On the projections of measures invariant under the geodesic flow

We consider general probability measures μ on SM, the unit tangent bundle of a Riemannian surface M, invariant under the geodesic flow, and study their projection to M. We show that if dim μ ≤ dim M(= 2) the projected measure has the same dimension as μ, and if dim μ > dim M the projection is regular with respect to the Lebesgue measure on M.