Continuity of the Complex Monge-Ampère Operator

Let Ω be an open subset in C. PSH(Ω) will stand for the set of all plurisubharmonic (psh) functions on Ω. We use the standard notations d = ∂ + ∂ and d = i (∂ − ∂). The complex Monge-Ampère operator (dd) is, via integrations by parts, well defined on PSH(Ω) ∩ Lloc(Ω) and is continuous under monotone limits, that is, (dd uj) n → (ddu) in the sense of currents if the monotone sequence of functions uj converges to u almost everywhere in Ω, see [B-T2]. This basic fact implies an important property that all psh functons are quasi-continuous with respect to the capacity Cn defined by