Finite products of probability kernels are probability kernels

If $n=0$, use the point mass on the empty-product singleton; if $n=1$, use $K_0(s)$. For $n\ge 2$, fix $s\in S$ and build the product recursively: start with $K_0(s)$, and at stage $j+1$ apply [F1] to the probability measure already constructed on the first $j$ factors and the probability measure $K_j(s)$. This constructs a probability measure on the recursively defined first $j+1$ factors, with the product rectangle formula. Finite induction gives $K(s,\cdot )={⨂}_{i<n}{K}_i(s,\cdot )$ and its formula for every $s$.

For a rectangle $R={\prod }_{i<n}{B}_i$, step 1.1 gives $s\mapsto K(s,R)={\prod }_{i<n}{K}_i(s,B_i)$, a finite product of measurable real functions, hence measurable. Let $\mathcal{D}$ be the product-measurable sets $E$ for which $s\mapsto K(s,E)$ is measurable. The class contains the whole space and all rectangles. If $E\in \mathcal{D}$, then $K(s,E^c)=1-K(s,E)$ is measurable. For pairwise disjoint $E_j\in \mathcal{D}$, countable additivity gives $K(s,{\bigcup }_j{E}_j)={\lim }_{N\to \mathrm{\infty }}{\sum }_{j\le N}K(s,E_j)$, a pointwise limit of measurable functions. Thus $\mathcal{D}$ is a lambda-system. Rectangles form a pi-system and generate the product sigma-algebra by [F2], so [F4] gives $\mathcal{D}={⨂}_{i<n}{\mathrm{T}}_i$. Hence $K$ is a probability kernel.

Any other probability kernel with the rectangle formula agrees with $K(s,\cdot )$ on the generating rectangle pi-system for each fixed $s$. The measures are finite, so uniqueness [F3] gives equality on the full product sigma-algebra at every $s$. This proves uniqueness, including the empty product case.