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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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    Use [Fatou's Lemma](http://en.wikipedia.org/wiki/Fatou%27s_lemma).2012-08-06
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    Here is an approach. Appealing to the inequality $ |\,|a|-|b|\,| \geq | a | -|b| $ we have $$ |\, ||f||_1 - ||f_n||_1 \,| \geq ||f||_1 - ||f_n||_1 $$ $$ \Rightarrow ||f||_1 - ||f_n||_1 \leq ||f||_1 - ||f_n||_1 \leq ||f_n||_1 - ||f||_1 $$ $$ \Rightarrow ||f||_1 \leq 2 ||f_n||_1 - ||f||_1 \Rightarrow ||f||_1 \leq ||f_n||_1 \leq 1 $$ $$ ||f||_1 \leq 1$$2014-01-25
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    @MhenniBenghorbal You do realize that $||a|-|b||\geq|a|-|b|$ does not imply that $|a|-|b|\leq |b|-|a|$? One can see the mistake without having any idea about measure theory or functional analysis.2014-01-26
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    @MhenniBenghorbal It is perfectly legitimate to downvote answers one considers unhelpful or wrong. It is not legitimate to repost deleted posts. But if you want to discuss site moderation, you should as a question on [meta](http://meta.math.stackexchange.com/). My only "interference" here was pointing out a mathematical mistake in a comment. Every user here with a reputation of at least 50 has the right to do so, and that includes the moderators. I will not respond to any further comments here that do not pertain to the mathematics of the present problem.2014-01-27
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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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