I am having trouble making the so-called "Whitehead equivalence" explicit.

It is quite easy to draw a picture of what a Whitehead move is, given a foliation, as a kind of limit of isotopies of the foliation. What I would like to more carefully understand is precisely what is happening to the transverse measure. The goal is to understand why the map that takes a measured foliation to a map from the space of isotopy classes of closed curves to the real numbers is constant under Whitehead equivalence (which is why, I suppose, measured foliations are considered as Whitehead equivalence classes).

Can anyone point out a resource? The more explicit the better. With respect, I have been reading the book by A. Fathi, F. Laudenbach, and V. Po´enaru ("Thurston's Work on Surfaces"), and while it is quite useful, it does not make this point clear.

Thanks!