Sorry if I explain it in my own words, I don't know if you've studied it the same way... And maybe there are small typos, but not very important.
1) We know that if Z is -measurable, then for any rv X (in L^2 I think), we have
So since Y is obviously -measurable, E[E[XY|Y]]=E[YE[X|Y]] (*)
But there's something that says :
Let be a -algebra. For any -measurable Z (positive), E[ZX]=E[Z E[X|Y]], and where X is positive. (this comes from the fact that E[X|Y] is the orthogonal projection of X over , but you don't really need to know it if you haven't learnt this...)
Exact same reasoning, under the condition that h is -measurable.2) How about E[X h(Y)]=E[E(X h(Y)|Y)]? Is this a correct statement?
I hope this is clear enough
* is the smallest sigma-algebra that makes B measurable
** Note : a rv A is -measurable iff there exists which is -measurable such that