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Math Help - proof this proposition

  1. #1
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    proof this proposition

    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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  2. #2
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    We can't given any hints or suggestions without seeing what you already know about this kind of problem. Please tell us what you have already done and what thoughts you have about it.
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  3. #3
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    Smile

    correct me if i'm wrong
    from |f_{n}| \leq g_{n}, we have -g_{n} \leq f_{n} \leq g_{n} .

    i) consider g_{n} - f_{n} \geq 0
    because g_{n} \to g a.e and f_{n} \to f a.e, then g_{n} - f_{n} \to g - f a.e.
    by fatou's lemma, we have

    \int g - f \leq \lim \inf \int (g_{n} - f_{n})

    using the linearity of integral, then we get

    \int g - \int f \leq \lim \inf \int g_{n} - \lim \sup \int f_{n}

    because \int g = \lim \int g_{n} = \lim \sup \int g_{n} = \lim \inf \int g_{n}, then

    \int f \geq \lim \sup \int f_{n}

    ii) consider f_{n} + g_{n} \geq 0
    by fatou's lemma, we get

    \int f \leq \lim \sup \int f_{n}

    from (i) and (ii), we conclude that

    \int f = \lim \int f_{n}
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