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proof of Fatou-Lebesgue theorem

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How are we able to apply Fatou's lemma in this proof? The conditions of Fatou's lemma clearly state we need a non-negative sequence of measurable functions yet here we are not given the non-negativeness of fn. How is this reconciled?

This is reconciled by considering the sequence \phi+f_n, which must necessarily be nonnegative. I admit that I overlooked this and will edit the proof accordingly when I have more time.

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