In measure theory there are three fundamentally related theorems about exchanging limits and integrals: Fatou's lemma, Lebesgue's Dominated Convergence Theorem, and Monotone Convergence Theorem. It is difficult to prove any of these from scratch, but once you have one, the others are easier.
My question is, for those who have learned these theorems: which one do you prefer to prove first? Difficulty, length, and, perhaps most importantly, how enlightening each path is are the key considerations. I suppose you could also phrase the question: if you were teaching a class in what order would you prove these theorems.
I've read through all of the proofs and there doesn't seem to be a big difference, but perhaps someone can shed some new light on this question.
Best Answer
I've generally seen MCT -> Fatou -> DCT. MCT is nice if the integral is defined as the supremum of the integrals of all simple functions less than $f$. Fatou points out that you can lose mass when passing to the limit, but cannot gain it. And DCT is nice to prove with two applications of Fatou, since turning your head upside down shows that you cannot gain mass either positively or negatively.
I disagree with Jonas's idea that DCT is the "biggest" one, since it doesn't speak about functions not in $L^1$, which the others do; this is often very important. Also, I see the hypothesis of the DCT as somewhat ad hoc. To my mind, the "biggest" one is the Vitali convergence theorem, whose hypothesis is uniform integrability, which is necessary and sufficient. But since it is more complicated it is often skipped.