The question is, Given a probability space $(\Omega, F, P)$, a sigma field $G \subset F$, random variables $X,Y$ and a measurable function $\phi$ such that $X \in G$, $Y$ is independent with $G$ and $E[|\phi(X,Y)|] < \infty$. Then,
Show that $E[\phi(X,Y)|G]= g(X)$, where $g(x) = E[\phi(x,Y)]$ for all $x$.
I proved this result when $X,Y$ are independent. What is direction to prove this if independence condition is not given?
For functions of the product form $\phi(X,Y)=h(X)f(Y)$ the claim follows directly from $\mathbb E[f(Y)|{\cal G}]=\mathbb E[f(Y)]$ and $\mathbb E[h(X)|{\cal G}]= h(X)\,:$ $$ \mathbb E[h(X)f(Y)|{\cal G}]=h(X)E[f(Y)] $$ which is obviously equal to $E[h(x)f(Y)]|_{x=X}\,.$ To extend this to all measurable $\phi$ with $\mathbb E[\phi(X,X)]<+\infty$ I would look at the monotone class theorem.