Proposition 13.10.3E1938BA48A2F

Proof. The recursive construction here is the same as the one in the positive Bedford--Taylor recursion. That result proves that each step is a positive current of order zero, so multiplication by the next locally bounded potential is defined; applying $\mathrm{dd}^{\mathrm c}$ makes the next product closed. Symmetry is proved, together with monotone continuity, in the Bedford--Taylor continuity theorem. Its proof uses the common-collar integration-by-parts identity and the Chern--Levine--Nirenberg bound, not the incorrect claim that $\mathrm du\wedge\mathrm d^{\mathrm c}u\wedge T$ is closed. Since all constructions agree in overlapping charts, they define the stated global current.