PomeFEFFC91695E0Theorem 5.13.18.FEFFC91695E0 Let $\varphi \colon X\rightarrow [-\infty ,\infty )$ be a measurable function. Then the following are equivalent:
3BB9EAC495FC$\varphi $ is locally integrable and $\mathrm{dd}^{\mathrm{c}}\varphi \geq 0$;
FB5BA344B4A3$\varphi $ coincides almost everywhere with a plurisubharmonic function $\psi $ on $X$.
Moreover, the plurisubharmonic function $\psi $ in 2 is unique.
Proof. Restrict a locally integrable $\varphi$ with $\mathrm{dd}^{\mathrm c}\varphi\geq0$ to each coordinate chart and apply the local representative theorem. The resulting plurisubharmonic representatives agree almost everywhere on overlaps, hence everywhere there by the uniqueness part of the same theorem. They therefore glue to a plurisubharmonic function on $X$ representing $\varphi$.
Conversely, local integrability and positivity hold in every chart for a plurisubharmonic function and are preserved by almost-everywhere equality. The same overlap argument proves uniqueness globally. On a zero-dimensional component all assertions reduce to finiteness at each point.
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