So I'm doing Folland's "Real analysis: modern techniques and applications" and to assert we can't define a non-trivial measure on $P(\mathbb{R})$ he defined this set (which I assume isn't Lebesgue-measurable)
we define an equivalence relation on $[0, 1)$ by declaring that $x \sim y$ iff $x - y$ is rational. Let $N$ be a subset of $[0, 1)$ that contains precisely one member of each equivalence class.
but I'm failing to see how it breaks the measurability test (i.e. $A$ is measurable iff $\forall E \subset \mathbb{R}$ we have $$\mu^*_L(E)=\mu^*_L(E\cap A)+\mu^*_L(E\cap A^c)$$ I also wanna calculate its outer measure if possible. I'm basically trying to get the hang of non-measurable sets and how they break the measurability test (i.e. how the test eliminates all pathological subsets of $\mathbb{R}$) and how they prevent the outer measure from becoming a measure.
