Let $X, \ Y, \ Z \in L^1$ be random variables such as $X = Y $ a.s. Is it in general true that $EXZ=EYZ$?
It is surely true if Z is independent of X and Y, cause $EXZ=EXEZ=EYEZ=EYZ. $
But is it true in general? I tried counting $EZX-ZY=EZ(X-Y)$ I think that $X-Y=0$ a.s. hence the integral (expectation) is zero and $EZX=EZY $. Is it correct?
And another question if Z is independent of X, does it imply that Z is independent of Y? How can I prove it, or what is the counterexample?