Suppose that we have real-valued random variables $X$ and $Y$ whose joint and conditional densities are well-defined everywhere. For e.g., $f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}$ is used to the evaluate the conditional expectation where we condition on $Y$. Let's use the notation $\langle\ \cdot \ \rangle$ for the expectation and $\langle\ \cdot \ |Y=y\rangle$ for the conditional expectation.
Question: Is it true that
$$ \big\langle \langle \phi(X,Y)|Y=y\rangle \big\rangle = \langle \phi(X,Y)\rangle ? $$
On the one hand, the 'law of total expectation' and my calculation below suggests that it might be, but on [p. 70, 1], the author says that it isn't.
'Proof': We have $$ \begin{align*} \langle \phi(X,Y)|Y=y\rangle &= \int_\mathbb R \phi(x,y) f_{X|Y}(x|y)dx \\ &= \int_\mathbb R \phi(x,y) \frac{f_{X,Y}(x,y)}{f_Y(y)}dx \end{align*} $$
so that $$ \begin{align*} \big\langle \langle \phi(X,Y)|Y=y\rangle \big\rangle &= \int_{\mathbb R} \int_\mathbb R \phi(x,y) \frac{f_{X,Y}(x,y)}{f_Y(y)}dx f_Y(y) dy\\ &= \int_{\mathbb R} \int_\mathbb R \phi(x,y) f_{X,Y}(x,y)dx dy\\ &= \langle \phi(X,Y)\rangle \end{align*} $$
Also let me know if I can use better notation to clear up this sort of a confusion without getting too much into measure theory.
[1]: G.S. Chirikjian, Stochastic Models, Information Theory, and Lie Groups - Vol. 1
