outer measure for definition question

Question

I'm reading Pugh's "Real Mathematical Analysis" and just started the chapter on Lebesgue Theory. I guess you'd have to a copy available to answer this question, but on page 374, he defines the outer measure of $A\subseteq\mathbb{R^n}$ as the infimum of the measure of open sets that contain $A$. My question is, how do we know that any open sets in $\mathbb{R^n}$ are measurable from what he's already established? He shows this for $\mathbb{R}$ and $\mathbb{R^2}$, but he just seems to be stating it as given for $n>2$. Or does it follow from one of the previous theorems in the section?

Answer 1

The results from the whole section hold for $\mathbb R^n$. In the proof of Theorem $11$, he mentions "the same reasoning applies to bounded subsets of the plane or $\mathbb R^n$", and this holds for all other theorems of the section.

Specifically, all $F_\sigma$ and $G_\delta$ sets are Lebesgue-measurable, and in particular, all open sets of $\mathbb R^n$ are Lebesgue-measurable.