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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. – Michael Greinecker Oct 31 '12 at 0:17

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

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