Let ${\mathcal{L}}^{\#}$ be a residually metrized model of $K_{X}$, determined on a proper dlt model ${\mathcal{X}}$ of $X$. To ${\mathcal{L}}^{\#}$ we associate a positive measure $\mu_{{\mathcal{L}}^{\#}}$ on $\Delta({\mathcal{L}})\subset\Delta({\mathcal{X}})$ defined by

$$\mu_{{\mathcal{L}}^{\#}}=\sum_{\sigma}\left(\int_{Y_{\sigma}}\operatorname{Res}_{Y_{\sigma}}({\mathcal{L}}^{\#})\right)b_{\sigma}^{-1}\lambda_{\sigma},$$

where $\sigma$ runs over the top-dimensional faces of $\Delta({\mathcal{L}})$.