For any $A \subseteq \mathbb{R}^d$, there exists a $G_\delta$ set $H \supseteq A$ such that for every measurable E, $|A \cap E|_e = |H \cap E|$
I've done the case that $|A|_e < \infty$ using Caratheodory's criterion, but I'm having some trouble with the case that $|A|_e = \infty$. I already have a theorem that some $G_\delta$ set $H \supseteq A$ exists with $|H| = |A|_e = \infty$ (in this case), and whenever $|A \cap E|_e = \infty$ this is rather obvious, but that's not always true. I'm having trouble putting together the exact way to "split" this problem into the bounded case I've already proven since $G_\delta$ sets don't play well with countable unions and I have little to no information about the class of $G_\delta$ sets themselves besides their literal definition as intersections of countably many open sets.
So, for example, I can split A into intersections with a cube grid of $\mathbb{R}^d$, so each piece $A_k$ has finite outer measure, so it has its own $G_\delta$ set of the sort I want thanks to the earlier case, but I can't just union those together because it probably won't end up being a $G_\delta$ set.
What is the right approach to this last case?
EDIT: I've now actually managed to construct using an intersection of individual open sets (which union to a big open set around A) a specific bunch of $H_k$'s which are $G_\delta$ supersets of the $A_k$'s and which union to another $G_\delta$ set $H \supseteq A$ so that $|H_k| = |A_k|_e$ for every k and $|H| = |A|_e = \infty$, but I still can't seem to make an inequality go the direction I need it to; using the finite case I merely have $|H \cap E| = |\cup_{k \in \mathbb{N}}(H_k \cap E)| = \sum_{k \in \mathbb{N}}|H_k \cap E| = \sum_{k \in \mathbb{N}}|A_k \cap E|_e$ and no idea how to ensure that this is at most $|A \cap E|_e$, since of course subadditivity points the wrong way and I already know it's at least that by simple monotonicity.