Gaussian field existence from a mean and a positive-semidefinite kernel

For each nonempty finite $F\subseteq T$, use AC to choose an enumeration $e_F:\{1,\dots ,n_F\}\to F$. Put $t_i=e_F(i)$, $m_F=(m(t_i){)}_{i=1}^{n_F}$, and $K_F=[K(t_i,t_j){]}_{i,j=1}^{n_F}$. Let $r_F:{\mathbb{R}}^{n_F}\to {\mathbb{R}}^F$ relabel coordinates by $r_F(x)(e_F(i))=x_i$. The matrix $K_F$ is symmetric positive semidefinite by the hypothesis on $K$, so [F1] supplies $N_{n_F}(m_F,K_F)$; define the probability measure on the actual coordinate space by ${\mu }_F:=(r_F{)}_{\mathrm{\#}}{N}_{n_F}(m_F,K_F)$. If $F=\mathrm{\varnothing }$, use the unique probability law on the singleton empty product. Changing the enumeration permutes $m_F$ and both indices of $K_F$ by the same permutation. The characteristic-function formula in [F2] shows that the corresponding permutation of ${\mathbb{R}}^{n_F}$ carries one Gaussian law to the other; after relabelling, the resulting measure on ${\mathbb{R}}^F$ is unchanged. Thus ${\mu }_F$ is independent of the chosen enumeration.

The intervals $(p,q)$ with rational endpoints form a countable family by the countability of $\mathbb{Q}$ and [F4]. They form a base: if $U$ is open and $x\in U$, choose $r>0$ with $(x-r,x+r)\subseteq U$; rational density supplies $p,q\in \mathbb{Q}$ with $x-r<p<x<q<x+r$, so $x\in (p,q)\subseteq U$. Thus $\mathbb{R}$ is second-countable by [F4]. If $x\mathrm{\ne }y$, the open balls about $x$ and $y$ of radius $\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}3$ are disjoint: a point in both would give $\mathrm{\mid }x-y\mathrm{\mid }<2\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}3$ by the triangle inequality. Hence the metric topology is Hausdorff. It is locally compact since each $x$ has the compact interval $[x-1,x+1]$ as a neighborhood by [F4]. Therefore [F3] makes $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ standard Borel.

Let $F\subseteq H$ be finite and nonempty, and use the enumerations fixed in step 1.1. Write ${\pi }_{H,F}:{\mathbb{R}}^H\to {\mathbb{R}}^F$ for restriction. For $u\in {\mathbb{R}}^F$, define $\tilde{u}\in {\mathbb{R}}^{n_H}$ by ${\tilde{u}}_j=u(e_H(j))$ when $e_H(j)\in F$, and ${\tilde{u}}_j=0$ otherwise. Under the coordinate relabelling $r_H$, the characteristic function of $({\pi }_{H,F}{)}_{\mathrm{\#}}{\mu }_H$ at $u$ is the characteristic function of $N_{n_H}(m_H,K_H)$ at $\tilde{u}$. By [F2] this is $\exp \text{\hspace{0.17em}⁣}(i{\sum }_{t\in F}u(t)m(t)-\frac{1}{2}{\sum }_{s,t\in F}u(s)u(t)K(s,t))\mathrm{.}$ This is also the characteristic function of ${\mu }_F=(r_F{)}_{\mathrm{\#}}{N}_{n_F}(m_F,K_F)$ at $u$, by [F2]. Identifying ${\mathbb{R}}^F$ with ${\mathbb{R}}^{n_F}$ through $r_F$, uniqueness of characteristic functions in [F2] gives $({\pi }_{H,F}{)}_{\mathrm{\#}}{\mu }_H={\mu }_F$. Empty sets give the unique singleton marginal. Hence the family is consistent in the sense of [F5].

The consistent family from step 2.1 now satisfies the standard-Borel hypotheses of [F7]. Kolmogorov extension gives a unique probability measure $\mu$ on ${\mathcal{C}}_T$ with the specified finite-coordinate marginals.

On $({\mathbb{R}}^T,{\mathcal{C}}_T,\mu )$ define $X(t)(\omega ):=\omega (t)$. Each coordinate is measurable by [F6], and for every finite set $F$ its joint law is ${\mu }_F$ by [F7]. The means and covariances are therefore $m(t)$ and $K(s,t)$ by [F1]. For any finite list $t_1,\dots ,t_n$, possibly with repeated sites, and coefficients $a_1,\dots ,a_n$, collect the distinct sites in $F=\{t_1,\dots ,t_n\}$ and put $b_t:={\sum }_{j:t_j=t}{a}_j$. Then ${\sum }_j{a}_{j}X(t_j)={\sum }_{t\in F}{b}_{t}X(t)$ is a linear projection of the $F$-coordinate Gaussian vector, so it has a normal law by [F1]. This is the Gaussian-process condition [F8], and the coordinates form a real field by [F9]. The extension supplies only the cylinder sigma-algebra; no continuity or joint-measurability conclusion was used.