Lemma 18.28.1518E5130ACB80

Proof. We first prove the $P$-part of (18.28.16).

If $\eta \in \mathrm{PSH}(X,\theta )$ with $\eta \preceq \varphi $ and $\eta \leq 0$, then $\eta +\inf _ K f$ is a candidate in the definition (18.28.3), and hence,

It follows that

Conversely, let $\eta \in \mathrm{PSH}(X,\theta )$ be such that

Since $K$ is non-pluripolar, by 5.17.4 we can find $C=C(\sup _ K f,K, X)>0$ so that $\eta \leq C$. Hence $\eta -C$ is a candidate in the definition of $P_{\theta }[\varphi ]$, and hence

By 18.28.9, this implies that

The proof of the $\mathcal{I}$-part is similar, but we spell it out since this comparison will be used below. If $\eta \in \mathrm{PSH}(X,\theta )$ satisfies $\eta \leq 0$ and $\eta \preceq _{\mathcal{I}}\varphi $, then $\eta +\inf _ K f$ is a candidate in (18.28.4). Hence

Conversely, let $\eta \in \mathrm{PSH}(X,\theta )$ be a candidate in (18.28.4). Since $K$ is non-pluripolar and $f$ is bounded, the same compactness estimate gives a constant $C>0$, independent of $\eta $, such that $\eta \leq C$. Then $\eta -C\leq 0$ and $\eta -C\preceq _{\mathcal{I}}\varphi $. By corollary 16.8.16,

Taking the upper envelope over all such $\eta $ gives

This completes the proof. $\square$