Positive trace-class covariance defines a Hilbert Gaussian

By [F1], choose an orthonormal eigenbasis $(e_j{)}_{j\in J}$ of $\mathrm{\Gamma }$ with eigenvalues ${\lambda }_j\ge 0$. This basis is finite or countable: take a dense sequence $(h_k)$ in $H$ and assign to each unit basis vector the least index $k$ with $\mathrm{\parallel }{e}_j-h_k{\mathrm{\parallel }}_H<1\mathrm{/}4$. Distinct orthogonal unit vectors are separated by $\sqrt{2}$, so the assignment is injective. By [F2], ${\sum }_{j\in J}{\lambda }_j=\mathrm{tr}(\mathrm{\Gamma })<\mathrm{\infty }$.

By [A1] and [F3], take independent standard normal variables $(Z_j{)}_{j\in J}$ on a common probability space. Define $T={\sum }_{j\in J}{\lambda }_j{Z}_j^2$ as a nonnegative extended random variable. Tonelli [F5] gives $\mathbb{E}T={\sum }_j{\lambda }_j\mathbb{E}{Z}_j^2=\mathrm{tr}(\mathrm{\Gamma })<\mathrm{\infty }$, so $T<\mathrm{\infty }$ almost surely: if $\mathbb{P}(T=\mathrm{\infty })>0$, then $\mathbb{E}T\ge M\mathbb{P}(T=\mathrm{\infty })$ for every $M>0$, a contradiction.

On $\{T<\mathrm{\infty }\}$ the partial sums $G_N={\sum }_{j\in J,j\le N}\sqrt{{\lambda }_j}{Z}_j{e}_j$ are Cauchy in $H$, since orthonormality gives $\mathrm{\parallel }{G}_M-G_N{\mathrm{\parallel }}_H^2={\sum }_{N<j\le M}{\lambda }_j{Z}_j^2$. Let $G$ be their limit there, and set $G=0$ on the measurable null complement. The convergence set is measurable by the Cauchy criterion. For $F=\mathrm{\varnothing }$ the preimage is empty. For nonempty closed $F\subseteq H$, the set $\{G\in F\}$ is the union of $\{T<\mathrm{\infty },\mathrm{dist}(G_N,F)\to 0\}$ and, when $0\in F$, $\{T=\mathrm{\infty }\}$; the first set is measurable because each distance map is continuous and the second is measurable. Thus $G$ is Borel. To see strong measurability, let $(h_k)$ be a dense sequence in $H$ and, for each $N$, map $x$ to the first point among $h_1,\dots ,h_N$ minimizing distance to $x$. The finitely many tie-broken cells are Borel, so these are finite-valued Borel maps; their distances from $x$ tend to zero by density. Composing them with $G$ gives measurable simple maps converging pointwise to $G$, as required by [F6]. On $\{T<\mathrm{\infty }\}$, $\mathrm{\parallel }G-G_N{\mathrm{\parallel }}_H^2={\sum }_{j>N}{\lambda }_j{Z}_j^2$; [F5] gives $\mathbb{E}\mathrm{\parallel }G-G_N{\mathrm{\parallel }}_H^2={\sum }_{j>N}{\lambda }_j\to 0$ and $\mathbb{E}\mathrm{\parallel }G{\mathrm{\parallel }}_H^2={\sum }_j{\lambda }_j=\mathrm{tr}(\mathrm{\Gamma })<\mathrm{\infty }$.

Fix $h\in H$. The finite sum $⟨G_N,h{⟩}_H$ is a sum of independent centered normal variables, so its characteristic function is $\exp (-t^2{\sum }_{j\le N}{\lambda }_j⟨e_j,h{⟩}_H^2\mathrm{/}2)$ by [F3] and [F4]. The series of variances increases to ${\sum }_j{\lambda }_j⟨e_j,h{⟩}_H^2=⟨\mathrm{\Gamma }h,h{⟩}_H$, and $G_N\to G$ in $L^2$ by step 2.1. Since $\mathrm{\mid }{e}^{itx}-e^{ity}\mathrm{\mid }\le \mathrm{\mid }t\mathrm{\mid }\mathrm{\mid }x-y\mathrm{\mid }$, the characteristic functions converge to $\exp (-t^2⟨\mathrm{\Gamma }h,h{⟩}_H\mathrm{/}2)$; [F3]–[F4] identify $⟨G,h{⟩}_H$ as the centered normal with that variance, including variance zero.

For $h,k\in H$, the finite-sum covariances are $\mathbb{E}[⟨G_N,h{⟩}_H⟨G_N,k{⟩}_H]={\sum }_{j\le N}{\lambda }_j⟨h,e_j{⟩}_H⟨e_j,k{⟩}_H$ by independence, centering and [F3]–[F4]. The $L^2$ convergence from step 2.1 and [F7] pass the left side to the limit, while the eigen-expansion of $\mathrm{\Gamma }$ makes the right side tend to $⟨\mathrm{\Gamma }h,k{⟩}_H$. Hence $G$ has the asserted covariance.