Let $X,Y$ be independent rv's.
For $f:\mathbb{R}^2\to\mathbb{R}$ measurable we have $f_X:\mathbb{R}\to\mathbb{R}$ where
$$f_X(y)= \left\{
\begin{array}{lr}
\mathbb{E}[f(X,y)] & : \mathbb{E}|f(X,y)|<\infty\\
0 & : \text{else}
\end{array}
\right..$$
Let $M_b$ be the set of all bounded measurable functions $\mathbb{R}^2\to\mathbb{R}$ and
$$C=\{f\in M_b: \mathbb{E}[f(X,Y)|Y]=f_X(Y)\}.$$
How do we prove that $C=M_b$?
If we use the monotone class theorem, then for a $\pi-$system $\Pi$ that generates $\mathcal{B}^2$ we can show that $M_b\subset C$.
Therefore we want
1) $1_{\mathbb{R}^2}\in C$:
This is clear since for $A\in\sigma(Y)$ we have $$\int_A\mathbb{E}[1_{\mathbb{R}^2}(X,Y)|Y]d\mathbb{P}=\int_A1_{\mathbb{R}^2}(X,Y)d\mathbb{P}=\int_A1d\mathbb{P}=\int_A\mathbb{E}[1_{\mathbb{R}^2}(X,Y)]d\mathbb{P}.$$
2) for an increasing, non-negative sequence $(f_n)_n$ in $C$, where $f_n\to f$ with $f$ bounded, then $f\in C$:
So we want to show
$$\mathbb{E}[f(X,Y)|Y]=\mathbb{E}[f(X,Y)]$$
Can we do this by using conditional monotone convergence and say
$$\mathbb{E}[f_n(X,Y)|Y]\to\mathbb{E}[f(X,Y)|Y]$$
and since
$$\mathbb{E}[f_n(X,Y)]\to\mathbb{E}[f(X,Y)]$$
by regular MCT, it follows?
3) for all $B\in \Pi$, $1_B\in C$:
Can we take $\Pi=\{(a,b)\times (c,d):a<b,c<d\in\mathbb{R}\}$?
Then for $A\in\sigma(Y)$
$\begin{align*}
\int_A\mathbb{E}[1_{(a,b)\times (c,d)}(X,Y)|Y]d\mathbb{P}&=\int_A1_{(a,b)\times (c,d)}(X,Y)d\mathbb{P}\\
&=\mathbb{P}[A\cap(X,Y)\in(a,b)\times (c,d)]\\
&=\mathbb{P}[A\cap\{X\in(a,b)\}]\mathbb{P}[A\cap\{Y\in (c,d)\}]\\
&=\int_A1_{X\in(a,b)}d\mathbb{P}\cdot\int_A1_{Y\in(c,d)}d\mathbb{P}\\
&=?\\
&=\int_A\mathbb{E}[1_{(a,b)\times (c,d)}(X,Y)]d\mathbb{P}.
\end{align*}$
Is this correct? Which steps am I missing?