  Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix
$1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of
$\omega$-$m$-subharmonic functions is non-decreasing with respect to the
singularity type. We then solve complex Hessian equations with prescribed
singularity, and prove a Hodge index type inequality for positive currents.
