The purpose of these notes is to discuss the Monge-Ampère equation

$$\operatorname{MA}(\phi)=\mu$$

in both the complex and non-Archimedean setting. Here $\mu$ is a positive measure¹¹ 1 All measures in this paper will be assumed to be Radon measures. on the analytification of a smooth projective variety, $\phi$ is a semipositive metric on an ample line bundle on $X$, and $\operatorname{MA}$ is the Monge-Ampère operator. All these terms will be explained below.

In the non-Archimedean case, our presentation is based on the papers [BFJ12, BFJ15] to which we refer for details. In the complex case, we follow [BBGZ13] rather closely. Generally speaking, we avoid technicalities or detailed proofs.