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Measure Theory/Convergence theorems

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Theorem (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

.

Theorem (Fatou's lemma):

Let be a measure space, and let be a sequence of non-negative functions. Then

.

Proof: 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,

.