Vershik's adic transformations are defined on the path space of a Bratteli diagram, a nonstationary analogue of the graph of a subshift of finite type. They can be used to model many dynamical systems usually constructed in other ways, such as substitution dynamical systems, cutting and stacking constructions and interval exchange transformations. We classify the invariant measures which are finite for some subdiagram defined by erasing vertices and edges; these measures may be infinite on every open subset of the path space. Ingredients include nonstationary versions of the Frobenius decomposition into communicating states, and a nonstationary Frobenius-Victory theorem. This extends and completes work of Bezuglyi, Kwiatkowski, Medynets and Solomyak. (Joint with Marina Talet of Aix-Marseille University).