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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.

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What a great question! –  anon Oct 14 '10 at 17:48
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4 Answers

up vote 7 down vote accepted

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.

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You're right about the hypothesis, but I meant "biggest" one because I have the idea that it is used most often or is the most important since for example it implies the differentiation lemma. Then to me the idea of proving that one first is the most appealing. But probably I have way less experience than you with measure and integration theory. –  Jonas Teuwen Oct 14 '10 at 13:08
@ Jonas - I just took a first course in measure theory so I speak from a humble point of view, but I was struck by the importance of MCT because it doesn't require the assumption that $f$ is integrable, and therefore can be used to prove that $f$ is integrable. (In DCT, you need to know $f$ has an integrable dominator, so it is already integrable by hypothesis.) MCT was a key lemma for the rest of the class: central to proof of Tonelli's and Fubini's theorems, existence of Radon-Nikodym derivative, and any time we wanted to generalize results from finite measure spaces to $\sigma$-finite... –  Ben Blum-Smith Dec 24 '11 at 14:16
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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!

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I believe I learned the monotone convergence theorem first, then dominated convergence, then Fatou. You could argue that this sequence orders the theorems in increasing order of hypothesis complexity.

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Confused by the hypothesis complexity claim. Fatou seems least complex to me: $f_n\to f$ a.e., $f_n,f\geq 0$. You don't have to assume that there is an integrable dominator (as you do for LDCT) or that $f_n\uparrow f$ (as you do for MCT). What am I missing? –  Ben Blum-Smith Dec 24 '11 at 14:06
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In fact i would add something more. For me it would be the Bounded Convergence theorem, then the Fatou's lemma and then the Monotone convergence theorem followed by Dominated convergence theorem.

I think the book by Royden has a very good way proving these theorems

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