An exercise about Lebesgue measure on compact sets of $\mathbb{R}$

Question

Let $E$ and $F$ two compact subsets of $\mathbb{R}$ such tha $E$$\subset$$F$ and $\mu$$(E)$$<$$\mu$$(F)$. Prove that $\forall$$a$$\in$$($$\mu$$(E)$,$\mu$$(F)$$)$ we can find a compact set $K$ such that $E$$\subset$$K$$\subset$$F$ and $\mu$$(K)$$=a$.

I came up with the idea of constructing a continuous function on the interval $($$\mu$$(E)$,$\mu$$(F)$$)$ and using the intermediate value theorem. Maybe this is a wrong idea but I am stuck.

Can someone help me ?

Thank you in advance!

Answer 1

Hint: For $r \ge 0,$ let $E_r=\{x\in F: d(x,E)\le r\}.$ Show each $E_r$ is compact. Define $f(r) = \mu(E_r).$ Then $f(0) = \mu(E)$ and $f(r)=\mu(F)$ for large enough $r.$ If you show $f$ is continuous, you'll be done by the IVT. For continuity you'll want to use "continuity properties of measures", along with the fact that $\mu(\{x\in F: d(x,E)= r\}) =0$ for $r>0.$