Given $\varepsilon>0$ we apply Theorem 5.1 to $\varphi+\varepsilon$ and $\psi$. This gives $\MA(\max\{\varphi+\varepsilon,\psi\})=\MA(\varphi)$ on $G\subseteq\{\varphi+\varepsilon>\psi\}$. Letting $\varepsilon\to 0$ and using Theorem 3.1 we get $\MA(\max\{\varphi,\psi\})=\MA(\varphi)$ on $G$. Exchanging the roles of $\varphi$ and $\psi$ shows that $\MA(\varphi)=\MA(\psi)$ on $G$. ∎

As in [GZ07, Theorem 1.5] the result easily follows from the locality property by integration. More precisely, for any $\varepsilon>0$ we have

so we obtain the desired estimate by letting $\varepsilon\to 0$. ∎

We may assume $0\leq h\leq 1$, $-M\leq\varphi\leq 0$ and $-M\leq\varphi_{j}\leq 0$ for all $j$, where $M\geq 1$. Given $\varepsilon>0$, let $G$ be an open set such that $\Capa_{\omega}(G)<\varepsilon$ and $h$ is continuous on $G^{c}$, see Definition 4.4. Using the Tietze extension theorem, we extend $h|_{G^{c}}$ to a continuous function $\tilde{h}$ on all of $X$ such that $0\leq\tilde{h}\leq 1$. We then have

It follows from Lemma 4.6 that

Since $\tilde{h}$ is continuous, $\int\tilde{h}\MA(\varphi_{j})\to\int\tilde{h}\MA(\varphi)$ as $j\to\infty$, thus

Letting $\varepsilon$ tend to zero completes the proof. ∎

We prove the result for successively more general functions $\varphi$, $\psi$.

Pick an SNC model $\mathcal{X}$ on which $\varphi$, $\psi$ and $\max\{\varphi,\psi\}$ are determined by vertical divisors $A,B$ and $C$ respectively. These three functions are then affine on any face of the dual complex $\Delta_{\mathcal{X}}$. Further, $\MA(\varphi)$ and $\MA(\max\{\varphi,\psi\})$ are both atomic measures, supported on divisorial points corresponding to irreducible components of the special fiber, see §2.7. If $E$ is such a component for which $\varphi(x_{E})>\psi(x_{E})$, then $\varphi(x_{F})\geq\psi(x_{F})$ and hence $\max\{\varphi(x_{F}),\psi(x_{F})\}=\varphi(x_{F})$ for all irreducible components $F$ of the special fiber intersecting $E_{\omega}$, or else $\max\{\varphi,\psi\}$ would not be affine on the face $[x_{E},x_{F}]$ in $\Delta_{\mathcal{X}}$. We have thus shown $\ord_{F}(A)=\ord_{F}(C)$ for all components $F$ of $\mathcal{X}_{0}$ intersecting $E_{\omega}$. If follows that $A|_{E}=C|_{E}$ as numerical classes on $E_{\omega}$, and hence $\MA(\max\{\varphi,\psi\})\{x_{E}\}=\MA(\varphi)\{x_{E}\}$ by definition of Monge-Ampère measures of model functions.

We may assume $-M\leq\varphi,\psi<0$, where $M\geq 1$. Note that the set $\Omega:=\{\varphi>\psi\}$ is open since $\varphi$ is continuous and $\psi$ is usc. It suffices to prove that $\int h\MA(\max\{\varphi,\psi\})=\int h\MA(\varphi)$ for all model functions $h$ whose support is contained in $\Omega$ and such that $0\leq h\leq 1$.

Fix a small number $\delta>0$. By Proposition 4.3 there exists an open set $G\subseteq X$ and a decreasing sequence $(\psi_{j})_{j=1}^{\infty}$ of $\omega$-psh model functions on $X$ such that $\Capa_{\omega}(G)<\delta$ and such that $\psi_{j}$ converges uniformly to $\psi$ on $G^{c}$. Pick $\varepsilon>0$ small and rational and write $\Omega_{j}:=\{\varphi+\varepsilon>\psi_{j}\}$. For $j\gg 0$, we have $\Omega\cap G^{c}\subseteq\Omega_{j}$. Since $\varphi+\varepsilon$ and $\psi_{j}$ are both model functions, we have $\MA(\max\{\varphi+\varepsilon,\psi_{j}\})=\MA(\varphi)$ on $\Omega_{j}$ by Step 1. It follows from Lemma 4.6 that

where we have used $0\leq h\leq 1$ and $-M\leq\varphi+\varepsilon,\psi\leq 0$.

Since $h$ is a model function, it is the difference of two $\omega$-psh model functions by Proposition 2.6. Now $\max\{\varphi+\varepsilon,\psi_{j}\}$ decreases to $\max\{\varphi,\psi\}$ as $j\to\infty$ and $\varepsilon\to 0$, so Theorem 3.1 and the above inequality imply

We obtain the desired equality letting $\delta\to 0$.

Let $(\varphi_{j})_{1}^{\infty}$ be a decreasing net of $\omega$-psh model functions converging to $\varphi$. Write $\Omega_{j}:=\{\varphi_{j}>\psi\}$. This is an open set. Set $u:=\max\{0,\varphi-\psi\}$. Then

By what precedes, $\MA(\max\{\varphi_{j},\psi\})=\MA(\varphi_{j})$ on $\Omega_{j}$. Moreover $\max\{\varphi_{j},\psi\}$ decreases to $\max\{\varphi,\psi\}$ and so the measure $\MA(\max\{\varphi_{j},\psi\})$ converges weakly to $\MA(\max\{\varphi,\psi\})$. Let $f$ be a continuous function on $X$. By Proposition 4.3 $\varphi,\psi$ are quasicontinuous. It follows that $u$ and $fu$ are also quasicontinuous, and applying Lemma 5.4 twice we get that

This holds for every $f\in C^{0}(X)$, so $\one_{\{\varphi>\psi\}}\MA(\max\{\varphi,\psi\})=\one_{\{\varphi>\psi\}}\MA(\varphi)$, as was to be shown. ∎