Head and Tail of a Diagonal Gaussian Measure on a Hilbert Space are Independent

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinate map $p_{n}$, the synthesis map $p_{n}^{*}$ and the projection $Q_{n}=\mathrm{id}_{X}-P_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and push-forwards as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, let $c$ be a variance sequence, let $\gamma_{c}$ be the diagonal Gaussian measure on $X$ with variances $c$, and let $n\in\mathbb{N}$, with $\gamma_{c^{(n)}}$ the diagonal Gaussian measure on $\mathbb{R}^{n}$ of the truncation $c^{(n)}$. Let $\mathbb{R}^{n}\times X$ carry the product metric; its Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ by The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product, both factors being separable (Euclidean Space is a Separable Metric Space §separable and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space); product measures on it are those of Existence and Uniqueness of the Product Measure, which applies to the finite, hence $\sigma$-finite, Borel probability measures used below. For maps into or out of $\mathbb{R}^{n}\times X$, Borel means measurable with respect to these Borel $\sigma$-algebras, and push-forwards are the image measures of claim 1 of Image Measures, Measures with Densities, and Change of Variables. Let $h_{n}:X\to\mathbb{R}^{n}\times X$ be the map $h_{n}(x)=(p_{n}(x),Q_{n}x)$ and $\Phi_{n}:\mathbb{R}^{n}\times X\to X$ the map $\Phi_{n}(u,w)=p_{n}^{*}(u)+w$.

1. (Head and tail are independent)¶ $h_{n}$ is continuous, hence Borel, and

$$(h_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}}\otimes(Q_{n})_{\#}\gamma_{c}.$$

2. (Synthesis)¶ $\Phi_{n}$ is continuous, hence Borel, $\Phi_{n}\circ h_{n}=\mathrm{id}_{X}$, and

$$\gamma_{c}=(\Phi_{n})_{\#}\bigl(\gamma_{c^{(n)}}\otimes(Q_{n})_{\#}\gamma_{c}\bigr).$$