Theorem 5.13.82CB6F65A67EE

Proof. In positive dimension, regard a locally integrable $\varphi$ as a real distribution. If $\mathrm{dd}^{\mathrm c}\varphi\geq0$, the plurisubharmonic representative theorem supplies a plurisubharmonic function $\psi$ representing this distribution, hence equal to $\varphi$ almost everywhere. Conversely, that theorem proves local integrability and positivity of $\mathrm{dd}^{\mathrm c}\psi$ for every plurisubharmonic $\psi$; these properties are unchanged by almost-everywhere equality.

Uniqueness follows from the shrinking-ball formula for the subharmonic representative. This is essential: an arbitrary pointwise representative of a positive distribution need not itself be plurisubharmonic. In dimension zero the domain is a point, with its counting measure; local integrability is finiteness, so both conditions simply say that the value is finite.