Proper full conditional kernel

Let $I$ be a finite nonempty index set, let $(S_i,{\mathcal{S}}_i{)}_{i\in I}$ be measurable spaces, set $S={\prod }_{i\in I}{S}_i$ with its finite product sigma-algebra, and let $\pi$ be a probability measure on $S$. For a nonempty block $B\subseteq I$, write $S_B={\prod }_{i\in B}{S}_i$, $S_{-B}={\prod }_{i\in I\setminus B}{S}_i$, let $p_{-B}:S\to S_{-B}$ be the coordinate projection, and define ${\pi }_{-B}(D):=\pi (p_{-B}^{-1}(D))$ for measurable $D\subseteq S_{-B}$.

Products and product sigma-algebras use the empty-product convention in The product sigma-algebra and its finite iterates: if $B=I$, then $S_{-B}$ is the one-point space $\{()\}$ with its trivial sigma-algebra.

A proper full conditional kernel for block $B$ is a supplied probability kernel $K_B:S_{-B}\to S_B$ such that for every product-measurable $C\subseteq S$ and every measurable $D\subseteq S_{-B}$, $\pi (C\cap p_{-B}^{-1}(D))={\int }_D{K}_B(z,C_z){\pi }_{-B}(dz),C_z:=\{u\in S_B:{\mathrm{splice}}_B(z,u)\in C\}\mathrm{.}$ Here ${\mathrm{splice}}_B(z,u)$ is the unique state whose coordinates on $I\setminus B$ are $z$ and whose coordinates on $B$ are $u$. This is the event-identity definition of a regular conditional distribution of the block given the retained coordinates, with every kernel section a probability measure at every $z$. It makes no existence claim; the A-page density corollary gives one sufficient case.