Lemma 17.13.7BC74B8EA334F

Proof. We first observe that for each $x\in X$, we have

Let $R=\{ a+\mathrm{i}b\in \mathbb {C}:a>0,b\in \mathbb {R}\} $. We consider

Let $\pi \colon X\times R\rightarrow X$ be the natural projection. Observe that the upper semicontinuous regularization $G$ of $F$ is $\pi ^*\theta $-psh by proposition 5.14.2. It suffices to show that $F=G$. We let

We want to argue that $E=\varnothing $. Since $F$ is invariant under translations in the imaginary direction of $R$, the same is true for its upper semicontinuous regularization $G$. So $E$ can be written as $B\times \mathrm{i}\mathbb {R}$ for some set $B\subseteq X\times (0,\infty )$. Since $E$ is a pluripolar set by proposition 5.14.5, it has zero Lebesgue measure. Hence, $B$ has zero Lebesgue measure. For each $x\in X$, write

By Fubini's theorem, $B_ x$ has vanishing $1$-dimensional Lebesgue measure for all $x\in X\setminus Z$, where $Z\subseteq X$ is a subset of measure $0$. We may assume that $Z\supseteq \{ \Gamma _{\Gamma _{\max }}=-\infty \} $ so that for $x\in X\setminus Z$, $\Gamma ^*_ t(x)\neq -\infty $ for all $t>0$.

For any $x\in X\setminus Z$, both $t\mapsto F(x,t)$ and $G(x,t)$ are convex functions with values in $\mathbb {R}$ on $(0,\infty )$. They agree almost everywhere, hence everywhere by their continuity. It follows that for $x\in X\setminus Z$, we have $B_ x=\varnothing $.

By proposition 17.13.5, for any $x\in X$, we have

On the other hand, let

By Kiselman's principle proposition 5.14.15, $\chi _{\tau }\in \mathrm{PSH}(X,\theta )$. But on $X\setminus Z$, we already know that $\Gamma _{\tau }=\chi _{\tau }$ for all $\tau <\Gamma _{\max }$. By proposition 5.14.6,

Now we conclude that $F(x,t)=G(x,t)$ by corollary 21.11.26. $\square$