Measure Theory/Convergence theorems

monotone convergence theorem Let be a measure space, and let be an ascending (that is, pointwise) sequence of non-negative functions, that converges pointwise to a function . Then . Fatou's lemma Let be a measure space, and let be a sequence of non-negative functions. Then . Note that, upon defining , that the sequence of functions is strictly ascending and converges pointwise to as . Hence, the monotone convergence theorem is applicable and we obtain . Now for each , we have , and if we take the lim inf, .