Lebesgue-measure for closed subsets of [0,1]

Question

I have to proof that: for each number $\alpha$ with $0<\alpha<1$ there is a closed subset $C$ of $[0,1]$ that satisfies $\lambda(C)=\alpha$ and includes no non-empty open set.

I think that the construction of the Cantor set might has something to do with it. But I'm not sure where to start.

Answer 1

Thanks, I'll provide my short answer here: Remove the open intervalls in form of $\frac{(1-\alpha)}{3^n}$ out of the intervalls of $C_{n-1}$ like in the classic Cantor-Set, so that $\lambda(C)=1-\frac{1-\alpha}{3}-\frac{2\cdot (1-\alpha)}{9}- ... = 1-\frac{1-\alpha}{3} \sum\nolimits_{k=0}^\infty (\frac{2}{3})^k=1-(1-\alpha)=\alpha$

And because of the construction of $C$ as a Cantor-Set and the well known properties of it the other conditions should be proven.