How I can write a negligible compact set?

Question

My problem:

Suppose $E$ is a negligible compact set, i.e. $E \subset \mathbb{R}$ is compact and $|E|=0$ where $|.|$ denote the Lebesgue measure. Can I write $E=\bigcap_{k \in \mathbb{N}} U_k$ where $U_k$ is open and $|U_k| \leq 2^{-k}$?

My attempt:

I tried taking for every $n \in \mathbb{N}$ and for every $x \in E$ the ball $B(x,\frac 1 n )$, using the compactness and then taking the intersection of the finite union of these balls. But I am not sure if this work.

Answer 1

The answer is YES. There exist open sets $V_k$ such that $E \subseteq V_k$ and $|V_k| <\frac 1 {2^{k}}$. Also any closed is a countable intersection of open sets. Let $E =\cap_k W_k$ with $W_k$ open. Take $U_k =V_k \cap W_k$.

[We can take $W_k=\{x: d(x,E) <\frac 1 k\}$].