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 .
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
by fatou's lemma, we get
from (i) and (ii), we conclude that
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