The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and integrability as in Measure Spaces and the Lebesgue Integral: Standing Notation, let $c$ be a variance sequence, $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$, $H_{c}$ the Cameron-Martin space of $c$ with Cameron-Martin square $|h|_{c}^{2}$ ($h\in H_{c}$), and, for $h\in X$, let $\ell_{h,n}$ ($n\in\mathbb{N}$) be the Paley-Wiener partial sums and $\ell_{h}$ the Paley-Wiener functional of $h$ relative to $c$.

1. (Borel measurability)¶ For every $h\in X$, the set of those $x\in X$ for which $(\ell_{h,n}(x))_{n\in\mathbb{N}}$ converges is a Borel set, and the functions $\ell_{h,n}$ ($n\in\mathbb{N}$) and $\ell_{h}$ are Borel.

2. (Basis vectors)¶ For every $k\in\mathbb{N}$ and every $x\in X$, $\ell_{e_{k}}(x)=x_{k}/c_{k}$.

3. (Convergence)¶ Let $h\in H_{c}$. Then:¶ the set of those $x\in X$ for which $(\ell_{h,n}(x))_{n\in\mathbb{N}}$ converges has $\gamma_{c}$-measure $1$;¶ the functions $\ell_{h}$, $(\ell_{h})^{2}$, $\ell_{h,n}$ and $(\ell_{h,n})^{2}$ ($n\in\mathbb{N}$) are integrable with respect to $\gamma_{c}$; and¶ for every $n\in\mathbb{N}$

$$\int_{X}\bigl(\ell_{h}(x)-\ell_{h,n}(x)\bigr)^{2}\,\gamma_{c}(dx)=\sum_{k=n+1}^{\infty}\frac{h_{k}^{2}}{c_{k}},$$

which tends to $0$ as $n\to\infty$.

4. (Mean and variance)¶ For every $h\in H_{c}$,

$$\int_{X}\ell_{h}\,d\gamma_{c}=0\qquad\text{and}\qquad\int_{X}(\ell_{h})^{2}\,d\gamma_{c}=|h|_{c}^{2}.$$