The Gibbs block update is a probability kernel

For each fixed $x$, the map $u\mapsto {\mathrm{splice}}_B(x_{-B},u)$ is measurable by the finite product sigma-algebra. Thus $P_B(x,\mathrm{\varnothing })=0$, $P_B(x,S)=\int 1d{K}_B(x_{-B})=1$, and for pairwise disjoint $C_n$ the indicators of their inverse images sum increasingly to the indicator of the inverse image of ${\bigcup }_n{C}_n$. [F4] gives countable additivity, so $P_B(x,\cdot )$ is a probability measure.

For fixed measurable $C\subseteq S$, the function $(x,u)\mapsto {\mathbf{1}}_C({\mathrm{splice}}_B(x_{-B},u))$ is product-measurable because coordinate projection and splicing are measurable for the finite product sigma-algebra. The lifted kernel $x\mapsto K_B(x_{-B},\cdot )$ is a probability kernel by [F1] and measurability of the projection. Applying [F3] to this kernel gives measurability of $x\mapsto P_B(x,C)$. Thus every evaluation is measurable and every section is a probability measure.