  We study the cone of transverse measures to a fixed geodesic lamination on an
infinite type hyperbolic surface. Under simple hypotheses on the metric, we
give an explicit description of this cone as an inverse limit of
finite-dimensional cones. We study the problem of when the cone of transverse
measures admits a base and show that such a base exists for many laminations.
Moreover, the base is a (typically infinite-dimensional) simplex (called a
Choquet simplex) and can be described explicitly as an inverse limit of
finite-dimensional simplices. We show that on any fixed infinite type
hyperbolic surface, every Choquet simplex arises as a base for some lamination.
We use our inverse limit description and a new construction of geodesic
laminations to give other explicit examples of cones with exotic properties.
