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
You are overloading the $y$ term; using it both as a free variable, and the bound variable of integration.
$g(y)=\langle \phi(X,Y)\mid Y=y\rangle$ is a function of constant $y$, not of the random variable $Y$.
The expectation of a function of a constant is the function of the constant, which is not the expectation of a function of a random variable.
$\qquad\begin{align}\big\langle\langle \phi(X,Y)\mid Y=y\rangle\big\rangle &= \langle g(y)\rangle\\ &= g(y) &&= \int_\Bbb R f_{X\mid Y}(s\mid y)\,\phi(s,y) \,\mathrm d s\\\text{versus}\\ \langle \phi(X,Y)\rangle&= \iint_{\Bbb R^2} f_{X,Y}(s,t)\,\phi(s,t)\,\mathrm d(s,t)\\ &= \int_\Bbb R f_{Y}(t)\int_\Bbb R f_{X\mid Y}(s\mid t)\,\phi(s,t)\,\mathrm d s\,\mathrm d t\\ &= \int_\Bbb R f_Y(t)\, g(t)\,\mathrm d t\\ &=\big\langle g(Y)\big\rangle \end{align}$
What you want is $\langle \phi(X,Y)\mid Y\rangle$, which is interpreted as a random variable that is a function of random variable $Y$... (indeed, that is $g(Y)$ above). It is used thus:
$\qquad\begin{align}\langle\phi(X,Y)\rangle &=\iint_{\Bbb R^2} f_{X,Y}(s,t)\,\phi(s,t)\,\mathrm d (s,t)\\ &= \int_\Bbb R f_Y(t)\int_\Bbb R f_{X\mid Y}(s\mid t)\,\phi(s,t)\,\mathrm d s\,\mathrm d t\\ &= \int_\Bbb R f_Y(t)\,\langle \phi(X,Y)\mid Y=t\rangle\,\mathrm d t\\ &=\big\langle\langle \phi(X,Y)\mid Y\rangle\big\rangle \end{align}$
Measure theory would say $\langle\cdot\mid Y\rangle$ is the conditional expectation with respect to the sigma-algebra generated by $Y$ ... but you don't need to drill down into that.