# Is the conditional expectation of a Caratheodory function a Caratheodory function?

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!

• I don't know right now whether the answer here mathoverflow.net/questions/124589 solves your problem but it might be relevant. Nov 22 at 21:40

Edit: The below answer is invalid, since $$\Sigma_0$$ is a sub sigma algebra of $$Y$$, not $$X \times Y$$.
I think the answer is no - take $$X = Y = [0, 1]$$, and $$f(y, x) =y$$. Pick some Borel set $$E \subset [0, 1]$$ such that $$E$$ and its complement have nonzero measure in every open interval, and let $$\Sigma_0$$ be the sigma algebra generated by the sets $$[0, 1] \times E$$ and $$\mathcal B([0, 1]^2 \setminus [0, 1] \times E)$$ where $$\mathcal B$$ denotes the restriction of the Borel sigma algebra to the given set.
Then $$g(y, x) = \frac{1}{2}$$ for $$(y, x) \in [0, 1] \times E$$ and $$g(y, x) = y$$ otherwise.
This has no modification that is continuous in $$x$$ for every $$y$$, since for every $$y$$ except for $$y= \frac{1}{2},$$ $$g(y, \cdot)$$ is essentially discontinuous everywhere.
• In the question, $\Sigma_0$ is a sub-$\sigma$-algebra of the $\sigma$-algebra on $Y$, not $X\times Y$. Nov 24 at 8:19
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.