Let $u$ be a subharmonic function defined on some open set $G\subset \mathbb{C}$. BY definition, it is uppersemicontinuous and $\vartriangle u\geq 0$ in the sense of distribution. Therefore we can regard it as a positive Radon measure $\mu=\vartriangle u$ on $G$ and we can make it compactly supported if we restrict to some relatively compact open subset $G_1\subset G$. Hence, the logarithmic potential of $\mu$ is well-defined:$$ L_{\mu}(z)=\int_{G_1} \log|z-w|^{-1} d\mu(w) $$
I would like to show that $e^{L_\mu(z)}$ is integrable near $0$ if and only if $\mu(\{0\})<2$.