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- Jun 20, 2014

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Let $(X,\mathcal{M}, \mu)$ be a positive measure space, and let $\{E_n\}$ be a sequence of sets in $\mathcal{M}$ such that $\displaystyle\lim_n \mu(E_n) = 0$. Prove that if $1 \le p \le \infty$, then for all $f\in \mathscr{L}^p(X,\mathcal{M},\mu)$, $\displaystyle\lim_n \int_{E_n} f\, d\mu = 0$.

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