This key theorem showcases the full power of Lebesgue Integration Theory.
Generalized Lebesgue Dominated Convergence Theorem
Let and be sequences of measurable functions on satisfying a.e. in , a.e. in , and a.e. in . If and , then .
We have . Applying Fatou’s lemma to the non-negative sequence we get
Since , we get . Since , this implies .