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- #1

- Jan 26, 2012

- 995

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**Problem**: Suppose that $\{f_n(x)\}$ and $\{g_n(x)\}$ are two sequences of measurable functions (on $\mathbb{R}$) such that $|f_n(x)|\leq g_n(x)$ for each $n=1,2,\ldots$. And suppose that $f_n$ converges to $f$ and that $g_n$ converges to $g$ almost everywhere. Show that

\[\lim_{n\to\infty}\int_{\mathbb{R}} g_n\,dm = \int_{\mathbb{R}} g\,dm\]

implies that

\[\lim_{n\to\infty}\int_{\mathbb{R}} f_n\,dm = \int_{\mathbb{R}} f\,dm\]

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