Assume $V$ is compact with only log terminal singularities, has index $N$ and let $h^{N}$ be a smooth hermitian metric on $\omega_{V}^{[N]}$. Let $\beta$ be a local generator local of $\omega_{V}^{[N]}$. Define $v_{\beta}(h)$ to be the volume form on $V^{reg}$:

$$v_{\beta}(h)=\left(\sqrt{-1}^{Nn}(-1)^{N\frac{n(n+1)}{2}}\frac{\beta\wedge\bar{\beta}}{\|\beta\|^{2}_{h^{N}}}\right)^{\frac{1}{N}}$$

Since $v_{\beta}(h)$ is independent of $\beta$, this expression defines an adapted measure $v(h)$ with ${\mathcal{C}}^{\infty}$ density on $V$.

Now, let $h^{N}_{sing}=e^{-N\chi}h^{N}$ be a singular metric on $\omega_{V}^{[N]}$. The Chern-Weil form $c_{1}(\omega_{X}^{N},h_{sing}^{N})$ is then well defined as a quasipositive current. Since $h_{sing}^{N}$ has locally ${\mathcal{C}}^{\infty}$+psh potentials $\chi$ is locally bounded above and the above formula defines a measure $v(h_{sing})=e^{\chi}v(h)$ on $V$ such that $\frac{v(h_{sing})}{v(h)}\in L^{\infty}_{loc}$. In particular

$$\frac{v(h_{sing})}{\Omega^{n}}\in L^{1+\varepsilon}(V,\Omega^{n})\text{ for }\varepsilon>0\text{ small enough}.$$

We have $c_{1}(K_{X},h_{sing})=c_{1}(K_{X},h)+dd^{c}\chi$, where $c_{1}(K_{X},h):=\frac{1}{N}c_{1}(\omega^{N}_{X},h^{N})$.

The definition has the virtue of generalizing the usual equivalence between smooth metrics on the canonical sheaf of a manifold and positive definite volume forms to singular metrics and log terminal spaces. This suggests the following ad hoc: