The Conditional Law of the Head Given the Tail for a Measure with a Density Relative to 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 $p_{n}$, $p_{n}^{*}$ and $Q_{n}$ as in 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 with diagonal Gaussian measure $\gamma_{c}$, let $n\in\mathbb{N}$, with $\gamma_{c^{(n)}}$ the diagonal Gaussian measure on $\mathbb{R}^{n}$ of the truncation $c^{(n)}$ and $\lambda_{n}$ Lebesgue measure on $\mathcal{B}(\mathbb{R}^{n})$. Let $\mu\in\mathcal{P}(X)$ have a density $f$ with respect to $\gamma_{c}$, a Borel function $f:X\to\mathbb{R}$ with $f\ge0$, and let $\nu_{n}=(Q_{n})_{\#}\mu$. Let $X\times\mathbb{R}^{n}$ carry the product metric, whose Borel $\sigma$-algebra is $\mathcal{B}(X)\otimes\mathcal{B}(\mathbb{R}^{n})$ by The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product, both factors being separable (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space and Euclidean Space is a Separable Metric Space §separable); for maps into $X\times\mathbb{R}^{n}$, 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 $k_{n}:X\to X\times\mathbb{R}^{n}$ be the map $k_{n}(x)=(Q_{n}x,p_{n}(x))$. Let $W_{0}$ be the set of the $w\in X$ for which the function $u\mapsto f(p_{n}^{*}(u)+w)$ on $\mathbb{R}^{n}$ is integrable with respect to $\gamma_{c^{(n)}}$ with positive integral, and for $w\in W_{0}$ let $Z(w)=\int_{\mathbb{R}^{n}}f(p_{n}^{*}(u)+w)\,\gamma_{c^{(n)}}(du)$, a positive real number. Define $K:X\times\mathcal{B}(\mathbb{R}^{n})\to\mathbb{R}$ by

$$K(w,B)=\frac{1}{Z(w)}\int_{\mathbb{R}^{n}}\mathbf{1}_{B}(u)\,f\bigl(p_{n}^{*}(u)+w\bigr)\,\gamma_{c^{(n)}}(du)\quad(w\in W_{0}),\qquad K(w,B)=\gamma_{c^{(n)}}(B)\quad(w\notin W_{0}).$$

1. (Good tails)¶ $k_{n}$ is continuous, hence Borel; $W_{0}\in\mathcal{B}(X)$ and $\nu_{n}(W_{0})=1$.

2. (A probability kernel)¶ $K$ is a probability kernel from $(X,\mathcal{B}(X))$ to $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$.

3. (Disintegration)¶ $(k_{n})_{\#}\mu=\nu_{n}\otimes K$, the composite measure of Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite.

4. (Absolute continuity of the fibre laws)¶ For every $w\in X$ and every $B\in\mathcal{B}(\mathbb{R}^{n})$ with $\lambda_{n}(B)=0$, $K(w,B)=0$.