That $E_{\omega}$ is nondecreasing, concave and satisfies $E_{\omega}(\varphi+c)=E_{\omega}(\varphi)+c$ follows formally from Proposition 6.1 (using that $\PSH(X,\omega)\cap\mathcal{D}(X)$ is convex and invariant under addition of a constant).

Upper semicontinuity is also a direct consequence of these algebraic properties of $E_{\omega}$ and of Theorem 2.10. Indeed, pick $\varphi_{0}\in\PSH(X,\omega)$ and $t\in\mathbf{R}$ such that $E_{\omega}(\varphi_{0})<t$. We need to show that $E_{\omega}(\varphi)<t$ for $\varphi$ in a neighborhood $U$ of $\varphi_{0}$ in $\PSH(X,\omega)$. By definition, there exists $\psi_{0}\in\PSH(X,\omega)\cap\mathcal{D}(X)$ such that $\psi_{0}\geq\varphi_{0}$ and $E_{\omega}(\psi_{0})<t-\varepsilon$ for some $\varepsilon>0$. By Theorem 2.10, $U:=\{\varphi\in\PSH(X,\omega)\mid\sup_{X}(\varphi-\psi_{0})<\varepsilon\}$ is an open neighborhood of $\varphi_{0}$ in $\PSH(X,\omega)$. By (6.2) we have $E_{\omega}(\varphi)\leq E_{\omega}(\psi_{0})+\varepsilon<t$ for all $\varphi\in U$, which proves upper semicontinuity.

Finally, being usc and nondecreasing, $E_{\omega}$ is automatically continuous along decreasing nets. ∎