For the â€˜infinite staircaseâ€™ square tiled surface we classify the
Radon invariant measures for the straight line flow, obtaining an analogue
of the celebrated Veech dichotomy for an infinite genus lattice surface. The
ergodic Radon measures arise from Lebesgue measure on a one parameter family
of deformations of the surface. The staircase is a Z-cover of the torus, reducing
the question to the well-studied cylinder map.