Example 16.8.19325A49CBCF74

Example 16.8.19.325A49CBCF74 Let $X=\mathbb {P}^1$ and $\omega $ be the Fubini–Study metric. Let $K\subseteq \mathbb {P}^1$ be a polar Cantor set carrying an atom free probability measure $\mu $ supported on $K$ (see [Page 31, Car83]). Write $\mu =\omega +\mathrm{dd}^{\mathrm{c}}\varphi $ for some $\omega $-subharmonic function $\varphi $. Since $\mu $ is atom free, we know that all Lelong numbers of $\varphi $ are $0$. On the other hand, $\varphi $ has $0$ non-pluripolar mass since $K$ is pluripolar.

Then observe that $\varphi \sim _{\mathcal{I}} 0$ while $\varphi \not\sim _ P 0$.