A joint density supplies a proper full conditional kernel

Apply [F1] with the block coordinate as target and the retained coordinates as conditioning variable. Its formula gives probability sections on $D$ by normalization and uses $\rho$ on every exceptional fibre; the supplier proves measurability of evaluations and that the exceptional set has zero marginal mass. Thus $K_B(z,\cdot )$ is a probability measure for every $z$, not only almost every $z$.

Fix any measurable $F\subseteq S_{-B}$. Under the coordinate grouping $S\cong S_B\times S_{-B}$, the inverse image under ${\mathrm{splice}}_B$ of a measurable rectangle $A\times C$ is $C\times A$. Since such rectangles generate the finite-product sigma-algebra (The product sigma-algebra and its finite iterates), the function $(z,u)\mapsto {\mathbf{1}}_E({\mathrm{splice}}_B(z,u))$ is product-measurable for every product-measurable $E\subseteq S$. The kernel integration theorem [F4] therefore makes $z\mapsto K_B(z,E_z)$ measurable. Define ${\nu }_F(E)={\int }_F{K}_B(z,E_z){\pi }_{-B}(dz)$; this integral is defined for every such $E$.

The set function $E\mapsto \pi (E\cap p_{-B}^{-1}(F))$ is a finite measure by restriction of $\pi$. The set function ${\nu }_F$ is also a finite measure: it vanishes at $\mathrm{\varnothing }$, and ${\nu }_F(S)={\int }_{F}1d{\pi }_{-B}={\pi }_{-B}(F)\le 1$. If $(E_n)$ are pairwise disjoint product-measurable sets, their sections $(E_n{)}_z$ are pairwise disjoint, so $K_B(z,({\bigcup }_n{E}_n{)}_z)={\sum }_n{K}_B(z,(E_n{)}_z)$. The partial sums are measurable by step 1.2 and increase; applying [F5] to ${\mathbf{1}}_F(z){\sum }_{n=1}^N{K}_B(z,(E_n{)}_z)$ and [F6] to each finite sum gives ${\nu }_F({\bigcup }_n{E}_n)={\lim }_N{\sum }_{n=1}^N{\nu }_F(E_n)={\sum }_n{\nu }_F(E_n)$.

For a coordinate rectangle $E=A\times C$, the density event identity [F1] gives $\pi (E\cap p_{-B}^{-1}(F))=\pi (A\times (C\cap F))={\int }_{C\cap F}{K}_B(z,A){\pi }_{-B}(dz)={\nu }_F(E)$. The coordinate rectangles form a generating pi-system and include the whole space. The two finite measures agree on that system and have the common finite total mass ${\pi }_{-B}(F)$, so the constant sequence consisting of the whole space is a finite-measure exhaustion in [F3]. Measure uniqueness yields $\pi (E\cap p_{-B}^{-1}(F))={\nu }_F(E)$ for every product-measurable $E$. Since $F$ was arbitrary, [F2] holds for the displayed kernel. The kernel/event construction itself uses no AC; the optional conditional-expectation-class convention is not invoked.