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- Jan 26, 2012

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As with the POTWs for the Secondary/High School and University students, Jameson and I will post a problem each Monday around 12:00 AM Eastern Standard Time (EST), and you'll have till Saturday at 11:59 PM EST to submit your solutions. With that said, let's get this started!!

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**Problem**: Let $g$ be a Lebesgue integrable function on a measurable set $E\subset\mathbb{R}$ and suppose that $\{f_n\}$ is a sequence of measurable functions such that $|f_n(x)|\leq g(x)$ $m$-a.e. on $E$. Show that

\[\int_E \liminf_{n\to\infty}f_n\,dm \leq \liminf_{n\to\infty}\int_E f_n\,dm \leq \limsup_{n\to\infty}\int_E f_n\,dm \leq \int_E \limsup_{n\to\infty}f_n\,dm.\]

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Remember to read the POTW submission guidelines to find out how to submit your answers!