The most fundamental example which may serve as a guideline to everything that follows is the case where $X=\mathbb{C}\mathbb{P}^{n}$ is the complex projective space and $\omega=\omega_{FS}$ is the Fubini-Study Kähler form. There is then a 1-to-1 correspondence between $PSH(\mathbb{C}\mathbb{P}^{n},\omega_{FS})$ and the Lelong class

$${\mathcal{L}}(\mathbb{C}^{n}):=\left\{\psi\in PSH(\mathbb{C}^{n})\,/\,\psi(z)\leq\frac{1}{2}\log[1+|z|^{2}]+C_{\psi}\right\}$$

which is given by the natural mapping

$$\psi\in{\mathcal{L}}(\mathbb{C}^{n})\mapsto\varphi(x)=\left\{\begin{array}[]{ccc}\psi(x)-\frac{1}{2}\log[1+|x|^{2}]&\text{ if }&x\in\mathbb{C}^{n}\\ \overline{\lim}_{y\in\mathbb{C}^{n}\rightarrow x}(\psi(y)-\frac{1}{2}\log[1+|y|^{2}])&\text{ if }&x\in H_{\infty},\end{array}\right.$$

where $H_{\infty}$ denotes the hyperplane at infinity. One can easily show that this mapping is bicontinuous for the $L_{loc}^{1}$ topology.