First we assume that ${\mathscr{L}}$ is ample. We choose a positive integer $n$ such that ${\mathscr{L}}^{\otimes n}\in\operatorname{Pic}({\mathscr{X}})$ and ${\mathscr{L}}^{\otimes n}$ is very ample. Then we have an embedding $\iota:{\mathscr{X}}\to{\mathbb{P}}(H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n}))$ and ${\mathscr{L}}^{\otimes n}=\iota^{*}({\mathscr{O}}_{{\mathbb{P}}(H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n}))}(1))$. Let $(e_{1},\ldots,e_{r})$ be a free basis of $H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n})$. We define a norm $\|\raisebox{1.72218pt}{.}\|$ of $H^{0}(X,L^{\otimes n})$ to be

Note that $(H^{0}(X,L^{\otimes n}),\|\raisebox{1.72218pt}{.}\|)_{\leq 1}=H^{0}({\mathscr{X}},{\mathscr{L}}^{\otimes n})$, so that, by Proposition 3.8, we have $|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{(H,\|\raisebox{1.20552pt}{.}\|)}(x)=|\raisebox{1.72218pt}{.}|_{\mathscr{L}^{\otimes n}}(x)$ for $x\in X^{\operatorname{an}}$. Thus $h$ is semipositive.

In general, let ${\mathscr{A}}$ be an ample invertible sheaf on ${\mathscr{X}}$ and $A:=\left.{{\mathscr{A}}}\right|_{{X}}$. We choose $\delta\in{\mathbb{Q}}_{>0}$ such that $L\otimes A^{\otimes a}$ is ample for all $a\in(-\delta,\delta)\cap{\mathbb{Q}}$. Note that $\overline{L}\otimes\left(A,|\raisebox{1.72218pt}{.}|_{{\mathscr{A}}}\right)^{\otimes\epsilon}=\left(L\otimes A^{\otimes\epsilon},|\raisebox{1.72218pt}{.}|_{{\mathscr{L}}\otimes{\mathscr{A}}^{\otimes\epsilon}}\right)$, so that $\mu_{\overline{L}\otimes\left(A,|\raisebox{1.20552pt}{.}|_{{\mathscr{A}}}\right)^{\otimes\epsilon}}=0$ for $\epsilon\in(0,\delta)\cap{\mathbb{Q}}$ by the previous observation together with Proposition 3.16. On the other hand, by (3) in Lemma 3.15,

Therefore, $\mu_{\overline{L}}=0$, and hence $h$ is semipositive by Proposition 3.16. ∎