Let $P$ be the set of irrational numbers in $[0,1]$. The set $P$ has Lebesgue measure $1$ and by regularity, there is for every $\epsilon>0$ a compact set $C\subseteq P$ satisfying $\lambda(C)>1-\epsilon$. The proof I know for this fact gives no indication as to how the approximating compact sets might look like. This leaves a big gap in my intuition and I would therefore like to know:
Is there an explicit way to construct for a given $\epsilon>0$ a compact set $C$ of irrational numbers between $0$ and $1$ such that $\lambda(C)>1-\epsilon$?