In the complex case, the Monge-Ampère operator is a second order differential operator: we set $\operatorname{MA}(\phi)=(dd^{c}\phi)^{n}$ for a smooth metric $\phi$. It is a nonlinear operator if $n>1$. When $\phi$ is semipositive, $\operatorname{MA}(\phi)$ is a smooth positive measure on ${X^{\mathrm{an}}}$ of mass $(L^{n})$. It is a volume form, that is, equivalent to Lebesgue measure, if $\phi$ is positive.

Next we turn to the non-Archimedean setting. From now on we assume that $K$ is discretely valued. Pick a uniformizer $t$ of the maximal ideal in the valuation ring $R$ of $K$.

Consider a model metric $\phi_{\mathcal{L}}$, associated to a model $({\mathcal{X}},{\mathcal{L}})$ of $(X,L)$ over $\operatorname{Spec}R$. Write the special fiber as ${\mathcal{X}}_{0}=\operatorname{div}t=\sum_{i\in I}b_{i}E_{i}$, where $E_{i}$ are the irreducible components of ${\mathcal{X}}_{0}$ and $b_{i}\in{\mathbf{Z}}_{>0}$. To each $E_{i}$ is associated a unique (divisorial) point $x_{i}\in{X^{\mathrm{an}}}$. We then define

$$\operatorname{MA}(\phi):=\sum_{i\in I}b_{i}({\mathcal{L}}|_{E_{i}})^{n}\delta_{x_{i}}.$$

If $\phi_{\mathcal{L}}$ is semipositive, ${\mathcal{L}}|_{E_{i}}$ is nef; hence $({\mathcal{L}}|_{E_{i}})^{n}\geq 0$ and $\operatorname{MA}(\phi_{\mathcal{L}})$ is a positive measure. Its total mass is

$$\int_{{X^{\mathrm{an}}}}1\cdot\operatorname{MA}(\phi_{\mathcal{L}})=\sum_{i\in I}b_{i}({\mathcal{L}}|_{E_{i}})^{n}=({\mathcal{L}}^{n}\cdot{\mathcal{X}}_{0})=({\mathcal{L}}^{n}\cdot{\mathcal{X}}_{\eta})=(L^{n}).$$

Here the second to last equality follows from the flatness of ${\mathcal{X}}$ over $\operatorname{Spec}R$, and the last equality from ${\mathcal{X}}_{\eta}\simeq X$, ${\mathcal{L}}_{\eta}\simeq L$.

From now on assume that $L$ is ample, that is, we have a polarized pair $(X,L)$. In both the complex and non-Archimedean case we define $\operatorname{MA}(\phi)$ for a continuous semipositive metric by $\operatorname{MA}(\phi):=\lim_{m\to\infty}\operatorname{MA}(\phi_{m})$ for any sequence $(\phi_{m})_{1}^{\infty}$ converging uniformly to $\phi$. Of course, it is not obvious that the limit exists or independent of the sequence $(\phi_{m})_{1}^{\infty}$. In the complex case this is a very special case of the Bedford-Taylor theory developed in [BT82, BT87]. The analogous analysis in the non-Archimedean case is due to Chambert-Loir [CL06].