Suppose $X,Y$ independent random variables and $\phi$ be a function such that $E[\phi(X,Y)]<\infty$. Let $g(x)=E[\phi(x,Y)]$.
We need to show that $g(X)=E[\phi(X,Y)|X]$.
However I cant show $g(X)$ is $\sigma(X)$ measurable (the rest of the problem is easy). Any hint is appreciated.