What is the $E[X| X, Y]$ when $X$ and $Y$ are dependent? Is it simply $X$?
I was expecting this result because the knowledge about $X$ is already known (as we are conditioning on $X$ and $Y$).
\begin{align*} E[X| X, Y] = E[X|X] = X \end{align*}
I don't have much knowledge in measure-theory to prove it is indeed the case. Can anyone explain it with a bit more detail?
$E[X|\sigma(X,Y)]=X$ because of the "pulling out know factor" property of the conditional expectation (https://en.wikipedia.org/wiki/Conditional_expectation#Basic_properties).
Indeed, $X$ is $\sigma(X,Y)$-measurable because $\sigma(X) \subseteq \sigma(X,Y)$, so you can take it out of the conditional expection. What is left is $XE[1|\sigma(X,Y)]=X$. Dependence between $X$ and $Y$ does not play any role here.