Let $(F,\nu)$ be a Thurston's foliation on a surface $S$ with a non-zero transverse measure $\nu.$ Assume that $F$ has no closed leaves nor compact separatrices. Did anyone study such foliations?
More specifically, I believe that any other transverse measure on $F$ is a scalar multiple of $\nu$ and I am looking for a quick proof or a reference to this fact.
Such foliations were studied rather intensely in the early works on measured foliations that introduced them to the mathematical world. See for example "Thurston's work on surfaces" aka "Travaux de Thurston sur les Surface" by Fathi, Laudenbach, Poenaru et. al.
Your statement about scalar multiples of $\nu$ is false in general. Constructions of counterexamples, known as "non-uniquely ergodic" measured foliations, go back to work of Keane. For a pretty comprehensive list of references look at page 2 of this paper.
