If $\mu$ is a probability measure on $X=\mathbb{C}\mathbb{P}^{n}$ and $\omega$ denotes as before the Fubini-Study Kähler form, then

$$\varphi_{\mu}(x):=\int_{\mathbb{C}\mathbb{P}^{n}}\log\left(\frac{||x\wedge y||}{||x||\cdot||y||}\right)d\mu(y)$$

defines a $\omega$-psh function on $\mathbb{C}\mathbb{P}^{n}$. Such functions have been considered by Molzon, Shiffman and Sibony [32], [31] in order to define capacities on $\mathbb{C}\mathbb{P}^{n}$. However they do not characterize pluripolar sets.