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.
Categories: math.CV math.AP math.DG