# Conditional expectation and norm interchange.

Dealing with expectation as a projection is a quite useful idea when we work with applications of the complex interpolation theorems. Actually, we can prove that expectation contracts $L^p$ and we can have an interesting norm convergence theorem of nested expectations.

Another kind of important result is, once defined properly a mixed norm $L^p_x, L^q_y$ when we have two spaces $(X, \mathcal{B}_X, \mu_X)$ and  $(Y, \mathcal{B}_Y, \mu_Y)$, checking the dominance of each of the original norms on the mixed ones. Basically, it is solved using similar techniques on the proof of Stein interpolation theorem and applying Fubini-Tonelli many times. The slides of the lecture I gave on it can be found here.