If $M$ is a manifold, you have no volume form, and you cannot give a meaning to integrability of a function. To express integrability, you h

Too long for a comment: If $M$ is a manifold, you have no volume form, and you cannot give a meaning to integrability of a function. To express integrability, you have to use densities, which are smooth sections of the fiber bundle of densities over the manifold $M$. Note that you don't have this difficulty with your example, since you have then the form $dx\wedge d\theta$ on $\mathbb R\times\mathbb R /2π\mathbb Z$.