The Dirichlet problem for the Complex MongeAmpere equation on a bounded pseudoconvex domain in $\mathbb{C}^n$ was studied in BedfordTaylor's seminal paper wherein they defined $(dd^{c} u)^n$ for locally bounded plurisubharmonic $u$. But, they don't seem to use it in their Perrontype method, instead using a convexmeasuretheoretic construction claiming that the upper envelope is not wellbehaved. Now that we know more about psh functions, have people studied the Dirichlet problem without using the measuretheoretic construction of Goffman and Serrin? (and reproved BedfordTaylor's results)
There is a result in the paper Cegrell, Urban On the Dirichlet problem for the complex MongeAmpère operator. Math. Z. 185 (1984), no. 2, 247–251. Theorem. Assume that $\Omega$ is strictly pseudoconvex and that $H(t, z)$ is a measurable, bounded and nonnegative function on $(\infty,, \max h] \times \Omega$, where $h$ is a continuous function on $\partial \Omega$.. If $H(t,z)$ is continuous on $(\infty, \max h]$ for every fixed $z \in \partial \Omega$, then the Dirichlet problem $\phi \in P(\Omega), (dd^c \varphi)^n =H(\varphi, z)dV$ on $\partial \Omega$, $\lim_{z \to \zeta)\varphi(z) = h(\zeta)$ on $\partial \Omega$ has a solution. (Here, $dV$ denotes the Lebesgue measure and $P(\omega)$ is the class of bounded plurisubharmonic functions in $\Omega$.) This theorem is a generalization of Theorem A in Bedford and Taylor where $H^{1/n}$ is also required to be convex and increasing in $t$.(These properties are due to the fact that $H$ comes from the GoffmanSerrin operator.) The proof uses fixed point methods. Further results of this type were obtained (by Cegrell as well as his students and collaborators), mainly in the setting of hyperconvex domains. 

