I am looking for some clarification regarding integrating differential forms. Hopefully this is very basic.
Suppose that I have a compact Riemannian manifold of dimension $n$, say $M$, and a subvariety of codimension $k$, say $V$. Let $\alpha$ be a differential $p$-form defined on $M \setminus V$ which (for the sake of argument) admits no continuous extension to the whole of $M$. Suppose further that $\sup_{M \setminus V} |\alpha| \leq C < +\infty$.
If $$\int_{M \setminus V} \alpha \wedge d\beta =0, \quad \forall \beta \in\Omega^{n-p-1}(M),$$ is there a sensible way to make sense of the claim that $$\int_{M} \alpha \wedge d\beta =0, \quad \forall \beta \in \Omega^{n-p-1}(M)?$$
Apologies for how basic this question is but my geometric measure theory is not good! Answers and references would be greatly appreciated!