Changing the order of Conditional Expectation

Question

Is it true in general that the conditional expectations $E(E(f(X,Y)|X)|Y) = E(E(f(X,Y)|Y)|X)$ if X and Y are independent random variables and f measurable? I believe if $f$ is simply the product function, then this is obviously true through independence and the basic properties of conditional expectations (taking out what is known). However, what if we have the function $f$ composed with X and Y? Would the result still hold?

Answer 1

LHS is a measurable function of $Y$ and RHS is a measurable function of $X$. Since $X$ and $Y$ are independent the two sides can be equal only when both sides are constants. A simple counter-example is given by $X=Y, f(X,Y)=X$ since LHS is $X$, not a constant.