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If $f_n = 0$ a.e. for all $n \in \mathbb{N}$, does $\sum f_n \rightarrow 0$ uniformly a.e. to 0 as well?

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    @user1770201: do you mean that the series converges uniformly and the limit is$0$a.e., or that the series converges uniformly a.e. to 0?2012-12-20

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Hint: the set of $x$ such that there is at least one $k$ with $f_k(x) \not = 0$ has measure $0$.