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Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space, $(E, |\cdot|)$ a Banach space, and $p \in [1, \infty)$. In a previous thread, I proved that

Theorem: Let $f_n, f \in \mathcal L_p (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e. and $\|f_n\|_p \to \|f\|_p$ as $n \to \infty$. Then $\|f_n - f\|_p \to 0$.

By Theorem, if $f_n \to f$ $\mu$-a.e., then $\|f_n\|_1 \to \|f\|_1 \iff \|f_n - f\|_1 \to 0$. As a corollary, if $f_n \to f$ $\mu$-a.e. and $\|f_n\|_1 \to \|f\|_1$, then $\int f_n \to \int f$.

Is it true that $f_n \to f$ $\mu$-a.e. and $\int f_n \to \int f$ imply $\|f_n\|_1 \to \|f\|_1$? If not, please provide a counter-example.

Akira
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1 Answers1

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On $\Bbb R$, $f_n(x):=\frac xn\mathbf 1_{\{|x|\le n\}}$ tends pointwise to $f(x):=0$ as $n\to\infty$, with $\int f_n=0=\int f$. However $\|f_n\|_1=n$.

nejimban
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