The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with push-forwards as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward and integrals 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 $|h|_{c}^{2}$, and $\exp$ the exponential function. Let $h\in H_{c}$, let $\ell_{h}$ be the Paley-Wiener functional of $h$ relative to $c$, let $\tau_{h}:X\to X$ be the translation $\tau_{h}(x)=x+h$, and define

$$\rho_{h}:X\to\mathbb{R},\qquad\rho_{h}(x)=\exp\Bigl(\ell_{h}(x)-\tfrac12|h|_{c}^{2}\Bigr).$$

1. (The density)¶ $\tau_{h}$ is Borel, $\rho_{h}$ is positive and Borel, and $\int_{X}\rho_{h}\,d\gamma_{c}=1$.

2. (Translation formula)¶ For every Borel set $B\subseteq X$,

$$\bigl((\tau_{h})_{\#}\gamma_{c}\bigr)(B)=\int_{X}\mathbf{1}_{B}\,\rho_{h}\,d\gamma_{c};$$

that is, $(\tau_{h})_{\#}\gamma_{c}$ is the measure with density $\rho_{h}$ with respect to $\gamma_{c}$.

3. (Integrals)¶¶ For every Borel function $f:X\to\mathbb{R}$ that is nonnegative,

$$\int_{X}f(x+h)\,\gamma_{c}(dx)=\int_{X}f(x)\,\rho_{h}(x)\,\gamma_{c}(dx),$$

where both sides lie in $[0,\infty]$.¶ For every Borel function $f:X\to\mathbb{R}$, the function $x\mapsto f(x+h)$ is integrable with respect to $\gamma_{c}$ if and only if $f\rho_{h}$ is, and then the same identity holds.