Let $\|\cdot\|$ be a toric semipositive algebraic metric on $L^{{\text{\rm an}}}$ and let $\psi=\psi_{\|\cdot\|}$ be the associated function on $N_{\mathbb{R}}$. Let $c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}$ be the associated measure. Then

(5.71)

$$({\operatorname{val}}_{K})_{\ast}(c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}})=n!{\overline{\mathcal{M}}}_{M}(\psi),$$

where ${\overline{\mathcal{M}}}(\psi)$ is the measure of Definition 5.32. Moreover,

(5.72)

$$c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}({\operatorname{\mathbf{e}}}_{K})_{\ast}n!{\overline{\mathcal{M}}}_{M}(\psi).$$