Solution
= Solution
A finite measure $\mu$ can index the field when its <Green energy>
$$
\iint G(x,y)d\mu(x)d\mu(y)
$$
is finite. On the two-dimensional square, the singularity is locally integrable because polar area contributes $r\,dr$ while $G$ contributes $1/r$. The near-diagonal integral is therefore proportional to $\int_0^\varepsilon dr<\infty$, and the mean $\Gamma(\mu)$ is a well-defined centered Gaussian random variable.