The Head Given the Tail is Absolutely Continuous for a Measure with a Density Relative to a Diagonal Gaussian Measure

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the maps $p_{n}$ and $Q_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, which are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, 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 $\gamma_{c}$, and let $\mu\in\mathcal{P}(X)$ have a density $f$ with respect to $\gamma_{c}$, a Borel function $f:X\to\mathbb{R}$ with $f\ge0$. Let $n\in\mathbb{N}$, let $\nu_{n}=(Q_{n})_{\#}\mu$, and let $\lambda$ be a conditional kernel of $\mu$ given $Q_{n}$, with $\lambda_{w}=\lambda(w,\cdot)$.

1. (Absolutely continuous heads)¶ There is a set $W_{1}\in\mathcal{B}(X)$ with $\nu_{n}(W_{1})=1$ such that, for every $w\in W_{1}$, the probability measure $(p_{n})_{\#}\lambda_{w}$ on $\mathbb{R}^{n}$ is absolutely continuous.