Degenerate complex Hessian equations with arbitrary measure in bounded domains

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