The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(H)

We prove that, in a $m$-hyperconvex domain in $\mathbb{C}^{n},$ if a non-negative Borel measure is dominated by a complex Hessian measure, then it is a complex Hessian measure of a function in the class $\mathcal{N}_m(H)$. This is an extension of P. {\AA}hg, U. Cegrell, R. Czy\.z and P.H. Hiep's result in \cite{ACCH}.