If is a Lebesgue measurable set and is a sequence of non-negative measurable functions, then

**Proof**: Let and . Let be a simple function with . Assume on , as the case where is trivial. Now define as follows.

Where is a value that ensures . Now note that and , so there must exist an such that for all .

Let and notice that . Also, . For we have the following.

So we have

Now notice that is a simple function, and so we can write it as

Now we have

Letting this gives us

This leaves us with

If , then we have

This gives the desired result if we let as is an arbitrary simple function with .

**Note**: This is an awesomely useful lemma, but I hate this proof.

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