Conditional KL is measurable for supplied kernels

Let $\rho =P+Q$ for arbitrary probability measures P,Q on E. [given, choose] For the algebra $\mathcal{A}$ in [F1], the class of measurable $B$ approximable in $\rho$-measure by members of $\mathcal{A}$ is a sigma-algebra: complements preserve symmetric-difference measure, finite unions preserve approximation, and a countable union is approximated by first truncating the union using finiteness of $\rho$ and then approximating that finite union. It contains $\mathcal{A}$, so it is all of $\mathcal{E}$. Hence any bounded measurable real function can be approximated in $L^1(\rho )$ by bounded simple functions from a countable family $\mathcal{G}$ of rational-valued simple functions on $\mathcal{A}$. On a fixed bound $M$, the functional ${\mathrm{\Phi }}_{P,Q}(g)=\int gdP-\ln \int e^{g}dQ$ is continuous under $L^1(\rho )$ approximation, because $e^t$ is Lipschitz on $[-M,M]$ and $\int e^{g}dQ\ge e^{-M}$. Thus [F3] gives $D(P\mathrm{\parallel }Q)={\sup }_{g\in \mathcal{G}}(\int gdP-\ln \int e^{g}dQ)\mathrm{.}$ The family $\mathcal{G}$ is countable and does not depend on P,Q.

For fixed $g\in \mathcal{G}$, the functions $z\mapsto \int g(x)K(z,dx)$ and $z\mapsto \int e^{g(x)}L(z,dx)$ are measurable by [F2], applying it to positive and negative parts for the first function. If $\mathrm{\mid }g\mathrm{\mid }\le M$, then $\int e^{g}dL(z)\ge e^{-M}>0$, so its logarithm is measurable. Therefore each candidate ${\mathrm{\Phi }}_z(g)$ is measurable.

Apply the identity in step 1.1 to $P=K(z,\cdot )$ and $Q=L(z,\cdot )$ for each fixed z. Then $D(K(z,\cdot )\mathrm{\parallel }L(z,\cdot ))={\sup }_{g\in \mathcal{G}}{\mathrm{\Phi }}_z(g)$, a countable supremum of measurable functions by step 2.1. This proves measurability and makes its nonnegative integral against any supplied base probability well-defined.