Let 𝒩 ⊆ ℝ be a non-measurable set with respect to the Lebesgue measure on ℝ.

Consider the following statements:

𝑃: If 𝑀 = { 𝑥 ∈ 𝒩 ∶ 𝑥 is irrational }, then 𝑀 is Lebesgue measurable.

𝑄: The boundary of 𝒩 has positive Lebesgue outer measure.

Then

Concept -

Explanation -

For Statement (P) -

If N is a non-measurable set with respect to the Lebesgue measure, then any subset of N,

such as M = { x ∈ N : x is irrational },

can also be non-measurable under the Lebesgue measure.

It's important to note that the irrationals themselves form a set of full measure so we can't assume that a subcollection is necessarily measurable.

So, P is not necessarily true.

For Statement (Q) -

The boundary of 𝒩 has positive Lebesgue outer measure.

This is ingeneral true. 

So Statement Q is true.

Hence option (2) is true.