To extend the theory to the case where $K$ is not algebraically closed, choose an algebraic closure of $K$ and denote its completion by $\mathbb{C}_{K}$. Then we define the Monge-Ampère measure as the push-forward of the previously defined Monge-Ampère measure on the base change to $\mathbb{C}_{K}$. We explain it here in the situation of Definition 4.11. Let $V$ be a strictly $K$-analytic Hausdorff space of dimension $n$, $\overline{L}_{1},...,\overline{L}_{n}$ potentially semipositive piecewise linear metrized line bundles on $V$ (i.e. metrized line bundles on $V$ which become semipositive piecewise linear metrized line bundles after base change to $\mathbb{C}_{K}$) and $\pi:V_{\mathbb{C}_{K}}\rightarrow V$ the base change. We can then define a measure on $V_{\mathbb{C}_{K}}$ with respect to the pull-backs of the line bundles $\overline{L}_{1},...,\overline{L}_{n}$ by Definition 4.11 and push the resulting measure forward to $V$ via $\pi$. To make this well defined we show that $\pi$ is a proper map of topological spaces. So let $C\subseteq V$ be compact. Then we can cover $C$ by finitely many affinoid subdomains $U_{1},...,U_{r}$. Then $\pi^{-1}(C)=\pi^{-1}\left(\bigcup C\cap U_{i}\right)=\bigcup\pi^{-1}(C\cap U_{i})$ and it is enough to show that $\pi^{-1}(C\cap U_{i})$ is compact for any $i$ so we may assume that $V$ is affinoid. But then $\pi$ is a continuous map between compact Hausdorff spaces and hence proper which yields the claim. We denote this measure again by $c_{1}(\overline{L}_{1})\wedge...\wedge c_{1}(\overline{L}_{n})$. One can check that all the results of this section remain true in this more general situation.