  We extend the Besicovitch-Federer projection theorem to transversal families
of mappings. As an application we show that on a certain class of Riemann
surfaces with constant negative curvature and with boundary, there exist
natural 2-dimensional measures invariant under the geodesic flow having
2-dimensional supports such that their projections to the base manifold are
2-dimensional but the supports of the projections are Lebesgue negligible. In
particular, the union of complete geodesics has Hausdorff dimension 2 and is
Lebesgue negligible.
