Following the work of Mirzakhani and Souto-Erlandsson, we consider the mapping class group orbit of a filling geodesic current and show that a certain average of these measures converges to Thurston's measure on the space of measured laminations. We use this result to count the number of lattice points in the ball of radius R in Teichmuller space equipped with Thurston's asymmetric metric. This is a joint work with Juan Souto. Speaker(s): Kasra Rafi (University of Toronto)