Let $X: \Omega \to U$ and $Y: \Omega \to \mathbb{R}^n$ be random variables. Let $$E[Y | X] := E(Y | \sigma(X )): \Omega \to \mathbb{R}^n$$ be the conditional expectation with respect to the sub-$\sigma$-algebra $\sigma(X) \subseteq \Sigma$ generated by $X$. By Doob-Dynkin lemma (source: wikipedia) there is a function $e_Y: U \to \mathbb{R}^n $, such that $e_Y \circ X = E[Y|X]$.
On the other hand (assuming $X,Y$ are $L^2$-random var's), we could ask for a measurable function $g:U \to \mathbb{R}^n$ that minimizes $E((Y-g \circ X)^2)$. If it exists, we could define $E(Y|X) := g \circ X$.
In this case,
- do we have $e_Y = g$ a.e.?
- If both $X$ and $Y$ are continuous, do the following identities hold a.e.? $$E(Y|X)(\omega) = E[Y|X](\omega) = \frac{1}{f_X(X(\omega))}\int y f_{X,Y}(X(\omega),y)dy$$