Could you tell me how to prove the equality for the volume form? Where can I find the proof?

@PeterMichor I have one doubt. When it comes to $\varepsilon-$ neighbourhood of $C$, $C$ consists of corners of $M$, so by $\\varepsilon-$ neighb. we mean balls of radius $\varepsilon$ around each point of $C$? A bump function has compact support. What would be the compact support of the function mentioned above? It is zero on the neighbourhood of $C$ and nonzero on the complement of $2 \varepsilon-$ neighb of $C$, isn't it?

@PeterMichor Thank you. Could you tell me if there is a way to link the theorem I presented in my post(where we assume that the manifold is orierntable, so we do not need to consider odd forms there) and the chapter about Stokes' theorem for chains in a manifold? For example, in the main proposition on page 27 we assume that a form is odd, because otherwise the integrals vanish.

@PeterMichor I mean, could you tell me how to link this representation of an $n$-form: $\omega = \sum_{\lambda \in \Lambda(m,n)} \omega_{\lambda} dx_{\lambda(1)} \wedge ... \wedge dx_{\lambda(n)}$ with odd and even differential forms defined in the book in paragraphs $4$ and $5$?

@PeterMichor I have one more question. I've been reading de Rham's book and I've read the chapters about even and odd forms a couple of time. Still, I don't understand the difference between odd and even forms. The definition of differential form of degree $n$ upon which I've been working so far is this: it is a map $U \ni x \to \omega(x) \in \mathcal{A}_n(\mathbb{R}^m)$, $U \subset \mathbb{R}^m$ and $\mathcal{A}_n(\mathbb{R}^m)$ are $n$ linear antisymmetric maps $(\mathbb{R}^m)^n \to \mathbb{R}$. Is there a way to make the definitions of odd and even diff. forms agree with the above?

@PeterMichor I see that now. Thank you.

@Peter Michor I'm afraid I do not understand what you mean by "step on a differential form on $M$ with a smooth bump function". Could you write it down more formally? I'm not familiar with this term.

Thank you. Could you explain to me how we can approximate forms on the manifold $M$ with compact support by forms on the manfifold with singularities removed: $M \setminus C$? (I suppose this is the key observation when it comes to that remark about Hausdorff measure zero).