example for Lebesgue measure

Question

Construct a closed, uncountable, perfect, nowhere dense subset of $[0,1]$ which has Lebesgue measure $\frac{1}{2}$.

(Hint: Find the Cantor subset of $[0, 1]$ with Lebesgue measure $\frac{1}{2}$)

Answer 1

At the $n$'th stage, if instead of the middle thirds you remove the middle $p$'ths, you multiply the Lebesgue measure by $1 - p$. Can you find a sequence $p_n$ such that $$ \prod_{n=1}^\infty (1 - p_n) = 1/2\ ?$$ Try a "telescoping product", analogous to a telescoping sum.