We follow the notation and terminology of §2.6. Consider an ample line bundle $L$ on $X$ and equip $L$ with a model metric $\|\cdot\|$. Any continuous metric on $L$ is then of the form $\|\cdot\|\,e^{-\varphi}$ where $\varphi\in C^{0}(X)$. Recall that this metric is semipositive iff the function $\varphi$ is $\theta$-psh, where $\theta:=c_{1}(L,\|\cdot\|)$. In this case, set

$$c_{1}(L,\|\cdot\|e^{-\varphi})^{n}:=(\theta+dd^{c}\varphi)^{n},$$

where the right hand side is the positive Radon measure in Theorem 3.1.

This is the same measure as the one defined by Chambert-Loir in [CL06]. Indeed, this is certainly true when $\varphi$ is a model function, as seen by comparing (2.2) and [CL06, Définition 2.4]. In general, Corollary 2.12 yields a sequence $(\varphi_{m})_{m=1}^{\infty}$ of $\theta$-psh model functions converging uniformly to $\varphi$ on $X$. The measure $\mu$ associated to $(L,\|\cdot\|e^{-\varphi})$ by Chambert-Loir is the limit of the measures $\mu_{m}:=(\theta+dd^{c}\varphi_{m})^{n}$, see [CL06, Proposition 2.7]. But after replacing $\varphi_{m}$ by $\varphi_{m}+\varepsilon_{m}$ with a suitable sequence $\varepsilon_{m}\searrow 0$, we may assume that the sequence $\varphi_{m}$ is decreasing, hence $\mu=(\theta+dd^{c}\varphi)^{n}$ by Theorem 3.1.