this prop. in royden page 270 the proof must be by Fatous lema" applying fatoues lemma to fn+gn>0 and gn-fn>0 "
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correct me if i'm wrong from , we have . i) consider because a.e and a.e, then a.e. by fatou's lemma, we have using the linearity of integral, then we get because , then ii) consider by fatou's lemma, we get from (i) and (ii), we conclude that
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