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Thread: Convergence

  1. #1
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    Convergence

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

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

    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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