Theorems in Measure Theory: Fatou's Lemma, Lebesgue DCT, Monotone CT

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.

I prefer the direction: Dominated convergence theorem -> Beppo-Levi (monotone convergence) -> Fatou.

This direction requires more machinery (like Egoroff and the absolute continuity of the Lebesgue integral) but it is in my opinion more tidy. A book that uses this approach is Bogachev - Measure Theory.

Edit: This way the later theorems are just "corollaries" of DCT, the "biggest" one!