Proposition 16.28.10B1F342A8772A

Proof. By lemma 16.28.9, the map $u\mapsto u-\delta \chi /2$ is a bijection between the two families of competitors, which proves the identity for $V_\delta $. Constants below $\inf _ K(\psi -\delta \chi /2)$ are competitors for $G_\delta $.

Since all competitors for $G_\delta $ lie below the constant $\sup _ K(\psi -\delta \chi /2)$ on $K$, lemma 5.40.1 gives a common upper bound for them. Together with a constant competitor, the upper-envelope theorem shows that $G_\delta ^*$ is bounded and $\theta _\delta $-plurisubharmonic. Addition of the smooth function $\delta \chi /2$ in (16.28.11) commutes with regularization, and the two-sided bounds for $G_\delta ^*$ give $F_\delta \in \mathcal L_\delta ^+$.

Locally, adding a potential of $\theta _\delta $ makes the competitors plurisubharmonic. The upper-envelope theorem recalled in section 5.21 shows that regularization changes their supremum only on a pluripolar set. Thus $G_\delta =G_\delta ^*$ almost everywhere, and adding $\delta \chi /2$ gives the same assertion for $V_\delta $. $\square$