Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. define two numbers $r(G)$ and $m(G)$ associated with the roots system algebra a group $G$. If dimension manifold is smaller than $r(G)$, then we show action preserves Borel probability measure. at most $m(G)$, there quasi-invariant measure such that measurably isomorphic to relatively measure-preserving over standard boundary action.