X ω = 1. An upper semicontinuous function φ : X → [−∞,+∞) is called ω-plurisubharmonic (ω-psh) if φ ∈ L(X) and ωφ := ω + dd φ ≥ 0. By PSH(X,ω) (resp. PSH(X,ω)) we denote the set of ω-psh (resp. negative ω-psh) functions on X . The complex Monge-Ampère equation ω u = fω n was solved for smooth positive f in the fundamental work of S. T. Yau (see [Yau]). Later S. Kolodziej showed that there exist...