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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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5  
Use Fatou's Lemma. –  David Mitra Aug 6 '12 at 2:55
    
Here is an approach. Appealing to the inequality $ |\,|a|-|b|\,| \geq | a | -|b| $ we have $$ |\, ||f||_1 - ||f_n||_1 \,| \geq ||f||_1 - ||f_n||_1 $$ $$ \Rightarrow ||f||_1 - ||f_n||_1 \leq ||f||_1 - ||f_n||_1 \leq ||f_n||_1 - ||f||_1 $$ $$ \Rightarrow ||f||_1 \leq 2 ||f_n||_1 - ||f||_1 \Rightarrow ||f||_1 \leq ||f_n||_1 \leq 1 $$ $$ ||f||_1 \leq 1$$ –  Mhenni Benghorbal Jan 25 at 23:07
    
@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. –  Michael Greinecker Jan 26 at 23:56
    
@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. 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. –  Michael Greinecker Jan 27 at 1:08
    
@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. –  Mhenni Benghorbal Jan 27 at 1:16

1 Answer 1

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?

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