Assertions 1),2),3) follow straightforwardly from the definition. Observe that 1.3.4 says that $PSH(X,\omega)$ is a convex set which is stable under taking maximum and also under the operation $(\varphi,\psi)\mapsto\log[e^{\varphi}+e^{\psi}]$. These are all consequences of the corresponding local properties of psh functions. We nevertheless give a proof, in the spirit of this article. That $(\varphi+\psi)/2\in PSH(X,\omega)$ follows by linearity. The latter assertion is a consequence of the following computation

$$dd^{c}\log[e^{\varphi}+e^{\psi}]=\frac{e^{\varphi}dd^{c}\varphi+e^{\psi}dd^{c}\psi}{e^{\varphi}+e^{\psi}}+\frac{e^{\varphi+\psi}d(\varphi-\psi)\wedge d^{c}(\varphi-\psi)}{[e^{\varphi}+e^{\psi}]^{2}},$$

using that $df\wedge d^{c}f\geq 0$. This computation makes sense if for instance $\varphi,\psi$ are smooth. The general case follows then by regularizing $\varphi,\psi$ (see Appendix). Finally observe that $\max(\varphi,\psi)=\lim j^{-1}\log[e^{j\varphi}+e^{j\psi}]\in PSH(X,\omega)$. ∎