$X$ and $Y$ are just random variables and we know $X|Y$ follows some distribution.
I am trying to show that $\text{E}[X|X,Y]=X$.
I know a bit from measure theory that if X is $\sigma(X,Y)$-measurable then we are done. So I know X is $\sigma(X)$-measurable, and $\sigma(X)\subset \sigma(X,Y)$. Is this enough to show the result?
Also are their distributions the same: i.e. $X|X,Y=X$?
Thanks.