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Math Help - Convergence

  1. #1
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    Convergence

    Let {g_n} be a sequence of nonnegative integrable functions defined on a measurable set E such that g_n \rightarrow g almost everywhere on E, where g is integrable on E. Assume that {f_n} is a sequence of measurable functions such that |f_n| \leq g_n and f_n \rightarrow f almost everywhere on E.
    If lim_{n \rightarrow \infty}  \int_E g_n = \int_E g, prove lin_{n \rightarrow \infty \int f_n \rightarrow \int_E f}.

    i think that i can use monotone convergence thm on this questions but do not really know how to go about this. please help me.
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  2. #2
    Senior Member
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    Smile

    Quote Originally Posted by Archi View Post
    Let {g_n} be a sequence of nonnegative integrable functions defined on a measurable set E such that g_n \rightarrow g almost everywhere on E, where g is integrable on E. Assume that {f_n} is a sequence of measurable functions such that |f_n| \leq g_n and f_n \rightarrow f almost everywhere on E.
    If lim_{n \rightarrow \infty}  \int_E g_n = \int_E g, prove lin_{n \rightarrow \infty \int f_n \rightarrow \int_E f}.

    i think that i can use monotone convergence thm on this questions but do not really know how to go about this. please help me.
    see http://www.mathhelpforum.com/math-he...oposition.html
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