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 .

## Proof

We have . Applying Fatou’s lemma to the non-negative sequence we get

That is,

Since , we get . Since , this implies .

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