Suppose $X,Y$ are independent random variable, and $X,Y \in L^1$, Then the conditional expectation is $E[Y|\sigma(X)] = E[Y]$
My question is that when I check the definition of conditional expectation, I don't see why the candidate $E[Y]$ is $\sigma(X)$-measurable? (Why NOT $\sigma(Y)$-measurable actually?) This confuses me a lot since every book I consult always say this step is obvious.
Indeed, I have some naive intuitions, since $X,Y$ are independent, the corresponding sigma-algebra generated by $X$ and $Y$ are also independent; i.e., $\sigma(X)$ is independent of $\sigma(Y)$; and $Y$ also independent of $\sigma(X)$ hence we can drop the condition intuitively.
E(Y) is a constant and a constant function is measurable with respect to any sigma algebra.