Let $X$ and $Y$ be random vectors defined on a common probability space. $X$ takes values in a finite-dimensional space $\mathcal{X} \subset \mathbb{R}^p$, while $Y$ takes values in $\mathbb{R}$. The conditional expectation $E(Y \mid X)$ is then a random variable that is uniquely defined up to null sets.

I am seeking a set of sufficient conditions on the joint distribution of $(X,Y)$ for the following statement to be true:

Given any point $x_0 \in \mathcal{X}$, there exists a function $f \colon \mathcal{X} \to \mathbb{R}$ such that (i) $f(X)$ is a version of $E(Y \mid X)$, and (ii) $f(\cdot)$ is continuous at $x_0$.

Obviously, the continuity part is the non-trivial one. By Lusin's theorem, any measurable function (such as any version of the conditional expectation function) is "nearly continuous", but this is not quite enough for me.

Ideally the sufficient conditions for the above statement would not involve restrictions on the densities or conditional densities of $X$ and $Y$. The problem that motivates this question has a complicated geometry, so it is difficult to characterize densities with respect to fixed dominating measures.

If you require more structure to the problem (but ideally the question would be answered in more generality), you may assume: $Y = g(A)$ for a continuous function $g(\cdot)$, $X = A + B$, and the random vectors $A$ and $B$ are independent. However, $A$ and $B$ may concentrate on different subspaces of $\mathbb{R}^p$, each of which is a complicated manifold.

Thank you in advance for your time!


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