This [paper](http://arxiv.org/pdf/math/0110227v7.pdf), "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the book ["Differential Geometry of Foliations: The Fundamental Integrability Problem"](https://rads.stackoverflow.com/amzn/click/com/3642690173) by Reinhart places a restriction on the transversely oriented foliation to have trivial holonomy. On the other hand, the latter result pertains to $C^k$-foliations, while the first one refers to the smooth case.


Is the claim in the paper false?


EDIT: "Smooth" is a bit of misnomer. What I had in mind is foliations suited to train tracks (so the singular leaves only have cusps as singularities.)



Are all transversely oriented, transversely measured foliations given by closed forms?
The claim is true and you find it on page 319 of Farb-Margalit's book.


The point is that the transition maps between foliation charts (with the leaves as y-level sets) are of the form $(x,y)\to (f(x,y),\pm y+c)$. (This is because the transverse measure has to be preserved. So the transverse measure seems essential for the claim to be true.)


Clearly $dy$ or $-dy$ define the singular foliation in a chart, and by coorientability of the foliation you can then consistently choose one of the two for each chart.


