# Does a series of functions that are 0 a.e. converge uniformly to the function 0 a.e. as well?

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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Uniformly, no. Say for example all $f_n$ are $1$ on the rationals and $0$ on the irrationals. – GEdgar Dec 20 '12 at 20:01
@GEdgar: the asker just asked for uniformly a.e. (in the text of the question; the title seems to ask something else). – Carl Mummert Dec 20 '12 at 20:02
@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? – Carl Mummert Dec 20 '12 at 20:04

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