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Given a sequence of real valued functions in a measure space, such that $\sum_{n=1}^{\infty}\lVert f_n \rVert_1<\infty$, I want to know whether the series $\sum_{n=1}^{\infty}f_n$ converges in measure and almost everywhere. It seems like it should converge at least in measure, as we have that $\lim_{n\to\infty}\sum_{k=n}^{\infty}\lVert f_k \rVert_1=0$ , but I'm struggling to find a candidate to limit. I've tried to find a pointwise limit almost everywhere, which would give both almost everywhere and hence also in measure convergence, but unsuccessfully. Any advice on how to find the limit candidate is appreciated so that I can work out the convergence myself afterwards.

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    $\begingroup$ The partial sums of the infinite series of the $f_n$ are Cauchy with respect to the $L^1$ norm, so they have a limit in $L^1$. $\endgroup$ May 16, 2023 at 19:05

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