Function-space priors and sample regularity

Write $m=\mathrm{\mid }{I}_{\ast }\mathrm{\mid }$ and enumerate the basis by placing the finite family first, followed by $(\mathrm{\ell },k)$ at index $m+(2^{\mathrm{\ell }}-1)+k$, since $2^{\mathrm{\ell }}-1$ is the number of level indices below $\mathrm{\ell }$. For $x\in H_r$, define $T_{r}x$ by $(T_{r}x{)}_t=⟨x,e_{i_t}{⟩}_{H_0}$ for the finite coordinates and $(T_{r}x{)}_{m+(2^{\mathrm{\ell }}-1)+k}=2^{r\mathrm{\ell }}⟨x,e_{\mathrm{\ell }k}{⟩}_{H_0}$. Parseval [F1] gives $\mathrm{\parallel }{T}_{r}x{\mathrm{\parallel }}_{{\mathrm{\ell }}^2}^2=\mathrm{\parallel }x{\mathrm{\parallel }}_{H_r}^2$, so $T_r$ is an isometry into real ${\mathrm{\ell }}^2(\mathbb{N})$.

Given $b\in {\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$, set the finite-family coefficients equal to its first $m$ coordinates and put $a_{\mathrm{\ell }k}=2^{-r\mathrm{\ell }}{b}_{m+(2^{\mathrm{\ell }}-1)+k}$. Since $r\ge 0$, the sum of squares of all these coefficients is at most ${\sum }_n\mathrm{\mid }{b}_n{\mathrm{\mid }}^2<\mathrm{\infty }$. By [F3], their orthonormal expansion converges in $H_0$ to a vector $x$ with those coefficients. Thus $x\in H_r$ and $T_{r}x=b$, so $T_r$ is onto. By [F2], the isometric pullback of the real ${\mathrm{\ell }}^2$ inner product makes $H_r$ complete, and hence a real Hilbert space.

The finite-support rational sequences form a countable subset of real ${\mathrm{\ell }}^2$: each is encoded by a finite list of coordinate indices, integer numerators, and positive denominators, and such finite integer lists have an explicit natural-number encoding. Given $b\in {\mathrm{\ell }}^2$ and $\epsilon >0$, choose $N\ge 1$ so that $({\sum }_{n\ge N}\mathrm{\mid }{b}_n{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}<\epsilon \mathrm{/}2$, then choose rationals $q_0,\dots ,q_{N-1}$ with $\mathrm{\mid }{b}_n-q_n\mathrm{\mid }<\epsilon \mathrm{/}(2\sqrt{N})$ for each $0\le n<N$, and set $q_n=0$ for $n\ge N$. Then $\mathrm{\parallel }b-q{\mathrm{\parallel }}_2^2={\sum }_{n<N}\mathrm{\mid }{b}_n-q_n{\mathrm{\mid }}^2+{\sum }_{n\ge N}\mathrm{\mid }{b}_n{\mathrm{\mid }}^2<{\epsilon }^2\mathrm{/}2<{\epsilon }^2$, so these finite-support rational sequences are dense. Because $T_r$ is an isometric bijection by steps 1.1 and 2.1, their inverse images are a countable dense subset of $H_r$. Hence $H_r$ is separable by [F4].

Each basis vector has a single nonzero coefficient, so its squared $H_r$ norm is $1$ on $I_{\ast }$ and $2^{2r\mathrm{\ell }}$ on level $\mathrm{\ell }$; distinct basis vectors are orthogonal under the weighted inner product. The finite-support coefficient vectors are dense by step 3.1, so the orthogonal family is complete in $H_r$. Finally Parseval [F1] gives $\mathrm{\parallel }x{\mathrm{\parallel }}_{H_0}^2={\sum }_{i\in I_{\ast }}\mathrm{\mid }⟨x,e_i⟩{\mathrm{\mid }}^2+{\sum }_{\mathrm{\ell },k}\mathrm{\mid }⟨x,e_{\mathrm{\ell }k}⟩{\mathrm{\mid }}^2\le \mathrm{\parallel }x{\mathrm{\parallel }}_{H_r}^2$ for $x\in H_r$, because $2^{2r\mathrm{\ell }}\ge 1$. Thus the inclusion has operator norm at most $1$.

For all $i,j$, the canonical inner product gives $⟨e_i,e_j⟩={\delta }_{ij}$, so the coordinate family is orthonormal.

Let $x=(x_j)\in H$ and $\epsilon >0$. By [F1], a finite set of coordinates has complement square-sum less than ${\epsilon }^2$. Every finite subset of $\mathbb{N}$ is contained in $\{j:j<N\}$ for some $N\in \mathbb{N}$, so choose $N$ such that ${\sum }_{j\ge N}\mathrm{\mid }{x}_j{\mathrm{\mid }}^2<{\epsilon }^2$. The coordinate formula gives $\mathrm{\parallel }x-{\sum }_{j<N}{x}_j{e}_j{\mathrm{\parallel }}_2^2={\sum }_{j\ge N}\mathrm{\mid }{x}_j{\mathrm{\mid }}^2<{\epsilon }^2$. Thus finite linear combinations of the $e_j$ are dense in $H$; by [F3], the orthonormal family is a complete orthonormal basis. The same tail estimate proves the stated truncation convergence, and the norm identity is [F1].

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 $\mathbb{E}\exp (a\mathrm{\parallel }S{\mathrm{\parallel }}_{H_1}^2)<\mathrm{\infty }$.