If $\varphi\in\mathcal{E}^{1}(X,\omega)$ and $(\varphi_{j})_{j}$ is a decreasing net of $\omega$-psh functions converging to $\varphi$, then $\varphi_{j}\in\mathcal{E}^{1}(X,\omega)$ for all $j$ and $\MA(\varphi_{j})\to\MA(\varphi)$ as $j\to\infty$ in the weak sense of measures.