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Suppose $f_n$ is a sequence of Lebesgue measurable functions defined on E. Suppose $f_n$ converges only on the set $E_0$ which is a subset of E. Can $E_0$ be non-measurable?

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    [No.](http://math.stackexchange.com/questions/104503/a-question-concerning-a-set-connected-to-a-sequence-of-measurable-functions)2012-10-31

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No.

$\{x : f_n(x) \rightarrow f(x)\} = \bigcap_k \bigcup_{N}\bigcap_{n,m\geq N} \{x : \left|f_n(x) - f_m(x)\right| \leq \frac{1}{k}\}$