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Fix a bounded planar domain $D$ and explicitly choose the Dirichlet form normalizationIts Dirichlet Green function of the Laplacian satisfies $-{\Delta }_x{G}_D(x,y)=2\pi {\delta }_y(x)$. Let $(e_n)$ be an orthonormal basis of the corresponding Sobolev space $H_0^1(D)$. The formal expansionis interpreted as a weak kernel identity. Concretely, for smooth compactly supported test functions $\varphi ,\psi$,The identity follows by applying Parseval's identity to the two Riesz representatives for the integration functionals. It is not a claim that arbitrary Sobolev space basis functions have pointwise values everywhere or that the series converges on the singular diagonal.

For a finite measure $\rho$ with finite Green energy, the same Hilbert-space identity gives ${\sum }_n(\int e_{n}d\rho {)}^2=\iint G_D(x,y)\rho (dx)\rho (dy)$. This is the square-summability condition for a Finite-Green-energy measure pairing with a Gaussian free field. An alternative energy normalization rescales both the basis and the Green kernel; specifying the factor prevents an unnoticed $2\pi$ mismatch.^[ref]

Choose an orthonormal basis $(e_n)$ of $H_0^1(D)$ and write the GFF formally as $h={\sum }_n{\xi }_n{e}_n$, where the ${\xi }_n$ are independent standard normal variables. The Green-kernel expansion in the Dirichlet space givesConsequently the seriesconverges in $L^2$. Its partial sums are centered Gaussian and their variances converge to the displayed Green energy. The $L^2$ limit is therefore Gaussian with mean zero and varianceThis constructs the Finite-Green-energy measure pairing with a Gaussian free field independently of the chosen orthonormal basis.