If is a sequence of non-negative measurable functions and increases monotonically to for each , pointwise (this can actually be relaxed to just having a.e.), then

.

**Proof**:

As a.e., we have that a.e. for each , and so for each . Now also notice that since , that forms a bounded monotonic sequence sequence of real numbers, so it must converge. So we have

By Fatou’s lemma we have that

Since we have inequalities both ways, we must have equality.

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