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Let $\rho$ be a finite measure on $D$ with finite Green energy, in the normalization of the Green-kernel expansion in the Dirichlet space. For an orthonormal basis $(e_n)$ and independent standard Gaussian random variables ${\xi }_n$, defineThe sum converges in L2 space, since the sum of the coefficient squares equals the Green energy. It is centered Gaussian withFor several measures, the same coefficients give their cross-covariances by the corresponding mixed Green integrals.

The joint law is independent of the chosen basis. A stronger pathwise statement needs consistent coefficients: realize the field through one isonormal Gaussian process $W$ on the Dirichlet Hilbert space and take ${\xi }_n=W(e_n)$ for each basis. Then Parseval's identity makes both expansions equal to the same $W$-pairing in L2 space. Drawing a fresh independent coefficient family for a different basis produces another field with the same law, not the same outcome-by-outcome field. This distinguishes basis invariance of a construction from an unspecified coupling of two constructions.^[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.