Function-space priors and sample regularity — Examples and Counterexamples

The interval $D=[1,2]$ is closed and hence Borel by [F4]. For every Borel $E\subseteq \mathbb{R}$, $G^{-1}(E)=E\cap D$ is in the trace Borel sigma-algebra on $D$ by [F3], so $G$ is an admissible measurable forward map.

The density in [F1] is continuous and strictly positive on $[0,\frac{1}{2}]$, so it has a positive minimum there and $\mu ((-\mathrm{\infty },1))\ge {\int }_0^{1\mathrm{/}2}\varphi (t)dt>0$.

Since $D\subseteq (-\mathrm{\infty },1{)}^c$, [F5] gives $\mu (D)\le \mu ((-\mathrm{\infty },1{)}^c)=1-\mu ((-\mathrm{\infty },1))<1$ by [F6] and step 1.2. This violates the full-measure condition in [F2], so $\mu$ is not a prior on the declared domain.

Group the positive terms by dyadic blocks. For each $m\ge 0$, there are $2^m$ integers $j$ with $2^m\le j<2^{m+1}$, and each has $j^{-2}\le 2^{-2m}$. Hence ${\sum }_{j\ge 1}{j}^{-2}\le {\sum }_{m\ge 0}{2}^m{2}^{-2m}={\sum }_{m\ge 0}{2}^{-m}<\mathrm{\infty }$.

By [F10], the coordinate vectors are a complete orthonormal basis of $H$. Apply [F1] using step 1.1 under the stated AC and countable-choice assumptions [A1]–[A2]. This gives the centered Borel Gaussian law $\mu$ on $H$, with covariance operator eigenvalues $j^{-2}$. The domain is $D=H$ and the identity map is measurable, so $\mu (D)=1$ and [F2] makes $\mu$ a prior on $D$.

By [F5], every $\mathrm{\ell }\in H^{\ast }$ has the form ${\mathrm{\ell }}_z(x)=⟨x,z⟩$ for a unique $z\in {\mathrm{\ell }}^2$. The covariance operator supplied by [F1] in step 2.1 is diagonal under this Riesz identification. By the covariance definition [F3] and commutation of bounded functionals with the Bochner integral [F8], $Q({\mathrm{\ell }}_z,{\mathrm{\ell }}_y)={\mathrm{\ell }}_y(C{\mathrm{\ell }}_z)={\sum }_j{j}^{-2}{z}_j{y}_j$; this sum converges absolutely by Cauchy–Schwarz and $j^{-2}\le 1$. The well-definedness theorem [F4] supplies $C{\mathrm{\ell }}_z$ and the quotient completion. In particular, applying the coordinate functional $m_j(x)=⟨x,e_j⟩$ commutes with the Bochner integral by [F8], so $⟨C{\mathrm{\ell }}_z,e_j⟩=m_j(C{\mathrm{\ell }}_z)=Q({\mathrm{\ell }}_z,m_j)=j^{-2}{z}_j$. Thus the canonical covariance map sends $z$ to the vector with coordinates $x_j=j^{-2}{z}_j$. Its quotient norm satisfies $\mathrm{\parallel }z{\mathrm{\parallel }}_Q^2=Q({\mathrm{\ell }}_z,{\mathrm{\ell }}_z)={\sum }_j{j}^{-2}{z}_j^2={\sum }_j{j}^2{x}_j^2$. The nullspace is zero because every $j^{-2}>0$. Finite truncations of Riesz vectors are dense in the quotient completion since their weighted-square tails tend to zero; conversely, for any $x\in {\mathrm{\ell }}^2$ with ${\sum }_j{j}^2{x}_j^2<\mathrm{\infty }$, the truncations of $z_j=j^2{x}_j$ converge in that norm and their images converge to $x$ in ${\mathrm{\ell }}^2$. Therefore the covariance RKHS is exactly the displayed weighted subspace.

By [F9], ${\mathrm{\ell }}^2$ is a complete real Hilbert space, and [F10] gives a countable complete orthonormal basis. Its finite real linear span is dense; approximating each finite list of real coefficients by rationals shows that finite rational combinations are dense as well. They form a countable set, so [F11] makes ${\mathrm{\ell }}^2$ Polish. Step 2.1 gives a Borel probability $\mu$ on this space; [F6] makes $\mu$ tight under AC [A1].

By [F10], finite-support sequences are dense in ${\mathrm{\ell }}^2$; each lies in the RKHS identified in step 3.1, so the RKHS is dense. The vector $x=(1\mathrm{/}j{)}_{j\ge 1}$ belongs to ${\mathrm{\ell }}^2$ by step 1.1 but not to the RKHS, since ${\sum }_j{j}^2(1\mathrm{/}j{)}^2={\sum }_{j}1=\mathrm{\infty }$. Therefore the RKHS is proper.

The hypotheses of [F7] hold: step 2.1 gives a centered Borel Gaussian, step 3.2 gives tightness, and AC and countable choice [A1]–[A2] are assumed. By step 4.1 the covariance RKHS has closure $H$, so [F7] gives $\mathrm{supp}\mu =H$.

Let $H_0=L^2([0,1])$. By [F1] under countable choice [A2], it is a real Hilbert space. The complete basis [F10] has a countable index set, so its finite rational linear combinations form a countable set. Its finite real linear span is dense by [F10], and approximating each finite list of real coefficients by rationals makes the rational span dense as well. Hence $H_0$ is separable by [F11]. For the first application of [F3], take $H_1=H_0$ with the identity inclusion; the supplied wavelet family is a complete orthonormal basis in both spaces, so $w_j=1$ for every index. Enumerate the finite scaling family first and the detail wavelets by increasing $\mathrm{\ell }$ and then $k$, set ${\lambda }_i=0$ on the scaling family, and set ${\lambda }_{\mathrm{\ell }k}=2^{-2\mathrm{\ell }(s+1\mathrm{/}2)}$ and retain $w_j=1$ on all indices. Then ${\sum }_j{\lambda }_j{w}_j={\sum }_{\mathrm{\ell }\ge 0}{2}^{\mathrm{\ell }}{2}^{-2\mathrm{\ell }(s+1\mathrm{/}2)}={\sum }_{\mathrm{\ell }\ge 0}{2}^{-2s\mathrm{\ell }}<\mathrm{\infty }$ because $s>0$. Apply [F3] under AC and countable choice [A1]–[A2] to obtain the Borel centered Gaussian limit $W$ in $H_0$. The identity on $D=H_0$ is measurable and $\mathbb{P}(W\in D)=1$, so [F4] makes its law a prior on $D$.

Enumerate the detail coefficients by increasing level and then by $k$, write the corresponding standard normals as $({\eta }_n{)}_{n\ge 1}$, and set $Z_n={\eta }_n^2$. By [F9] under countable choice [A2], the square map is measurable. Since each ${\eta }_n$ has the same standard normal law, the $Z_n$ have the same pushforward law under this map; [F6] preserves independence, so [F7] makes $(Z_n)$ IID. By [F5] under AC [A1], their mean is $1$ and their variance is finite; [F8] gives $R_N\mathrm{/}N\to 1$ almost surely for $R_N={\sum }_{n=1}^N{Z}_n$. For every $\mathrm{\ell }\ge 0$, set $A_{\mathrm{\ell }}=2^{-\mathrm{\ell }}{\sum }_{k<2^{\mathrm{\ell }}}{\xi }_{\mathrm{\ell }k}^2$. For $\mathrm{\ell }\ge 1$, put $N_{\mathrm{\ell }}=2^{\mathrm{\ell }}-1$, so $A_{\mathrm{\ell }}=(R_{N_{\mathrm{\ell }+1}}-R_{N_{\mathrm{\ell }}})\mathrm{/}{2}^{\mathrm{\ell }}$. Since $N_{\mathrm{\ell }+1}\mathrm{/}{2}^{\mathrm{\ell }}\to 2$ and $N_{\mathrm{\ell }}\mathrm{/}{2}^{\mathrm{\ell }}\to 1$ as $\mathrm{\ell }\to \mathrm{\infty }$, the strong-law limits give $A_{\mathrm{\ell }}\to 2-1=1$ almost surely for the same variables used in step 1.1.

Fix $0\le r<s$ and take $H_1=B_{2,2}^r([0,1])$. By [F2], it is a separable real Hilbert space continuously included in $H_0$, and the same wavelet family is complete and orthogonal in $H_1$, with squared basis norms $1$ on the finite scaling family and $w_{\mathrm{\ell }k}=2^{2r\mathrm{\ell }}$ on detail levels. The weighted eigenvalue sum is ${\sum }_{\mathrm{\ell }\ge 0}{2}^{\mathrm{\ell }}{2}^{-2\mathrm{\ell }(s+1\mathrm{/}2)}{2}^{2r\mathrm{\ell }}={\sum }_{\mathrm{\ell }\ge 0}{2}^{-2(s-r)\mathrm{\ell }}<\mathrm{\infty }$. Apply [F3] under AC and countable choice [A1]–[A2] to the same coefficients as in step 1.1; the partial wavelet sums converge almost surely in $H_1$. Their $H_0$ limit is $W$ by step 1.1, and continuity of the inclusion makes the $H_1$ limit the same element. Hence $W\in B_{2,2}^r([0,1])$ almost surely for this fixed $r$.

For each detail wavelet, its coordinate functional is continuous on $H_0$; after the term enters the partial sums, the coefficient is $2^{-\mathrm{\ell }(s+1\mathrm{/}2)}{\xi }_{\mathrm{\ell }k}$, so convergence in step 1.1 gives the same coefficient for $W$. The scaling coefficients vanish by construction. Therefore $\mathrm{\parallel }W{\mathrm{\parallel }}_{B_{2,2}^s}^2={\sum }_{\mathrm{\ell }\ge 0}{2}^{2\mathrm{\ell }s}{\sum }_{k<2^{\mathrm{\ell }}}{2}^{-2\mathrm{\ell }(s+1\mathrm{/}2)}{\xi }_{\mathrm{\ell }k}^2={\sum }_{\mathrm{\ell }\ge 0}{A}_{\mathrm{\ell }}$. By step 2.1, $A_{\mathrm{\ell }}\to 1$ almost surely, so the nonnegative series diverges and $W\mathrm{\notin }{B}_{2,2}^s([0,1])$ almost surely.