# Is this conditional expectation identity true?

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TL;DR Summary
I am working an exercise to show a conditional expectation identity, but I'm not sure if it is true at all.
I'm working through an exercise to prove various identities of the conditional expectation. One of the identities I need to show is the following $$E(f(X,Y)\mid Y=y)=E(f(X,y)\mid Y=y).$$ But I am a little concerned about this identity from things I've read elsewhere. I am paraphrasing from Breiman's book Probability:
Proposition 4.36: Let ##X## and ##Y## be random variables. Let ##f(X,Y)## be a real valued, random variable such that the expectation of the absolute value of ##f(X,Y)## is finite. If ##Q(\cdot\mid Y=y)## is a regular conditional distribution for ##X## given ##Y=y##, then, $$E(f(X,Y)\mid Y=y)=\int f(x,y) \, dQ(\cdot \mid Y=y)\quad \text{a.s. with respect to the law of }Y.\tag{4.37}$$
Again, paraphrasing Breiman in section 4.3, page 80:
It is tempting to replace the right-hand side of (4.37) by ##E(f(X,y)\mid Y=y)##. But this object cannot be defined through the standard definition of conditional expectation (4.18) [defined below].
Definition 4.18: ##E(X\mid Y=y)## is any random variable on ##\mathbb R##, where ##Q(B)=P(Y \in B)##, satisfying $$\int_B E(X\mid Y=y) dQ = \int_{Y \in B} X dP,$$ for all Borel sets ##B##.
Thus, does ##E(f(X,Y)\mid Y=y)= E(f(X,y)\mid Y=y)## make sense? Breiman seems to suggest that ##E(f(X,y)\mid Y=y)## is not well defined, so I'm not sure how to proceed.

I think this pdf clarified things a bit.

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