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A few days I skimmed over some book and found a few extensions/generalizations of the convergence theorems for Lebesgue integral, namely dominated convergence theorem, Fatou's lemma, and monotone convergence theorem. I tried to find the book for the whole day today but without luck.

Would appreciate if you know some of these generalizations and the references for the proof/origin.

Thank you.

EDIT

For example, one of the variations of dominated convergence is to replace the point-wise convergence with convergence in measure. Another one for monotone convergence is to replace non-negativity with boundedness from below by some integrable function. I was looking for variations on the theme like those. I know some of these maybe very small modifications from the original theorems, but they appear to provide wider applicability and to give some insight about the tightness of these assumptions in the theorems.

Maybe this can stir some interest to my original question?

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Could you please provide more information on what sorts of generalizations you are looking for? –  Jonas Meyer Jun 7 '11 at 5:43
    
@Jonas, for example, in the link www.math.umass.edu/~lr7q/ps_files/teaching/math623/PS3.pdf in Problem 2. Basically, I just want to know what conditions in these theorems can be loosened or extended. –  Qiang Li Jun 7 '11 at 5:48
    
There are lots of possible variations here. I'd suggest picking up a few different measure theory books and looking at the relevant chapters. Often such extensions are left as exercises, so look at those as well. –  Nate Eldredge Jun 8 '11 at 12:38
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1 Answer

Are you looking for the Vitali convergence theorem? It's a generalization of Lebesgue's Dominated Convergence Theorem.

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thank you. I knew about this theorem. But this is not what I read the other day. I am more interested in the examples given in the link in my last comment. –  Qiang Li Jun 7 '11 at 14:33
    
@Qiang Li: Are you talking of the Lusin's theorem, which is a generalization of the Egorov's theorem. –  user9413 Jul 8 '11 at 8:20
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