. $\lambda(E_\epsilon)=O(\epsilon^\delta)$?

This question seems quite standard, but I was surprised to not find an answer in previous discussions; nor was I able to devise a proof on my own.

Thanks ahead

Answer 1

$HINT$

Let $\epsilon_n \to 0$

Then $\bigcap_n{E_n}=E$ where $E_n=\{x:d(x,E)<\epsilon_n\}$

Also use the fact that,for a compact set $E$ that if $d(x,E)=0$, then $x \in E$

Can you continue?