For each $t\in\mathbf{R}^{N}$ let $f_{t}$ be the upper envelope of the family of piecewise $\mathbf{Q}$-affine convex functions $f$ on $N_{\mathbf{R}}$ such that $f=g_{\Delta}+O(1)$ and $f(x_{i})\leq t_{i}$ for all $i$, and let $\|\cdot\|_{t}$ be the corresponding continuous toric semipositive metric. By Proposition 8.6, each measure $\mu_{w}$ with $w\in\mathbf{R}_{+}^{N}\cap\{\sum_{i}w_{i}=\deg L\}$ is of the form $c_{1}(L,\|\cdot\|_{t})^{n}$ for some $t\in\mathbf{R}^{N}$. Now elementary Newton polytope considerations show that $f_{t}$ is piecewise $\mathbf{Q}$-affine when all $t_{i}$ are rational, and the result follows by continuity of $t\mapsto c_{1}(L,\|\cdot\|_{t})^{n}$. ∎