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Let X,Y be random variables defined on a probability space $(\Omega,S,P)$ and let $\phi(X,Y)$ be a borel function of X and Y.

I want to prove the statement $$E\{ \phi(X,Y)|Y=y \} = E\{\phi(X,y)|Y=y\} $$

It seems intuitively easy, but I can't prove it rigorously in the continuous case (if X, Y are continuous) .

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    It would help you to formulate precisely how $$E\{ \phi(X,Y)|Y=y \}$$ and $$E\{\phi(X,y)|Y=y\} $$ are defined, in the non discrete case that interests you. – 2017-02-28

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