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I have the following problem:

Suppose that $f,f_1,...,f_n,...\in L^1(\Omega)$ and $f_n\stackrel{a.e.}{\rightarrow}f$. Show that $$f_n\stackrel{L^1}{\rightarrow}f\Leftrightarrow ||f_n||_1\rightarrow ||f||_1$$If possible use the functions $g_n=|f|+|f_n|-|f_n-f|$.

Till now I have the following:

Proof

$\Rightarrow$ Let us assume that $f_n\stackrel{L^1}{\rightarrow}f\Leftrightarrow \lim_{n\rightarrow \infty}\int_\Omega |f_n-f|d\mu=0$. This is equivalent to say that $$\forall \epsilon >0 \,\,\exists N\in \mathbb{N}\,\,\,s.t.\,\,\,\forall n\geq N\,\,\,\int_\Omega |f_n-f|d\mu<\epsilon \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ Let $\epsilon >0$ satisfying (1), then $$\left|\int_\Omega |f_n|d\mu-\int_\Omega |f|d\mu\right|=\left|\int_\Omega |f_n|-|f|d\mu\right|\leq \int_\Omega \left|\left|f_n\right|-\left|f\right|\right|d\mu\leq \int_\Omega |f_n-f|d\mu<\epsilon$$ Therefore $\forall \epsilon >0 \,\,\exists N\in \mathbb{N}\,\,\,s.t.\,\,\,\forall n\geq N\,\,\,\left|{||f_n||_1-||f||_1}\right|<\epsilon$.

Is this correct until here?

$\Leftarrow$ I think that in this direction I need do use my $g_n$ but I don't see where.

If the first direction is correct could someone give me a hint for the second one?

Thank you.

user123234
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    Hint: Use Fatou's lemma on $g_n$. – Evangelopoulos Foivos Oct 31 '21 at 20:51
  • So only for $\Leftarrow$ this direction? i.e. is the other one correct or is it also wrong? – user123234 Oct 31 '21 at 20:52
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    The other direction looks fine. In general, convergence in norm implies the convergence of norms, i.e., $||u_n-u||\to 0 \implies ||u_n||\to||u||$. This is a consequence of the reverse triangle inequality. – Evangelopoulos Foivos Oct 31 '21 at 21:04
  • okey perfect, so I will try to use fatou's lemma, but could you maybe explain why we use exactly this $g_n$ because I don't see the connection yet. – user123234 Oct 31 '21 at 21:06
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  • $g_n$ is positive so you can, in fact, apply Fatou.
  • You want to somehow connect $|f_n|, |f| $ and $|f_n-f|$. Remember, you know $|f_n| \to |f|$, and want to show $|f_n-f| \to0$. With a bit of fiddling you end up with $g_n$.
  • You may need to use the fact that $\liminf(-a_n) = - \limsup (a_n)$.
    • – Evangelopoulos Foivos Oct 31 '21 at 21:12
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    There is also a useful lemma of Brezis and Lieb that solves your problem, but it's a bit of an overkill. – Evangelopoulos Foivos Oct 31 '21 at 21:21
  • Hahah thanks but we only hat fatou's lemma. I will try it and maybe ask again if I don't see how to procede. – user123234 Oct 31 '21 at 21:25
  • This has been answered many times here, but is hard to find. Try https://math.stackexchange.com/a/51503/27978 – copper.hat Nov 01 '21 at 19:27