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Suppose that $(X,\mathcal{A},\mu)$ is a measure space. Let $f,f_1,f_2,\ldots:X\rightarrow\mathbb{R}$ be $\mathcal{A}$-measurable functions that are $\mu$-integrable.

Show that if \begin{align} \sum_{n=1}^{\infty} \int |f_n-f|d\mu < +\infty \end{align} then $\{f_n\}$ converges to $f$ $\mu$-a.e.

My idea is to prove the contrapositive, i.e. that if $\{f_n\}$ does not converges to $f$ $\mu$-a.e., then $\sum_{n=1}^{\infty} \int |f_n-f|d\mu = +\infty$. However, I am a bit stuck. Any hints?

BasicUser
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  • If the infinite sum is finite, this implies the terms converge to $0$ – FShrike Feb 13 '22 at 14:34
  • Yes, this is a repost then @TheSilverDoe. I guess interchanging the sum and the integral follows from Beppo Levi's Theorem (and thus MCT) and then we get what we want. – BasicUser Feb 13 '22 at 14:38

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