Degenerate complex Hessian equations with arbitrary measure in bounded domains

Let $\Omega$ be a bounded strictly $m$-pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Hessian equations of the form $(\omega + dd^c \varphi)^m\wedge\beta^{n-m} = \mu$ in the generalized Cegrell classes $\mathcal{K}_m(\Omega,\omega,\phi)$, where $\phi \in \mathcal{E}_m(\Omega)$ is a $m$-maximal function, $\omega$ is a smooth real $(1,1)$-form defined in a neighborhood of $\bar\Omega$ and $\mu$ is a positive Radon measure which is dominated by a Hessian measure of $m$-subharnomic functions in Cegrell class.