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...