  We prove a general inequality for mixed Hessian measures by global arguments.
Our method also yields a simplification for the case of complex Monge-Amp\`ere
equation. Exploiting this and using Ko{\l}odziej's mass concentration technique
we also prove the uniqueness of the solutions to the complex Hessian equation
on compact K\"ahler manifolds in the case of probability measures vanishing on
$m$-polar sets.
