This paper, "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" 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.)