The Range of the Monge-Ampère operator (ω+ ddᶜ .)ⁿ in bounded domains

Let $Ω$ be a bounded strictly pseudoconvex domain of Cⁿ. We solve degenerate complex Monge-Ampère equations of the form (ω+ ddᶜ φ)ⁿ = μ in the generalized Cegrell classes K(Ω,ω,H), where H ∈ E(Ω) is maximal, $ω$ is a smooth real $(1,1)$-form defined in a neighborhood of Ω and $μ$ is a positive Radon measure. This generalizes the previous work of the last author {Sal25} to the case of non-continuous functions H and also to the case of measures $μ$ which do not vanish on pluripolar sets.