The lemma 4.26 states the the following:
Suppose G is a nonempty open subset of $\mathbb{R}$. Then there exists a closed set $F \subset G \setminus \mathbb{Q}$ such that $\left|F\right| > 0$.
In his proof he constructs such a set by removing from a closed interval $J$ contained in G exponentially decreasing open balls around the rationals of $J$.
When I tried to prove this theorem before reading his proof, I simply said, that due to the (inner) regularity of the Lebesgue measure, and due to the fact that $\mathbb{Q}$ is a null set, there exists a compact (and thus closed, since $\mathbb{R}$ is Hausdorff) set, that has measure larger than say $\frac{\left|G\right|}{2}= \frac{\left|G \setminus \mathbb{Q}\right|}{2} > 0$.
Does this sound right?