For $\overline{L},\overline{L}^{\prime}\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)_{{\mathbb{Q}}}$, we have the following:

1. (1)

   $\mu_{\overline{L}\otimes\overline{L}^{\prime}}(x)\leq\mu_{\overline{L}}(x)+\mu_{\overline{L}^{\prime}}(x)$ for $x\in X^{\operatorname{an}}$.

2. (2)

   For $a\in{\mathbb{Q}}_{\geq 0}$, $\mu_{\overline{L}^{\otimes a}}=a\mu_{\overline{L}}$ on $X^{\operatorname{an}}$.

3. (3)

   Let $\overline{L}_{1},\ldots,\overline{L}_{r}$ be elements of $\widehat{\operatorname{Pic}}_{C_{0}}(X)_{{\mathbb{Q}}}$. We assume that there are open intervals $I_{1},\ldots,I_{r}$ of ${\mathbb{R}}$ such that

   $$\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}\in\widehat{\operatorname{Pic}}^{+}_{C^{0}}(X)_{{\mathbb{Q}}}$$

   for all $(t_{1},\ldots,t_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$. Then, for a fixed $x\in X^{\operatorname{an}}$, there is a continuous function $f:I_{1}\times\cdots\times I_{r}\to{\mathbb{R}}$ such that

   $$f(t_{1},\ldots,t_{r})=\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}}(x)$$

   for all $(t_{1},\ldots,t_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$.