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- Jun 20, 2014

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Let $(X,\mu)$ be a measure space, $f\in \mathcal{L}^1(\mu)$, and $\phi_n\in \mathcal{L}^1(\mu)$ such that $\sup_{n,t}\lvert \phi_n(t)\rvert \le 1$ and $\|\phi_n\|_1 \to 0$ as $n\to \infty$. Show that $\|f\phi_n\|_1 \to 0$ as $n\to \infty$.

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