Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation $-\chi \left(u\right)\left(d{d}^{c}u\right)ⁿ=d\mu$. Under some additional assumption the solution u is uniquely determined.

Our aim in this article is the study of subextension and approximation of plurisubharmonic functions in ${}_{\chi }(\Omega ,H)$, the class of functions with finite χ-energy and given boundary values. We show that, under certain conditions, one can approximate any function in ${}_{\chi }(\Omega ,H)$ by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

Let μ be a non-negative measure with finite mass given by $\phi \left(d{d}^c\psi \right)ⁿ$, where ψ is a bounded plurisubharmonic function with zero boundary values and $\phi \in L^q\left(\left(d{d}^c\psi \right)ⁿ\right)$, φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

We give a pluripotential-theoretic proof of the product property for the transfinite diameter originally shown by Bloom and Calvi. The main tool is the Rumely formula expressing the transfinite diameter in terms of the global extremal function.