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 and we can have an interesting norm convergence theorem of nested expectations.
Another kind of important result is, once defined properly a mixed norm when we have two spaces and , 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.