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?