I am currently working on a project, where I would (ideally) like to apply Stokes theorem on a Manifold with corners. I have found various sources, which justify this application. Except one thing:
In the book "Stokes's Theorem and Whitney Manifolds" by Anthony W. Knapp, on page 108, the author uses an indicator function $I_k(x)$ which is 1 on the subset $D(x,E)\geq2^{-k}$ and 0 elsewhere, where E is the set of "exceptional points" such as corners, and D is the distance function (exceptional points are the points on the boundary, which make the boundary non-differentiable). He later arrives at the conclusion, that, if E has the condition$$\operatorname*{lim}_{\delta\downarrow0}\;\delta^{-1}\vert\{x\in\mathbb{R}^{m}\mid D(x,E)\lt \delta\}=0,$$ then Stokes theorem $\int_{B-E}\omega=\int_{U}d\omega$ holds. The problem is that this proof doesn't work, if E is dense, due to the fact that the set of points which satisfies $D(x,E)\geq2^{-k}$ is empty. So, my question is: Is Stokes theorem applicable on manifolds with dense corners (or better, with a dense set of exceptional points)? If so, why/why not?