I'm trying to formulate a totally rigorous proof of Green's Theorem, say on a set $S$ bounded by a piecewise smooth simple closed curve. This is not the most general statement, but let's stick with it for now.
Typical proofs usually begin by assuming your shape has a relatively simple description, such as $$ S = \{ a \leq x \leq b, f(x) \leq y \leq g(x) \},$$ for piecewise smooth $f$ and $g$, and $a,b \in \mathbb R$. From here the proof amounts to the grinding out the FTC and Fubini. After this, the proof is said to follow more generally by `crosscutting' your set in such a way that you can partition it into subregions of the form above (or with the $x/y$ coordinates interchanged). The line integrals where these partitions overlap cancel each other out due to orientation.
This last step is what is bothering me -- the existence of the crosscut partition. It seems intuitively clear, but the absurdity of $\mathbb R^n$ leaves me wanting something more rigorous. I even checked Apostol (Calc Vol II), which is my go-to for these sorts of things, and his proof handwaves the existence of the partition.
A couple of thoughts: The boundary curve itself is the only problem. As the boundary is piecewise smooth, it is measurable with measure zero. Perhaps it could be covered by sufficiently small rectangles in an appropriate way ensure that the intersection of those rectangles with $S$ are of the type described above. Alternatively, I could try to adapt the proof of the generalized Stokes' theorem, but this seems like a fundamentally different proof, as I'd be using coordinate charts which could overlap, versus a more strict partition.