The statement that I am trying the proof of is:
$f \,dm$ is regular iff $f \in L^1_{loc}(\mathbb{R}^n, m)$, where $m$ is the Lebesgue measure.
Here is the proof provided by Folland:
In the proof, Folland utilized Corollary 3.6: 
However, $f$ is required in Corollary 3.6 to be integrable instead of only locally integrable. So why is it appropriate to use the corollary here?
One argument that I have seen is that as both $U$ and $E$ are bounded, they are both contained in some compact sets $K$, and thus we can just apply the corollary to $f\chi_K$. However, I am not very convinced by this argument as in my mind, $K$ has to be independent of the choice of $\delta$ here to use corollary for the following reason: Given $\epsilon > 0$, in order for us to determine which $\delta > 0$ to use, we need to invoke corollary 3.6 already. But how do we invoke corollary 3.6 without knowing $U$, which is based on the choice of $\delta > 0$? Maybe I am just confused about the quantifier here, but it seems like then the logic is circular here.
Definition of Regular Borel Measures:
