Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ for each $y\in Y$, measurable and bounded by a $L^1$ function that does not depend on $x$). Let ${\Sigma}\_0$ be sub $\sigma$-algebra of $\Sigma$, and let $g(\cdot,x)=E(f(\cdot,x)|\Sigma\_0)$ denote the conditional expectation with respect to ${\Sigma}\_0$. Does $g$ have a version that is a Carathéodory function as well?


Thanks!



Is the conditional expectation of a Caratheodory function a Caratheodory function?
Here is a positive answer for the case that $\Sigma\_0$ is generated by a random variable with values in a Polish space, so that we can use regular conditional probabilities and for some kernel $\kappa:Y\to\Delta(Y)$, we can let
$$\mathbb{E}(f(\cdot,x)|\Sigma\_0)\_y=\int f(,\cdot,x)~\mathrm d\kappa\_y.$$
Then continuity follows from the assumption that there is a dominating integrable function and the dominated convergence theorem.


