Q:

Does integration wrt to a differential form always come from a measure?

More precisely, is there an $n$-manifold $M$ with an $n$-form $\omega$ such that there is no measure $\nu$ on $M$ satisfying $$\int f \omega = \int f d\mu $$ for all compactly supported smooth functions $f$?
EDIT: More generally, assuming the answer to the above is yes, what if we have a $k$-form  $\omega$ ($k\leqslant n$)? Then for every oriented $k$-submanifold $S$ we have the functional $$f\mapsto \int _S f \omega,$$ so for every such $S$ there is a measure such that this is integration wrt it. But is there a single measure for all such $S$? I guess you'd have to say something about the orientation here...

A:

I am answering my own question because Aloizio Macedo answered it in the comments, and I don't want it to remain unanswered. (I hope this is OK).
$M$ is locally compact and Hausdorff and either $\int (\cdot) \omega$ or $-\int (\cdot) \omega$ is a positive linear functional*, so by Riesz Representation Theorem one of them comes from a measure.
*I'm not entirely sure I understand why one of them is positive, but I think it's probably an elementary fact of differential forms. 
I still don't know if there's an answer to the more general question I asked.

