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Let $(\Omega,\mathcal{F},P)$ denote a probability space and let $X:\Omega\to \mathbb{R}$ and $Y:\Omega\to \mathbb{R}$ denote two random variables with finite second moments. Let $g_0: \mathbb{R} \to \mathbb{R}$ be a measurable function that minimizes $$ E(X - g(Y))^2 $$ over all measurable functions $g$. $g_0(Y)$ is the conditional expectation of $X$ given $Y$.

Are there general conditions under which $g_0$ is known to be absolutely continuous?

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    $\begingroup$ Far from a general condition but if $X$ and $Y$ are jointly Gaussian, the conditional expectation reduces to projection of $X$ onto the linear span of $Y$. A linear function is absolutely continuous so this constitutes an example I guess. $\endgroup$
    – Calculon
    Oct 19, 2016 at 12:29

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