Every measurable set in a locally compact, $\sigma$-compact, Hausdorff space $X$, under a suitably nice regular measure, is the union of an $F_\sigma$ and a set of measure zero; this is well-known. However, I am struggling to see how one can write $\mathbb{R}\setminus\mathbb{Q}$ (under the Lebesgue measure on $\mathbb{R}$) as such a union- is there a known such construction? I'm stuck on where to begin. This is certainly a measurable set; $\mathbb R$ is $\sigma$-compact and Hausdorff; and the Lebesgue measure can be obtained via the Riesz-Markov-Kakutani theorem, and is hence "nice" enough, so certainly this is possible, but I can't think of a suitable measure zero subset of $\mathbb R\setminus\mathbb Q$ to remove, or conversely of a suitable $F_\sigma$ to remove.
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