Any finite-dimensional real vector space $V$ comes equipped with a Lebesgue (or Haar) measure $\lambda$, uniquely defined up to a multiplicative constant. Any lattice $\Lambda\subset V$ allows us to normalize $\lambda$ by $\lambda(V/\Lambda)=1$.

To any top-dimensional differential form $\omega$ on a $C^{\infty}$ manifold $X$ is associated a positive measure $|\omega|$ on $X$. For example, if $\Lambda\subset V$ is a lattice as above, $m_{1},\dots,m_{n}$ is a basis of the dual lattice, then $|dm_{1}\wedge\dots\wedge dm_{n}|$ is Lebesgue measure on $V$ normalized by $\Lambda$.

If $X$ is a complex manifold of dimension $n$, and $\Omega$ is a section of $K_{X}$, that is, a holomorphic $n$-form, we define $|\Omega|^{2}$ as the positive measure

$$|\Omega|^{2}:=\frac{i^{n^{2}}}{2^{n}}|\Omega\wedge\bar{\Omega}|.$$

The normalization is chosen so that the measure associated to the form $dz=dx+idy$ on ${\mathbb{C}}$ is Lebesgue measure $|dz|^{2}=|dx\wedge dy|$ on ${\mathbb{C}}\simeq{\mathbb{R}}^{2}$.

This construction induces a natural bijection between smooth metrics on the canonical bundle $K_{X}$ and (smooth, positive) volume forms on $X$, which associates to a smooth metric $\psi$ on $K_{X}$ the volume form $e^{2\psi}$ locally defined by

$$e^{2\psi}:=\frac{i^{n^{2}}|\Omega\wedge\bar{\Omega}|}{2^{n}|\Omega|^{2}_{\psi}}=\frac{|\Omega|^{2}}{|\Omega|^{2}e^{-2\psi}}$$

for any local section $\Omega$ of $K_{X}$. If $\psi^{\prime}$ is another metric on $K_{X}$, then

$$e^{2\psi^{\prime}}=e^{2(\psi^{\prime}-\psi)}e^{2\psi},$$

where $e^{2(\psi^{\prime}-\psi)}$ is the usual exponential of the smooth function $2(\psi^{\prime}-\psi)\in C^{\infty}(X)$. This can be used to make sense of $e^{2\psi}$ as a positive measure for any (possibly singular) metric $\psi$ on $K_{X}$. Similarly, $e^{2\psi/m}$ is a volume form for every metric $\psi$ on $mK_{X}$, $m\in{\mathbb{Z}}$.

Now assume $(X,B)$ is a pair in the sense of the Minimal Model Program, i.e. $X$ is a normal complex space and $B$ is a (not necessarily effective) ${\mathbb{Q}}$-Weil divisor on $X$ such that

$$K_{(X,B)}:=K_{X}+B$$

is a ${\mathbb{Q}}$-line bundle. Denote by $\phi_{B}$ the canonical singular metric on $B|_{X_{\mathrm{reg}}}$, viewed as a ${\mathbb{Q}}$-line bundle. If $\psi$ is smooth metric on the ${\mathbb{Q}}$-line bundle $K_{(X,B)}$, then $\psi-\phi_{B}$ is a smooth metric on $K_{X_{\mathrm{reg}}\setminus B}$, and $e^{2(\psi-\phi_{B})}$ is thus a volume form on $X_{\mathrm{reg}}\setminus B$.³³ 3 Here and in what follows, we write $X\setminus D$ for the complement of the support of a (not necessarily reduced) divisor $D$ in a complex space $X$.

A pair $(X,B)$ is subklt if for some (or, equivalently, any) log resolution $\rho\colon X^{\prime}\to X$ of $(X,B)$, the unique ${\mathbb{Q}}$-divisor $B^{\prime}$ such that $\rho^{*}K_{(X,B)}=K_{(X^{\prime},B^{\prime})}$ and $\rho_{*}B^{\prime}=B$ has coefficients $<1$. The pair $(X,B)$ is klt if $B$ is further effective.

When $(X,B)$ is subklt, we may thus view $e^{2(\psi-\phi_{B})}$ as a finite positive (Radon) measure on $X$, putting no mass on Zariski closed subsets. Such measures are called adapted in [EGZ09, BBEGZ11].