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- Jan 26, 2012

- 995

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**Problem**: Suppose that $\{f_n\}$ is a collection of non-negative measurable functions with $f_1\geq f_2\geq\cdots\geq 0$ and $f_n(x)\rightarrow f(x)$ for every $x\in X$. Furthermore, suppose that $f_1\in L_{\mu}^1(X)$. Prove that $f\in L_{\mu}^1(X)$ and

\[\int_X f\,d\mu = \lim_{n\to\infty} \int_X f_n\,d\mu.\]

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