Theorem 13.12.49F9E57CC1422

Proof. By capacity comparison, zero $\theta$-capacity is equivalent to zero Kähler capacity. The comparison with coordinate-ball capacities makes this equivalent to zero local Bedford--Taylor capacity in every member of a finite coordinate cover. The local zero-capacity theorem identifies these zero sets with the pluripolar sets. This proves both implications. For a Borel set, one may first apply the local statement to compact subsets and then use the capacitability and zero-capacity characterization proved with that theorem; a mere measure-zero argument would not suffice.