In this paper we study the range of the complex Hessian operator on some weighted classes . For a given measure $\nu$ and a negative real function $\chi$ we prove that $\chi(\mathcal{E}_{m,\chi})\subset \mathbb{L}^1(\Omega, d\nu)$ if and only if the measure $\nu$ can be written as the Hessian of a unique function in $\mathcal{E}_{m,\chi}$. We extend also this result to an energy class with a given boundary data.