The Range of the Monge-Ampère operator $(\omega +d{d}^c\mathrm{.}{)}^n$ in bounded domains

Let $\mathrm{\Omega }$ be a bounded strictly pseudoconvex domain of ${\mathbb{C}}^n$. We solve degenerate complex Monge-Ampère equations of the form $(\omega +d{d}^c\phi {)}^n=\mu$ in the generalized Cegrell classes $\mathcal{K}(\mathrm{\Omega },\omega ,H)$, where $H\in \mathcal{E}(\mathrm{\Omega })$ is maximal, $\omega$ is a smooth real $(1,1)$-form defined in a neighborhood of $\bar{\mathrm{\Omega }}$ and $\mu$ is a positive Radon measure. This generalizes the previous work of the last author \cite{Sal25} to the case of non-continuous functions $H$ and also to the case of measures $\mu$ which do not vanish on pluripolar sets.