Good point. Though the Wikipedia article on it gives the most general form as requiring our region to have closure with finite Lebesgue measure. This is necessa

Good point. Though the Wikipedia article on it gives the most general form as requiring our region to have closure with finite Lebesgue measure. This is necessary of course by the example another corner brought up about swapping the inside and outside. But that means it exactly can't rule out the existence of regions with infinite volume (outside and inside) and finite surface area!