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Please see the error page for more details.Q:
Convergence in $L^1$
I’m having a problem with a test on convergence in $L^1$. It goes like this:
Let $f_n \ge 0$ and $f_n \to f $ in $L^1(E)$ for all $E \subseteq \mathbb R$ and $\int_E f_n \to \int_E f $. Show that $f_n \to f$ almost everywhere.
I’m only able to prove that $f_n \to f$ almost everywhere since $\int_E f_n \to \int_E f $.
Could someone give me a hint on how to prove that $f_n \to f$?
Thanks.
A:
To prove convergence in $L^1$, you can argue as follows:
1) For each $E$, $\int_E |f_n-f| \leq \int_E |f_n-f| 1_E$ by the triangle inequality. So, we just need to estimate $\int_E |f_n-f| 1_E$.
2) The monotone convergence theorem (MCT) implies that $\lim_{n \to \infty} \int_E |f_n-f| 1_E = \int_E |f_\infty-f| 1_E$. Since $\int_E |f_\infty-f| 1_E$ is finite, it is clear that $\lim_{n \to \infty} \int_E |f_n-f| 1_E$ exists.
3) For each $E$, $\int_E |f_n-f| 1_E
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