Just routine conditional expectation. For random variables X and Y (under very meager assumptions, E[X|Y] is the almost sure unique random variable that int

Just routine conditional expectation. For random variables X and Y (under very meager assumptions, E[X|Y] is the almost sure unique random variable that integrates like X over all events in the sigma algebra generated by Y. In particular E[X] = E[E[X|Y]] Details follow from the Radon-Nikodym theorem in, say, Rudin, Real and Complex Analysis where Rudin give the von Neumann proof. Details are also in the standard texts on graduate probability by Loeve, Neveu, Breiman, Chung, and a few more.