Weighted Hilbert Gaussian series and prior regularity

Put $Y_N={\sum }_{j=1}^N{\lambda }_j{w}_j{\xi }_j^2$ and $Y={\lim }_N{Y}_N$. The supplied second moments give $\mathbb{E}{Y}_N={\sum }_{j\le N}{\lambda }_j{w}_j$, so [F4] gives $\mathbb{E}Y={\sum }_j{\lambda }_j{w}_j<\mathrm{\infty }$. Therefore $Y<\mathrm{\infty }$ almost surely: if $\mathbb{P}(Y=\mathrm{\infty })>0$, then $\mathbb{E}Y\ge M\mathbb{P}(Y=\mathrm{\infty })$ for every $M>0$, contradicting finiteness. On the full-measure event ${\mathrm{\Omega }}_0=\{Y<\mathrm{\infty }\}$, orthogonality gives $\mathrm{\parallel }{S}_m-S_n{\mathrm{\parallel }}_{H_1}^2={\sum }_{n<j\le m}{\lambda }_j{w}_j{\xi }_j^2\to 0$ as $m,n\to \mathrm{\infty }$. Completeness of $H_1$ gives an almost-sure limit. This convergence argument uses no choice axiom.

Define $S$ to be this limit on ${\mathrm{\Omega }}_0$ and $0$ on its measurable null complement. Each $S_N$ is Borel measurable. For any proper open $O\subset H_1$, let $C=H_1\setminus O$. The scalar functions $d(S_N,C)$ are measurable and on ${\mathrm{\Omega }}_0$ converge to $d(S,C)$; because $S\in O$ exactly when $d(S,C)>0$, the set $S^{-1}(O)$ is ${\mathrm{\Omega }}_0\cap \{{\mathrm{lim\hspace{0.17em}inf}}_{N}d(S_N,C)>0\}$, with ${\mathrm{\Omega }}_0^c$ added exactly when $0\in O$. The liminf condition is a countable combination of measurable scalar inequalities, so $S^{-1}(O)$ is measurable. The cases $O=\mathrm{\varnothing }$ and $O=H_1$ are immediate. Thus $S$ is Borel measurable.

On ${\mathrm{\Omega }}_0$, $\mathrm{\parallel }S-S_N{\mathrm{\parallel }}_{H_1}^2=Y-Y_N$. Since $\mathbb{P}({\mathrm{\Omega }}_0^c)=0$, [F4] and step 1.1 give $\mathbb{E}\mathrm{\parallel }S-S_N{\mathrm{\parallel }}_{H_1}^2={\sum }_{j>N}{\lambda }_j{w}_j\to 0$ and $\mathbb{E}\mathrm{\parallel }S{\mathrm{\parallel }}_{H_1}^2={\sum }_j{\lambda }_j{w}_j$. Hence $S_N\to S$ in $L^2(\mathrm{\Omega };H_1)$.

For any $\mathrm{\ell }\in H_1^{\ast }$, let $z\in H_1$ be its Riesz vector and write ${\mathrm{\ell }}_z(x)=⟨x,z⟩$, so $\mathrm{\ell }={\mathrm{\ell }}_z$. Set $c_j=\sqrt{{\lambda }_j}{\mathrm{\ell }}_z(e_j)=\sqrt{{\lambda }_j{w}_j}⟨f_j,z⟩$. Since $\mathrm{\mid }⟨f_j,z⟩\mathrm{\mid }\le \mathrm{\parallel }z\mathrm{\parallel }$, ${\sum }_j{c}_j^2\le \mathrm{\parallel }z{\mathrm{\parallel }}^2{\sum }_j{\lambda }_j{w}_j<\mathrm{\infty }$. By step 3.1, ${\mathrm{\ell }}_z(S_N)\to {\mathrm{\ell }}_z(S)$ in $L^2$, hence in probability; [F3] makes ${\mathrm{\ell }}_z(S)$ centered normal with variance ${\sum }_j{c}_j^2$. Together with step 2.1 and [F2], this proves that $S$ is a centered Borel Gaussian random element. For Riesz vectors $z,y\in H_1$, independence, centering, and unit variances give $\mathbb{E}[{\mathrm{\ell }}_z(S_N){\mathrm{\ell }}_y(S_N)]={\sum }_{j\le N}{\lambda }_j{w}_j⟨f_j,z⟩⟨f_j,y⟩$. The $L^2$ convergence from step 3.1 and Cauchy–Schwarz let these expectations pass to the limit, so the covariance form is $Q({\mathrm{\ell }}_z,{\mathrm{\ell }}_y)={\sum }_j{\lambda }_j{w}_j⟨f_j,z⟩⟨f_j,y⟩$. The series $v_z:={\sum }_j{\lambda }_j{w}_j⟨f_j,z⟩f_j$ converges in $H_1$: the finite Pythagorean identity gives ${\sum }_{j\le N}\mathrm{\mid }⟨f_j,z⟩{\mathrm{\mid }}^2\le \mathrm{\parallel }z{\mathrm{\parallel }}^2$, and ${\sup }_j{\lambda }_j{w}_j\le {\sum }_j{\lambda }_j{w}_j<\mathrm{\infty }$, so ${\sum }_j({\lambda }_j{w}_j{)}^2\mathrm{\mid }⟨f_j,z⟩{\mathrm{\mid }}^2<\mathrm{\infty }$. Thus $Q({\mathrm{\ell }}_z,{\mathrm{\ell }}_y)=⟨v_z,y⟩$. By [F6]–[F7] the covariance Bochner integral exists, and [F9] shows that applying any Riesz coordinate functional commutes with that integral. Hence $v_z$ is the Riesz vector of $C{\mathrm{\ell }}_z$, so the covariance operator transported to $H_1$ by Riesz is diagonal in $(f_j)$ with eigenvalues ${\lambda }_j{w}_j$, whose sum is finite.

If $D\subset H_1$ is a Borel forward-map domain with $\mathbb{P}(S\in D)=1$, then the law $\mu$ of $S$ satisfies $\mu (D)=\mathbb{P}(S\in D)=1$. The definition [F1] therefore makes $\mu$ a prior on $D$.

The space $H_1$ is a real separable Banach space, and step 4.1 supplies a centered Borel Gaussian random element in it. Under AC, [F5] applies and gives $a>0$ with $\mathbb{E}\exp (a\mathrm{\parallel }S{\mathrm{\parallel }}_{H_1}^2)<\mathrm{\infty }$.