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I know that in generally that $L^1$ convergence does not imply pointwise convergence, but I want to check for this special case.

Suppose $\mu$ is a counting measure, and $f_n \to f$ in $L^1(\mu$). That is $\sum_{m\in\mathbb{N}} | f_n(m) - f(m)| \to 0$. Since for every $m\in \mathbb{N}, | f_n(m) - f(m)| \le \sum_{m\in\mathbb{N}} | f_n(m) - f(m)| \to 0$, I have $| f_n(m) - f(m)| \to 0$. Therefore, $f_n\to f$ pointwise.

May I ask if the above argument holds?

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    $\begingroup$ Yes, for $g$ in $L^1(\mu)$ and all $n$: $$|g(n)|\leq\| g\|_1.$$ That is not true for all measures. $\endgroup$ Aug 5 at 23:48

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This is correct. $\forall n$ and for $f \in L^{1}(\mu)$ we know $|f(n)| \leq ||f||_{1}$

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