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.
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?