Bayesian posterior as a regular conditional law

Let a proper Bayesian model be given, and let $J$ be its joint prior-predictive probability on $(\mathrm{\Theta }\times \mathsf{Y},\mathcal{T}\otimes \mathcal{Y})$, supplied by A prior and model kernel define a joint law. More generally, let $(\mathrm{\Omega },\mathcal{F},Q)$ be any probability space carrying parameter and data random elements $X:\mathrm{\Omega }\to \mathrm{\Theta }$ and $Y:\mathrm{\Omega }\to \mathsf{Y}$ whose joint law is $J$ (Probability measures and probability spaces, Random elements and real random variables, Law or distribution of a random element). A posterior regular conditional distribution (posterior RCD) is a regular conditional distribution of $X$ given $\sigma (Y)$, in the sense of Regular conditional distribution. It is a kernel indexed by the sample point $\omega \in \mathrm{\Omega }$. The canonical realization is $\mathrm{\Omega }=\mathrm{\Theta }\times \mathsf{Y}$ with $Q=J$ and coordinate maps $X(\theta ,y)=\theta$, $Y(\theta ,y)=y$.

A data-indexed posterior kernel is a probability kernel ${\mathrm{\Pi }}_{\cdot }$ from $(\mathsf{Y},\mathcal{Y})$ to $(\mathrm{\Theta },\mathcal{T})$ (Measure kernel and probability kernel) for which $\omega \mapsto {\mathrm{\Pi }}_{Y(\omega )}$ is a conditional law of $X$ given $Y$ (Conditional law given a random element). Equivalently, for every $A\in \mathcal{T}$ and $B\in \mathcal{Y}$, $Q(X\in A,Y\in B)={\int }_B{\mathrm{\Pi }}_y(A)Q_Y(dy),$ where $Q_Y$ is the data marginal (equal to $J_Y$). This kernel is indexed by data values, not sample points.

The definition asserts no existence of either conditional kernel. Values of a data-indexed version on a measurable $J_Y$-null set are not fixed by the conditional-law identities; replacing them there by a fixed probability preserves those identities. A dominated-density construction is a separate route: Bayes formula for dominated kernels requires a proper prior, a sigma-finite data measure, and a specified jointly measurable sampling density, and fills fibres whose marginal density is zero or infinite with the prior. It does not require the data space to be standard Borel.