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Suppose $\{f_n\}\subset L^1(\mathbb{R})$ with $||f_n||_1\leq 1$ $\forall n$ and $f_n \to f$ a.e. How can I show that $||f||_1 \leq 1$? This will be easy once we know $f\in L^1(\mathbb{R})$ so I guess that is my question.

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    @MichaelGreinecker: I answered this question $18$ months ago. I still have not had a closer look at it since not all the time we are ready to do so.2014-01-27

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Just following the David Mitra's hint, this is the Fatou's Lemma from Zygmund & Whedeen Measure and Integral:

fatou

You know that $|f_n|\to |f|$ pointwise a.e., this says that $\liminf |f_n|=|f|$ a.e. So in order to conclude what you want, by Fatou's Lemma, it's enough to show that $\liminf \int |f_n|\leq 1$.

Remember that: $\liminf \int |f_n|=\sup\left\{\inf\left\{\int |f_n|,\int |f_{n+1}|,\int |f_{n+2}|,\ldots,\right\}:n\in \Bbb N\right\}.$

Can you catch it from here?