Let $\Sigma$ be a complete fan of $N_{\mathbb{R}}$, let $\Psi$ be a support function on $\Sigma$ and let $L=\mathcal{O}(D_{\Psi})$. Let $\|\cdot\|$ be an approachable metric on $L^{{\text{\rm an}}}$ and let $\psi=\psi_{\|\cdot\|}$ be the corresponding concave function. Then

(5.82)

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

Moreover, the measure $c_{1}(\overline{L})^{n}\land\delta_{X_{\Sigma}}$ is characterized, in the Archimedean case, by equation (5.82) and the fact of being toric, while in the non-Archimedean case it is given by

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