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

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

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.

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