Proof.
Step 1. First we construct $(v^-_ j)_ j$.
Let
\[ C_{K,v} \coloneqq \sup \left\{ \sup _ X \eta : \eta \in \mathrm{PSH}(X,\theta +\omega ), \eta |_ K \leq v\right\} . \]
Since $K$ is non-pluripolar, we have $C_{K,v} \in \mathbb {R}$ by 5.17.4. Set
\[ M_{K,v}\coloneqq \max \{ C_{K,v}+1,\sup _ K v\} . \]
Now define $\widetilde v \colon X \to \mathbb {R}$ as
\[ \widetilde{v}(x)=\left\{ \begin{aligned} v(x),& \quad x\in K;\\ M_{K,v},& \quad x\in X\setminus K. \end{aligned} \right. \]
Since $M_{K,v}\geq \sup _ K v$, the function $\widetilde v$ is lower semicontinuous. Therefore there exists an increasing and uniformly bounded sequence $(v^-_ j)_ j$ in $C^\infty (X)$ such that $v^-_ j \nearrow \widetilde v$ and
\[ \| v_ j^-\| _{C^0(X)}\leq \| \tilde{v}\| _{L^{\infty }(X)}+1. \]
The right-hand side can clearly be bounded by a constant as in (18.28.32).
Step 2. We construct $(v_ j^+)_ j$.
Since
\[ h \coloneqq P_{\theta + \omega ,K}[V_{\theta +\omega }](v) \lor \left(\inf _ K v -1\right) \]
is usc, there exists a decreasing and uniformly bounded sequence $(v^+_ j)_ j$ in $C^\infty (X)$ such that $v^+_ j \searrow h$. We may guarantee that
\[ \| v^+_ j\| _{C^0(X)}\leq \| h\| _{L^{\infty }(X)}+1. \]
Note that $\inf _ K v-1\leq h\leq C_{K,v} \lor (\inf _ K v-1)$. So (18.28.32) is satisfied.
Step 3. We finish the proof. Fix $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ and $\delta \in [0,1]$.
Note that 2 follows from corollary 18.28.14, 18.28.15, proposition 16.5.37 and 1. Let us prove 1.
Step 3.1. We prove 1 for $(v^-_ j)_ j$.
Observe that $P_{\theta + \delta \omega ,X}[\varphi ](v^-_ j)$ is increasing in $j\geq 1$, and
\[ P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right)\leq P_{\theta + \delta \omega ,K}[\varphi ](v). \]
We claim that
A58ACF533BAE
18.28.33
\begin{equation} P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right) \nearrow P_{\theta + \delta \omega ,K}[\varphi ](v)\quad \textup{a.e.}. \end{equation}
Let $\eta \in \mathrm{PSH}(X,\theta +\delta \omega )$ be a potential such that $\eta \preceq \varphi $ and $\eta |_ K\leq v$. Take some $\epsilon >0$. Then, since $\eta $ is upper semicontinuous, $\widetilde v$ is lower semicontinuous and $\eta < \epsilon +\widetilde v$, compactness gives $j_0>0$ such that $\eta < \epsilon + v^-_ j$ for $j \geq j_0$. Hence,
\[ \eta \leq \epsilon +P_{\theta + \delta \omega ,X}[\varphi ](v^-_ j) \]
for $j\geq j_0$. Therefore,
\[ \eta \leq \operatorname *{sup}_{j\geq 1}{}\! {}^{*} P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right)+\epsilon . \]
Letting $\epsilon \to 0+$, we find
\[ \eta \leq \operatorname *{sup}_{j\geq 1}{}\! {}^{*} P_{\theta + \delta \omega ,X}[\varphi ]\left(v^-_ j\right). \]
Since $\eta $ is arbitrary, (18.28.33) follows.
Step 3.2. We prove 1 for $(v^+_ j)_ j$.
Note that $P_{\theta + \delta \omega ,X}[\varphi ](v^+_ j)$ is decreasing in $j$, and
11BA40C76F9B
18.28.34
\begin{equation} \chi \coloneqq \lim _{j\to \infty } P_{\theta + \delta \omega ,X}[\varphi ]\left(v^+_ j\right) \geq P_{\theta + \delta \omega ,K}[\varphi ](v). \end{equation}
In particular, $\chi \in \mathrm{PSH}(X,\theta +\delta \omega )_{>0}$, since $P_{\theta + \delta \omega ,K}[\varphi ](v)$ has the same singularity types as $P_{\theta + \delta \omega }[\varphi ]$ by 18.28.15. Observe that
\[ P_{\theta + \delta \omega ,X}[\varphi ]\left(v_ j^+\right)\leq v_ j^+ \]
for all $j\geq 1$, since $v_ j^+$ is continuous. Thus, $\chi \leq h$. On the other hand, $h|_ K\leq v$ quasi-everywhere. So in particular, $\chi |_{K}\leq v$ quasi-everywhere. On the other hand,
\[ \chi \leq P_{\theta + \delta \omega ,X}[\varphi ]\left(v^+_1\right)\sim _ P \varphi \]
by 18.28.15. Hence, by corollary 18.28.13,
\[ \chi \leq P_{\theta + \delta \omega ,K}[\varphi ](v). \]
Therefore, (18.28.34) is an equality.
$\square$
Pome
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