Polygonal approximations of closed parametric varifolds

We define holonomic measures to be certain analogues of varifolds that keep track of the local parameterization and orientation of the submanifold they represent. They are Borel measures on the direct sum of several copies of the tangent bundle. We show that there is an approximation to these by smooth singular chains whose boundaries and Lagrangian actions are controlled. As an illustration of the usefulness of this result, we show how this can be applied to study foliations on the torus. We give other applications elsewhere.