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I'm a bit stuck on this problem, and I was hoping someone could point me in the right direction.

Suppose $f, f_1, f_2,\ldots \in L^{1}(\Omega,A,\mu)$ , and further suppose that $\lim_{n \to \infty} \int_{\Omega} |f-f_n| \, d\mu = 0$. Show that $f_n \rightarrow f$ in measure $\mu$.

In case you aren't sure, $L^1(\Omega,A,\mu)$ is the complex Lebesgue integrable functions on $\Omega$ with measure $\mu$.

I believe I have to use the Dominated convergence theorem to get this result, and they usually do some trick like taking a new function $g$ that relates to $f$ and $f_n$ in some way, but I'm not seeing it. Any advice?

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