A trace-class Gaussian with full support and a proper Cameron–Martin space

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 $\mathrm{supp}\mu =H$.