= Finite-Green-energy measure pairing with a Gaussian free field
{c}
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> $\xi_n$, define
$$
(h,\rho)=\sum_{n\geq1}\xi_n\int_De_n\,d\rho.
$$
The sum converges in <L2 space>, since the sum of the coefficient squares equals the <Green energy>. It is centered Gaussian with
$$
\operatorname{Var}(h,\rho)=\iint G_D(x,y)\rho(dx)\rho(dy).
$$
For 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.<Gaussian free fields for mathematicians by Scott Sheffield>[Sections 2.1–2.3: Gaussian Hilbert spaces and series construction]{ref}
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