Theorem 16.28.445D3A8D823A8

Proof. First suppose that $\theta =\omega $ is Hermitian. Choose bounded lower semicontinuous functions $f_ j\downarrow f$ quasi-everywhere, with common bounds, as in lemma 5.29.15. The bounded functions $p_ j=P_\omega (f_ j)$ decrease to $p$. Indeed, their limit is a competitor for $p$, since it lies below $f$ quasi-everywhere, and the reverse inequality follows from monotonicity.

For fixed $k$, put $H_ k=\min \{ 1,(f-p_ k)_+\} $, a bounded quasi-continuous function. Where $H_ k>0$ and $j\geq k$, we have $p_ j\leq p_ k<f\leq f_ j$ outside a pluripolar set. The lower semicontinuous case therefore gives

The convergence of bounded products against quasi-continuous weights in section 5.29 gives $\int _ XH_ k\, \omega _ p^ n=0$. Since the sets $\{ p_ k<f\} $ increase to $\{ p<f\} $, this proves the assertion for Hermitian $\theta $.

For general semipositive $\theta $, set $\theta _ j=\theta +j^{-1}\omega $. The functions $p_ j=P_{\theta _ j}(f)$ have the common bounds $\inf _ Xf\leq p_ j\leq \sup _ Xf$ and decrease. Their limit is $\theta $-plurisubharmonic by distributional convergence and lies below $f$ quasi-everywhere, so it is at most $p=P_\theta (f)$; since $p\leq p_ j$ for all $j$, it equals $p$. For each fixed $k$, the same coefficient $H_ k=\min \{ 1,(f-p_ k)_+\} $ gives

All $p_ j$ are bounded potentials for the fixed Hermitian form $\theta +\omega $. Expanding the products with background $\theta _ j$ with respect to this fixed background, we may use convergence of bounded products and uniform capacity bounds to pass the last identity to the limit, with the quasi-continuous weight $H_ k$; this gives $\int _ XH_ k\, \mathrm{MA}_\theta (p)=0$. Exhausting $\{ p<f\} $ as before completes the proof. $\square$