## $[0, 1]\setminus \mathbb Q$ can’t be exhausted by Jordan sets

We wish to show that $$[0, 1]\setminus \mathbb Q$$ can’t be exhausted by Jordan sets. What I mean by this is that there’s no nested sequence of Jordan sets $$J_1 \subseteq J_2 \subseteq \dots$$ such that $$\bigcup_{k=1}^{\infty}J_k = [0,1]\setminus\mathbb Q$$

Sadly, the fact that the boundary of the union is a subset of the union of the boundaries is only true for finite union, else this would be piece of cake.

How would you tackle this? It may be related to improper integration