I have to construct for every $\varepsilon > 0$ a nowhere dense perfect set $\mathit K \subseteq [0,1]$ with $\lambda(K) > 1 - \varepsilon$. A hint is to do it similar as constructing the Cantor set.
I know that the Lebesgue measure of the Cantor set is zero, so $\lambda(C) = 0$.
I also know that the Cantor set can be constructed as follows:
$\mathcal C = [0,1]\backslash \cup_{n=0}^{\infty} \cup_{k=0}^{3^n-1} \left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1}} \right)$.
Unfortunately the hint doesn't help me too much. What would be a good way to start tackling this?
Edit: Do I understand correctly that because $\mathit K \subseteq [0,1]$ the Lebesgue measure $\lambda (K)$ can at mostly be 1?