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Measurability of sets with null interior?

This is an excersise from Chung, Diaconis, Holmes
Let $F$ be a subset of $\mathbb{R}^2$, and suppose that if $x\in F$ then there exists a set $E_x$ (defined in the exercise) such that $E_x\cap F$ has null interior. (Null interior means set is unb https://www.sidestreetstudioarts.org/profile/puflocomgoapromber/profile
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