The purpose of this question is to gather one/multiple statements of the divergence theorem that can cover most of the cases that one might encounter, say, in a standard PDE course (see the bottom of the post for details). The statement should thoroughly clarify the terms it uses, and should include a reference too, if possible.
The reasons for this question are two:
I couldn't find neither online or on MSE a precise and general enough statement, but not so advanced it would take me hours to understand it (for instance, geometric measure theory versions with sets of finite perimeter and such)
Even in the books I've read (Evans' "Partial differential equations" to mention one), only the classic case is presented ($C^1$ boundary, $C^1$ functions), but then different versions are used.
Summing up, let's remove as many regularity assumptions as possible, making the statement as general as possible, with the constraint of not ending up in something not accessible (see below). Thank you!
By "statement covering the cases one might encounter while in a standard PDE course" I mean something powerful enough to hold on an emisphere for example(I guess we are talking about Lipschitz open bounded sets), and with weak derivatives (even distributional ones are welcome, if there exists a statement with it).
Lastly, I'm comfortable with Sobolev spaces and Hausdorff measures, and I guess a measure theoretic approach is necessary. I will provide feedback to communicate when a reply is "too advanced".