Discussion
Let \(\mathcal{N} \subseteq \mathbb{R}\) be a non-measurable set with respect to the Lebesgue measure on \(\mathbb{R}\). Consider the following statements:
P: If \(M = \{x \in \mathcal{N} : x \text{ is irrational}\}\), then M is Lebesgue measurable.
Q: The boundary of \(\mathcal{N}\) has positive Lebesgue outer measure.
Then
Answer: B
P is FALSE and Q is TRUE.
P: \(\mathcal{N}\) is the union of M and \(\mathcal{N} \cap \mathbb{Q}\), which is countable and so measurable with measure zero. If M were measurable, \(\mathcal{N}\) would be a union of two measurable sets and hence measurable, a contradiction. So M is not measurable.
Q: write \(\mathcal{N}\) as the union of its interior and \(\mathcal{N} \cap \partial\mathcal{N}\). The interior is open and so measurable. If the boundary had outer measure zero, \(\mathcal{N} \cap \partial\mathcal{N}\) would be a subset of a null set, hence measurable because Lebesgue measure is complete, and \(\mathcal{N}\) would be measurable. So the boundary must have positive outer measure.
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