Proposition 13.2.51FAE60CC76F9

Proof. First let all potentials be locally bounded. Near each point they are limits of decreasing sequences of smooth plurisubharmonic functions, for instance of their mollifications (proposition 5.9.2(2)). The complex Hessians of these functions are semipositive $(1,1)$-forms (proposition 5.9.2(1)), hence strongly positive, since the two notions of positivity coincide in bidegree $(1,1)$; by proposition 5.3.2, their exterior products are strongly positive, and the product of such a product with a weakly positive form of complementary bidegree is a nonnegative multiple of the volume form. Thus the smooth products pair nonnegatively with every smooth compactly supported weakly positive form. By theorem 7.3.7, the Bedford–Taylor product is their weak limit, and therefore it pairs nonnegatively with these forms as well.

By lemma 13.2.1, applied to the bounded products in the definition of $S_ k$, every $S_ k$ is strongly positive. Since $S_ k\to S$ weakly by lemma 13.2.2, so is $S$. $\square$