It is well known that a finite measure on the Baire $\sigma$-algebra of a, say, compact Hausdorff space can be extended to a unique regular measure defined on the Borel $\sigma$-algebra. The Baire $\sigma$-algebra is the smallest $\sigma$-algebra that makes every real continuous function measurable, and the Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing all open sets. I would like to know if one can generalize this to kernels.
Let $X$ and $Y$ be compact Hausdorff spaces, $Ba_X$ and $Ba_Y$ the Baire $\sigma$-algebras on $X$ and $Y$ respectively. Let $B_X$ and $B_Y$ be the corresponding Borel $\sigma$-algebras.
Let $\kappa:X\times Ba_Y\to\mathbb{R}$ a kernel, so that $\kappa(x,\cdot)$ is a Baire measure for each $x\in X$ and $\kappa(\cdot,E)$ a Baire-measurable function for each Baire set $E\in Ba_Y$.
Let $\kappa^*:X\times B_Y\to\mathbb{R}$ be defined by letting $\kappa^*(x,\cdot)$ be the unique extension of $\kappa(x,\cdot)$ to a regular Borel measure for each $x\in X$.
Is $\kappa^*(\cdot,F)$ a Borel-measurable function for each Borel set $F\in B_Y$?
The problem came up in understanding some constructions in statistical decision theory. If it helps, one can restrict the kernels to be probability kernels.
