Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} \sum_{x\in \mathbb{Z}^{d}}|\psi(x)|^{2}<\infty\}$ with inner product $\langle \psi, \varphi\rangle_{l^{2}}:= \sum_{x\in \mathbb{Z}^{d}}\overline{\psi(x)}\varphi(x)$. We can introduce in $l^{2}(\mathbb{Z}^{d})$ the discrete Laplacian as the linear operator: $$(\Delta \psi)(x) := \sum_{k=1}^{d}[-2\psi(x)+\psi(x+e_{k})+\psi(x-e_{k})]$$ where $\{e_{1},...,e_{d}\}$ is the canonical basis of $\mathbb{R}^{d}$. Because $(-\Delta+m^{2})$ has a resolvent for every $m\in \mathbb{R}$, we can consider its inverse $(-\Delta+m^{2})^{-1}$. It's integral Kernel or Green's function $G(x,y)$ is given by: \begin{eqnarray} G(x,y) = \frac{1}{(2\pi)^{d}}\int_{[-\pi,\pi]^{d}}d^{d}p \frac{1}{\lambda_{p}+m^{2}}e^{ip\cdot(x-y)} \tag{1}\label{1} \end{eqnarray} where $p\cdot (x-y) = \sum_{i=1}^{d}p_{i}(x_{i}-y_{i})$ and $\lambda_{p} :=2\sum_{i=1}^{d}(1-\cos p_{i})$ is the eigenvalue of $-\Delta$ associated to its eigenvector $e^{ip\cdot x}$.

[First Approach] If $m \in \mathbb{Z}$, let $s_{m} :=\{\phi\in \mathbb{R}^{\mathbb{N}}: \hspace{0.1cm} \sum_{n=1}^{\infty}n^{2m}|\phi_{n}|^{2} \equiv ||\phi||_{m}^{2}<+\infty\}$, $s:=\bigcap_{m\in \mathbb{Z}}s_{m}$ and $s':=\bigcup_{m\in \mathbb{Z}}s_{m}$. Note that $s$ is a Fréchet space when its topology is given by the family of semi-norms $||\cdot||_{m}$ and $s'$ is the dual space of $s$ if $l_{\psi}$ is a continuous linear map on $s$ with $l_{\psi}(\phi) =( \psi,\phi) := \sum_{n=1}^{\infty}\psi_{n}\phi_{n}$. Let $C=(C_{xy})_{x,y \in \mathbb{Z}^{d}}$ be an 'infinite matrix' with entries $C_{xy}:= G(x,y)$. We can consider $C_{xy}$ to be a matrix $C=(C_{ij})_{i,j \in \mathbb{N}}$ by enumerating $\mathbb{Z}^{d}$. Now, let us define the bilinear map: \begin{eqnarray} s\times s \ni (\phi, \psi) \mapsto \sum_{n=1}^{\infty}\phi_{i}C_{ij}\psi_{j} \equiv (\phi, C\psi) \tag{2}\label{2} \end{eqnarray} Thus, $\phi \mapsto (\phi, C\phi)$ is a quadratic form and we can define: $$W_{C}:=e^{-\frac{1}{2}(\phi,C\phi)}$$ Using Minlo's Theorem for $s$, there exists a Gaussian measure $d\mu_{C}$ on $s'$ (or $\mathbb{R}^{\mathbb{Z}^{d}})$ satisfying: \begin{eqnarray} W_{C}(\psi) = \int_{s'}e^{i(\psi,\phi)}d\mu_{C}(\phi) \tag{3}\label{3} \end{eqnarray}

[Second Approach] For each finite $\Lambda \subset \mathbb{Z}^{d}$, set $C_{\Lambda}$ to be the matrix $C_{\Lambda} =(C_{xy})_{x,y \in \Lambda}$ where $C_{xy}$ are defined as before. Then, these matrices $C_{\Lambda}$ are all positive-definite, so that they define Gaussian measures $\mu_{\Lambda}$ on $\mathbb{R}^{\Lambda}$. Besides, these are compatible in the sense that if $\Lambda \subset \Lambda'$ are both finite and $E$ is a Borel set in $\mathbb{R}^{\Lambda}$ then $\mu_{\Lambda}(E) = \mu_{\Lambda'}(E\times \mathbb{R}^{\Lambda'\setminus\Lambda})$. By Kolmogorov's Extension Theorem, there exists a Gaussian measure $\nu_{C}$ with covariance $C$ on $l^{2}(\mathbb{Z}^{d})$ which is compatible with $\mu_{\Lambda}$ for every finite $\Lambda$.

Now, It seems that these two constructions occur when the so-called thermodynamics limit is taken in QFT and Statistical Mechanics. Both Gaussian measures $\mu_{C}$ and $\nu_{C}$ are measures on $\mathbb{R}^{\mathbb{Z}^{d}}\cong \mathbb{R}^{\mathbb{N}}$. I don't know if this is true but I'd expect these two constructions to be equivalent in some sense, but it is not obvious to me that they are. For instance, the first construction provides a Gaussian measure on $s'$ and the second one on $l^{2}(\mathbb{Z}^{d})$. Are there any relation between these two measures? Are they equal? Maybe the Fourier transform of $\nu_{C}$ would give $W_{C}$, proving these two are the same. Anyway, I'm very lost here and any help would be appreciated.